Optimal. Leaf size=161 \[ -\frac {a^2 e^{7/2} (2 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{24 b^{5/2}}+\frac {a e^2 (e x)^{3/2} \sqrt {a+b x^3} (2 A b-a B)}{24 b^2}+\frac {(e x)^{9/2} \sqrt {a+b x^3} (2 A b-a B)}{12 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{9 b e} \]
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Rubi [A] time = 0.11, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 7, 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^{7/2} (2 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{24 b^{5/2}}+\frac {a e^2 (e x)^{3/2} \sqrt {a+b x^3} (2 A b-a B)}{24 b^2}+\frac {(e x)^{9/2} \sqrt {a+b x^3} (2 A b-a B)}{12 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{9 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} \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx &=\frac {B (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{9 b e}-\frac {\left (-9 A b+\frac {9 a B}{2}\right ) \int (e x)^{7/2} \sqrt {a+b x^3} \, dx}{9 b}\\ &=\frac {(2 A b-a B) (e x)^{9/2} \sqrt {a+b x^3}}{12 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{9 b e}+\frac {(a (2 A b-a B)) \int \frac {(e x)^{7/2}}{\sqrt {a+b x^3}} \, dx}{8 b}\\ &=\frac {a (2 A b-a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{24 b^2}+\frac {(2 A b-a B) (e x)^{9/2} \sqrt {a+b x^3}}{12 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{9 b e}-\frac {\left (a^2 (2 A b-a B) e^3\right ) \int \frac {\sqrt {e x}}{\sqrt {a+b x^3}} \, dx}{16 b^2}\\ &=\frac {a (2 A b-a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{24 b^2}+\frac {(2 A b-a B) (e x)^{9/2} \sqrt {a+b x^3}}{12 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{9 b e}-\frac {\left (a^2 (2 A b-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 )}{8 b^2}\\ &=\frac {a (2 A b-a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{24 b^2}+\frac {(2 A b-a B) (e x)^{9/2} \sqrt {a+b x^3}}{12 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{9 b e}-\frac {\left (a^2 (2 A b-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 )}{24 b^2}\\ &=\frac {a (2 A b-a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{24 b^2}+\frac {(2 A b-a B) (e x)^{9/2} \sqrt {a+b x^3}}{12 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{9 b e}-\frac {\left (a^2 (2 A b-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 )}{24 b^2}\\ &=\frac {a (2 A b-a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{24 b^2}+\frac {(2 A b-a B) (e x)^{9/2} \sqrt {a+b x^3}}{12 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{9 b e}-\frac {a^2 (2 A b-a B) e^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{24 b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 145, normalized size = 0.90 \begin {gather*} \frac {e^3 \sqrt {e x} \sqrt {a+b x^3} \left (3 a^{3/2} (a B-2 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 (-3 a^2 B+2 a b \left (3 A+B x^3\right )+4 b^2 x^3 \left (3 A+2 B x^3\right )\right )\right )}{72 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 = 0.60, size = 162, normalized size = 1.01 \begin {gather*} \frac {\sqrt {a+b x^3} \left (-3 a^2 B e^6 (e x)^{3/2}+6 a A b e^6 (e x)^{3/2}+2 a b B e^3 (e x)^{9/2}+12 A b^2 e^3 (e x)^{9/2}+8 b^2 B (e x)^{15/2}\right )}{72 b^2 e^4}+\frac {e^5 \sqrt {\frac {b}{e^3}} \left (2 a^2 A b-a^3 B\right ) \log \left (\sqrt {a+b x^3}-\sqrt {\frac {b}{e^3}} (e x)^{3/2}\right )}{24 b^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.26, size = 295, normalized size = 1.83 \begin {gather*} \left [-\frac {3 \, {\left (B a^{3} - 2 \, A a^{2} 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 (8 \, B b^{2} e^{3} x^{7} + 2 \, {\left (B a b + 6 \, A b^{2}\right )} e^{3} x^{4} - 3 \, {\left (B a^{2} - 2 \, A a b\right )} e^{3} x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{288 \, b^{2}}, -\frac {3 \, {\left (B a^{3} - 2 \, A a^{2} 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 (8 \, B b^{2} e^{3} x^{7} + 2 \, {\left (B a b + 6 \, A b^{2}\right )} e^{3} x^{4} - 3 \, {\left (B a^{2} - 2 \, A a b\right )} e^{3} x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{144 \, b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.75, size = 251, normalized size = 1.56 \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 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 x^{\frac {3}{2}} e^{\frac {5}{2}} - \frac {{\left (B^{2} a^{6} e^{7} - 4 \, A B a^{5} b e^{7} + 4 \, A^{2} a^{4} b^{2} e^{7}\right )} e^{\left (-\frac {1}{2}\right )} \log \left ({\left | -{\left (B a^{3} x^{\frac {3}{2}} e^{\frac {11}{2}} - 2 \, A a^{2} b x^{\frac {3}{2}} e^{\frac {11}{2}}\right )} \sqrt {b} e^{\frac {1}{2}} + \sqrt {B^{2} a^{7} e^{12} - 4 \, A B a^{6} b e^{12} + 4 \, A^{2} a^{5} b^{2} e^{12} + {\left (B a^{3} x^{\frac {3}{2}} e^{\frac {11}{2}} - 2 \, A a^{2} b x^{\frac {3}{2}} e^{\frac {11}{2}}\right )}^{2} b e} \right |}\right )}{24 \, b^{\frac {5}{2}} {\left | -B a^{3} e^{3} + 2 \, A a^{2} b e^{3} \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 8.02, size = 7293, normalized size = 45.30 \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 )} \sqrt {b x^{3} + a} \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 = -0.01 \begin {gather*} \int \left (B\,x^3+A\right )\,{\left (e\,x\right )}^{7/2}\,\sqrt {b\,x^3+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 130.86, size = 292, normalized size = 1.81 \begin {gather*} \frac {A a^{\frac {3}{2}} e^{\frac {7}{2}} x^{\frac {3}{2}}}{12 b \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {A \sqrt {a} e^{\frac {7}{2}} x^{\frac {9}{2}}}{4 \sqrt {1 + \frac {b x^{3}}{a}}} - \frac {A a^{2} e^{\frac {7}{2}} \operatorname {asinh}{\left (\frac {\sqrt {b} x^{\frac {3}{2}}}{\sqrt {a}} \right )}}{12 b^{\frac {3}{2}}} + \frac {A b e^{\frac {7}{2}} x^{\frac {15}{2}}}{6 \sqrt {a} \sqrt {1 + \frac {b x^{3}}{a}}} - \frac {B a^{\frac {5}{2}} e^{\frac {7}{2}} x^{\frac {3}{2}}}{24 b^{2} \sqrt {1 + \frac {b x^{3}}{a}}} - \frac {B a^{\frac {3}{2}} e^{\frac {7}{2}} x^{\frac {9}{2}}}{72 b \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {5 B \sqrt {a} e^{\frac {7}{2}} x^{\frac {15}{2}}}{36 \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {B a^{3} e^{\frac {7}{2}} \operatorname {asinh}{\left (\frac {\sqrt {b} x^{\frac {3}{2}}}{\sqrt {a}} \right )}}{24 b^{\frac {5}{2}}} + \frac {B b e^{\frac {7}{2}} x^{\frac {21}{2}}}{9 \sqrt {a} \sqrt {1 + \frac {b x^{3}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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