Optimal. Leaf size=241 \[ -\frac {a^4 e^{7/2} (10 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{384 b^{5/2}}+\frac {a^3 e^2 (e x)^{3/2} \sqrt {a+b x^3} (10 A b-3 a B)}{384 b^2}+\frac {a^2 (e x)^{9/2} \sqrt {a+b x^3} (10 A b-3 a B)}{192 b e}+\frac {(e x)^{9/2} \left (a+b x^3\right )^{5/2} (10 A b-3 a B)}{120 b e}+\frac {a (e x)^{9/2} \left (a+b x^3\right )^{3/2} (10 A b-3 a B)}{144 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e} \]
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Rubi [A] time = 0.16, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 9, 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^3 e^2 (e x)^{3/2} \sqrt {a+b x^3} (10 A b-3 a B)}{384 b^2}-\frac {a^4 e^{7/2} (10 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{384 b^{5/2}}+\frac {a^2 (e x)^{9/2} \sqrt {a+b x^3} (10 A b-3 a B)}{192 b e}+\frac {(e x)^{9/2} \left (a+b x^3\right )^{5/2} (10 A b-3 a B)}{120 b e}+\frac {a (e x)^{9/2} \left (a+b x^3\right )^{3/2} (10 A b-3 a B)}{144 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 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 )^{5/2} \left (A+B x^3\right ) \, dx &=\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}-\frac {\left (-15 A b+\frac {9 a B}{2}\right ) \int (e x)^{7/2} \left (a+b x^3\right )^{5/2} \, dx}{15 b}\\ &=\frac {(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}+\frac {(a (10 A b-3 a B)) \int (e x)^{7/2} \left (a+b x^3\right )^{3/2} \, dx}{16 b}\\ &=\frac {a (10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{144 b e}+\frac {(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}+\frac {\left (a^2 (10 A b-3 a B)\right ) \int (e x)^{7/2} \sqrt {a+b x^3} \, dx}{32 b}\\ &=\frac {a^2 (10 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{192 b e}+\frac {a (10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{144 b e}+\frac {(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}+\frac {\left (a^3 (10 A b-3 a B)\right ) \int \frac {(e x)^{7/2}}{\sqrt {a+b x^3}} \, dx}{128 b}\\ &=\frac {a^3 (10 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{384 b^2}+\frac {a^2 (10 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{192 b e}+\frac {a (10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{144 b e}+\frac {(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}-\frac {\left (a^4 (10 A b-3 a B) e^3\right ) \int \frac {\sqrt {e x}}{\sqrt {a+b x^3}} \, dx}{256 b^2}\\ &=\frac {a^3 (10 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{384 b^2}+\frac {a^2 (10 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{192 b e}+\frac {a (10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{144 b e}+\frac {(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}-\frac {\left (a^4 (10 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 )}{128 b^2}\\ &=\frac {a^3 (10 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{384 b^2}+\frac {a^2 (10 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{192 b e}+\frac {a (10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{144 b e}+\frac {(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}-\frac {\left (a^4 (10 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 )}{384 b^2}\\ &=\frac {a^3 (10 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{384 b^2}+\frac {a^2 (10 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{192 b e}+\frac {a (10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{144 b e}+\frac {(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}-\frac {\left (a^4 (10 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 )}{384 b^2}\\ &=\frac {a^3 (10 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{384 b^2}+\frac {a^2 (10 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{192 b e}+\frac {a (10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{144 b e}+\frac {(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}-\frac {a^4 (10 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 )}{384 b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 