Optimal. Leaf size=296 \[ \frac{288\ 3^{3/4} \sqrt{2+\sqrt{3}} a^4 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{21505 b^{7/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{432 a^3 x \sqrt{a+b x^3}}{21505 b^2}+\frac{54 a^2 x^4 \sqrt{a+b x^3}}{4301 b}+\frac{2}{23} x^7 \left (a+b x^3\right )^{3/2}+\frac{18}{391} a x^7 \sqrt{a+b x^3} \]
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Rubi [A] time = 0.128033, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {279, 321, 218} \[ -\frac{432 a^3 x \sqrt{a+b x^3}}{21505 b^2}+\frac{288\ 3^{3/4} \sqrt{2+\sqrt{3}} a^4 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{21505 b^{7/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{54 a^2 x^4 \sqrt{a+b x^3}}{4301 b}+\frac{2}{23} x^7 \left (a+b x^3\right )^{3/2}+\frac{18}{391} a x^7 \sqrt{a+b x^3} \]
Antiderivative was successfully verified.
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Rule 279
Rule 321
Rule 218
Rubi steps
\begin{align*} \int x^6 \left (a+b x^3\right )^{3/2} \, dx &=\frac{2}{23} x^7 \left (a+b x^3\right )^{3/2}+\frac{1}{23} (9 a) \int x^6 \sqrt{a+b x^3} \, dx\\ &=\frac{18}{391} a x^7 \sqrt{a+b x^3}+\frac{2}{23} x^7 \left (a+b x^3\right )^{3/2}+\frac{1}{391} \left (27 a^2\right ) \int \frac{x^6}{\sqrt{a+b x^3}} \, dx\\ &=\frac{54 a^2 x^4 \sqrt{a+b x^3}}{4301 b}+\frac{18}{391} a x^7 \sqrt{a+b x^3}+\frac{2}{23} x^7 \left (a+b x^3\right )^{3/2}-\frac{\left (216 a^3\right ) \int \frac{x^3}{\sqrt{a+b x^3}} \, dx}{4301 b}\\ &=-\frac{432 a^3 x \sqrt{a+b x^3}}{21505 b^2}+\frac{54 a^2 x^4 \sqrt{a+b x^3}}{4301 b}+\frac{18}{391} a x^7 \sqrt{a+b x^3}+\frac{2}{23} x^7 \left (a+b x^3\right )^{3/2}+\frac{\left (432 a^4\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{21505 b^2}\\ &=-\frac{432 a^3 x \sqrt{a+b x^3}}{21505 b^2}+\frac{54 a^2 x^4 \sqrt{a+b x^3}}{4301 b}+\frac{18}{391} a x^7 \sqrt{a+b x^3}+\frac{2}{23} x^7 \left (a+b x^3\right )^{3/2}+\frac{288\ 3^{3/4} \sqrt{2+\sqrt{3}} a^4 \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 )}{21505 b^{7/3} \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 = 0.0722159, size = 79, normalized size = 0.27 \[ \frac{2 x \sqrt{a+b x^3} \left (\frac{8 a^3 \, _2F_1\left (-\frac{3}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{\sqrt{\frac{b x^3}{a}+1}}-\left (8 a-17 b x^3\right ) \left (a+b x^3\right )^2\right )}{391 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 355, normalized size = 1.2 \begin{align*}{\frac{2\,b{x}^{10}}{23}\sqrt{b{x}^{3}+a}}+{\frac{52\,a{x}^{7}}{391}\sqrt{b{x}^{3}+a}}+{\frac{54\,{a}^{2}{x}^{4}}{4301\,b}\sqrt{b{x}^{3}+a}}-{\frac{432\,{a}^{3}x}{21505\,{b}^{2}}\sqrt{b{x}^{3}+a}}-{\frac{{\frac{288\,i}{21505}}{a}^{4}\sqrt{3}}{{b}^{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{\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 ({\left (b x^{9} + a x^{6}\right )} \sqrt{b x^{3} + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.8252, size = 39, normalized size = 0.13 \begin{align*} \frac{a^{\frac{3}{2}} x^{7} \Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{10}{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{\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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