\(\int \frac {(d+c x^2) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-d+c x^2} \, dx\) [2426]

Optimal result

Integrand size = 42, antiderivative size = 196 \[ \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-d+c x^2} \, dx=-\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}+a d \text {RootSum}\left [b^4 c-2 b^2 c \text {$\#$1}^4-4 a^2 d \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {-b^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-\log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{b^2 c \text {$\#$1}+2 a^2 d \text {$\#$1}-c \text {$\#$1}^5}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(454\) vs. \(2(196)=392\).

Time = 0.93 (sec) , antiderivative size = 454, normalized size of antiderivative = 2.32, number of steps used = 21, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6857, 2142, 14, 2144, 1642, 840, 1180, 214, 211} \[ \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-d+c x^2} \, dx=\frac {2 \sqrt {d} \sqrt {a \sqrt {d}-\sqrt {a^2 d+b^2 c}} \arctan \left (\frac {\sqrt [4]{c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {a \sqrt {d}-\sqrt {a^2 d+b^2 c}}}\right )}{c^{3/4}}+\frac {2 \sqrt {d} \sqrt {\sqrt {a^2 d+b^2 c}+a \sqrt {d}} \arctan \left (\frac {\sqrt [4]{c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {\sqrt {a^2 d+b^2 c}+a \sqrt {d}}}\right )}{c^{3/4}}-\frac {2 \sqrt {d} \sqrt {a \sqrt {d}-\sqrt {a^2 d+b^2 c}} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {a \sqrt {d}-\sqrt {a^2 d+b^2 c}}}\right )}{c^{3/4}}-\frac {2 \sqrt {d} \sqrt {\sqrt {a^2 d+b^2 c}+a \sqrt {d}} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {\sqrt {a^2 d+b^2 c}+a \sqrt {d}}}\right )}{c^{3/4}}-\frac {b^2}{a \sqrt {\sqrt {a^2 x^2+b^2}+a x}}+\frac {\left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}{3 a} \]

[In]

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

[Out]

-(b^2/(a*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])) + (a*x + Sqrt[b^2 + a^2*x^2])^(3/2)/(3*a) + (2*Sqrt[d]*Sqrt[a*Sqrt[
d] - Sqrt[b^2*c + a^2*d]]*ArcTan[(c^(1/4)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqrt[a*Sqrt[d] - Sqrt[b^2*c + a^2*d
]]])/c^(3/4) + (2*Sqrt[d]*Sqrt[a*Sqrt[d] + Sqrt[b^2*c + a^2*d]]*ArcTan[(c^(1/4)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]
])/Sqrt[a*Sqrt[d] + Sqrt[b^2*c + a^2*d]]])/c^(3/4) - (2*Sqrt[d]*Sqrt[a*Sqrt[d] - Sqrt[b^2*c + a^2*d]]*ArcTanh[
(c^(1/4)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqrt[a*Sqrt[d] - Sqrt[b^2*c + a^2*d]]])/c^(3/4) - (2*Sqrt[d]*Sqrt[a*
Sqrt[d] + Sqrt[b^2*c + a^2*d]]*ArcTanh[(c^(1/4)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqrt[a*Sqrt[d] + Sqrt[b^2*c +
a^2*d]]])/c^(3/4)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 214

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

Rule 840

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

Rule 1180

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

Rule 1642

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 2142

Int[((g_.) + (h_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_))^(p_.), x_Symbol] :> Dist[1/(2*
e), Subst[Int[(g + h*x^n)^p*((d^2 + a*f^2 - 2*d*x + x^2)/(d - x)^2), x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /
; FreeQ[{a, c, d, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[p]

