Optimal. Leaf size=299 \[ \frac{18\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (17 A b-2 a B) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{935 b^{4/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2 x \left (a+b x^3\right )^{3/2} (17 A b-2 a B)}{187 b}+\frac{18 a x \sqrt{a+b x^3} (17 A b-2 a B)}{935 b}+\frac{2 B x \left (a+b x^3\right )^{5/2}}{17 b} \]
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Rubi [A] time = 0.123566, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {388, 195, 218} \[ \frac{18\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} (17 A b-2 a B) F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{935 b^{4/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2 x \left (a+b x^3\right )^{3/2} (17 A b-2 a B)}{187 b}+\frac{18 a x \sqrt{a+b x^3} (17 A b-2 a B)}{935 b}+\frac{2 B x \left (a+b x^3\right )^{5/2}}{17 b} \]
Antiderivative was successfully verified.
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Rule 388
Rule 195
Rule 218
Rubi steps
\begin{align*} \int \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx &=\frac{2 B x \left (a+b x^3\right )^{5/2}}{17 b}-\frac{\left (2 \left (-\frac{17 A b}{2}+a B\right )\right ) \int \left (a+b x^3\right )^{3/2} \, dx}{17 b}\\ &=\frac{2 (17 A b-2 a B) x \left (a+b x^3\right )^{3/2}}{187 b}+\frac{2 B x \left (a+b x^3\right )^{5/2}}{17 b}+\frac{(9 a (17 A b-2 a B)) \int \sqrt{a+b x^3} \, dx}{187 b}\\ &=\frac{18 a (17 A b-2 a B) x \sqrt{a+b x^3}}{935 b}+\frac{2 (17 A b-2 a B) x \left (a+b x^3\right )^{3/2}}{187 b}+\frac{2 B x \left (a+b x^3\right )^{5/2}}{17 b}+\frac{\left (27 a^2 (17 A b-2 a B)\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{935 b}\\ &=\frac{18 a (17 A b-2 a B) x \sqrt{a+b x^3}}{935 b}+\frac{2 (17 A b-2 a B) x \left (a+b x^3\right )^{3/2}}{187 b}+\frac{2 B x \left (a+b x^3\right )^{5/2}}{17 b}+\frac{18\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 (17 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 )}{935 b^{4/3} \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 = 0.056973, size = 77, normalized size = 0.26 \[ \frac{2 x \sqrt{a+b x^3} \left (B \left (a+b x^3\right )^2-\frac{a \left (a B-\frac{17 A b}{2}\right ) \, _2F_1\left (-\frac{3}{2},\frac{1}{3};\frac{4}{3};-\frac{b x^3}{a}\right )}{\sqrt{\frac{b x^3}{a}+1}}\right )}{17 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 654, 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{\left (B x^{3} + A\right )}{\left (b x^{3} + a\right )}^{\frac{3}{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 ({\left (B b x^{6} +{\left (B a + A b\right )} x^{3} + A a\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 = 4.0534, size = 170, normalized size = 0.57 \begin{align*} \frac{A a^{\frac{3}{2}} x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} + \frac{A \sqrt{a} b x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{7}{3}\right )} + \frac{B a^{\frac{3}{2}} x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{7}{3}\right )} + \frac{B \sqrt{a} b x^{7} \Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{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 )}{\left (b x^{3} + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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