Optimal. Leaf size=99 \[ \frac{a \sqrt{c+d x^3}}{3 b \left (a+b x^3\right ) (b c-a d)}-\frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{3/2} (b c-a d)^{3/2}} \]
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Rubi [A] time = 0.0844353, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {446, 78, 63, 208} \[ \frac{a \sqrt{c+d x^3}}{3 b \left (a+b x^3\right ) (b c-a d)}-\frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{3/2} (b c-a d)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 446
Rule 78
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{x^5}{\left (a+b x^3\right )^2 \sqrt{c+d x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x}{(a+b x)^2 \sqrt{c+d x}} \, dx,x,x^3\right )\\ &=\frac{a \sqrt{c+d x^3}}{3 b (b c-a d) \left (a+b x^3\right )}+\frac{(2 b c-a d) \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,x^3\right )}{6 b (b c-a d)}\\ &=\frac{a \sqrt{c+d x^3}}{3 b (b c-a d) \left (a+b x^3\right )}+\frac{(2 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 d (b c-a d)}\\ &=\frac{a \sqrt{c+d x^3}}{3 b (b c-a d) \left (a+b x^3\right )}-\frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{3/2} (b c-a d)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0916238, size = 98, normalized size = 0.99 \[ \frac{\frac{a \sqrt{b} \sqrt{c+d x^3}}{\left (a+b x^3\right ) (b c-a d)}+\frac{(a d-2 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{(b c-a d)^{3/2}}}{3 b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.01, size = 892, normalized size = 9. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A] time = 1.63511, size = 722, normalized size = 7.29 \begin{align*} \left [\frac{{\left ({\left (2 \, b^{2} c - a b d\right )} x^{3} + 2 \, a b c - a^{2} d\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 (a b^{2} c - a^{2} b d\right )} \sqrt{d x^{3} + c}}{6 \,{\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2} +{\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} x^{3}\right )}}, \frac{{\left ({\left (2 \, b^{2} c - a b d\right )} x^{3} + 2 \, a b c - a^{2} d\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 (a b^{2} c - a^{2} b d\right )} \sqrt{d x^{3} + c}}{3 \,{\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2} +{\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} x^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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.
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Giac [A] time = 1.10433, size = 157, normalized size = 1.59 \begin{align*} \frac{\frac{\sqrt{d x^{3} + c} a d^{2}}{{\left (b^{2} c - a b d\right )}{\left ({\left (d x^{3} + c\right )} b - b c + a d\right )}} + \frac{{\left (2 \, b c d - a d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{{\left (b^{2} c - a b d\right )} \sqrt{-b^{2} c + a b d}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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