Optimal. Leaf size=189 \[ -\frac{a^2 \left (c+d x^3\right )^{5/2}}{3 b^2 \left (a+b x^3\right ) (b c-a d)}-\frac{a \left (c+d x^3\right )^{3/2} (4 b c-7 a d)}{9 b^3 (b c-a d)}-\frac{a \sqrt{c+d x^3} (4 b c-7 a d)}{3 b^4}+\frac{a (4 b c-7 a d) \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{9/2}}+\frac{2 \left (c+d x^3\right )^{5/2}}{15 b^2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.237007, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {446, 89, 80, 50, 63, 208} \[ -\frac{a^2 \left (c+d x^3\right )^{5/2}}{3 b^2 \left (a+b x^3\right ) (b c-a d)}-\frac{a \left (c+d x^3\right )^{3/2} (4 b c-7 a d)}{9 b^3 (b c-a d)}-\frac{a \sqrt{c+d x^3} (4 b c-7 a d)}{3 b^4}+\frac{a (4 b c-7 a d) \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{9/2}}+\frac{2 \left (c+d x^3\right )^{5/2}}{15 b^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 446
Rule 89
Rule 80
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^8 \left (c+d x^3\right )^{3/2}}{\left (a+b x^3\right )^2} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2 (c+d x)^{3/2}}{(a+b x)^2} \, dx,x,x^3\right )\\ &=-\frac{a^2 \left (c+d x^3\right )^{5/2}}{3 b^2 (b c-a d) \left (a+b x^3\right )}+\frac{\operatorname{Subst}\left (\int \frac{(c+d x)^{3/2} \left (-\frac{1}{2} a (2 b c-5 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 )^{5/2}}{15 b^2 d}-\frac{a^2 \left (c+d x^3\right )^{5/2}}{3 b^2 (b c-a d) \left (a+b x^3\right )}-\frac{(a (4 b c-7 a d)) \operatorname{Subst}\left (\int \frac{(c+d x)^{3/2}}{a+b x} \, dx,x,x^3\right )}{6 b^2 (b c-a d)}\\ &=-\frac{a (4 b c-7 a d) \left (c+d x^3\right )^{3/2}}{9 b^3 (b c-a d)}+\frac{2 \left (c+d x^3\right )^{5/2}}{15 b^2 d}-\frac{a^2 \left (c+d x^3\right )^{5/2}}{3 b^2 (b c-a d) \left (a+b x^3\right )}-\frac{(a (4 b c-7 a d)) \operatorname{Subst}\left (\int \frac{\sqrt{c+d x}}{a+b x} \, dx,x,x^3\right )}{6 b^3}\\ &=-\frac{a (4 b c-7 a d) \sqrt{c+d x^3}}{3 b^4}-\frac{a (4 b c-7 a d) \left (c+d x^3\right )^{3/2}}{9 b^3 (b c-a d)}+\frac{2 \left (c+d x^3\right )^{5/2}}{15 b^2 d}-\frac{a^2 \left (c+d x^3\right )^{5/2}}{3 b^2 (b c-a d) \left (a+b x^3\right )}-\frac{(a (4 b c-7 a d) (b c-a d)) \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,x^3\right )}{6 b^4}\\ &=-\frac{a (4 b c-7 a d) \sqrt{c+d x^3}}{3 b^4}-\frac{a (4 b c-7 a d) \left (c+d x^3\right )^{3/2}}{9 b^3 (b c-a d)}+\frac{2 \left (c+d x^3\right )^{5/2}}{15 b^2 d}-\frac{a^2 \left (c+d x^3\right )^{5/2}}{3 b^2 (b c-a d) \left (a+b x^3\right )}-\frac{(a (4 b c-7 a d) (b c-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^4 d}\\ &=-\frac{a (4 b c-7 a d) \sqrt{c+d x^3}}{3 b^4}-\frac{a (4 b c-7 a d) \left (c+d x^3\right )^{3/2}}{9 b^3 (b c-a d)}+\frac{2 \left (c+d x^3\right )^{5/2}}{15 b^2 d}-\frac{a^2 \left (c+d x^3\right )^{5/2}}{3 b^2 (b c-a d) \left (a+b x^3\right )}+\frac{a (4 b c-7 a d) \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.162128, size = 162, normalized size = 0.86 \[ \frac{\sqrt{c+d x^3} \left (5 a^2 b d \left (14 d x^3-19 c\right )+105 a^3 d^2+2 a b^2 \left (3 c^2-34 c d x^3-7 d^2 x^6\right )+6 b^3 x^3 \left (c+d x^3\right )^2\right )}{45 b^4 d \left (a+b x^3\right )}+\frac{a (4 b c-7 a d) \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{9/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.039, size = 1003, normalized size = 5.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.18314, size = 944, normalized size = 4.99 \begin{align*} \left [-\frac{15 \,{\left (4 \, a^{2} b c d - 7 \, a^{3} d^{2} +{\left (4 \, a b^{2} c d - 7 \, a^{2} b d^{2}\right )} x^{3}\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x^{3} + 2 \, b c - a d - 2 \, \sqrt{d x^{3} + c} b \sqrt{\frac{b c - a d}{b}}}{b x^{3} + a}\right ) - 2 \,{\left (6 \, b^{3} d^{2} x^{9} + 2 \,{\left (6 \, b^{3} c d - 7 \, a b^{2} d^{2}\right )} x^{6} + 6 \, a b^{2} c^{2} - 95 \, a^{2} b c d + 105 \, a^{3} d^{2} + 2 \,{\left (3 \, b^{3} c^{2} - 34 \, a b^{2} c d + 35 \, a^{2} b d^{2}\right )} x^{3}\right )} \sqrt{d x^{3} + c}}{90 \,{\left (b^{5} d x^{3} + a b^{4} d\right )}}, \frac{15 \,{\left (4 \, a^{2} b c d - 7 \, a^{3} d^{2} +{\left (4 \, a b^{2} c d - 7 \, a^{2} b d^{2}\right )} x^{3}\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (-\frac{\sqrt{d x^{3} + c} b \sqrt{-\frac{b c - a d}{b}}}{b c - a d}\right ) +{\left (6 \, b^{3} d^{2} x^{9} + 2 \,{\left (6 \, b^{3} c d - 7 \, a b^{2} d^{2}\right )} x^{6} + 6 \, a b^{2} c^{2} - 95 \, a^{2} b c d + 105 \, a^{3} d^{2} + 2 \,{\left (3 \, b^{3} c^{2} - 34 \, a b^{2} c d + 35 \, a^{2} b d^{2}\right )} x^{3}\right )} \sqrt{d x^{3} + c}}{45 \,{\left (b^{5} d x^{3} + a b^{4} d\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15928, size = 285, normalized size = 1.51 \begin{align*} -\frac{{\left (4 \, a b^{2} c^{2} - 11 \, a^{2} b c d + 7 \, a^{3} d^{2}\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^{4}} - \frac{\sqrt{d x^{3} + c} a^{2} b c d - \sqrt{d x^{3} + c} a^{3} d^{2}}{3 \,{\left ({\left (d x^{3} + c\right )} b - b c + a d\right )} b^{4}} + \frac{2 \,{\left (3 \,{\left (d x^{3} + c\right )}^{\frac{5}{2}} b^{8} d^{4} - 10 \,{\left (d x^{3} + c\right )}^{\frac{3}{2}} a b^{7} d^{5} - 30 \, \sqrt{d x^{3} + c} a b^{7} c d^{5} + 45 \, \sqrt{d x^{3} + c} a^{2} b^{6} d^{6}\right )}}{45 \, b^{10} d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]