Optimal. Leaf size=201 \[ -\frac {a^3 e^{7/2} (8 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{192 b^{5/2}}+\frac {a^2 e^2 (e x)^{3/2} \sqrt {a+b x^3} (8 A b-3 a B)}{192 b^2}+\frac {(e x)^{9/2} \left (a+b x^3\right )^{3/2} (8 A b-3 a B)}{72 b e}+\frac {a (e x)^{9/2} \sqrt {a+b x^3} (8 A b-3 a B)}{96 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e} \]
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Rubi [A] time = 0.14, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {459, 279, 321, 329, 275, 217, 206} \begin {gather*} \frac {a^2 e^2 (e x)^{3/2} \sqrt {a+b x^3} (8 A b-3 a B)}{192 b^2}-\frac {a^3 e^{7/2} (8 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{192 b^{5/2}}+\frac {(e x)^{9/2} \left (a+b x^3\right )^{3/2} (8 A b-3 a B)}{72 b e}+\frac {a (e x)^{9/2} \sqrt {a+b x^3} (8 A b-3 a B)}{96 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 275
Rule 279
Rule 321
Rule 329
Rule 459
Rubi steps
\begin {align*} \int (e x)^{7/2} \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx &=\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}-\frac {\left (-12 A b+\frac {9 a B}{2}\right ) \int (e x)^{7/2} \left (a+b x^3\right )^{3/2} \, dx}{12 b}\\ &=\frac {(8 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{72 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}+\frac {(a (8 A b-3 a B)) \int (e x)^{7/2} \sqrt {a+b x^3} \, dx}{16 b}\\ &=\frac {a (8 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{96 b e}+\frac {(8 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{72 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}+\frac {\left (a^2 (8 A b-3 a B)\right ) \int \frac {(e x)^{7/2}}{\sqrt {a+b x^3}} \, dx}{64 b}\\ &=\frac {a^2 (8 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{192 b^2}+\frac {a (8 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{96 b e}+\frac {(8 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{72 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}-\frac {\left (a^3 (8 A b-3 a B) e^3\right ) \int \frac {\sqrt {e x}}{\sqrt {a+b x^3}} \, dx}{128 b^2}\\ &=\frac {a^2 (8 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{192 b^2}+\frac {a (8 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{96 b e}+\frac {(8 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{72 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}-\frac {\left (a^3 (8 A b-3 a B) e^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^6}{e^3}}} \, dx,x,\sqrt {e x}\right )}{64 b^2}\\ &=\frac {a^2 (8 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{192 b^2}+\frac {a (8 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{96 b e}+\frac {(8 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{72 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}-\frac {\left (a^3 (8 A b-3 a B) e^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^2}{e^3}}} \, dx,x,(e x)^{3/2}\right )}{192 b^2}\\ &=\frac {a^2 (8 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{192 b^2}+\frac {a (8 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{96 b e}+\frac {(8 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{72 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}-\frac {\left (a^3 (8 A b-3 a B) e^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {b x^2}{e^3}} \, dx,x,\frac {(e x)^{3/2}}{\sqrt {a+b x^3}}\right )}{192 b^2}\\ &=\frac {a^2 (8 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{192 b^2}+\frac {a (8 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{96 b e}+\frac {(8 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{72 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}-\frac {a^3 (8 A b-3 a B) e^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{192 b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 167, normalized size = 0.83 \begin {gather*} \frac {e^3 \sqrt {e x} \sqrt {a+b x^3} \left (3 a^{5/2} (3 a B-8 A b) \sinh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )+\sqrt {b} x^{3/2} \sqrt {\frac {b x^3}{a}+1} \left (-9 a^3 B+6 a^2 b \left (4 A+B x^3\right )+8 a b^2 x^3 \left (14 A+9 B x^3\right )+16 b^3 x^6 \left (4 A+3 B x^3\right )\right )\right )}{576 b^{5/2} \sqrt {x} \sqrt {\frac {b x^3}{a}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.84, size = 200, normalized size = 1.00 \begin {gather*} \frac {e^5 \sqrt {\frac {b}{e^3}} \left (8 a^3 A b-3 a^4 B\right ) \log \left (\sqrt {a+b x^3}-\sqrt {\frac {b}{e^3}} (e x)^{3/2}\right )}{192 b^3}+\frac {\sqrt {a+b x^3} \left (-9 a^3 B e^9 (e x)^{3/2}+24 a^2 A b e^9 (e x)^{3/2}+6 a^2 b B e^6 (e x)^{9/2}+112 a A b^2 e^6 (e x)^{9/2}+72 a b^2 B e^3 (e x)^{15/2}+64 A b^3 e^3 (e x)^{15/2}+48 b^3 B (e x)^{21/2}\right )}{576 b^2 e^7} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.20, size = 355, normalized size = 1.77 \begin {gather*} \left [-\frac {3 \, {\left (3 \, B a^{4} - 8 \, A a^{3} b\right )} e^{3} \sqrt {\frac {e}{b}} \log \left (-8 \, b^{2} e x^{6} - 8 \, a b e x^{3} - a^{2} e + 4 \, {\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt {b x^{3} + a} \sqrt {e x} \sqrt {\frac {e}{b}}\right ) - 4 \, {\left (48 \, B b^{3} e^{3} x^{10} + 8 \, {\left (9 \, B a b^{2} + 8 \, A b^{3}\right )} e^{3} x^{7} + 2 \, {\left (3 \, B a^{2} b + 56 \, A a b^{2}\right )} e^{3} x^{4} - 3 \, {\left (3 \, B a^{3} - 8 \, A a^{2} b\right )} e^{3} x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{2304 \, b^{2}}, -\frac {3 \, {\left (3 \, B a^{4} - 8 \, A a^{3} b\right )} e^{3} \sqrt {-\frac {e}{b}} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {e x} b x \sqrt {-\frac {e}{b}}}{2 \, b e x^{3} + a e}\right ) - 2 \, {\left (48 \, B b^{3} e^{3} x^{10} + 8 \, {\left (9 \, B a b^{2} + 8 \, A b^{3}\right )} e^{3} x^{7} + 2 \, {\left (3 \, B a^{2} b + 56 \, A a b^{2}\right )} e^{3} x^{4} - 3 \, {\left (3 \, B a^{3} - 8 \, A a^{2} b\right )} e^{3} x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{1152 \, b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.45, size = 387, normalized size = 1.93 \begin {gather*} \frac {1}{12} \, \sqrt {b x^{3} e^{4} + a e^{4}} {\left (2 \, x^{3} e^{\left (-1\right )} + \frac {a e^{\left (-1\right )}}{b}\right )} A a x^{\frac {3}{2}} e^{\frac {5}{2}} + \frac {1}{72} \, \sqrt {b x^{3} e^{4} + a e^{4}} {\left (2 \, {\left (4 \, x^{3} e^{\left (-4\right )} + \frac {a e^{\left (-4\right )}}{b}\right )} x^{3} e^{3} - \frac {3 \, a^{2} e^{\left (-1\right )}}{b^{2}}\right )} B a x^{\frac {3}{2}} e^{\frac {5}{2}} + \frac {1}{72} \, \sqrt {b x^{3} e^{4} + a e^{4}} {\left (2 \, {\left (4 \, x^{3} e^{\left (-4\right )} + \frac {a e^{\left (-4\right )}}{b}\right )} x^{3} e^{3} - \frac {3 \, a^{2} e^{\left (-1\right )}}{b^{2}}\right )} A b x^{\frac {3}{2}} e^{\frac {5}{2}} + \frac {1}{576} \, \sqrt {b x^{3} e^{4} + a e^{4}} {\left (2 \, {\left (4 \, {\left (6 \, x^{3} e^{\left (-7\right )} + \frac {a e^{\left (-7\right )}}{b}\right )} x^{3} e^{3} - \frac {5 \, a^{2} e^{\left (-4\right )}}{b^{2}}\right )} x^{3} e^{3} + \frac {15 \, a^{3} e^{\left (-1\right )}}{b^{3}}\right )} B b x^{\frac {3}{2}} e^{\frac {5}{2}} - \frac {{\left (9 \, B^{2} a^{8} e^{7} - 48 \, A B a^{7} b e^{7} + 64 \, A^{2} a^{6} b^{2} e^{7}\right )} e^{\left (-\frac {1}{2}\right )} \log \left ({\left | -{\left (3 \, B a^{4} x^{\frac {3}{2}} e^{\frac {11}{2}} - 8 \, A a^{3} b x^{\frac {3}{2}} e^{\frac {11}{2}}\right )} \sqrt {b} e^{\frac {1}{2}} + \sqrt {9 \, B^{2} a^{9} e^{12} - 48 \, A B a^{8} b e^{12} + 64 \, A^{2} a^{7} b^{2} e^{12} + {\left (3 \, B a^{4} x^{\frac {3}{2}} e^{\frac {11}{2}} - 8 \, A a^{3} b x^{\frac {3}{2}} e^{\frac {11}{2}}\right )}^{2} b e} \right |}\right )}{192 \, b^{\frac {5}{2}} {\left | -3 \, B a^{4} e^{3} + 8 \, A a^{3} b e^{3} \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.14, size = 7705, normalized size = 38.33 \begin {gather*} \text {output too large to display} \end {gather*}
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 \begin {gather*} \int {\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {7}{2}}\,{d x} \end {gather*}
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 \begin {gather*} \int \left (B\,x^3+A\right )\,{\left (e\,x\right )}^{7/2}\,{\left (b\,x^3+a\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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