3.27.100 \(\int \frac {\sqrt {b+a x} (-g+f x^2)}{(e+d x^2) \sqrt {c+\sqrt {b+a x}}} \, dx\)

Optimal. Leaf size=245 \[ -\frac {a (d g+e f) \text {RootSum}\left [\text {$\#$1}^8 d-4 \text {$\#$1}^6 c d-2 \text {$\#$1}^4 b d+6 \text {$\#$1}^4 c^2 d+4 \text {$\#$1}^2 b c d-4 \text {$\#$1}^2 c^3 d+a^2 e+b^2 d-2 b c^2 d+c^4 d\& ,\frac {\text {$\#$1}^2 \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )-c \log \left (\sqrt {\sqrt {a x+b}+c}-\text {$\#$1}\right )}{\text {$\#$1}^5-2 \text {$\#$1}^3 c-\text {$\#$1} b+\text {$\#$1} c^2}\& \right ]}{2 d^2}+\frac {4 f \left (3 a x+3 b+8 c^2\right ) \sqrt {\sqrt {a x+b}+c}}{15 a d}-\frac {16 c f \sqrt {a x+b} \sqrt {\sqrt {a x+b}+c}}{15 a d} \]

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Rubi [B]  time = 23.54, antiderivative size = 2139, normalized size of antiderivative = 8.73, number of steps used = 31, number of rules used = 10, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {6740, 194, 6688, 12, 1988, 1094, 634, 618, 206, 628}

result too large to display

Warning: Unable to verify antiderivative.

[In]

Int[(Sqrt[b + a*x]*(-g + f*x^2))/((e + d*x^2)*Sqrt[c + Sqrt[b + a*x]]),x]

[Out]

