Optimal. Leaf size=163 \[ \frac{\left (c+d x^3\right )^{3/2} (2 b c-5 a d)}{9 b^2 (b c-a d)}+\frac{\sqrt{c+d x^3} (2 b c-5 a d)}{3 b^3}-\frac{(2 b c-5 a d) \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{7/2}}+\frac{a \left (c+d x^3\right )^{5/2}}{3 b \left (a+b x^3\right ) (b c-a d)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.136803, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {446, 78, 50, 63, 208} \[ \frac{\left (c+d x^3\right )^{3/2} (2 b c-5 a d)}{9 b^2 (b c-a d)}+\frac{\sqrt{c+d x^3} (2 b c-5 a d)}{3 b^3}-\frac{(2 b c-5 a d) \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{7/2}}+\frac{a \left (c+d x^3\right )^{5/2}}{3 b \left (a+b x^3\right ) (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 446
Rule 78
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^5 \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 (c+d x)^{3/2}}{(a+b x)^2} \, dx,x,x^3\right )\\ &=\frac{a \left (c+d x^3\right )^{5/2}}{3 b (b c-a d) \left (a+b x^3\right )}+\frac{(2 b c-5 a d) \operatorname{Subst}\left (\int \frac{(c+d x)^{3/2}}{a+b x} \, dx,x,x^3\right )}{6 b (b c-a d)}\\ &=\frac{(2 b c-5 a d) \left (c+d x^3\right )^{3/2}}{9 b^2 (b c-a d)}+\frac{a \left (c+d x^3\right )^{5/2}}{3 b (b c-a d) \left (a+b x^3\right )}+\frac{(2 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}\\ &=\frac{(2 b c-5 a d) \sqrt{c+d x^3}}{3 b^3}+\frac{(2 b c-5 a d) \left (c+d x^3\right )^{3/2}}{9 b^2 (b c-a d)}+\frac{a \left (c+d x^3\right )^{5/2}}{3 b (b c-a d) \left (a+b x^3\right )}+\frac{((2 b c-5 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^3}\\ &=\frac{(2 b c-5 a d) \sqrt{c+d x^3}}{3 b^3}+\frac{(2 b c-5 a d) \left (c+d x^3\right )^{3/2}}{9 b^2 (b c-a d)}+\frac{a \left (c+d x^3\right )^{5/2}}{3 b (b c-a d) \left (a+b x^3\right )}+\frac{((2 b c-5 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^3 d}\\ &=\frac{(2 b c-5 a d) \sqrt{c+d x^3}}{3 b^3}+\frac{(2 b c-5 a d) \left (c+d x^3\right )^{3/2}}{9 b^2 (b c-a d)}+\frac{a \left (c+d x^3\right )^{5/2}}{3 b (b c-a d) \left (a+b x^3\right )}-\frac{(2 b c-5 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^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.101708, size = 125, normalized size = 0.77 \[ \frac{\sqrt{c+d x^3} \left (-15 a^2 d+a b \left (11 c-10 d x^3\right )+2 b^2 x^3 \left (4 c+d x^3\right )\right )}{9 b^3 \left (a+b x^3\right )}-\frac{(2 b c-5 a d) \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.009, size = 983, normalized size = 6. \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 = 1.6323, size = 679, normalized size = 4.17 \begin{align*} \left [-\frac{3 \,{\left ({\left (2 \, b^{2} c - 5 \, a b d\right )} x^{3} + 2 \, a b c - 5 \, a^{2} d\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 (2 \, b^{2} d x^{6} + 2 \,{\left (4 \, b^{2} c - 5 \, a b d\right )} x^{3} + 11 \, a b c - 15 \, a^{2} d\right )} \sqrt{d x^{3} + c}}{18 \,{\left (b^{4} x^{3} + a b^{3}\right )}}, -\frac{3 \,{\left ({\left (2 \, b^{2} c - 5 \, a b d\right )} x^{3} + 2 \, a b c - 5 \, a^{2} d\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 (2 \, b^{2} d x^{6} + 2 \,{\left (4 \, b^{2} c - 5 \, a b d\right )} x^{3} + 11 \, a b c - 15 \, a^{2} d\right )} \sqrt{d x^{3} + c}}{9 \,{\left (b^{4} x^{3} + a b^{3}\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 = 1.11715, size = 234, normalized size = 1.44 \begin{align*} \frac{{\left (2 \, b^{2} c^{2} - 7 \, a b c d + 5 \, a^{2} 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^{3}} + \frac{\sqrt{d x^{3} + c} a b c d - \sqrt{d x^{3} + c} a^{2} d^{2}}{3 \,{\left ({\left (d x^{3} + c\right )} b - b c + a d\right )} b^{3}} + \frac{2 \,{\left ({\left (d x^{3} + c\right )}^{\frac{3}{2}} b^{4} + 3 \, \sqrt{d x^{3} + c} b^{4} c - 6 \, \sqrt{d x^{3} + c} a b^{3} d\right )}}{9 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]