Optimal. Leaf size=88 \[ \frac{\sqrt{a+b x^3} \sqrt{c+d x^3}}{3 b d}-\frac{(a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^3}}{\sqrt{b} \sqrt{c+d x^3}}\right )}{3 b^{3/2} d^{3/2}} \]
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Rubi [A] time = 0.0933625, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {446, 80, 63, 217, 206} \[ \frac{\sqrt{a+b x^3} \sqrt{c+d x^3}}{3 b d}-\frac{(a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^3}}{\sqrt{b} \sqrt{c+d x^3}}\right )}{3 b^{3/2} d^{3/2}} \]
Antiderivative was successfully verified.
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Rule 446
Rule 80
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^5}{\sqrt{a+b x^3} \sqrt{c+d x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,x^3\right )\\ &=\frac{\sqrt{a+b x^3} \sqrt{c+d x^3}}{3 b d}-\frac{(b c+a d) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,x^3\right )}{6 b d}\\ &=\frac{\sqrt{a+b x^3} \sqrt{c+d x^3}}{3 b d}-\frac{(b c+a d) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{a d}{b}+\frac{d x^2}{b}}} \, dx,x,\sqrt{a+b x^3}\right )}{3 b^2 d}\\ &=\frac{\sqrt{a+b x^3} \sqrt{c+d x^3}}{3 b d}-\frac{(b c+a d) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{b}} \, dx,x,\frac{\sqrt{a+b x^3}}{\sqrt{c+d x^3}}\right )}{3 b^2 d}\\ &=\frac{\sqrt{a+b x^3} \sqrt{c+d x^3}}{3 b d}-\frac{(b c+a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^3}}{\sqrt{b} \sqrt{c+d x^3}}\right )}{3 b^{3/2} d^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.180121, size = 123, normalized size = 1.4 \[ \frac{b \sqrt{d} \sqrt{a+b x^3} \left (c+d x^3\right )-\sqrt{b c-a d} (a d+b c) \sqrt{\frac{b \left (c+d x^3\right )}{b c-a d}} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^3}}{\sqrt{b c-a d}}\right )}{3 b^2 d^{3/2} \sqrt{c+d x^3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{{x}^{5}{\frac{1}{\sqrt{b{x}^{3}+a}}}{\frac{1}{\sqrt{d{x}^{3}+c}}}}\, dx \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 = 1.40686, size = 597, normalized size = 6.78 \begin{align*} \left [\frac{4 \, \sqrt{b x^{3} + a} \sqrt{d x^{3} + c} b d +{\left (b c + a d\right )} \sqrt{b d} \log \left (8 \, b^{2} d^{2} x^{6} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x^{3} - 4 \,{\left (2 \, b d x^{3} + b c + a d\right )} \sqrt{b x^{3} + a} \sqrt{d x^{3} + c} \sqrt{b d}\right )}{12 \, b^{2} d^{2}}, \frac{2 \, \sqrt{b x^{3} + a} \sqrt{d x^{3} + c} b d +{\left (b c + a d\right )} \sqrt{-b d} \arctan \left (\frac{{\left (2 \, b d x^{3} + b c + a d\right )} \sqrt{b x^{3} + a} \sqrt{d x^{3} + c} \sqrt{-b d}}{2 \,{\left (b^{2} d^{2} x^{6} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x^{3}\right )}}\right )}{6 \, b^{2} d^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\sqrt{a + b x^{3}} \sqrt{c + d x^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17714, size = 140, normalized size = 1.59 \begin{align*} \frac{\frac{{\left (b c + a d\right )} \log \left ({\left | -\sqrt{b x^{3} + a} \sqrt{b d} + \sqrt{b^{2} c +{\left (b x^{3} + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} d} + \frac{\sqrt{b x^{3} + a} \sqrt{b^{2} c +{\left (b x^{3} + a\right )} b d - a b d}}{b d}}{3 \,{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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