Optimal. Leaf size=650 \[ -\frac {27\ 3^{3/4} \left (1-\sqrt {3}\right ) a^{7/3} \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} (a B+20 A b) F\left (\cos ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{896 b^{2/3} e^2 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {81 \sqrt [4]{3} a^{7/3} \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} (a B+20 A b) E\left (\cos ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{448 b^{2/3} e^2 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {81 \left (1+\sqrt {3}\right ) a^2 \sqrt {e x} \sqrt {a+b x^3} (a B+20 A b)}{448 b^{2/3} e^2 \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )}+\frac {(e x)^{5/2} \left (a+b x^3\right )^{5/2} (a B+20 A b)}{10 a e^4}+\frac {3 (e x)^{5/2} \left (a+b x^3\right )^{3/2} (a B+20 A b)}{28 e^4}+\frac {27 a (e x)^{5/2} \sqrt {a+b x^3} (a B+20 A b)}{224 e^4}-\frac {2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt {e x}} \]
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Rubi [A] time = 0.70, antiderivative size = 650, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {453, 279, 329, 308, 225, 1881} \[ \frac {81 \left (1+\sqrt {3}\right ) a^2 \sqrt {e x} \sqrt {a+b x^3} (a B+20 A b)}{448 b^{2/3} e^2 \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )}-\frac {27\ 3^{3/4} \left (1-\sqrt {3}\right ) a^{7/3} \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} (a B+20 A b) F\left (\cos ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{896 b^{2/3} e^2 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {81 \sqrt [4]{3} a^{7/3} \sqrt {e x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} (a B+20 A b) E\left (\cos ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x+\sqrt [3]{a}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{448 b^{2/3} e^2 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {(e x)^{5/2} \left (a+b x^3\right )^{5/2} (a B+20 A b)}{10 a e^4}+\frac {3 (e x)^{5/2} \left (a+b x^3\right )^{3/2} (a B+20 A b)}{28 e^4}+\frac {27 a (e x)^{5/2} \sqrt {a+b x^3} (a B+20 A b)}{224 e^4}-\frac {2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt {e x}} \]
Antiderivative was successfully verified.
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Rule 225
Rule 279
Rule 308
Rule 329
Rule 453
Rule 1881
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{5/2} \left (A+B x^3\right )}{(e x)^{3/2}} \, dx &=-\frac {2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt {e x}}+\frac {(20 A b+a B) \int (e x)^{3/2} \left (a+b x^3\right )^{5/2} \, dx}{a e^3}\\ &=\frac {(20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 a e^4}-\frac {2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt {e x}}+\frac {(3 (20 A b+a B)) \int (e x)^{3/2} \left (a+b x^3\right )^{3/2} \, dx}{4 e^3}\\ &=\frac {3 (20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{28 e^4}+\frac {(20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 a e^4}-\frac {2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt {e x}}+\frac {(27 a (20 A b+a B)) \int (e x)^{3/2} \sqrt {a+b x^3} \, dx}{56 e^3}\\ &=\frac {27 a (20 A b+a B) (e x)^{5/2} \sqrt {a+b x^3}}{224 e^4}+\frac {3 (20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{28 e^4}+\frac {(20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 a e^4}-\frac {2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt {e x}}+\frac {\left (81 a^2 (20 A b+a B)\right ) \int \frac {(e x)^{3/2}}{\sqrt {a+b x^3}} \, dx}{448 e^3}\\ &=\frac {27 a (20 A b+a B) (e x)^{5/2} \sqrt {a+b x^3}}{224 e^4}+\frac {3 (20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{28 e^4}+\frac {(20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 a e^4}-\frac {2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt {e x}}+\frac {\left (81 a^2 (20 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 e^4}\\ &=\frac {27 a (20 A b+a B) (e x)^{5/2} \sqrt {a+b x^3}}{224 e^4}+\frac {3 (20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{28 e^4}+\frac {(20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 a e^4}-\frac {2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt {e x}}-\frac {\left (81 a^2 (20 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^{2/3} e^4}-\frac {\left (81 \left (1-\sqrt {3}\right ) a^{8/3} (20 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 )}{448 b^{2/3} e^2}\\ &=\frac {27 a (20 A b+a B) (e x)^{5/2} \sqrt {a+b x^3}}{224 e^4}+\frac {81 \left (1+\sqrt {3}\right ) a^2 (20 A b+a B) \sqrt {e x} \sqrt {a+b x^3}}{448 b^{2/3} e^2 \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )}+\frac {3 (20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{3/2}}{28 e^4}+\frac {(20 A b+a B) (e x)^{5/2} \left (a+b x^3\right )^{5/2}}{10 a e^4}-\frac {2 A \left (a+b x^3\right )^{7/2}}{a e \sqrt {e x}}-\frac {81 \sqrt [4]{3} a^{7/3} (20 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 )}{448 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 {27\ 3^{3/4} \left (1-\sqrt {3}\right ) a^{7/3} (20 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 )}{896 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*}
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Mathematica [C] time = 0.05, size = 87, normalized size = 0.13 \[ \frac {2 x \sqrt {a+b x^3} \left (\frac {a^2 x^3 (a B+20 A b) \, _2F_1\left (-\frac {5}{2},\frac {5}{6};\frac {11}{6};-\frac {b x^3}{a}\right )}{\sqrt {\frac {b x^3}{a}+1}}-5 A \left (a+b x^3\right )^3\right )}{5 a (e x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B b^{2} x^{9} + {\left (2 \, B a b + A b^{2}\right )} x^{6} + {\left (B a^{2} + 2 \, A a b\right )} x^{3} + A a^{2}\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{e^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {5}{2}}}{\left (e x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.15, size = 6530, normalized size = 10.05 \[ \text {output too large to display} \]
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 \[ \int \frac {{\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {5}{2}}}{\left (e x\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\left (B\,x^3+A\right )\,{\left (b\,x^3+a\right )}^{5/2}}{{\left (e\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 39.74, size = 311, normalized size = 0.48 \[ \frac {A a^{\frac {5}{2}} \Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{6} \\ \frac {5}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {5}{6}\right )} + \frac {2 A a^{\frac {3}{2}} b x^{\frac {5}{2}} \Gamma \left (\frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{6} \\ \frac {11}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 e^{\frac {3}{2}} \Gamma \left (\frac {11}{6}\right )} + \frac {A \sqrt {a} b^{2} x^{\frac {11}{2}} \Gamma \left (\frac {11}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{6} \\ \frac {17}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 e^{\frac {3}{2}} \Gamma \left (\frac {17}{6}\right )} + \frac {B a^{\frac {5}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{6} \\ \frac {11}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 e^{\frac {3}{2}} \Gamma \left (\frac {11}{6}\right )} + \frac {2 B a^{\frac {3}{2}} b x^{\frac {11}{2}} \Gamma \left (\frac {11}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{6} \\ \frac {17}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 e^{\frac {3}{2}} \Gamma \left (\frac {17}{6}\right )} + \frac {B \sqrt {a} b^{2} x^{\frac {17}{2}} \Gamma \left (\frac {17}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {17}{6} \\ \frac {23}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 e^{\frac {3}{2}} \Gamma \left (\frac {23}{6}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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