(4*c^2*f*Sqrt[c + Sqrt[b + a*x]])/(a*d) - (8*c*f*(c + Sqrt[b + a*x])^(3/2))/(3*a*d) + (4*f*(c + Sqrt[b + a*x])
^(5/2))/(5*a*d) + ((b*Sqrt[-d] - a*Sqrt[e])*(e*f + d*g)*ArcTanh[(Sqrt[c*(-d)^(1/4) + Sqrt[-(b*Sqrt[-d]) + c^2*
Sqrt[-d] + a*Sqrt[e]]] - Sqrt[2]*(-d)^(1/8)*Sqrt[c + Sqrt[b + a*x]])/Sqrt[c*(-d)^(1/4) - Sqrt[-(b*Sqrt[-d]) +
c^2*Sqrt[-d] + a*Sqrt[e]]]])/(Sqrt[2]*(-d)^(13/8)*Sqrt[c*(-d)^(1/4) - Sqrt[-(b*Sqrt[-d]) + c^2*Sqrt[-d] + a*Sq
rt[e]]]*Sqrt[-(b*Sqrt[-d]) + c^2*Sqrt[-d] + a*Sqrt[e]]*Sqrt[e]) - ((b*Sqrt[-d] - a*Sqrt[e])*(e*f + d*g)*ArcTan
h[(Sqrt[c*(-d)^(1/4) + Sqrt[-(b*Sqrt[-d]) + c^2*Sqrt[-d] + a*Sqrt[e]]] + Sqrt[2]*(-d)^(1/8)*Sqrt[c + Sqrt[b +
a*x]])/Sqrt[c*(-d)^(1/4) - Sqrt[-(b*Sqrt[-d]) + c^2*Sqrt[-d] + a*Sqrt[e]]]])/(Sqrt[2]*(-d)^(13/8)*Sqrt[c*(-d)^
(1/4) - Sqrt[-(b*Sqrt[-d]) + c^2*Sqrt[-d] + a*Sqrt[e]]]*Sqrt[-(b*Sqrt[-d]) + c^2*Sqrt[-d] + a*Sqrt[e]]*Sqrt[e]
) - ((b + (a*Sqrt[e])/Sqrt[-d])*(e*f + d*g)*ArcTanh[(Sqrt[c*(-d)^(3/4) + Sqrt[d*(b*Sqrt[-d] - c^2*Sqrt[-d] + a
*Sqrt[e])]] - Sqrt[2]*(-d)^(3/8)*Sqrt[c + Sqrt[b + a*x]])/Sqrt[c*(-d)^(3/4) - Sqrt[d*(b*Sqrt[-d] - c^2*Sqrt[-d
] + a*Sqrt[e])]]])/(Sqrt[2]*(-d)^(3/8)*Sqrt[c*(-d)^(3/4) - Sqrt[d*(b*Sqrt[-d] - c^2*Sqrt[-d] + a*Sqrt[e])]]*Sq
rt[d*(b*Sqrt[-d] - c^2*Sqrt[-d] + a*Sqrt[e])]*Sqrt[e]) + ((b + (a*Sqrt[e])/Sqrt[-d])*(e*f + d*g)*ArcTanh[(Sqrt
[c*(-d)^(3/4) + Sqrt[d*(b*Sqrt[-d] - c^2*Sqrt[-d] + a*Sqrt[e])]] + Sqrt[2]*(-d)^(3/8)*Sqrt[c + Sqrt[b + a*x]])
/Sqrt[c*(-d)^(3/4) - Sqrt[d*(b*Sqrt[-d] - c^2*Sqrt[-d] + a*Sqrt[e])]]])/(Sqrt[2]*(-d)^(3/8)*Sqrt[c*(-d)^(3/4)
- Sqrt[d*(b*Sqrt[-d] - c^2*Sqrt[-d] + a*Sqrt[e])]]*Sqrt[d*(b*Sqrt[-d] - c^2*Sqrt[-d] + a*Sqrt[e])]*Sqrt[e]) -
((b*Sqrt[-d] - a*Sqrt[e])*(e*f + d*g)*Log[Sqrt[-(b*Sqrt[-d]) + c^2*Sqrt[-d] + a*Sqrt[e]] - Sqrt[2]*(-d)^(1/8)*
Sqrt[c*(-d)^(1/4) + Sqrt[-(b*Sqrt[-d]) + c^2*Sqrt[-d] + a*Sqrt[e]]]*Sqrt[c + Sqrt[b + a*x]] + (-d)^(1/4)*(c +
Sqrt[b + a*x])])/(2*Sqrt[2]*(-d)^(13/8)*Sqrt[c*(-d)^(1/4) + Sqrt[-(b*Sqrt[-d]) + c^2*Sqrt[-d] + a*Sqrt[e]]]*Sq
rt[-(b*Sqrt[-d]) + c^2*Sqrt[-d] + a*Sqrt[e]]*Sqrt[e]) + ((b*Sqrt[-d] - a*Sqrt[e])*(e*f + d*g)*Log[Sqrt[-(b*Sqr
t[-d]) + c^2*Sqrt[-d] + a*Sqrt[e]] + Sqrt[2]*(-d)^(1/8)*Sqrt[c*(-d)^(1/4) + Sqrt[-(b*Sqrt[-d]) + c^2*Sqrt[-d]
+ a*Sqrt[e]]]*Sqrt[c + Sqrt[b + a*x]] + (-d)^(1/4)*(c + Sqrt[b + a*x])])/(2*Sqrt[2]*(-d)^(13/8)*Sqrt[c*(-d)^(1
/4) + Sqrt[-(b*Sqrt[-d]) + c^2*Sqrt[-d] + a*Sqrt[e]]]*Sqrt[-(b*Sqrt[-d]) + c^2*Sqrt[-d] + a*Sqrt[e]]*Sqrt[e])
+ ((b + (a*Sqrt[e])/Sqrt[-d])*(e*f + d*g)*Log[Sqrt[d*(b*Sqrt[-d] - c^2*Sqrt[-d] + a*Sqrt[e])] - Sqrt[2]*(-d)^(
3/8)*Sqrt[c*(-d)^(3/4) + Sqrt[d*(b*Sqrt[-d] - c^2*Sqrt[-d] + a*Sqrt[e])]]*Sqrt[c + Sqrt[b + a*x]] + (-d)^(3/4)
*(c + Sqrt[b + a*x])])/(2*Sqrt[2]*(-d)^(3/8)*Sqrt[c*(-d)^(3/4) + Sqrt[d*(b*Sqrt[-d] - c^2*Sqrt[-d] + a*Sqrt[e]
)]]*Sqrt[d*(b*Sqrt[-d] - c^2*Sqrt[-d] + a*Sqrt[e])]*Sqrt[e]) - ((b + (a*Sqrt[e])/Sqrt[-d])*(e*f + d*g)*Log[Sqr
t[d*(b*Sqrt[-d] - c^2*Sqrt[-d] + a*Sqrt[e])] + Sqrt[2]*(-d)^(3/8)*Sqrt[c*(-d)^(3/4) + Sqrt[d*(b*Sqrt[-d] - c^2
*Sqrt[-d] + a*Sqrt[e])]]*Sqrt[c + Sqrt[b + a*x]] + (-d)^(3/4)*(c + Sqrt[b + a*x])])/(2*Sqrt[2]*(-d)^(3/8)*Sqrt
[c*(-d)^(3/4) + Sqrt[d*(b*Sqrt[-d] - c^2*Sqrt[-d] + a*Sqrt[e])]]*Sqrt[d*(b*Sqrt[-d] - c^2*Sqrt[-d] + a*Sqrt[e]
)]*Sqrt[e])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1094

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}
, Dist[1/(2*c*q*r), Int[(r - x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(r + x)/(q + r*x + x^2), x], x
]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]

Rule 1988

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && TrinomialQ[u, x] &&  !TrinomialMatch
Q[u, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6740

Int[(v_)/((a_) + (b_.)*(u_)^(n_.)), x_Symbol] :> Int[ExpandIntegrand[PolynomialInSubst[v, u, x]/(a + b*x^n), x
] /. x -> u, x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && PolynomialInQ[v, u, x]

