Optimal. Leaf size=487 \[ \frac{8 \sqrt{2} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{4 \sqrt{2-\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{8 \sqrt{a+b x^3}}{3 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{2 x^2}{3 b \sqrt{a+b x^3}} \]
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Rubi [A] time = 0.14887, antiderivative size = 487, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {288, 303, 218, 1877} \[ \frac{8 \sqrt{2} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{4 \sqrt{2-\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{8 \sqrt{a+b x^3}}{3 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{2 x^2}{3 b \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Rule 288
Rule 303
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int \frac{x^4}{\left (a+b x^3\right )^{3/2}} \, dx &=-\frac{2 x^2}{3 b \sqrt{a+b x^3}}+\frac{4 \int \frac{x}{\sqrt{a+b x^3}} \, dx}{3 b}\\ &=-\frac{2 x^2}{3 b \sqrt{a+b x^3}}+\frac{4 \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt{a+b x^3}} \, dx}{3 b^{4/3}}+\frac{\left (4 \sqrt{2 \left (2-\sqrt{3}\right )} \sqrt [3]{a}\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{3 b^{4/3}}\\ &=-\frac{2 x^2}{3 b \sqrt{a+b x^3}}+\frac{8 \sqrt{a+b x^3}}{3 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{4 \sqrt{2-\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{8 \sqrt{2} \sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}\\ \end{align*}
Mathematica [C] time = 0.018426, size = 55, normalized size = 0.11 \[ -\frac{2 x^2 \left (\sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{2}{3},\frac{3}{2};\frac{5}{3};-\frac{b x^3}{a}\right )-1\right )}{b \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 457, normalized size = 0.9 \begin{align*} -{\frac{2\,{x}^{2}}{3\,b}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{a}{b}} \right ) b}}}}-{\frac{{\frac{8\,i}{9}}\sqrt{3}}{{b}^{2}}\sqrt [3]{-{b}^{2}a}\sqrt{{i\sqrt{3}b \left ( x+{\frac{1}{2\,b}\sqrt [3]{-{b}^{2}a}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}} \right ){\frac{1}{\sqrt [3]{-{b}^{2}a}}}}}\sqrt{{ \left ( x-{\frac{1}{b}\sqrt [3]{-{b}^{2}a}} \right ) \left ( -{\frac{3}{2\,b}\sqrt [3]{-{b}^{2}a}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}} \right ) ^{-1}}}\sqrt{{-i\sqrt{3}b \left ( x+{\frac{1}{2\,b}\sqrt [3]{-{b}^{2}a}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}} \right ){\frac{1}{\sqrt [3]{-{b}^{2}a}}}}} \left ( \left ( -{\frac{3}{2\,b}\sqrt [3]{-{b}^{2}a}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}} \right ){\it EllipticE} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}b \left ( x+{\frac{1}{2\,b}\sqrt [3]{-{b}^{2}a}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}} \right ){\frac{1}{\sqrt [3]{-{b}^{2}a}}}}}},\sqrt{{\frac{i\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a} \left ( -{\frac{3}{2\,b}\sqrt [3]{-{b}^{2}a}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}} \right ) ^{-1}}} \right ) +{\frac{1}{b}\sqrt [3]{-{b}^{2}a}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}b \left ( x+{\frac{1}{2\,b}\sqrt [3]{-{b}^{2}a}}-{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}} \right ){\frac{1}{\sqrt [3]{-{b}^{2}a}}}}}},\sqrt{{\frac{i\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a} \left ( -{\frac{3}{2\,b}\sqrt [3]{-{b}^{2}a}}+{\frac{{\frac{i}{2}}\sqrt{3}}{b}\sqrt [3]{-{b}^{2}a}} \right ) ^{-1}}} \right ) } \right ){\frac{1}{\sqrt{b{x}^{3}+a}}}} \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^{4}}{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}\,{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} x^{4}}{b^{2} x^{6} + 2 \, a b x^{3} + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.989794, size = 37, normalized size = 0.08 \begin{align*} \frac{x^{5} \Gamma \left (\frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{3}{2}} \Gamma \left (\frac{8}{3}\right )} \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^{4}}{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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