3.25.58 \(\int \frac {d+c x^2}{(-d+c x^2) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\) [2458]

Optimal. Leaf size=199 \[ -\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}+a d \text {RootSum}\left [b^4 c-2 b^2 c \text {$\#$1}^4-4 a^2 d \text {$\#$1}^4+c \text {$\#$1}^8\& ,\frac {-b^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-\log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{b^2 c \text {$\#$1}^3+2 a^2 d \text {$\#$1}^3-c \text {$\#$1}^7}\& \right ] \]

[Out]

Unintegrable

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(453\) vs. \(2(199)=398\).
time = 1.23, antiderivative size = 453, normalized size of antiderivative = 2.28, number of steps used = 23, number of rules used = 10, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6857, 2142, 14, 2144, 1642, 842, 840, 1180, 214, 211} \begin {gather*} -\frac {2 \sqrt {d} \text {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 )}{\sqrt [4]{c} \sqrt {a \sqrt {d}-\sqrt {a^2 d+b^2 c}}}-\frac {2 \sqrt {d} \text {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 )}{\sqrt [4]{c} \sqrt {\sqrt {a^2 d+b^2 c}+a \sqrt {d}}}-\frac {2 \sqrt {d} \tanh ^{-1}\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 )}{\sqrt [4]{c} \sqrt {a \sqrt {d}-\sqrt {a^2 d+b^2 c}}}-\frac {2 \sqrt {d} \tanh ^{-1}\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 )}{\sqrt [4]{c} \sqrt {\sqrt {a^2 d+b^2 c}+a \sqrt {d}}}-\frac {b^2}{3 a \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}+\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 842

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e
*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(d +
e*x)^(m + 1)*(Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c,
d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]

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*} \int \frac {d+c x^2}{\left (-d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx &=\int \left (\frac {1}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {2 d}{\left (-d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}\right ) \, dx\\ &=(2 d) \int \frac {1}{\left (-d+c x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx+\int \frac {1}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\\ &=\frac {\text {Subst}\left (\int \frac {b^2+x^2}{x^{5/2}} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 a}+(2 d) \int \left (-\frac {1}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {c} x\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}-\frac {1}{2 \sqrt {d} \left (\sqrt {d}+\sqrt {c} x\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}\right ) \, dx\\ &=\frac {\text {Subst}\left (\int \left (\frac {b^2}{x^{5/2}}+\frac {1}{\sqrt {x}}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 a}-\sqrt {d} \int \frac {1}{\left (\sqrt {d}-\sqrt {c} x\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx-\sqrt {d} \int \frac {1}{\left (\sqrt {d}+\sqrt {c} x\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}-\sqrt {d} \text {Subst}\left (\int \frac {b^2+x^2}{x^{3/2} \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}{x^{3/2} \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}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}-\sqrt {d} \text {Subst}\left (\int \left (-\frac {1}{\sqrt {c} x^{3/2}}+\frac {2 \left (b^2 \sqrt {c}+a \sqrt {d} x\right )}{\sqrt {c} x^{3/2} \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} x^{3/2}}+\frac {2 \left (b^2 \sqrt {c}-a \sqrt {d} x\right )}{\sqrt {c} x^{3/2} \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}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}-\frac {\left (2 \sqrt {d}\right ) \text {Subst}\left (\int \frac {b^2 \sqrt {c}+a \sqrt {d} x}{x^{3/2} \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}{x^{3/2} \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}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}-\frac {\left (2 \sqrt {d}\right ) \text {Subst}\left (\int \frac {-a b^2 \sqrt {c} \sqrt {d}+b^2 c 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 )}{b^2 c}+\frac {\left (2 \sqrt {d}\right ) \text {Subst}\left (\int \frac {-a b^2 \sqrt {c} \sqrt {d}-b^2 c 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 )}{b^2 c}\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}-\frac {\left (4 \sqrt {d}\right ) \text {Subst}\left (\int \frac {-a b^2 \sqrt {c} \sqrt {d}+b^2 c 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 )}{b^2 c}+\frac {\left (4 \sqrt {d}\right ) \text {Subst}\left (\int \frac {-a b^2 \sqrt {c} \sqrt {d}-b^2 c 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 )}{b^2 c}\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}-\left (2 \sqrt {d}\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 )-\left (2 \sqrt {d}\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 )-\left (2 \sqrt {d}\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 )-\left (2 \sqrt {d}\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 )\\ &=-\frac {b^2}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{a}-\frac {2 \sqrt {d} \tan ^{-1}\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 )}{\sqrt [4]{c} \sqrt {a \sqrt {d}-\sqrt {b^2 c+a^2 d}}}-\frac {2 \sqrt {d} \tan ^{-1}\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 )}{\sqrt [4]{c} \sqrt {a \sqrt {d}+\sqrt {b^2 c+a^2 d}}}-\frac {2 \sqrt {d} \tanh ^{-1}\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 )}{\sqrt [4]{c} \sqrt {a \sqrt {d}-\sqrt {b^2 c+a^2 d}}}-\frac {2 \sqrt {d} \tanh ^{-1}\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 )}{\sqrt [4]{c} \sqrt {a \sqrt {d}+\sqrt {b^2 c+a^2 d}}}\\ \end {align*}