Rubi steps

\begin {align*} \int \frac {\sqrt {b+a x} \left (-g+f x^2\right )}{\left (e+d x^2\right ) \sqrt {c+\sqrt {b+a x}}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {x^2 \left (-a^2 g+f \left (b-x^2\right )^2\right )}{\sqrt {c+x} \left (e+\frac {d \left (b-x^2\right )^2}{a^2}\right )} \, dx,x,\sqrt {b+a x}\right )}{a^3}\\ &=\frac {4 \operatorname {Subst}\left (\int \frac {\left (c-x^2\right )^2 \left (-a^2 g+f \left (b-\left (c-x^2\right )^2\right )^2\right )}{e+\frac {d \left (b-\left (c-x^2\right )^2\right )^2}{a^2}} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a^3}\\ &=\frac {4 \operatorname {Subst}\left (\int \left (\frac {a^2 b f}{d}-\frac {a^2 f \left (b-\left (c-x^2\right )^2\right )}{d}-\frac {a^2 b (e f+d g)-a^2 (e f+d g) \left (b-\left (c-x^2\right )^2\right )}{d \left (e+\frac {d \left (b-\left (c-x^2\right )^2\right )^2}{a^2}\right )}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a^3}\\ &=\frac {4 b f \sqrt {c+\sqrt {b+a x}}}{a d}-\frac {4 \operatorname {Subst}\left (\int \frac {a^2 b (e f+d g)-a^2 (e f+d g) \left (b-\left (c-x^2\right )^2\right )}{e+\frac {d \left (b-\left (c-x^2\right )^2\right )^2}{a^2}} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a^3 d}-\frac {(4 f) \operatorname {Subst}\left (\int \left (b-\left (c-x^2\right )^2\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a d}\\ &=-\frac {4 \operatorname {Subst}\left (\int \frac {a^2 (e f+d g) \left (c-x^2\right )^2}{e+\frac {d \left (b-\left (c-x^2\right )^2\right )^2}{a^2}} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a^3 d}+\frac {(4 f) \operatorname {Subst}\left (\int \left (c-x^2\right )^2 \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a d}\\ &=\frac {(4 f) \operatorname {Subst}\left (\int \left (c^2-2 c x^2+x^4\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a d}-\frac {(4 (e f+d g)) \operatorname {Subst}\left (\int \frac {\left (c-x^2\right )^2}{e+\frac {d \left (b-\left (c-x^2\right )^2\right )^2}{a^2}} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a d}\\ &=\frac {4 c^2 f \sqrt {c+\sqrt {b+a x}}}{a d}-\frac {8 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}-\frac {(4 (e f+d g)) \operatorname {Subst}\left (\int \left (\frac {a b \sqrt {e}-\frac {a^2 e}{\sqrt {-d}}}{2 e \left (a \sqrt {e}-\sqrt {-d} \left (b-\left (c-x^2\right )^2\right )\right )}+\frac {a b \sqrt {e}+\frac {a^2 e}{\sqrt {-d}}}{2 e \left (a \sqrt {e}+\sqrt {-d} \left (b-\left (c-x^2\right )^2\right )\right )}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a d}\\ &=\frac {4 c^2 f \sqrt {c+\sqrt {b+a x}}}{a d}-\frac {8 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}-\frac {\left (2 \left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{a \sqrt {e}+\sqrt {-d} \left (b-\left (c-x^2\right )^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{d \sqrt {e}}-\frac {\left (2 \left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{a \sqrt {e}-\sqrt {-d} \left (b-\left (c-x^2\right )^2\right )} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{d \sqrt {e}}\\ &=\frac {4 c^2 f \sqrt {c+\sqrt {b+a x}}}{a d}-\frac {8 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}-\frac {\left (2 \left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}+2 c \sqrt {-d} x^2-\sqrt {-d} x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{d \sqrt {e}}-\frac {\left (2 \left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}-2 c \sqrt {-d} x^2+\sqrt {-d} x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{d \sqrt {e}}\\ &=\frac {4 c^2 f \sqrt {c+\sqrt {b+a x}}}{a d}-\frac {8 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}-\frac {\left (\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}}}{(-d)^{3/8}}-x}{\frac {\sqrt {-b (-d)^{3/2}+c^2 (-d)^{3/2}+a d \sqrt {e}}}{(-d)^{3/4}}-\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}} x}{(-d)^{3/8}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{\sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}-\frac {\left (\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}}}{(-d)^{3/8}}+x}{\frac {\sqrt {-b (-d)^{3/2}+c^2 (-d)^{3/2}+a d \sqrt {e}}}{(-d)^{3/4}}+\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}} x}{(-d)^{3/8}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{\sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}+\frac {\left (\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}}{\sqrt [8]{-d}}-x}{\frac {\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}{\sqrt [4]{-d}}-\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} x}{\sqrt [8]{-d}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{\sqrt {2} (-d)^{9/8} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}+\frac {\left (\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}}{\sqrt [8]{-d}}+x}{\frac {\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}{\sqrt [4]{-d}}+\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} x}{\sqrt [8]{-d}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{\sqrt {2} (-d)^{9/8} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}\\ &=\frac {4 c^2 f \sqrt {c+\sqrt {b+a x}}}{a d}-\frac {8 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}-\frac {\left (\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {-b (-d)^{3/2}+c^2 (-d)^{3/2}+a d \sqrt {e}}}{(-d)^{3/4}}-\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}} x}{(-d)^{3/8}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{2 (-d)^{3/4} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}-\frac {\left (\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {-b (-d)^{3/2}+c^2 (-d)^{3/2}+a d \sqrt {e}}}{(-d)^{3/4}}+\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}} x}{(-d)^{3/8}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{2 (-d)^{3/4} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}+\frac {\left (\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}}}{(-d)^{3/8}}+2 x}{\frac {\sqrt {-b (-d)^{3/2}+c^2 (-d)^{3/2}+a d \sqrt {e}}}{(-d)^{3/4}}-\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}} x}{(-d)^{3/8}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{2 \sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}-\frac {\left (\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}}}{(-d)^{3/8}}+2 x}{\frac {\sqrt {-b (-d)^{3/2}+c^2 (-d)^{3/2}+a d \sqrt {e}}}{(-d)^{3/4}}+\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}} x}{(-d)^{3/8}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{2 \sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}+\frac {\left (\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}{\sqrt [4]{-d}}-\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} x}{\sqrt [8]{-d}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{2 (-d)^{5/4} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}+\frac {\left (\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}{\sqrt [4]{-d}}+\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} x}{\sqrt [8]{-d}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{2 (-d)^{5/4} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}-\frac {\left (\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}}{\sqrt [8]{-d}}+2 x}{\frac {\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}{\sqrt [4]{-d}}-\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} x}{\sqrt [8]{-d}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{2 \sqrt {2} (-d)^{9/8} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}+\frac {\left (\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}}{\sqrt [8]{-d}}+2 x}{\frac {\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}{\sqrt [4]{-d}}+\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} x}{\sqrt [8]{-d}}+x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{2 \sqrt {2} (-d)^{9/8} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}\\ &=\frac {4 c^2 f \sqrt {c+\sqrt {b+a x}}}{a d}-\frac {8 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}-\frac {\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g) \log \left (\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}-\sqrt {2} \sqrt [8]{-d} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {c+\sqrt {b+a x}}+\sqrt [4]{-d} \left (c+\sqrt {b+a x}\right )\right )}{2 \sqrt {2} (-d)^{9/8} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}+\frac {\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g) \log \left (\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}+\sqrt {2} \sqrt [8]{-d} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {c+\sqrt {b+a x}}+\sqrt [4]{-d} \left (c+\sqrt {b+a x}\right )\right )}{2 \sqrt {2} (-d)^{9/8} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}+\frac {\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g) \log \left (\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}-\sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {c+\sqrt {b+a x}}+(-d)^{3/4} \left (c+\sqrt {b+a x}\right )\right )}{2 \sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}-\frac {\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g) \log \left (\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}+\sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {c+\sqrt {b+a x}}+(-d)^{3/4} \left (c+\sqrt {b+a x}\right )\right )}{2 \sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}+\frac {\left (\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{2 \left (c-\frac {\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}}{(-d)^{3/4}}\right )-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}}}{(-d)^{3/8}}+2 \sqrt {c+\sqrt {b+a x}}\right )}{(-d)^{3/4} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}+\frac {\left (\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{2 \left (c-\frac {\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}}{(-d)^{3/4}}\right )-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {c (-d)^{3/4}+\sqrt {-d \left (-b \sqrt {-d}+c^2 \sqrt {-d}-a \sqrt {e}\right )}}}{(-d)^{3/8}}+2 \sqrt {c+\sqrt {b+a x}}\right )}{(-d)^{3/4} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}-\frac {\left (\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{2 \left (c+\frac {d \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}{(-d)^{5/4}}\right )-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}}{\sqrt [8]{-d}}+2 \sqrt {c+\sqrt {b+a x}}\right )}{(-d)^{5/4} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}-\frac {\left (\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g)\right ) \operatorname {Subst}\left (\int \frac {1}{2 \left (c+\frac {d \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}{(-d)^{5/4}}\right )-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}}{\sqrt [8]{-d}}+2 \sqrt {c+\sqrt {b+a x}}\right )}{(-d)^{5/4} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}\\ &=\frac {4 c^2 f \sqrt {c+\sqrt {b+a x}}}{a d}-\frac {8 c f \left (c+\sqrt {b+a x}\right )^{3/2}}{3 a d}+\frac {4 f \left (c+\sqrt {b+a x}\right )^{5/2}}{5 a d}-\frac {\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g) \tanh ^{-1}\left (\frac {(-d)^{3/8} \left (\frac {\sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}}}{(-d)^{3/8}}-\sqrt {2} \sqrt {c+\sqrt {b+a x}}\right )}{\sqrt {c (-d)^{3/4}-\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}}}\right )}{\sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}-\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}+\frac {\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g) \tanh ^{-1}\left (\frac {\sqrt [8]{-d} \left (\frac {\sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}}{\sqrt [8]{-d}}-\sqrt {2} \sqrt {c+\sqrt {b+a x}}\right )}{\sqrt {c \sqrt [4]{-d}-\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}}\right )}{\sqrt {2} (-d)^{9/8} \sqrt {c \sqrt [4]{-d}-\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}+\frac {\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g) \tanh ^{-1}\left (\frac {(-d)^{3/8} \left (\frac {\sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}}}{(-d)^{3/8}}+\sqrt {2} \sqrt {c+\sqrt {b+a x}}\right )}{\sqrt {c (-d)^{3/4}-\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}}}\right )}{\sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}-\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}-\frac {\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g) \tanh ^{-1}\left (\frac {\sqrt [8]{-d} \left (\frac {\sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}}{\sqrt [8]{-d}}+\sqrt {2} \sqrt {c+\sqrt {b+a x}}\right )}{\sqrt {c \sqrt [4]{-d}-\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}}}\right )}{\sqrt {2} (-d)^{9/8} \sqrt {c \sqrt [4]{-d}-\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}-\frac {\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g) \log \left (\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}-\sqrt {2} \sqrt [8]{-d} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {c+\sqrt {b+a x}}+\sqrt [4]{-d} \left (c+\sqrt {b+a x}\right )\right )}{2 \sqrt {2} (-d)^{9/8} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}+\frac {\left (b+\frac {a d \sqrt {e}}{(-d)^{3/2}}\right ) (e f+d g) \log \left (\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}+\sqrt {2} \sqrt [8]{-d} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {c+\sqrt {b+a x}}+\sqrt [4]{-d} \left (c+\sqrt {b+a x}\right )\right )}{2 \sqrt {2} (-d)^{9/8} \sqrt {c \sqrt [4]{-d}+\sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}}} \sqrt {-b \sqrt {-d}+c^2 \sqrt {-d}+a \sqrt {e}} \sqrt {e}}+\frac {\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g) \log \left (\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}-\sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {c+\sqrt {b+a x}}+(-d)^{3/4} \left (c+\sqrt {b+a x}\right )\right )}{2 \sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}-\frac {\left (b+\frac {a \sqrt {e}}{\sqrt {-d}}\right ) (e f+d g) \log \left (\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}+\sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {c+\sqrt {b+a x}}+(-d)^{3/4} \left (c+\sqrt {b+a x}\right )\right )}{2 \sqrt {2} (-d)^{3/8} \sqrt {c (-d)^{3/4}+\sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )}} \sqrt {d \left (b \sqrt {-d}-c^2 \sqrt {-d}+a \sqrt {e}\right )} \sqrt {e}}\\ \end {align*}

