Optimal. Leaf size=64 \[ \frac{x^4 \sqrt{\frac{d x^6}{c}+1} F_1\left (\frac{2}{3};2,\frac{1}{2};\frac{5}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{4 a^2 \sqrt{c+d x^6}} \]
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Rubi [A] time = 0.0730317, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {465, 511, 510} \[ \frac{x^4 \sqrt{\frac{d x^6}{c}+1} F_1\left (\frac{2}{3};2,\frac{1}{2};\frac{5}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{4 a^2 \sqrt{c+d x^6}} \]
Antiderivative was successfully verified.
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Rule 465
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a+b x^6\right )^2 \sqrt{c+d x^6}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{\left (a+b x^3\right )^2 \sqrt{c+d x^3}} \, dx,x,x^2\right )\\ &=\frac{\sqrt{1+\frac{d x^6}{c}} \operatorname{Subst}\left (\int \frac{x}{\left (a+b x^3\right )^2 \sqrt{1+\frac{d x^3}{c}}} \, dx,x,x^2\right )}{2 \sqrt{c+d x^6}}\\ &=\frac{x^4 \sqrt{1+\frac{d x^6}{c}} F_1\left (\frac{2}{3};2,\frac{1}{2};\frac{5}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{4 a^2 \sqrt{c+d x^6}}\\ \end{align*}
Mathematica [B] time = 0.16944, size = 168, normalized size = 2.62 \[ -\frac{x^4 \left (b d x^6 \left (a+b x^6\right ) \sqrt{\frac{d x^6}{c}+1} F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )-5 \left (a+b x^6\right ) \sqrt{\frac{d x^6}{c}+1} (b c-3 a d) F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )-10 a b \left (c+d x^6\right )\right )}{60 a^2 \left (a+b x^6\right ) \sqrt{c+d x^6} (b c-a d)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.04, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{ \left ( b{x}^{6}+a \right ) ^{2}}{\frac{1}{\sqrt{d{x}^{6}+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^{3}}{{\left (b x^{6} + a\right )}^{2} \sqrt{d x^{6} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (b x^{6} + a\right )}^{2} \sqrt{d x^{6} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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