Optimal. Leaf size=304 \[ \frac{7 \sqrt{2+\sqrt{3}} b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (13 A b-10 a B) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{60 \sqrt [4]{3} a^3 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{7 \sqrt{a+b x^3} (13 A b-10 a B)}{60 a^3 x^2}-\frac{13 A b-10 a B}{15 a^2 x^2 \sqrt{a+b x^3}}-\frac{A}{5 a x^5 \sqrt{a+b x^3}} \]
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Rubi [A] time = 0.135799, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {453, 290, 325, 218} \[ \frac{7 \sqrt{2+\sqrt{3}} b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (13 A b-10 a B) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{60 \sqrt [4]{3} a^3 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{7 \sqrt{a+b x^3} (13 A b-10 a B)}{60 a^3 x^2}-\frac{13 A b-10 a B}{15 a^2 x^2 \sqrt{a+b x^3}}-\frac{A}{5 a x^5 \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Rule 453
Rule 290
Rule 325
Rule 218
Rubi steps
\begin{align*} \int \frac{A+B x^3}{x^6 \left (a+b x^3\right )^{3/2}} \, dx &=-\frac{A}{5 a x^5 \sqrt{a+b x^3}}-\frac{\left (\frac{13 A b}{2}-5 a B\right ) \int \frac{1}{x^3 \left (a+b x^3\right )^{3/2}} \, dx}{5 a}\\ &=-\frac{A}{5 a x^5 \sqrt{a+b x^3}}-\frac{13 A b-10 a B}{15 a^2 x^2 \sqrt{a+b x^3}}-\frac{(7 (13 A b-10 a B)) \int \frac{1}{x^3 \sqrt{a+b x^3}} \, dx}{30 a^2}\\ &=-\frac{A}{5 a x^5 \sqrt{a+b x^3}}-\frac{13 A b-10 a B}{15 a^2 x^2 \sqrt{a+b x^3}}+\frac{7 (13 A b-10 a B) \sqrt{a+b x^3}}{60 a^3 x^2}+\frac{(7 b (13 A b-10 a B)) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{120 a^3}\\ &=-\frac{A}{5 a x^5 \sqrt{a+b x^3}}-\frac{13 A b-10 a B}{15 a^2 x^2 \sqrt{a+b x^3}}+\frac{7 (13 A b-10 a B) \sqrt{a+b x^3}}{60 a^3 x^2}+\frac{7 \sqrt{2+\sqrt{3}} b^{2/3} (13 A b-10 a B) \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 = 0.0337289, size = 72, normalized size = 0.24 \[ \frac{x^3 \sqrt{\frac{b x^3}{a}+1} (13 A b-10 a B) \, _2F_1\left (-\frac{2}{3},\frac{3}{2};\frac{1}{3};-\frac{b x^3}{a}\right )-4 a A}{20 a^2 x^5 \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.026, size = 667, normalized size = 2.2 \begin{align*} \text{result too large to display} \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{B x^{3} + A}{{\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{{\left (B x^{3} + A\right )} \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 = 139.688, size = 90, normalized size = 0.3 \begin{align*} \frac{A \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 )} + \frac{B \Gamma \left (- \frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{3}{2} \\ \frac{1}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{3}{2}} x^{2} \Gamma \left (\frac{1}{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{B x^{3} + A}{{\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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