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Mathematica [A]  time = 27.37, size = 373, normalized size = 1.52 \begin {gather*} -\frac {\left (\sqrt {a x+b}+c\right )^{7/2} \left (\frac {c}{\sqrt {a x+b}+c}-1\right ) \left (\frac {15 a^2 (d g+e f) \text {RootSum}\left [\text {$\#$1}^8 a^2 e+\text {$\#$1}^8 b^2 d-2 \text {$\#$1}^8 b c^2 d+\text {$\#$1}^8 c^4 d+4 \text {$\#$1}^6 b c d-4 \text {$\#$1}^6 c^3 d-2 \text {$\#$1}^4 b d+6 \text {$\#$1}^4 c^2 d-4 \text {$\#$1}^2 c d+d\&,\frac {\text {$\#$1}^5 c^2 \log \left (\frac {1}{\sqrt {\sqrt {a x+b}+c}}-\text {$\#$1}\right )-2 \text {$\#$1}^3 c \log \left (\frac {1}{\sqrt {\sqrt {a x+b}+c}}-\text {$\#$1}\right )+\text {$\#$1} \log \left (\frac {1}{\sqrt {\sqrt {a x+b}+c}}-\text {$\#$1}\right )}{\text {$\#$1}^6 a^2 e+\text {$\#$1}^6 b^2 d-2 \text {$\#$1}^6 b c^2 d+\text {$\#$1}^6 c^4 d+3 \text {$\#$1}^4 b c d-3 \text {$\#$1}^4 c^3 d-\text {$\#$1}^2 b d+3 \text {$\#$1}^2 c^2 d-c d}\&\right ]}{\left (\sqrt {a x+b}+c\right )^{5/2}}+\frac {8 f \left (-4 c \sqrt {a x+b}+3 a x+3 b+8 c^2\right )}{\left (\sqrt {a x+b}+c\right )^2}\right )}{30 a d \sqrt {a x+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[b + a*x]*(-g + f*x^2))/((e + d*x^2)*Sqrt[c + Sqrt[b + a*x]]),x]

[Out]

-1/30*((c + Sqrt[b + a*x])^(7/2)*(-1 + c/(c + Sqrt[b + a*x]))*((8*f*(3*b + 8*c^2 + 3*a*x - 4*c*Sqrt[b + a*x]))
/(c + Sqrt[b + a*x])^2 + (15*a^2*(e*f + d*g)*RootSum[d - 4*c*d*#1^2 - 2*b*d*#1^4 + 6*c^2*d*#1^4 + 4*b*c*d*#1^6
- 4*c^3*d*#1^6 + b^2*d*#1^8 - 2*b*c^2*d*#1^8 + c^4*d*#1^8 + a^2*e*#1^8 & , (Log[1/Sqrt[c + Sqrt[b + a*x]] - #
1]*#1 - 2*c*Log[1/Sqrt[c + Sqrt[b + a*x]] - #1]*#1^3 + c^2*Log[1/Sqrt[c + Sqrt[b + a*x]] - #1]*#1^5)/(-(c*d) -
b*d*#1^2 + 3*c^2*d*#1^2 + 3*b*c*d*#1^4 - 3*c^3*d*#1^4 + b^2*d*#1^6 - 2*b*c^2*d*#1^6 + c^4*d*#1^6 + a^2*e*#1^6
) & ])/(c + Sqrt[b + a*x])^(5/2)))/(a*d*Sqrt[b + a*x])

