Optimal. Leaf size=161 \[ \frac {a^2 \sqrt {e} (6 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{24 b^{3/2}}+\frac {(e x)^{3/2} \left (a+b x^3\right )^{3/2} (6 A b-a B)}{36 b e}+\frac {a (e x)^{3/2} \sqrt {a+b x^3} (6 A b-a B)}{24 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 b e} \]
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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 = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {459, 279, 329, 275, 217, 206} \begin {gather*} \frac {a^2 \sqrt {e} (6 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^{3/2}}+\frac {(e x)^{3/2} \left (a+b x^3\right )^{3/2} (6 A b-a B)}{36 b e}+\frac {a (e x)^{3/2} \sqrt {a+b x^3} (6 A b-a B)}{24 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 b e} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 275
Rule 279
Rule 329
Rule 459
Rubi steps
\begin {align*} \int \sqrt {e x} \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx &=\frac {B (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 b e}-\frac {\left (-9 A b+\frac {3 a B}{2}\right ) \int \sqrt {e x} \left (a+b x^3\right )^{3/2} \, dx}{9 b}\\ &=\frac {(6 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{36 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 b e}+\frac {(a (6 A b-a B)) \int \sqrt {e x} \sqrt {a+b x^3} \, dx}{8 b}\\ &=\frac {a (6 A b-a B) (e x)^{3/2} \sqrt {a+b x^3}}{24 b e}+\frac {(6 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{36 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 b e}+\frac {\left (a^2 (6 A b-a B)\right ) \int \frac {\sqrt {e x}}{\sqrt {a+b x^3}} \, dx}{16 b}\\ &=\frac {a (6 A b-a B) (e x)^{3/2} \sqrt {a+b x^3}}{24 b e}+\frac {(6 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{36 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 b e}+\frac {\left (a^2 (6 A b-a B)\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^6}{e^3}}} \, dx,x,\sqrt {e x}\right )}{8 b e}\\ &=\frac {a (6 A b-a B) (e x)^{3/2} \sqrt {a+b x^3}}{24 b e}+\frac {(6 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{36 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 b e}+\frac {\left (a^2 (6 A b-a B)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^2}{e^3}}} \, dx,x,(e x)^{3/2}\right )}{24 b e}\\ &=\frac {a (6 A b-a B) (e x)^{3/2} \sqrt {a+b x^3}}{24 b e}+\frac {(6 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{36 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 b e}+\frac {\left (a^2 (6 A b-a B)\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 e}\\ &=\frac {a (6 A b-a B) (e x)^{3/2} \sqrt {a+b x^3}}{24 b e}+\frac {(6 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{36 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 b e}+\frac {a^2 (6 A b-a B) \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{24 b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 143, normalized size = 0.89 \begin {gather*} \frac {\sqrt {e x} \sqrt {a+b x^3} \left (\sqrt {b} x^{3/2} \sqrt {\frac {b x^3}{a}+1} \left (3 a^2 B+2 a b \left (15 A+7 B x^3\right )+4 b^2 x^3 \left (3 A+2 B x^3\right )\right )-3 a^{3/2} (a B-6 A b) \sinh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )\right )}{72 b^{3/2} \sqrt {x} \sqrt {\frac {b x^3}{a}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.71, 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}+30 a A b e^6 (e x)^{3/2}+14 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 e^7}-\frac {e^2 \sqrt {\frac {b}{e^3}} \left (6 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^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.56, size = 273, normalized size = 1.70 \begin {gather*} \left [-\frac {3 \, {\left (B a^{3} - 6 \, A a^{2} b\right )} \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} x^{7} + 2 \, {\left (7 \, B a b + 6 \, A b^{2}\right )} x^{4} + 3 \, {\left (B a^{2} + 10 \, A a b\right )} x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{288 \, b}, \frac {3 \, {\left (B a^{3} - 6 \, A a^{2} b\right )} \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} x^{7} + 2 \, {\left (7 \, B a b + 6 \, A b^{2}\right )} x^{4} + 3 \, {\left (B a^{2} + 10 \, A a b\right )} x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{144 \, b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.06, size = 7290, normalized size = 45.28 \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}} \sqrt {e x}\,{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 )\,\sqrt {e\,x}\,{\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 [B] time = 25.46, size = 335, normalized size = 2.08 \begin {gather*} \frac {A a^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}} \sqrt {1 + \frac {b x^{3}}{a}}}{3 e} + \frac {A a^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}}}{12 e \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {A \sqrt {a} b \left (e x\right )^{\frac {9}{2}}}{4 e^{4} \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {A a^{2} \sqrt {e} \operatorname {asinh}{\left (\frac {\sqrt {b} \left (e x\right )^{\frac {3}{2}}}{\sqrt {a} e^{\frac {3}{2}}} \right )}}{4 \sqrt {b}} + \frac {A b^{2} \left (e x\right )^{\frac {15}{2}}}{6 \sqrt {a} e^{7} \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {B a^{\frac {5}{2}} \left (e x\right )^{\frac {3}{2}}}{24 b e \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {17 B a^{\frac {3}{2}} \left (e x\right )^{\frac {9}{2}}}{72 e^{4} \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {11 B \sqrt {a} b \left (e x\right )^{\frac {15}{2}}}{36 e^{7} \sqrt {1 + \frac {b x^{3}}{a}}} - \frac {B a^{3} \sqrt {e} \operatorname {asinh}{\left (\frac {\sqrt {b} \left (e x\right )^{\frac {3}{2}}}{\sqrt {a} e^{\frac {3}{2}}} \right )}}{24 b^{\frac {3}{2}}} + \frac {B b^{2} \left (e x\right )^{\frac {21}{2}}}{9 \sqrt {a} e^{10} \sqrt {1 + \frac {b x^{3}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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