Optimal. Leaf size=88 \[ \frac{x^2 \sqrt{\frac{b x^3}{a}+1} \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 \sqrt{a+b x^3} \sqrt{c+d x^3}} \]
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Rubi [A] time = 0.0706296, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {511, 510} \[ \frac{x^2 \sqrt{\frac{b x^3}{a}+1} \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 \sqrt{a+b x^3} \sqrt{c+d x^3}} \]
Antiderivative was successfully verified.
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Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{a+b x^3} \sqrt{c+d x^3}} \, dx &=\frac{\sqrt{1+\frac{b x^3}{a}} \int \frac{x}{\sqrt{1+\frac{b x^3}{a}} \sqrt{c+d x^3}} \, dx}{\sqrt{a+b x^3}}\\ &=\frac{\left (\sqrt{1+\frac{b x^3}{a}} \sqrt{1+\frac{d x^3}{c}}\right ) \int \frac{x}{\sqrt{1+\frac{b x^3}{a}} \sqrt{1+\frac{d x^3}{c}}} \, dx}{\sqrt{a+b x^3} \sqrt{c+d x^3}}\\ &=\frac{x^2 \sqrt{1+\frac{b x^3}{a}} \sqrt{1+\frac{d x^3}{c}} F_1\left (\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 \sqrt{a+b x^3} \sqrt{c+d x^3}}\\ \end{align*}
Mathematica [A] time = 0.0430736, size = 90, normalized size = 1.02 \[ \frac{x^2 \sqrt{\frac{a+b x^3}{a}} \sqrt{\frac{c+d x^3}{c}} F_1\left (\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 \sqrt{a+b x^3} \sqrt{c+d x^3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.04, size = 0, normalized size = 0. \begin{align*} \int{x{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{b x^{3} + a} \sqrt{d x^{3} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x^{3} + a} \sqrt{d x^{3} + c} x}{b d x^{6} +{\left (b c + a d\right )} x^{3} + a c}, x\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}{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{b x^{3} + a} \sqrt{d x^{3} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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