Optimal. Leaf size=580 \[ -\frac{5 \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (11 A b-8 a B) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{12 \sqrt{2} \sqrt [4]{3} a^{8/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{5 \sqrt{2-\sqrt{3}} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (11 A b-8 a B) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{16\ 3^{3/4} a^{8/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{5 \sqrt{a+b x^3} (11 A b-8 a B)}{24 a^3 x}-\frac{5 \sqrt [3]{b} \sqrt{a+b x^3} (11 A b-8 a B)}{24 a^3 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{11 A b-8 a B}{12 a^2 x \sqrt{a+b x^3}}-\frac{A}{4 a x^4 \sqrt{a+b x^3}} \]
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Rubi [A] time = 0.303617, antiderivative size = 580, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {453, 290, 325, 303, 218, 1877} \[ -\frac{5 \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (11 A b-8 a B) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{12 \sqrt{2} \sqrt [4]{3} a^{8/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{5 \sqrt{2-\sqrt{3}} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (11 A b-8 a B) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{16\ 3^{3/4} a^{8/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{5 \sqrt{a+b x^3} (11 A b-8 a B)}{24 a^3 x}-\frac{5 \sqrt [3]{b} \sqrt{a+b x^3} (11 A b-8 a B)}{24 a^3 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{11 A b-8 a B}{12 a^2 x \sqrt{a+b x^3}}-\frac{A}{4 a x^4 \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Rule 453
Rule 290
Rule 325
Rule 303
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int \frac{A+B x^3}{x^5 \left (a+b x^3\right )^{3/2}} \, dx &=-\frac{A}{4 a x^4 \sqrt{a+b x^3}}-\frac{\left (\frac{11 A b}{2}-4 a B\right ) \int \frac{1}{x^2 \left (a+b x^3\right )^{3/2}} \, dx}{4 a}\\ &=-\frac{A}{4 a x^4 \sqrt{a+b x^3}}-\frac{11 A b-8 a B}{12 a^2 x \sqrt{a+b x^3}}-\frac{(5 (11 A b-8 a B)) \int \frac{1}{x^2 \sqrt{a+b x^3}} \, dx}{24 a^2}\\ &=-\frac{A}{4 a x^4 \sqrt{a+b x^3}}-\frac{11 A b-8 a B}{12 a^2 x \sqrt{a+b x^3}}+\frac{5 (11 A b-8 a B) \sqrt{a+b x^3}}{24 a^3 x}-\frac{(5 b (11 A b-8 a B)) \int \frac{x}{\sqrt{a+b x^3}} \, dx}{48 a^3}\\ &=-\frac{A}{4 a x^4 \sqrt{a+b x^3}}-\frac{11 A b-8 a B}{12 a^2 x \sqrt{a+b x^3}}+\frac{5 (11 A b-8 a B) \sqrt{a+b x^3}}{24 a^3 x}-\frac{\left (5 b^{2/3} (11 A b-8 a B)\right ) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt{a+b x^3}} \, dx}{48 a^3}-\frac{\left (5 \sqrt{\frac{1}{2} \left (2-\sqrt{3}\right )} b^{2/3} (11 A b-8 a B)\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{24 a^{8/3}}\\ &=-\frac{A}{4 a x^4 \sqrt{a+b x^3}}-\frac{11 A b-8 a B}{12 a^2 x \sqrt{a+b x^3}}+\frac{5 (11 A b-8 a B) \sqrt{a+b x^3}}{24 a^3 x}-\frac{5 \sqrt [3]{b} (11 A b-8 a B) \sqrt{a+b x^3}}{24 a^3 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{5 \sqrt{2-\sqrt{3}} \sqrt [3]{b} (11 A b-8 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}} 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 )}{16\ 3^{3/4} a^{8/3} \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{5 \sqrt [3]{b} (11 A b-8 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 )}{12 \sqrt{2} \sqrt [4]{3} a^{8/3} \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 = 0.0331659, size = 72, normalized size = 0.12 \[ \frac{x^3 \sqrt{\frac{b x^3}{a}+1} (11 A b-8 a B) \, _2F_1\left (-\frac{1}{3},\frac{3}{2};\frac{2}{3};-\frac{b x^3}{a}\right )-2 a A}{8 a^2 x^4 \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.024, size = 975, normalized size = 1.7 \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^{5}}\,{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^{11} + 2 \, a b x^{8} + a^{2} x^{5}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 107.752, size = 88, normalized size = 0.15 \begin{align*} \frac{A \Gamma \left (- \frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{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^{4} \Gamma \left (- \frac{1}{3}\right )} + \frac{B \Gamma \left (- \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{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 \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{B x^{3} + A}{{\left (b x^{3} + a\right )}^{\frac{3}{2}} x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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