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IntegrateAlgebraic [A]  time = 0.60, size = 223, normalized size = 0.91 \begin {gather*} \frac {4 \sqrt {c+\sqrt {b+a x}} \left (8 c^2 f-4 c f \sqrt {b+a x}+3 f (b+a x)\right )}{15 a d}-\frac {a (e f+d g) \text {RootSum}\left [b^2 d-2 b c^2 d+c^4 d+a^2 e+4 b c d \text {$\#$1}^2-4 c^3 d \text {$\#$1}^2-2 b d \text {$\#$1}^4+6 c^2 d \text {$\#$1}^4-4 c d \text {$\#$1}^6+d \text {$\#$1}^8\&,\frac {-c \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right )+\log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-b \text {$\#$1}+c^2 \text {$\#$1}-2 c \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]}{2 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[b + a*x]*(-g + f*x^2))/((e + d*x^2)*Sqrt[c + Sqrt[b + a*x]]),x]

[Out]

(4*Sqrt[c + Sqrt[b + a*x]]*(8*c^2*f - 4*c*f*Sqrt[b + a*x] + 3*f*(b + a*x)))/(15*a*d) - (a*(e*f + d*g)*RootSum[
b^2*d - 2*b*c^2*d + c^4*d + a^2*e + 4*b*c*d*#1^2 - 4*c^3*d*#1^2 - 2*b*d*#1^4 + 6*c^2*d*#1^4 - 4*c*d*#1^6 + d*#
1^8 & , (-(c*Log[Sqrt[c + Sqrt[b + a*x]] - #1]) + Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^2)/(-(b*#1) + c^2*#1 -
2*c*#1^3 + #1^5) & ])/(2*d^2)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+b)^(1/2)*(f*x^2-g)/(d*x^2+e)/(c+(a*x+b)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 146.73, size = 2731, normalized size = 11.15

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+b)^(1/2)*(f*x^2-g)/(d*x^2+e)/(c+(a*x+b)^(1/2))^(1/2),x, algorithm="giac")

[Out]

