Optimal. Leaf size=48 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^3}}{\sqrt{b} \sqrt{c+d x^3}}\right )}{3 \sqrt{b} \sqrt{d}} \]
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Rubi [A] time = 0.0594079, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {444, 63, 217, 206} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^3}}{\sqrt{b} \sqrt{c+d x^3}}\right )}{3 \sqrt{b} \sqrt{d}} \]
Antiderivative was successfully verified.
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Rule 444
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{a+b x^3} \sqrt{c+d x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} \sqrt{c+d x}} \, dx,x,x^3\right )\\ &=\frac{2 \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}\\ &=\frac{2 \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}\\ &=\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^3}}{\sqrt{b} \sqrt{c+d x^3}}\right )}{3 \sqrt{b} \sqrt{d}}\\ \end{align*}
Mathematica [A] time = 0.0652979, size = 85, normalized size = 1.77 \[ \frac{2 \sqrt{c+d x^3} \sinh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^3}}{\sqrt{b c-a d}}\right )}{3 \sqrt{d} \sqrt{b c-a d} \sqrt{\frac{b \left (c+d x^3\right )}{b c-a d}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.04, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2}{\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 [B] time = 1.29587, size = 440, normalized size = 9.17 \begin{align*} \left [\frac{\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 )}{6 \, b d}, -\frac{\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 )}{3 \, b d}\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^{2}}{\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 = 1.14752, size = 73, normalized size = 1.52 \begin{align*} -\frac{2 \, b \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 )}{3 \, \sqrt{b d}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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