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Mathematica [A]
time = 0.25, size = 194, normalized size = 0.97 \begin {gather*} \frac {2 \left (b^2+3 a x \left (a x+\sqrt {b^2+a^2 x^2}\right )\right )}{3 a \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}+a d \text {RootSum}\left [b^4 c-2 b^2 c \text {$\#$1}^4-4 a^2 d \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {b^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )+\log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^4}{-b^2 c \text {$\#$1}^3-2 a^2 d \text {$\#$1}^3+c \text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b^2 + 3*a*x*(a*x + Sqrt[b^2 + a^2*x^2])))/(3*a*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2)) + 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*#1^3) - 2*a^2*d*#1^3 + c*#1^7) & ]

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {c \,x^{2}+d}{\left (c \,x^{2}-d \right ) \sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.45, size = 1894, normalized size = 9.52 \begin {gather*} -\frac {12 \, a b^{2} \left (\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}} \arctan \left (\frac {{\left (\sqrt {a^{3} d^{6} x + \sqrt {a^{2} x^{2} + b^{2}} a^{2} d^{6} - {\left (2 \, a^{2} b^{4} c^{3} d^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - a^{2} b^{2} c d^{5} - 2 \, a^{4} d^{6}\right )} \sqrt {\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}}} {\left (b^{4} c^{4} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - a^{2} c d^{3}\right )} \sqrt {\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}} - {\left (a b^{4} c^{4} d^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - a^{3} c d^{6}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \sqrt {\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}}\right )} \left (\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}}}{a^{2} d^{7}}\right ) - 12 \, a b^{2} \left (-\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {a^{3} d^{6} x + \sqrt {a^{2} x^{2} + b^{2}} a^{2} d^{6} + {\left (2 \, a^{2} b^{4} c^{3} d^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + a^{2} b^{2} c d^{5} + 2 \, a^{4} d^{6}\right )} \sqrt {-\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}}} {\left (b^{4} c^{4} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + a^{2} c d^{3}\right )} \left (-\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {3}{4}} - {\left (a b^{4} c^{4} d^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + a^{3} c d^{6}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \left (-\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {3}{4}}}{a^{2} d^{7}}\right ) - 3 \, a b^{2} \left (\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}} \log \left (8 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a d^{3} + 8 \, {\left (b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - a^{2} d^{3}\right )} \left (\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}}\right ) + 3 \, a b^{2} \left (\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}} \log \left (8 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a d^{3} - 8 \, {\left (b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - a^{2} d^{3}\right )} \left (\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + b^{2} c d^{2} + 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}}\right ) + 3 \, a b^{2} \left (-\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}} \log \left (8 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a d^{3} + 8 \, {\left (b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + a^{2} d^{3}\right )} \left (-\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}}\right ) - 3 \, a b^{2} \left (-\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}} \log \left (8 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a d^{3} - 8 \, {\left (b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} + a^{2} d^{3}\right )} \left (-\frac {2 \, b^{4} c^{3} \sqrt {\frac {a^{2} b^{2} c d^{5} + a^{4} d^{6}}{b^{8} c^{6}}} - b^{2} c d^{2} - 2 \, a^{2} d^{3}}{b^{4} c^{3}}\right )^{\frac {1}{4}}\right ) + 2 \, {\left (a^{2} x^{2} - \sqrt {a^{2} x^{2} + b^{2}} a x - b^{2}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{3 \, a b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(12*a*b^2*((2*b^4*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/(b^8*c^6)) + b^2*c*d^2 + 2*a^2*d^3)/(b^4*c^3))^(1/4)