Rule 2144

Int[((g_.) + (h_.)*(x_))^(m_.)*((e_.)*(x_) + (f_.)*Sqrt[(a_.) + (c_.)*(x_)^2])^(n_.), x_Symbol] :> Dist[1/(2^(
m + 1)*e^(m + 1)), Subst[Int[x^(n - m - 2)*(a*f^2 + x^2)*((-a)*f^2*h + 2*e*g*x + h*x^2)^m, x], x, e*x + f*Sqrt
[a + c*x^2]], x] /; FreeQ[{a, c, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[m]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}+\frac {2 d \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-d+c x^2}\right ) \, dx \\ & = (2 d) \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-d+c x^2} \, dx+\int \sqrt {a x+\sqrt {b^2+a^2 x^2}} \, dx \\ & = \frac {\text {Subst}\left (\int \frac {b^2+x^2}{x^{3/2}} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 a}+(2 d) \int \left (-\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {c} x\right )}-\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{2 \sqrt {d} \left (\sqrt {d}+\sqrt {c} x\right )}\right ) \, dx \\ & = \frac {\text {Subst}\left (\int \left (\frac {b^2}{x^{3/2}}+\sqrt {x}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 a}-\sqrt {d} \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {d}-\sqrt {c} x} \, dx-\sqrt {d} \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {d}+\sqrt {c} x} \, dx \\ & = -\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}-\sqrt {d} \text {Subst}\left (\int \frac {b^2+x^2}{\sqrt {x} \left (b^2 \sqrt {c}+2 a \sqrt {d} x-\sqrt {c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )-\sqrt {d} \text {Subst}\left (\int \frac {b^2+x^2}{\sqrt {x} \left (-b^2 \sqrt {c}+2 a \sqrt {d} x+\sqrt {c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right ) \\ & = -\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}-\sqrt {d} \text {Subst}\left (\int \left (-\frac {1}{\sqrt {c} \sqrt {x}}+\frac {2 \left (b^2 \sqrt {c}+a \sqrt {d} x\right )}{\sqrt {c} \sqrt {x} \left (b^2 \sqrt {c}+2 a \sqrt {d} x-\sqrt {c} x^2\right )}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )-\sqrt {d} \text {Subst}\left (\int \left (\frac {1}{\sqrt {c} \sqrt {x}}+\frac {2 \left (b^2 \sqrt {c}-a \sqrt {d} x\right )}{\sqrt {c} \sqrt {x} \left (-b^2 \sqrt {c}+2 a \sqrt {d} x+\sqrt {c} x^2\right )}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right ) \\ & = -\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}-\frac {\left (2 \sqrt {d}\right ) \text {Subst}\left (\int \frac {b^2 \sqrt {c}+a \sqrt {d} x}{\sqrt {x} \left (b^2 \sqrt {c}+2 a \sqrt {d} x-\sqrt {c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{\sqrt {c}}-\frac {\left (2 \sqrt {d}\right ) \text {Subst}\left (\int \frac {b^2 \sqrt {c}-a \sqrt {d} x}{\sqrt {x} \left (-b^2 \sqrt {c}+2 a \sqrt {d} x+\sqrt {c} x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{\sqrt {c}} \\ & = -\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}-\frac {\left (4 \sqrt {d}\right ) \text {Subst}\left (\int \frac {b^2 \sqrt {c}+a \sqrt {d} x^2}{b^2 \sqrt {c}+2 a \sqrt {d} x^2-\sqrt {c} x^4} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {c}}-\frac {\left (4 \sqrt {d}\right ) \text {Subst}\left (\int \frac {b^2 \sqrt {c}-a \sqrt {d} x^2}{-b^2 \sqrt {c}+2 a \sqrt {d} x^2+\sqrt {c} x^4} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {c}} \\ & = -\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}-\frac {\left (2 \sqrt {d} \left (a \sqrt {d}-\sqrt {b^2 c+a^2 d}\right )\right ) \text {Subst}\left (\int \frac {1}{a \sqrt {d}-\sqrt {b^2 c+a^2 d}-\sqrt {c} x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {c}}+\frac {\left (2 \sqrt {d} \left (a \sqrt {d}-\sqrt {b^2 c+a^2 d}\right )\right ) \text {Subst}\left (\int \frac {1}{a \sqrt {d}-\sqrt {b^2 c+a^2 d}+\sqrt {c} x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {c}}-\frac {\left (2 \sqrt {d} \left (a \sqrt {d}+\sqrt {b^2 c+a^2 d}\right )\right ) \text {Subst}\left (\int \frac {1}{a \sqrt {d}+\sqrt {b^2 c+a^2 d}-\sqrt {c} x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {c}}+\frac {\left (2 \sqrt {d} \left (a \sqrt {d}+\sqrt {b^2 c+a^2 d}\right )\right ) \text {Subst}\left (\int \frac {1}{a \sqrt {d}+\sqrt {b^2 c+a^2 d}+\sqrt {c} x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{\sqrt {c}} \\ & = -\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}+\frac {2 \sqrt {d} \sqrt {a \sqrt {d}-\sqrt {b^2 c+a^2 d}} \arctan \left (\frac {\sqrt [4]{c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a \sqrt {d}-\sqrt {b^2 c+a^2 d}}}\right )}{c^{3/4}}+\frac {2 \sqrt {d} \sqrt {a \sqrt {d}+\sqrt {b^2 c+a^2 d}} \arctan \left (\frac {\sqrt [4]{c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a \sqrt {d}+\sqrt {b^2 c+a^2 d}}}\right )}{c^{3/4}}-\frac {2 \sqrt {d} \sqrt {a \sqrt {d}-\sqrt {b^2 c+a^2 d}} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a \sqrt {d}-\sqrt {b^2 c+a^2 d}}}\right )}{c^{3/4}}-\frac {2 \sqrt {d} \sqrt {a \sqrt {d}+\sqrt {b^2 c+a^2 d}} \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a \sqrt {d}+\sqrt {b^2 c+a^2 d}}}\right )}{c^{3/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-d+c x^2} \, dx=-\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}+a d \text {RootSum}\left [b^4 c-2 b^2 c \text {$\#$1}^4-4 a^2 d \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {-b^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-\log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{b^2 c \text {$\#$1}+2 a^2 d \text {$\#$1}-c \text {$\#$1}^5}\&\right ] \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.19

