Optimal. Leaf size=563 \[ -\frac{2 \sqrt{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (4 a B+5 A b) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{27 \sqrt [4]{3} a^{5/3} b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{2 \sqrt{a+b x^3} (4 a B+5 A b)}{27 a^2 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{\sqrt{2-\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (4 a B+5 A b) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{9\ 3^{3/4} a^{5/3} b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2 x^2 (4 a B+5 A b)}{27 a^2 b \sqrt{a+b x^3}}+\frac{2 x^2 (A b-a B)}{9 a b \left (a+b x^3\right )^{3/2}} \]
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Rubi [A] time = 0.248184, antiderivative size = 563, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {457, 290, 303, 218, 1877} \[ -\frac{2 \sqrt{a+b x^3} (4 a B+5 A b)}{27 a^2 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{2 \sqrt{2} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (4 a B+5 A b) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{27 \sqrt [4]{3} a^{5/3} b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{\sqrt{2-\sqrt{3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (4 a B+5 A b) E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{9\ 3^{3/4} a^{5/3} b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2 x^2 (4 a B+5 A b)}{27 a^2 b \sqrt{a+b x^3}}+\frac{2 x^2 (A b-a B)}{9 a b \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 457
Rule 290
Rule 303
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int \frac{x \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx &=\frac{2 (A b-a B) x^2}{9 a b \left (a+b x^3\right )^{3/2}}+\frac{\left (2 \left (\frac{5 A b}{2}+2 a B\right )\right ) \int \frac{x}{\left (a+b x^3\right )^{3/2}} \, dx}{9 a b}\\ &=\frac{2 (A b-a B) x^2}{9 a b \left (a+b x^3\right )^{3/2}}+\frac{2 (5 A b+4 a B) x^2}{27 a^2 b \sqrt{a+b x^3}}-\frac{(5 A b+4 a B) \int \frac{x}{\sqrt{a+b x^3}} \, dx}{27 a^2 b}\\ &=\frac{2 (A b-a B) x^2}{9 a b \left (a+b x^3\right )^{3/2}}+\frac{2 (5 A b+4 a B) x^2}{27 a^2 b \sqrt{a+b x^3}}-\frac{(5 A b+4 a B) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt{a+b x^3}} \, dx}{27 a^2 b^{4/3}}-\frac{\left (\sqrt{2 \left (2-\sqrt{3}\right )} (5 A b+4 a B)\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{27 a^{5/3} b^{4/3}}\\ &=\frac{2 (A b-a B) x^2}{9 a b \left (a+b x^3\right )^{3/2}}+\frac{2 (5 A b+4 a B) x^2}{27 a^2 b \sqrt{a+b x^3}}-\frac{2 (5 A b+4 a B) \sqrt{a+b x^3}}{27 a^2 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{\sqrt{2-\sqrt{3}} (5 A b+4 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 )}{9\ 3^{3/4} a^{5/3} b^{5/3} \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{2 \sqrt{2} (5 A b+4 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^{5/3} b^{5/3} \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 = 0.0709847, size = 81, normalized size = 0.14 \[ \frac{x^2 \left (\left (a+b x^3\right ) \sqrt{\frac{b x^3}{a}+1} (4 a B+5 A b) \, _2F_1\left (\frac{2}{3},\frac{5}{2};\frac{5}{3};-\frac{b x^3}{a}\right )-4 a^2 B\right )}{10 a^2 b \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.019, size = 986, normalized size = 1.8 \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{{\left (B x^{3} + A\right )} x}{{\left (b x^{3} + a\right )}^{\frac{5}{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^{4} + A x\right )} \sqrt{b x^{3} + a}}{b^{3} x^{9} + 3 \, a b^{2} x^{6} + 3 \, a^{2} b x^{3} + a^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 170.762, size = 80, normalized size = 0.14 \begin{align*} \frac{A x^{2} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{5}{2} \\ \frac{5}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{5}{2}} \Gamma \left (\frac{5}{3}\right )} + \frac{B x^{5} \Gamma \left (\frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{3}, \frac{5}{2} \\ \frac{8}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{5}{2}} \Gamma \left (\frac{8}{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{{\left (B x^{3} + A\right )} x}{{\left (b x^{3} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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