*arctan((sqrt(a^3*d^6*x + sqrt(a^2*x^2 + b^2)*a^2*d^6 - (2*a^2*b^4*c^3*d^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/(b^8
*c^6)) - a^2*b^2*c*d^5 - 2*a^4*d^6)*sqrt((2*b^4*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/(b^8*c^6)) + b^2*c*d^2 + 2*
a^2*d^3)/(b^4*c^3)))*(b^4*c^4*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/(b^8*c^6)) - a^2*c*d^3)*sqrt((2*b^4*c^3*sqrt((a^2
*b^2*c*d^5 + a^4*d^6)/(b^8*c^6)) + b^2*c*d^2 + 2*a^2*d^3)/(b^4*c^3)) - (a*b^4*c^4*d^3*sqrt((a^2*b^2*c*d^5 + a^
4*d^6)/(b^8*c^6)) - a^3*c*d^6)*sqrt(a*x + sqrt(a^2*x^2 + b^2))*sqrt((2*b^4*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/
(b^8*c^6)) + b^2*c*d^2 + 2*a^2*d^3)/(b^4*c^3)))*((2*b^4*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/(b^8*c^6)) + b^2*c*
d^2 + 2*a^2*d^3)/(b^4*c^3))^(1/4)/(a^2*d^7)) - 12*a*b^2*(-(2*b^4*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/(b^8*c^6))
- b^2*c*d^2 - 2*a^2*d^3)/(b^4*c^3))^(1/4)*arctan((sqrt(a^3*d^6*x + sqrt(a^2*x^2 + b^2)*a^2*d^6 + (2*a^2*b^4*c
^3*d^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/(b^8*c^6)) + a^2*b^2*c*d^5 + 2*a^4*d^6)*sqrt(-(2*b^4*c^3*sqrt((a^2*b^2*c
*d^5 + a^4*d^6)/(b^8*c^6)) - b^2*c*d^2 - 2*a^2*d^3)/(b^4*c^3)))*(b^4*c^4*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/(b^8*c
^6)) + a^2*c*d^3)*(-(2*b^4*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/(b^8*c^6)) - b^2*c*d^2 - 2*a^2*d^3)/(b^4*c^3))^(
3/4) - (a*b^4*c^4*d^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/(b^8*c^6)) + a^3*c*d^6)*sqrt(a*x + sqrt(a^2*x^2 + b^2))*(
-(2*b^4*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/(b^8*c^6)) - b^2*c*d^2 - 2*a^2*d^3)/(b^4*c^3))^(3/4))/(a^2*d^7)) -
3*a*b^2*((2*b^4*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/(b^8*c^6)) + b^2*c*d^2 + 2*a^2*d^3)/(b^4*c^3))^(1/4)*log(8*
sqrt(a*x + sqrt(a^2*x^2 + b^2))*a*d^3 + 8*(b^4*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/(b^8*c^6)) - a^2*d^3)*((2*b^
4*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/(b^8*c^6)) + b^2*c*d^2 + 2*a^2*d^3)/(b^4*c^3))^(1/4)) + 3*a*b^2*((2*b^4*c
^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/(b^8*c^6)) + b^2*c*d^2 + 2*a^2*d^3)/(b^4*c^3))^(1/4)*log(8*sqrt(a*x + sqrt(a
^2*x^2 + b^2))*a*d^3 - 8*(b^4*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/(b^8*c^6)) - a^2*d^3)*((2*b^4*c^3*sqrt((a^2*b
^2*c*d^5 + a^4*d^6)/(b^8*c^6)) + b^2*c*d^2 + 2*a^2*d^3)/(b^4*c^3))^(1/4)) + 3*a*b^2*(-(2*b^4*c^3*sqrt((a^2*b^2
*c*d^5 + a^4*d^6)/(b^8*c^6)) - b^2*c*d^2 - 2*a^2*d^3)/(b^4*c^3))^(1/4)*log(8*sqrt(a*x + sqrt(a^2*x^2 + b^2))*a
*d^3 + 8*(b^4*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/(b^8*c^6)) + a^2*d^3)*(-(2*b^4*c^3*sqrt((a^2*b^2*c*d^5 + a^4*
d^6)/(b^8*c^6)) - b^2*c*d^2 - 2*a^2*d^3)/(b^4*c^3))^(1/4)) - 3*a*b^2*(-(2*b^4*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^
6)/(b^8*c^6)) - b^2*c*d^2 - 2*a^2*d^3)/(b^4*c^3))^(1/4)*log(8*sqrt(a*x + sqrt(a^2*x^2 + b^2))*a*d^3 - 8*(b^4*c
^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/(b^8*c^6)) + a^2*d^3)*(-(2*b^4*c^3*sqrt((a^2*b^2*c*d^5 + a^4*d^6)/(b^8*c^6))
- b^2*c*d^2 - 2*a^2*d^3)/(b^4*c^3))^(1/4)) + 2*(a^2*x^2 - sqrt(a^2*x^2 + b^2)*a*x - b^2)*sqrt(a*x + sqrt(a^2*
x^2 + b^2)))/(a*b^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {c\,x^2+d}{\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}\,\left (d-c\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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