Optimal. Leaf size=277 \[ \frac{91 \sqrt{2+\sqrt{3}} b^{5/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{60 \sqrt [4]{3} a^3 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{91 b \sqrt{a+b x^3}}{60 a^3 x^2}-\frac{13 \sqrt{a+b x^3}}{15 a^2 x^5}+\frac{2}{3 a x^5 \sqrt{a+b x^3}} \]
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Rubi [A] time = 0.0939251, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {290, 325, 218} \[ \frac{91 \sqrt{2+\sqrt{3}} b^{5/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{60 \sqrt [4]{3} a^3 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{91 b \sqrt{a+b x^3}}{60 a^3 x^2}-\frac{13 \sqrt{a+b x^3}}{15 a^2 x^5}+\frac{2}{3 a x^5 \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 218
Rubi steps
\begin{align*} \int \frac{1}{x^6 \left (a+b x^3\right )^{3/2}} \, dx &=\frac{2}{3 a x^5 \sqrt{a+b x^3}}+\frac{13 \int \frac{1}{x^6 \sqrt{a+b x^3}} \, dx}{3 a}\\ &=\frac{2}{3 a x^5 \sqrt{a+b x^3}}-\frac{13 \sqrt{a+b x^3}}{15 a^2 x^5}-\frac{(91 b) \int \frac{1}{x^3 \sqrt{a+b x^3}} \, dx}{30 a^2}\\ &=\frac{2}{3 a x^5 \sqrt{a+b x^3}}-\frac{13 \sqrt{a+b x^3}}{15 a^2 x^5}+\frac{91 b \sqrt{a+b x^3}}{60 a^3 x^2}+\frac{\left (91 b^2\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{120 a^3}\\ &=\frac{2}{3 a x^5 \sqrt{a+b x^3}}-\frac{13 \sqrt{a+b x^3}}{15 a^2 x^5}+\frac{91 b \sqrt{a+b x^3}}{60 a^3 x^2}+\frac{91 \sqrt{2+\sqrt{3}} b^{5/3} \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 )}{60 \sqrt [4]{3} a^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 = 0.0090248, size = 54, normalized size = 0.19 \[ -\frac{\sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (-\frac{5}{3},\frac{3}{2};-\frac{2}{3};-\frac{b x^3}{a}\right )}{5 a x^5 \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 342, normalized size = 1.2 \begin{align*}{\frac{2\,{b}^{2}x}{3\,{a}^{3}}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{a}{b}} \right ) b}}}}-{\frac{1}{5\,{x}^{5}{a}^{2}}\sqrt{b{x}^{3}+a}}+{\frac{17\,b}{20\,{x}^{2}{a}^{3}}\sqrt{b{x}^{3}+a}}-{\frac{{\frac{91\,i}{180}}b\sqrt{3}}{{a}^{3}}\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}}}}}{\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 ){\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{1}{{\left (b x^{3} + a\right )}^{\frac{3}{2}} x^{6}}\,{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}}{b^{2} x^{12} + 2 \, a b x^{9} + a^{2} x^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.3081, size = 44, normalized size = 0.16 \begin{align*} \frac{\Gamma \left (- \frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{3}, \frac{3}{2} \\ - \frac{2}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{3}{2}} x^{5} \Gamma \left (- \frac{2}{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{1}{{\left (b x^{3} + a\right )}^{\frac{3}{2}} x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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