Optimal. Leaf size=585 \[ -\frac{\left (1-\sqrt{3}\right ) \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (4 A b-a B) \text{EllipticF}\left (\cos ^{-1}\left (\frac{\sqrt [3]{a}+\left (1-\sqrt{3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x}\right ),\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{3 \sqrt [4]{3} a^{5/3} b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}+\frac{2 \left (1+\sqrt{3}\right ) \sqrt{e x} \sqrt{a+b x^3} (4 A b-a B)}{3 a^2 b^{2/3} e^2 \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )}-\frac{2 \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (4 A b-a B) E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{3^{3/4} a^{5/3} b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{2 (e x)^{5/2} (4 A b-a B)}{3 a^2 e^4 \sqrt{a+b x^3}}-\frac{2 A}{a e \sqrt{e x} \sqrt{a+b x^3}} \]
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Rubi [A] time = 0.570769, antiderivative size = 585, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {453, 290, 329, 308, 225, 1881} \[ \frac{2 \left (1+\sqrt{3}\right ) \sqrt{e x} \sqrt{a+b x^3} (4 A b-a B)}{3 a^2 b^{2/3} e^2 \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )}-\frac{\left (1-\sqrt{3}\right ) \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (4 A b-a B) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{3 \sqrt [4]{3} a^{5/3} b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{2 \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (4 A b-a B) E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{3^{3/4} a^{5/3} b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{2 (e x)^{5/2} (4 A b-a B)}{3 a^2 e^4 \sqrt{a+b x^3}}-\frac{2 A}{a e \sqrt{e x} \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Rule 453
Rule 290
Rule 329
Rule 308
Rule 225
Rule 1881
Rubi steps
\begin{align*} \int \frac{A+B x^3}{(e x)^{3/2} \left (a+b x^3\right )^{3/2}} \, dx &=-\frac{2 A}{a e \sqrt{e x} \sqrt{a+b x^3}}-\frac{(4 A b-a B) \int \frac{(e x)^{3/2}}{\left (a+b x^3\right )^{3/2}} \, dx}{a e^3}\\ &=-\frac{2 A}{a e \sqrt{e x} \sqrt{a+b x^3}}-\frac{2 (4 A b-a B) (e x)^{5/2}}{3 a^2 e^4 \sqrt{a+b x^3}}+\frac{(2 (4 A b-a B)) \int \frac{(e x)^{3/2}}{\sqrt{a+b x^3}} \, dx}{3 a^2 e^3}\\ &=-\frac{2 A}{a e \sqrt{e x} \sqrt{a+b x^3}}-\frac{2 (4 A b-a B) (e x)^{5/2}}{3 a^2 e^4 \sqrt{a+b x^3}}+\frac{(4 (4 A b-a B)) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{a+\frac{b x^6}{e^3}}} \, dx,x,\sqrt{e x}\right )}{3 a^2 e^4}\\ &=-\frac{2 A}{a e \sqrt{e x} \sqrt{a+b x^3}}-\frac{2 (4 A b-a B) (e x)^{5/2}}{3 a^2 e^4 \sqrt{a+b x^3}}-\frac{(2 (4 A b-a B)) \operatorname{Subst}\left (\int \frac{\left (-1+\sqrt{3}\right ) a^{2/3} e^2-2 b^{2/3} x^4}{\sqrt{a+\frac{b x^6}{e^3}}} \, dx,x,\sqrt{e x}\right )}{3 a^2 b^{2/3} e^4}-\frac{\left (2 \left (1-\sqrt{3}\right ) (4 A b-a B)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^6}{e^3}}} \, dx,x,\sqrt{e x}\right )}{3 a^{4/3} b^{2/3} e^2}\\ &=-\frac{2 A}{a e \sqrt{e x} \sqrt{a+b x^3}}-\frac{2 (4 A b-a B) (e x)^{5/2}}{3 a^2 e^4 \sqrt{a+b x^3}}+\frac{2 \left (1+\sqrt{3}\right ) (4 A b-a B) \sqrt{e x} \sqrt{a+b x^3}}{3 a^2 b^{2/3} e^2 \left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )}-\frac{2 (4 A b-a B) \sqrt{e x} \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 (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\cos ^{-1}\left (\frac{\sqrt [3]{a}+\left (1-\sqrt{3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{3^{3/4} a^{5/3} b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{\left (1-\sqrt{3}\right ) (4 A b-a B) \sqrt{e x} \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 (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} F\left (\cos ^{-1}\left (\frac{\sqrt [3]{a}+\left (1-\sqrt{3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{3 \sqrt [4]{3} a^{5/3} b^{2/3} e^2 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}\\ \end{align*}
Mathematica [C] time = 0.0420644, size = 77, normalized size = 0.13 \[ \frac{x \left (2 x^3 \sqrt{\frac{b x^3}{a}+1} (a B-4 A b) \, _2F_1\left (\frac{5}{6},\frac{3}{2};\frac{11}{6};-\frac{b x^3}{a}\right )-10 a A\right )}{5 a^2 (e x)^{3/2} \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.052, size = 5563, normalized size = 9.5 \begin{align*} \text{output 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}} \left (e x\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 (\frac{{\left (B x^{3} + A\right )} \sqrt{b x^{3} + a} \sqrt{e x}}{b^{2} e^{2} x^{8} + 2 \, a b e^{2} x^{5} + a^{2} e^{2} 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{3}{2}} \left (e x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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