4/15*(3*a^4*(c + sqrt(a*x + b))^(5/2)*d^4*f - 10*a^4*(c + sqrt(a*x + b))^(3/2)*c*d^4*f + 15*a^4*sqrt(c + sqrt(
a*x + b))*c^2*d^4*f)/(a^5*d^5) + 1/2*((a^7*(c + sqrt((b*d + sqrt(-d*e)*a)/d))^2*d^5*g - 2*a^7*(c + sqrt((b*d +
sqrt(-d*e)*a)/d))*c*d^5*g + a^7*c^2*d^5*g + a^7*(c + sqrt((b*d + sqrt(-d*e)*a)/d))^2*d^4*f*e - 2*a^7*(c + sqr
t((b*d + sqrt(-d*e)*a)/d))*c*d^4*f*e + a^7*c^2*d^4*f*e)*log(sqrt(c + sqrt(a*x + b)) + sqrt(c + sqrt((b*d + sqr
t(-d*e)*a)/d)))/((c + sqrt((b*d + sqrt(-d*e)*a)/d))^(7/2)*d - 3*(c + sqrt((b*d + sqrt(-d*e)*a)/d))^(5/2)*c*d +
(3*c^2*d - b*d)*(c + sqrt((b*d + sqrt(-d*e)*a)/d))^(3/2) - (c^3*d - b*c*d)*sqrt(c + sqrt((b*d + sqrt(-d*e)*a)
/d))) - (a^7*(c + sqrt((b*d + sqrt(-d*e)*a)/d))^2*d^5*g - 2*a^7*(c + sqrt((b*d + sqrt(-d*e)*a)/d))*c*d^5*g + a
^7*c^2*d^5*g + a^7*(c + sqrt((b*d + sqrt(-d*e)*a)/d))^2*d^4*f*e - 2*a^7*(c + sqrt((b*d + sqrt(-d*e)*a)/d))*c*d
^4*f*e + a^7*c^2*d^4*f*e)*log(sqrt(c + sqrt(a*x + b)) - sqrt(c + sqrt((b*d + sqrt(-d*e)*a)/d)))/((c + sqrt((b*
d + sqrt(-d*e)*a)/d))^(7/2)*d - 3*(c + sqrt((b*d + sqrt(-d*e)*a)/d))^(5/2)*c*d + (3*c^2*d - b*d)*(c + sqrt((b*
d + sqrt(-d*e)*a)/d))^(3/2) - (c^3*d - b*c*d)*sqrt(c + sqrt((b*d + sqrt(-d*e)*a)/d))) + (a^7*(c - sqrt((b*d +
sqrt(-d*e)*a)/d))^2*d^5*g - 2*a^7*(c - sqrt((b*d + sqrt(-d*e)*a)/d))*c*d^5*g + a^7*c^2*d^5*g + a^7*(c - sqrt((
b*d + sqrt(-d*e)*a)/d))^2*d^4*f*e - 2*a^7*(c - sqrt((b*d + sqrt(-d*e)*a)/d))*c*d^4*f*e + a^7*c^2*d^4*f*e)*log(
sqrt(c + sqrt(a*x + b)) + sqrt(c - sqrt((b*d + sqrt(-d*e)*a)/d)))/((c - sqrt((b*d + sqrt(-d*e)*a)/d))^(7/2)*d
- 3*(c - sqrt((b*d + sqrt(-d*e)*a)/d))^(5/2)*c*d + (3*c^2*d - b*d)*(c - sqrt((b*d + sqrt(-d*e)*a)/d))^(3/2) -
(c^3*d - b*c*d)*sqrt(c - sqrt((b*d + sqrt(-d*e)*a)/d))) - (a^7*(c - sqrt((b*d + sqrt(-d*e)*a)/d))^2*d^5*g - 2*
a^7*(c - sqrt((b*d + sqrt(-d*e)*a)/d))*c*d^5*g + a^7*c^2*d^5*g + a^7*(c - sqrt((b*d + sqrt(-d*e)*a)/d))^2*d^4*
f*e - 2*a^7*(c - sqrt((b*d + sqrt(-d*e)*a)/d))*c*d^4*f*e + a^7*c^2*d^4*f*e)*log(sqrt(c + sqrt(a*x + b)) - sqrt
(c - sqrt((b*d + sqrt(-d*e)*a)/d)))/((c - sqrt((b*d + sqrt(-d*e)*a)/d))^(7/2)*d - 3*(c - sqrt((b*d + sqrt(-d*e
)*a)/d))^(5/2)*c*d + (3*c^2*d - b*d)*(c - sqrt((b*d + sqrt(-d*e)*a)/d))^(3/2) - (c^3*d - b*c*d)*sqrt(c - sqrt(
(b*d + sqrt(-d*e)*a)/d))) + (a^7*(c + sqrt((b*d - sqrt(-d*e)*a)/d))^2*d^5*g - 2*a^7*(c + sqrt((b*d - sqrt(-d*e
)*a)/d))*c*d^5*g + a^7*c^2*d^5*g + a^7*(c + sqrt((b*d - sqrt(-d*e)*a)/d))^2*d^4*f*e - 2*a^7*(c + sqrt((b*d - s
qrt(-d*e)*a)/d))*c*d^4*f*e + a^7*c^2*d^4*f*e)*log(sqrt(c + sqrt(a*x + b)) + sqrt(c + sqrt((b*d - sqrt(-d*e)*a)
/d)))/((c + sqrt((b*d - sqrt(-d*e)*a)/d))^(7/2)*d - 3*(c + sqrt((b*d - sqrt(-d*e)*a)/d))^(5/2)*c*d + (3*c^2*d
- b*d)*(c + sqrt((b*d - sqrt(-d*e)*a)/d))^(3/2) - (c^3*d - b*c*d)*sqrt(c + sqrt((b*d - sqrt(-d*e)*a)/d))) - (a
^7*(c + sqrt((b*d - sqrt(-d*e)*a)/d))^2*d^5*g - 2*a^7*(c + sqrt((b*d - sqrt(-d*e)*a)/d))*c*d^5*g + a^7*c^2*d^5
*g + a^7*(c + sqrt((b*d - sqrt(-d*e)*a)/d))^2*d^4*f*e - 2*a^7*(c + sqrt((b*d - sqrt(-d*e)*a)/d))*c*d^4*f*e + a
^7*c^2*d^4*f*e)*log(sqrt(c + sqrt(a*x + b)) - sqrt(c + sqrt((b*d - sqrt(-d*e)*a)/d)))/((c + sqrt((b*d - sqrt(-
d*e)*a)/d))^(7/2)*d - 3*(c + sqrt((b*d - sqrt(-d*e)*a)/d))^(5/2)*c*d + (3*c^2*d - b*d)*(c + sqrt((b*d - sqrt(-
d*e)*a)/d))^(3/2) - (c^3*d - b*c*d)*sqrt(c + sqrt((b*d - sqrt(-d*e)*a)/d))) + (a^7*(c - sqrt((b*d - sqrt(-d*e)
*a)/d))^2*d^5*g - 2*a^7*(c - sqrt((b*d - sqrt(-d*e)*a)/d))*c*d^5*g + a^7*c^2*d^5*g + a^7*(c - sqrt((b*d - sqrt
(-d*e)*a)/d))^2*d^4*f*e - 2*a^7*(c - sqrt((b*d - sqrt(-d*e)*a)/d))*c*d^4*f*e + a^7*c^2*d^4*f*e)*log(sqrt(c + s
qrt(a*x + b)) + sqrt(c - sqrt((b*d - sqrt(-d*e)*a)/d)))/((c - sqrt((b*d - sqrt(-d*e)*a)/d))^(7/2)*d - 3*(c - s
qrt((b*d - sqrt(-d*e)*a)/d))^(5/2)*c*d + (3*c^2*d - b*d)*(c - sqrt((b*d - sqrt(-d*e)*a)/d))^(3/2) - (c^3*d - b
*c*d)*sqrt(c - sqrt((b*d - sqrt(-d*e)*a)/d))) - (a^7*(c - sqrt((b*d - sqrt(-d*e)*a)/d))^2*d^5*g - 2*a^7*(c - s
qrt((b*d - sqrt(-d*e)*a)/d))*c*d^5*g + a^7*c^2*d^5*g + a^7*(c - sqrt((b*d - sqrt(-d*e)*a)/d))^2*d^4*f*e - 2*a^
7*(c - sqrt((b*d - sqrt(-d*e)*a)/d))*c*d^4*f*e + a^7*c^2*d^4*f*e)*log(sqrt(c + sqrt(a*x + b)) - sqrt(c - sqrt(
(b*d - sqrt(-d*e)*a)/d)))/((c - sqrt((b*d - sqrt(-d*e)*a)/d))^(7/2)*d - 3*(c - sqrt((b*d - sqrt(-d*e)*a)/d))^(
5/2)*c*d + (3*c^2*d - b*d)*(c - sqrt((b*d - sqrt(-d*e)*a)/d))^(3/2) - (c^3*d - b*c*d)*sqrt(c - sqrt((b*d - sqr
t(-d*e)*a)/d))))/(a^6*d^5)

