Optimal. Leaf size=161 \[ -\frac {a^2 \left (c+d x^3\right )^{3/2}}{3 b^2 \left (a+b x^3\right ) (b c-a d)}+\frac {a (4 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 b^{7/2} \sqrt {b c-a d}}-\frac {a \sqrt {c+d x^3} (4 b c-5 a d)}{3 b^3 (b c-a d)}+\frac {2 \left (c+d x^3\right )^{3/2}}{9 b^2 d} \]
________________________________________________________________________________________
Rubi [A] time = 0.19, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {446, 89, 80, 50, 63, 208} \begin {gather*} -\frac {a^2 \left (c+d x^3\right )^{3/2}}{3 b^2 \left (a+b x^3\right ) (b c-a d)}-\frac {a \sqrt {c+d x^3} (4 b c-5 a d)}{3 b^3 (b c-a d)}+\frac {a (4 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 b^{7/2} \sqrt {b c-a d}}+\frac {2 \left (c+d x^3\right )^{3/2}}{9 b^2 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 63
Rule 80
Rule 89
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {x^8 \sqrt {c+d x^3}}{\left (a+b x^3\right )^2} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2 \sqrt {c+d x}}{(a+b x)^2} \, dx,x,x^3\right )\\ &=-\frac {a^2 \left (c+d x^3\right )^{3/2}}{3 b^2 (b c-a d) \left (a+b x^3\right )}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {c+d x} \left (-\frac {1}{2} a (2 b c-3 a d)+b (b c-a d) x\right )}{a+b x} \, dx,x,x^3\right )}{3 b^2 (b c-a d)}\\ &=\frac {2 \left (c+d x^3\right )^{3/2}}{9 b^2 d}-\frac {a^2 \left (c+d x^3\right )^{3/2}}{3 b^2 (b c-a d) \left (a+b x^3\right )}-\frac {(a (4 b c-5 a d)) \operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{a+b x} \, dx,x,x^3\right )}{6 b^2 (b c-a d)}\\ &=-\frac {a (4 b c-5 a d) \sqrt {c+d x^3}}{3 b^3 (b c-a d)}+\frac {2 \left (c+d x^3\right )^{3/2}}{9 b^2 d}-\frac {a^2 \left (c+d x^3\right )^{3/2}}{3 b^2 (b c-a d) \left (a+b x^3\right )}-\frac {(a (4 b c-5 a d)) \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^3\right )}{6 b^3}\\ &=-\frac {a (4 b c-5 a d) \sqrt {c+d x^3}}{3 b^3 (b c-a d)}+\frac {2 \left (c+d x^3\right )^{3/2}}{9 b^2 d}-\frac {a^2 \left (c+d x^3\right )^{3/2}}{3 b^2 (b c-a d) \left (a+b x^3\right )}-\frac {(a (4 b c-5 a d)) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{3 b^3 d}\\ &=-\frac {a (4 b c-5 a d) \sqrt {c+d x^3}}{3 b^3 (b c-a d)}+\frac {2 \left (c+d x^3\right )^{3/2}}{9 b^2 d}-\frac {a^2 \left (c+d x^3\right )^{3/2}}{3 b^2 (b c-a d) \left (a+b x^3\right )}+\frac {a (4 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 b^{7/2} \sqrt {b c-a d}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.29, size = 147, normalized size = 0.91 \begin {gather*} \frac {-\frac {a^2 \left (c+d x^3\right )^{3/2}}{a+b x^3}+\frac {a (5 a d-4 b c) \left (\sqrt {b} \sqrt {c+d x^3}-\sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )\right )}{b^{3/2}}+\frac {2 \left (c+d x^3\right )^{3/2} (b c-a d)}{3 d}}{3 b^2 (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.27, size = 142, normalized size = 0.88 \begin {gather*} \frac {\left (4 a b c-5 a^2 d\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3} \sqrt {a d-b c}}{b c-a d}\right )}{3 b^{7/2} \sqrt {a d-b c}}+\frac {\sqrt {c+d x^3} \left (-15 a^2 d+2 a b c-10 a b d x^3+2 b^2 c x^3+2 b^2 d x^6\right )}{9 b^3 d \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.92, size = 469, normalized size = 2.91 \begin {gather*} \left [-\frac {3 \, {\left (4 \, a^{2} b c d - 5 \, a^{3} d^{2} + {\left (4 \, a b^{2} c d - 5 \, a^{2} b d^{2}\right )} x^{3}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x^{3} + 2 \, b c - a d - 2 \, \sqrt {d x^{3} + c} \sqrt {b^{2} c - a b d}}{b x^{3} + a}\right ) - 2 \, {\left (2 \, {\left (b^{4} c d - a b^{3} d^{2}\right )} x^{6} + 2 \, a b^{3} c^{2} - 17 \, a^{2} b^{2} c d + 15 \, a^{3} b d^{2} + 2 \, {\left (b^{4} c^{2} - 6 \, a b^{3} c d + 5 \, a^{2} b^{2} d^{2}\right )} x^{3}\right )} \sqrt {d x^{3} + c}}{18 \, {\left (a b^{5} c d - a^{2} b^{4} d^{2} + {\left (b^{6} c d - a b^{5} d^{2}\right )} x^{3}\right )}}, -\frac {3 \, {\left (4 \, a^{2} b c d - 5 \, a^{3} d^{2} + {\left (4 \, a b^{2} c d - 5 \, a^{2} b d^{2}\right )} x^{3}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-b^{2} c + a b d}}{b d x^{3} + b c}\right ) - {\left (2 \, {\left (b^{4} c d - a b^{3} d^{2}\right )} x^{6} + 2 \, a b^{3} c^{2} - 17 \, a^{2} b^{2} c d + 15 \, a^{3} b d^{2} + 2 \, {\left (b^{4} c^{2} - 6 \, a b^{3} c d + 5 \, a^{2} b^{2} d^{2}\right )} x^{3}\right )} \sqrt {d x^{3} + c}}{9 \, {\left (a b^{5} c d - a^{2} b^{4} d^{2} + {\left (b^{6} c d - a b^{5} d^{2}\right )} x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 136, normalized size = 0.84 \begin {gather*} -\frac {\sqrt {d x^{3} + c} a^{2} d}{3 \, {\left ({\left (d x^{3} + c\right )} b - b c + a d\right )} b^{3}} - \frac {{\left (4 \, a b c - 5 \, a^{2} d\right )} \arctan \left (\frac {\sqrt {d x^{3} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{3 \, \sqrt {-b^{2} c + a b d} b^{3}} + \frac {2 \, {\left ({\left (d x^{3} + c\right )}^{\frac {3}{2}} b^{4} d^{2} - 6 \, \sqrt {d x^{3} + c} a b^{3} d^{3}\right )}}{9 \, b^{6} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.36, size = 917, normalized size = 5.70
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 6.84, size = 202, normalized size = 1.25 \begin {gather*} \frac {2\,x^3\,\sqrt {d\,x^3+c}}{9\,b^2}-\frac {\sqrt {d\,x^3+c}\,\left (\frac {4\,c}{3\,b^2}-\frac {2\,b^2\,c-2\,a\,b\,d}{b^4}+\frac {2\,a\,d}{b^3}\right )}{3\,d}+\frac {a^2\,\left (\frac {2\,a\,d}{3\,\left (2\,b^2\,c-2\,a\,b\,d\right )}-\frac {2\,b\,c}{3\,\left (2\,b^2\,c-2\,a\,b\,d\right )}\right )\,\sqrt {d\,x^3+c}}{b^2\,\left (b\,x^3+a\right )}+\frac {a\,\ln \left (\frac {2\,b\,c-a\,d+b\,d\,x^3+\sqrt {b}\,\sqrt {d\,x^3+c}\,\sqrt {a\,d-b\,c}\,2{}\mathrm {i}}{b\,x^3+a}\right )\,\left (5\,a\,d-4\,b\,c\right )\,1{}\mathrm {i}}{6\,b^{7/2}\,\sqrt {a\,d-b\,c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________