188, normalized size = 0.78 \begin {gather*} \frac {e^3 \sqrt {e x} \sqrt {a+b x^3} \left (15 a^{7/2} (3 a B-10 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 (-45 a^4 B+30 a^3 b \left (5 A+B x^3\right )+4 a^2 b^2 x^3 \left (295 A+186 B x^3\right )+16 a b^3 x^6 \left (85 A+63 B x^3\right )+96 b^4 x^9 \left (5 A+4 B x^3\right )\right )\right )}{5760 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 = 0.95, size = 238, normalized size = 0.99 \begin {gather*} \frac {e^5 \sqrt {\frac {b}{e^3}} \left (10 a^4 A b-3 a^5 B\right ) \log \left (\sqrt {a+b x^3}-\sqrt {\frac {b}{e^3}} (e x)^{3/2}\right )}{384 b^3}+\frac {\sqrt {a+b x^3} \left (-45 a^4 B e^{12} (e x)^{3/2}+150 a^3 A b e^{12} (e x)^{3/2}+30 a^3 b B e^9 (e x)^{9/2}+1180 a^2 A b^2 e^9 (e x)^{9/2}+744 a^2 b^2 B e^6 (e x)^{15/2}+1360 a A b^3 e^6 (e x)^{15/2}+1008 a b^3 B e^3 (e x)^{21/2}+480 A b^4 e^3 (e x)^{21/2}+384 b^4 B (e x)^{27/2}\right )}{5760 b^2 e^{10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.33, size = 409, normalized size = 1.70 \begin {gather*} \left [-\frac {15 \, {\left (3 \, B a^{5} - 10 \, A a^{4} 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 (384 \, B b^{4} e^{3} x^{13} + 48 \, {\left (21 \, B a b^{3} + 10 \, A b^{4}\right )} e^{3} x^{10} + 8 \, {\left (93 \, B a^{2} b^{2} + 170 \, A a b^{3}\right )} e^{3} x^{7} + 10 \, {\left (3 \, B a^{3} b + 118 \, A a^{2} b^{2}\right )} e^{3} x^{4} - 15 \, {\left (3 \, B a^{4} - 10 \, A a^{3} b\right )} e^{3} x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{23040 \, b^{2}}, -\frac {15 \, {\left (3 \, B a^{5} - 10 \, A a^{4} 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 (384 \, B b^{4} e^{3} x^{13} + 48 \, {\left (21 \, B a b^{3} + 10 \, A b^{4}\right )} e^{3} x^{10} + 8 \, {\left (93 \, B a^{2} b^{2} + 170 \, A a b^{3}\right )} e^{3} x^{7} + 10 \, {\left (3 \, B a^{3} b + 118 \, A a^{2} b^{2}\right )} e^{3} x^{4} - 15 \, {\left (3 \, B a^{4} - 10 \, A a^{3} b\right )} e^{3} x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{11520 \, b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.49, size = 563, normalized size = 2.34 \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^{2} 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^{2} x^{\frac {3}{2}} e^{\frac {5}{2}} + \frac {1}{36} \, \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 a b x^{\frac {3}{2}} e^{\frac {5}{2}} + \frac {1}{288} \, \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 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 )} A b^{2} x^{\frac {3}{2}} e^{\frac {5}{2}} + \frac {1}{5760} \, \sqrt {b x^{3} e^{4} + a e^{4}} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, x^{3} e^{\left (-10\right )} + \frac {a e^{\left (-10\right )}}{b}\right )} x^{3} e^{3} - \frac {7 \, a^{2} e^{\left (-7\right )}}{b^{2}}\right )} x^{3} e^{3} + \frac {35 \, a^{3} e^{\left (-4\right )}}{b^{3}}\right )} x^{3} e^{3} - \frac {105 \, a^{4} e^{\left (-1\right )}}{b^{4}}\right )} B b^{2} x^{\frac {3}{2}} e^{\frac {5}{2}} - \frac {{\left (9 \, B^{2} a^{10} e^{7} - 60 \, A B a^{9} b e^{7} + 100 \, A^{2} a^{8} b^{2} e^{7}\right )} e^{\left (-\frac {1}{2}\right )} \log \left ({\left | -{\left (3 \, B a^{5} x^{\frac {3}{2}} e^{\frac {11}{2}} - 10 \, A a^{4} b x^{\frac {3}{2}} e^{\frac {11}{2}}\right )} \sqrt {b} e^{\frac {1}{2}} + \sqrt {9 \, B^{2} a^{11} e^{12} - 60 \, A B a^{10} b e^{12} + 100 \, A^{2} a^{9} b^{2} e^{12} + {\left (3 \, B a^{5} x^{\frac {3}{2}} e^{\frac {11}{2}} - 10 \, A a^{4} b x^{\frac {3}{2}} e^{\frac {11}{2}}\right )}^{2} b e} \right |}\right )}{384 \, b^{\frac {5}{2}} {\left | -3 \, B a^{5} e^{3} + 10 \, A a^{4} b e^{3} \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.06, size = 8117, normalized size = 33.68 \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 {5}{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 )}^{5/2} \,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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