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maple [B]  time = 0.47, size = 226, normalized size = 0.92

method	result	size
derivativedivides	\(\frac {\frac {4 f \left (\frac {\left (c +\sqrt {a x +b}\right )^{\frac {5}{2}}}{5}-\frac {2 c \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+c^{2} \sqrt {c +\sqrt {a x +b}}\right )}{d}-\frac {a^{2} \left (\munderset {\textit {\_R} =\RootOf \left (d \,\textit {\_Z}^{8}-4 c d \,\textit {\_Z}^{6}+\left (6 c^{2} d -2 b d \right ) \textit {\_Z}^{4}+\left (-4 c^{3} d +4 b c d \right ) \textit {\_Z}^{2}+c^{4} d -2 b \,c^{2} d +a^{2} e +b^{2} d \right )}{\sum }\frac {\left (\textit {\_R}^{4} \left (d g +e f \right )+2 c \left (-d g -e f \right ) \textit {\_R}^{2}+c^{2} d g +c^{2} e f \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5} c +3 \textit {\_R}^{3} c^{2}-\textit {\_R}^{3} b -\textit {\_R} \,c^{3}+\textit {\_R} b c}\right )}{2 d^{2}}}{a}\)	\(226\)
default	\(\frac {\frac {4 f \left (\frac {\left (c +\sqrt {a x +b}\right )^{\frac {5}{2}}}{5}-\frac {2 c \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+c^{2} \sqrt {c +\sqrt {a x +b}}\right )}{d}-\frac {a^{2} \left (\munderset {\textit {\_R} =\RootOf \left (d \,\textit {\_Z}^{8}-4 c d \,\textit {\_Z}^{6}+\left (6 c^{2} d -2 b d \right ) \textit {\_Z}^{4}+\left (-4 c^{3} d +4 b c d \right ) \textit {\_Z}^{2}+c^{4} d -2 b \,c^{2} d +a^{2} e +b^{2} d \right )}{\sum }\frac {\left (\textit {\_R}^{4} \left (d g +e f \right )+2 c \left (-d g -e f \right ) \textit {\_R}^{2}+c^{2} d g +c^{2} e f \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5} c +3 \textit {\_R}^{3} c^{2}-\textit {\_R}^{3} b -\textit {\_R} \,c^{3}+\textit {\_R} b c}\right )}{2 d^{2}}}{a}\)	\(226\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+b)^(1/2)*(f*x^2-g)/(d*x^2+e)/(c+(a*x+b)^(1/2))^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/a*(2*f/d*(1/5*(c+(a*x+b)^(1/2))^(5/2)-2/3*c*(c+(a*x+b)^(1/2))^(3/2)+c^2*(c+(a*x+b)^(1/2))^(1/2))-1/4*a^2/d^2
*sum((_R^4*(d*g+e*f)+2*c*(-d*g-e*f)*_R^2+c^2*d*g+c^2*e*f)/(_R^7-3*_R^5*c+3*_R^3*c^2-_R^3*b-_R*c^3+_R*b*c)*ln((
c+(a*x+b)^(1/2))^(1/2)-_R),_R=RootOf(d*_Z^8-4*c*d*_Z^6+(6*c^2*d-2*b*d)*_Z^4+(-4*c^3*d+4*b*c*d)*_Z^2+c^4*d-2*b*
c^2*d+a^2*e+b^2*d)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (f x^{2} - g\right )} \sqrt {a x + b}}{{\left (d x^{2} + e\right )} \sqrt {c + \sqrt {a x + b}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+b)^(1/2)*(f*x^2-g)/(d*x^2+e)/(c+(a*x+b)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate((f*x^2 - g)*sqrt(a*x + b)/((d*x^2 + e)*sqrt(c + sqrt(a*x + b))), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {\left (g-f\,x^2\right )\,\sqrt {b+a\,x}}{\sqrt {c+\sqrt {b+a\,x}}\,\left (d\,x^2+e\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((g - f*x^2)*(b + a*x)^(1/2))/((c + (b + a*x)^(1/2))^(1/2)*(e + d*x^2)),x)

[Out]

int(-((g - f*x^2)*(b + a*x)^(1/2))/((c + (b + a*x)^(1/2))^(1/2)*(e + d*x^2)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+b)**(1/2)*(f*x**2-g)/(d*x**2+e)/(c+(a*x+b)**(1/2))**(1/2),x)

[Out]

Timed out

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