\[\int \frac {\left (c \,x^{2}+d \right ) \sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}{c \,x^{2}-d}d x\]

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.33 (sec) , antiderivative size = 1727, normalized size of antiderivative = 8.81 \[ \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-d+c x^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/3*(3*a*sqrt(-sqrt((b^2*c*d^2 + 2*a^2*d^3 + 2*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/c^6))/c^3))*log(16*sqrt(a*x
+ sqrt(a^2*x^2 + b^2))*a*b^2*d^4 + 16*(a^2*c*d^3 - c^4*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/c^6))*sqrt(-sqrt((b^2*c
*d^2 + 2*a^2*d^3 + 2*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/c^6))/c^3))*sqrt((b^2*c*d^2 + 2*a^2*d^3 + 2*c^3*sqrt((
a^2*b^2*c*d^5 + a^4*d^6)/c^6))/c^3)) - 3*a*sqrt(-sqrt((b^2*c*d^2 + 2*a^2*d^3 + 2*c^3*sqrt((a^2*b^2*c*d^5 + a^4
*d^6)/c^6))/c^3))*log(16*sqrt(a*x + sqrt(a^2*x^2 + b^2))*a*b^2*d^4 - 16*(a^2*c*d^3 - c^4*sqrt((a^2*b^2*c*d^5 +
a^4*d^6)/c^6))*sqrt(-sqrt((b^2*c*d^2 + 2*a^2*d^3 + 2*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/c^6))/c^3))*sqrt((b^2
*c*d^2 + 2*a^2*d^3 + 2*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/c^6))/c^3)) + 3*a*sqrt(-sqrt((b^2*c*d^2 + 2*a^2*d^3
- 2*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/c^6))/c^3))*log(16*sqrt(a*x + sqrt(a^2*x^2 + b^2))*a*b^2*d^4 + 16*(a^2*
c*d^3 + c^4*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/c^6))*sqrt(-sqrt((b^2*c*d^2 + 2*a^2*d^3 - 2*c^3*sqrt((a^2*b^2*c*d^5
+ a^4*d^6)/c^6))/c^3))*sqrt((b^2*c*d^2 + 2*a^2*d^3 - 2*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/c^6))/c^3)) - 3*a*s
qrt(-sqrt((b^2*c*d^2 + 2*a^2*d^3 - 2*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/c^6))/c^3))*log(16*sqrt(a*x + sqrt(a^2
*x^2 + b^2))*a*b^2*d^4 - 16*(a^2*c*d^3 + c^4*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/c^6))*sqrt(-sqrt((b^2*c*d^2 + 2*a^
2*d^3 - 2*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/c^6))/c^3))*sqrt((b^2*c*d^2 + 2*a^2*d^3 - 2*c^3*sqrt((a^2*b^2*c*d
^5 + a^4*d^6)/c^6))/c^3)) - 3*a*((b^2*c*d^2 + 2*a^2*d^3 + 2*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/c^6))/c^3)^(1/4
)*log(16*sqrt(a*x + sqrt(a^2*x^2 + b^2))*a*b^2*d^4 + 16*(a^2*c*d^3 - c^4*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/c^6))*
((b^2*c*d^2 + 2*a^2*d^3 + 2*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/c^6))/c^3)^(3/4)) + 3*a*((b^2*c*d^2 + 2*a^2*d^3
+ 2*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/c^6))/c^3)^(1/4)*log(16*sqrt(a*x + sqrt(a^2*x^2 + b^2))*a*b^2*d^4 - 16
*(a^2*c*d^3 - c^4*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/c^6))*((b^2*c*d^2 + 2*a^2*d^3 + 2*c^3*sqrt((a^2*b^2*c*d^5 + a
^4*d^6)/c^6))/c^3)^(3/4)) - 3*a*((b^2*c*d^2 + 2*a^2*d^3 - 2*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/c^6))/c^3)^(1/4
)*log(16*sqrt(a*x + sqrt(a^2*x^2 + b^2))*a*b^2*d^4 + 16*(a^2*c*d^3 + c^4*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/c^6))*
((b^2*c*d^2 + 2*a^2*d^3 - 2*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/c^6))/c^3)^(3/4)) + 3*a*((b^2*c*d^2 + 2*a^2*d^3
- 2*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/c^6))/c^3)^(1/4)*log(16*sqrt(a*x + sqrt(a^2*x^2 + b^2))*a*b^2*d^4 - 16
*(a^2*c*d^3 + c^4*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/c^6))*((b^2*c*d^2 + 2*a^2*d^3 - 2*c^3*sqrt((a^2*b^2*c*d^5 + a
^4*d^6)/c^6))/c^3)^(3/4)) - 2*(2*a*x - sqrt(a^2*x^2 + b^2))*sqrt(a*x + sqrt(a^2*x^2 + b^2)))/a

Sympy [N/A]

Not integrable

Time = 2.47 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.17 \[ \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-d+c x^2} \, dx=\int \frac {\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \left (c x^{2} + d\right )}{c x^{2} - d}\, dx \]

[In]

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

[Out]

Integral(sqrt(a*x + sqrt(a**2*x**2 + b**2))*(c*x**2 + d)/(c*x**2 - d), x)

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.20 \[ \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-d+c x^2} \, dx=\int { \frac {{\left (c x^{2} + d\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{c x^{2} - d} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.52 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.20 \[ \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-d+c x^2} \, dx=\int { \frac {{\left (c x^{2} + d\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{c x^{2} - d} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 6.67 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.20 \[ \int \frac {\left (d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-d+c x^2} \, dx=\int -\frac {\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}\,\left (c\,x^2+d\right )}{d-c\,x^2} \,d x \]

[In]

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

[Out]

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