Optimal. Leaf size=320 \[ -\frac{16 \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (14 A b-5 a B) \text{EllipticF}\left (\cos ^{-1}\left (\frac{\sqrt [3]{a}+\left (1-\sqrt{3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x}\right ),\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{135 \sqrt [4]{3} a^{10/3} e^4 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{16 \sqrt{e x} (14 A b-5 a B)}{135 a^3 e^4 \sqrt{a+b x^3}}-\frac{2 \sqrt{e x} (14 A b-5 a B)}{45 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac{2 A}{5 a e (e x)^{5/2} \left (a+b x^3\right )^{3/2}} \]
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Rubi [A] time = 0.257692, antiderivative size = 320, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {453, 290, 329, 225} \[ -\frac{16 \sqrt{e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} (14 A b-5 a B) F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{135 \sqrt [4]{3} a^{10/3} e^4 \sqrt{\frac{\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt{3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt{a+b x^3}}-\frac{16 \sqrt{e x} (14 A b-5 a B)}{135 a^3 e^4 \sqrt{a+b x^3}}-\frac{2 \sqrt{e x} (14 A b-5 a B)}{45 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac{2 A}{5 a e (e x)^{5/2} \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 453
Rule 290
Rule 329
Rule 225
Rubi steps
\begin{align*} \int \frac{A+B x^3}{(e x)^{7/2} \left (a+b x^3\right )^{5/2}} \, dx &=-\frac{2 A}{5 a e (e x)^{5/2} \left (a+b x^3\right )^{3/2}}-\frac{(14 A b-5 a B) \int \frac{1}{\sqrt{e x} \left (a+b x^3\right )^{5/2}} \, dx}{5 a e^3}\\ &=-\frac{2 A}{5 a e (e x)^{5/2} \left (a+b x^3\right )^{3/2}}-\frac{2 (14 A b-5 a B) \sqrt{e x}}{45 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac{(8 (14 A b-5 a B)) \int \frac{1}{\sqrt{e x} \left (a+b x^3\right )^{3/2}} \, dx}{45 a^2 e^3}\\ &=-\frac{2 A}{5 a e (e x)^{5/2} \left (a+b x^3\right )^{3/2}}-\frac{2 (14 A b-5 a B) \sqrt{e x}}{45 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac{16 (14 A b-5 a B) \sqrt{e x}}{135 a^3 e^4 \sqrt{a+b x^3}}-\frac{(16 (14 A b-5 a B)) \int \frac{1}{\sqrt{e x} \sqrt{a+b x^3}} \, dx}{135 a^3 e^3}\\ &=-\frac{2 A}{5 a e (e x)^{5/2} \left (a+b x^3\right )^{3/2}}-\frac{2 (14 A b-5 a B) \sqrt{e x}}{45 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac{16 (14 A b-5 a B) \sqrt{e x}}{135 a^3 e^4 \sqrt{a+b x^3}}-\frac{(32 (14 A b-5 a B)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b x^6}{e^3}}} \, dx,x,\sqrt{e x}\right )}{135 a^3 e^4}\\ &=-\frac{2 A}{5 a e (e x)^{5/2} \left (a+b x^3\right )^{3/2}}-\frac{2 (14 A b-5 a B) \sqrt{e x}}{45 a^2 e^4 \left (a+b x^3\right )^{3/2}}-\frac{16 (14 A b-5 a B) \sqrt{e x}}{135 a^3 e^4 \sqrt{a+b x^3}}-\frac{16 (14 A b-5 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 )}{135 \sqrt [4]{3} a^{10/3} e^4 \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 = 0.0750694, size = 121, normalized size = 0.38 \[ \frac{x \left (a^2 \left (110 B x^3-54 A\right )+32 x^3 \left (a+b x^3\right ) \sqrt{\frac{b x^3}{a}+1} (5 a B-14 A b) \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};-\frac{b x^3}{a}\right )+a \left (80 b B x^6-308 A b x^3\right )-224 A b^2 x^6\right )}{135 a^3 (e x)^{7/2} \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.066, size = 7299, normalized size = 22.8 \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{5}{2}} \left (e x\right )^{\frac{7}{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^{3} e^{4} x^{13} + 3 \, a b^{2} e^{4} x^{10} + 3 \, a^{2} b e^{4} x^{7} + a^{3} e^{4} x^{4}}, 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}} \left (e x\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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