Optimal. Leaf size=621 \[ -\frac {9\ 3^{3/4} \left (1-\sqrt {3}\right ) a^{7/3} e \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} (4 A b-a B) F\left (\cos ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{896 b^{5/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {27 \sqrt [4]{3} a^{7/3} e \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} (4 A b-a B) E\left (\cos ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{448 b^{5/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {27 \left (1+\sqrt {3}\right ) a^2 e \sqrt {e x} \sqrt {a+b x^3} (4 A b-a B)}{448 b^{5/3} \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )}+\frac {(e x)^{5/2} \left (a+b x^3\right )^{3/2} (4 A b-a B)}{28 b e}+\frac {9 a (e x)^{5/2} \sqrt {a+b x^3} (4 A b-a B)}{224 b e}+\frac {B (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 b e} \]
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Rubi [A] time = 0.66, antiderivative size = 621, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {459, 279, 329, 308, 225, 1881} \[ \frac {27 \left (1+\sqrt {3}\right ) a^2 e \sqrt {e x} \sqrt {a+b x^3} (4 A b-a B)}{448 b^{5/3} \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )}-\frac {9\ 3^{3/4} \left (1-\sqrt {3}\right ) a^{7/3} e \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} (4 A b-a B) F\left (\cos ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{896 b^{5/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {27 \sqrt [4]{3} a^{7/3} e \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} (4 A b-a B) E\left (\cos ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{448 b^{5/3} \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {(e x)^{5/2} \left (a+b x^3\right )^{3/2} (4 A b-a B)}{28 b e}+\frac {9 a (e x)^{5/2} \sqrt {a+b x^3} (4 A b-a B)}{224 b e}+\frac {B (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 b e} \]
Antiderivative was successfully verified.
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Rule 225
Rule 279
Rule 308
Rule 329
Rule 459
Rule 1881
Rubi steps
\begin {align*} \int (e x)^{3/2} \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx &=\frac {B (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 b e}-\frac {\left (-10 A b+\frac {5 a B}{2}\right ) \int (e x)^{3/2} \left (a+b x^3\right )^{3/2} \, dx}{10 b}\\ &=\frac {(4 A b-a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{28 b e}+\frac {B (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 b e}+\frac {(9 a (4 A b-a B)) \int (e x)^{3/2} \sqrt {a+b x^3} \, dx}{56 b}\\ &=\frac {9 a (4 A b-a B) (e x)^{5/2} \sqrt {a+b x^3}}{224 b e}+\frac {(4 A b-a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{28 b e}+\frac {B (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 b e}+\frac {\left (27 a^2 (4 A b-a B)\right ) \int \frac {(e x)^{3/2}}{\sqrt {a+b x^3}} \, dx}{448 b}\\ &=\frac {9 a (4 A b-a B) (e x)^{5/2} \sqrt {a+b x^3}}{224 b e}+\frac {(4 A b-a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{28 b e}+\frac {B (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 b e}+\frac {\left (27 a^2 (4 A b-a B)\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {a+\frac {b x^6}{e^3}}} \, dx,x,\sqrt {e x}\right )}{224 b e}\\ &=\frac {9 a (4 A b-a B) (e x)^{5/2} \sqrt {a+b x^3}}{224 b e}+\frac {(4 A b-a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{28 b e}+\frac {B (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 b e}-\frac {\left (27 a^2 (4 A b-a B)\right ) \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 )}{448 b^{5/3} e}-\frac {\left (27 \left (1-\sqrt {3}\right ) a^{8/3} (4 A b-a B) e\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^6}{e^3}}} \, dx,x,\sqrt {e x}\right )}{448 b^{5/3}}\\ &=\frac {9 a (4 A b-a B) (e x)^{5/2} \sqrt {a+b x^3}}{224 b e}+\frac {27 \left (1+\sqrt {3}\right ) a^2 (4 A b-a B) e \sqrt {e x} \sqrt {a+b x^3}}{448 b^{5/3} \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )}+\frac {(4 A b-a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{28 b e}+\frac {B (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 b e}-\frac {27 \sqrt [4]{3} a^{7/3} (4 A b-a B) e \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 )}{448 b^{5/3} \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 {9\ 3^{3/4} \left (1-\sqrt {3}\right ) a^{7/3} (4 A b-a B) e \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 )}{896 b^{5/3} \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*}
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Mathematica [C] time = 0.13, size = 96, normalized size = 0.15 \[ \frac {x (e x)^{3/2} \sqrt {a+b x^3} \left (a (4 A b-a B) \, _2F_1\left (-\frac {3}{2},\frac {5}{6};\frac {11}{6};-\frac {b x^3}{a}\right )+B \sqrt {\frac {b x^3}{a}+1} \left (a+b x^3\right )^2\right )}{10 b \sqrt {\frac {b x^3}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B b e x^{7} + {\left (B a + A b\right )} e x^{4} + A a e x\right )} \sqrt {b x^{3} + a} \sqrt {e x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.96, size = 5790, normalized size = 9.32 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (B\,x^3+A\right )\,{\left (e\,x\right )}^{3/2}\,{\left (b\,x^3+a\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 45.76, size = 199, normalized size = 0.32 \[ \frac {A a^{\frac {3}{2}} e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{6} \\ \frac {11}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {11}{6}\right )} + \frac {A \sqrt {a} b e^{\frac {3}{2}} x^{\frac {11}{2}} \Gamma \left (\frac {11}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{6} \\ \frac {17}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {17}{6}\right )} + \frac {B a^{\frac {3}{2}} e^{\frac {3}{2}} x^{\frac {11}{2}} \Gamma \left (\frac {11}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{6} \\ \frac {17}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {17}{6}\right )} + \frac {B \sqrt {a} b e^{\frac {3}{2}} x^{\frac {17}{2}} \Gamma \left (\frac {17}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {17}{6} \\ \frac {23}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {23}{6}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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