Optimal. Leaf size=578 \[ \frac{5 \sqrt{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (11 A b-2 a B) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{27 \sqrt [4]{3} a^{8/3} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{5 \sqrt{a+b x^3} (11 A b-2 a B)}{27 a^3 b^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{5 \sqrt{2-\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (11 A b-2 a B) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{18\ 3^{3/4} a^{8/3} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{5 x^2 (11 A b-2 a B)}{27 a^3 \sqrt{a+b x^3}}-\frac{x^2 (11 A b-2 a B)}{9 a^2 \left (a+b x^3\right )^{3/2}}-\frac{A}{a x \left (a+b x^3\right )^{3/2}} \]
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Rubi [A] time = 0.299794, antiderivative size = 578, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {453, 290, 303, 218, 1877} \[ \frac{5 \sqrt{a+b x^3} (11 A b-2 a B)}{27 a^3 b^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{5 \sqrt{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (11 A b-2 a B) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{27 \sqrt [4]{3} a^{8/3} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{5 \sqrt{2-\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (11 A b-2 a B) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{18\ 3^{3/4} a^{8/3} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{5 x^2 (11 A b-2 a B)}{27 a^3 \sqrt{a+b x^3}}-\frac{x^2 (11 A b-2 a B)}{9 a^2 \left (a+b x^3\right )^{3/2}}-\frac{A}{a x \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 453
Rule 290
Rule 303
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int \frac{A+B x^3}{x^2 \left (a+b x^3\right )^{5/2}} \, dx &=-\frac{A}{a x \left (a+b x^3\right )^{3/2}}-\frac{\left (\frac{11 A b}{2}-a B\right ) \int \frac{x}{\left (a+b x^3\right )^{5/2}} \, dx}{a}\\ &=-\frac{A}{a x \left (a+b x^3\right )^{3/2}}-\frac{(11 A b-2 a B) x^2}{9 a^2 \left (a+b x^3\right )^{3/2}}-\frac{(5 (11 A b-2 a B)) \int \frac{x}{\left (a+b x^3\right )^{3/2}} \, dx}{18 a^2}\\ &=-\frac{A}{a x \left (a+b x^3\right )^{3/2}}-\frac{(11 A b-2 a B) x^2}{9 a^2 \left (a+b x^3\right )^{3/2}}-\frac{5 (11 A b-2 a B) x^2}{27 a^3 \sqrt{a+b x^3}}+\frac{(5 (11 A b-2 a B)) \int \frac{x}{\sqrt{a+b x^3}} \, dx}{54 a^3}\\ &=-\frac{A}{a x \left (a+b x^3\right )^{3/2}}-\frac{(11 A b-2 a B) x^2}{9 a^2 \left (a+b x^3\right )^{3/2}}-\frac{5 (11 A b-2 a B) x^2}{27 a^3 \sqrt{a+b x^3}}+\frac{(5 (11 A b-2 a B)) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt{a+b x^3}} \, dx}{54 a^3 \sqrt [3]{b}}+\frac{\left (5 \sqrt{\frac{1}{2} \left (2-\sqrt{3}\right )} (11 A b-2 a B)\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{27 a^{8/3} \sqrt [3]{b}}\\ &=-\frac{A}{a x \left (a+b x^3\right )^{3/2}}-\frac{(11 A b-2 a B) x^2}{9 a^2 \left (a+b x^3\right )^{3/2}}-\frac{5 (11 A b-2 a B) x^2}{27 a^3 \sqrt{a+b x^3}}+\frac{5 (11 A b-2 a B) \sqrt{a+b x^3}}{27 a^3 b^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{5 \sqrt{2-\sqrt{3}} (11 A b-2 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 )}{18\ 3^{3/4} a^{8/3} b^{2/3} \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} (11 A b-2 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 )}{27 \sqrt [4]{3} a^{8/3} b^{2/3} \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 = 0.0384735, size = 86, normalized size = 0.15 \[ -\frac{x^2 \sqrt{\frac{b x^3}{a}+1} \left (\frac{11 A b}{2}-a B\right ) \, _2F_1\left (\frac{2}{3},\frac{5}{2};\frac{5}{3};-\frac{b x^3}{a}\right )}{2 a^3 \sqrt{a+b x^3}}-\frac{A}{a x \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.026, size = 1001, 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{5}{2}} x^{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{{\left (B x^{3} + A\right )} \sqrt{b x^{3} + a}}{b^{3} x^{11} + 3 \, a b^{2} x^{8} + 3 \, a^{2} b x^{5} + a^{3} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{5}{2}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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