Optimal. Leaf size=201 \[ \frac {5 a^3 \sqrt {e} (8 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{192 b^{3/2}}+\frac {5 a^2 (e x)^{3/2} \sqrt {a+b x^3} (8 A b-a B)}{192 b e}+\frac {(e x)^{3/2} \left (a+b x^3\right )^{5/2} (8 A b-a B)}{72 b e}+\frac {5 a (e x)^{3/2} \left (a+b x^3\right )^{3/2} (8 A b-a B)}{288 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{7/2}}{12 b e} \]
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Rubi [A] time = 0.13, antiderivative size = 201, 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 = {459, 279, 329, 275, 217, 206} \begin {gather*} \frac {5 a^3 \sqrt {e} (8 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{192 b^{3/2}}+\frac {5 a^2 (e x)^{3/2} \sqrt {a+b x^3} (8 A b-a B)}{192 b e}+\frac {(e x)^{3/2} \left (a+b x^3\right )^{5/2} (8 A b-a B)}{72 b e}+\frac {5 a (e x)^{3/2} \left (a+b x^3\right )^{3/2} (8 A b-a B)}{288 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{7/2}}{12 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 )^{5/2} \left (A+B x^3\right ) \, dx &=\frac {B (e x)^{3/2} \left (a+b x^3\right )^{7/2}}{12 b e}-\frac {\left (-12 A b+\frac {3 a B}{2}\right ) \int \sqrt {e x} \left (a+b x^3\right )^{5/2} \, dx}{12 b}\\ &=\frac {(8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{72 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{7/2}}{12 b e}+\frac {(5 a (8 A b-a B)) \int \sqrt {e x} \left (a+b x^3\right )^{3/2} \, dx}{48 b}\\ &=\frac {5 a (8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{288 b e}+\frac {(8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{72 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{7/2}}{12 b e}+\frac {\left (5 a^2 (8 A b-a B)\right ) \int \sqrt {e x} \sqrt {a+b x^3} \, dx}{64 b}\\ &=\frac {5 a^2 (8 A b-a B) (e x)^{3/2} \sqrt {a+b x^3}}{192 b e}+\frac {5 a (8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{288 b e}+\frac {(8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{72 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{7/2}}{12 b e}+\frac {\left (5 a^3 (8 A b-a B)\right ) \int \frac {\sqrt {e x}}{\sqrt {a+b x^3}} \, dx}{128 b}\\ &=\frac {5 a^2 (8 A b-a B) (e x)^{3/2} \sqrt {a+b x^3}}{192 b e}+\frac {5 a (8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{288 b e}+\frac {(8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{72 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{7/2}}{12 b e}+\frac {\left (5 a^3 (8 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 )}{64 b e}\\ &=\frac {5 a^2 (8 A b-a B) (e x)^{3/2} \sqrt {a+b x^3}}{192 b e}+\frac {5 a (8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{288 b e}+\frac {(8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{72 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{7/2}}{12 b e}+\frac {\left (5 a^3 (8 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 )}{192 b e}\\ &=\frac {5 a^2 (8 A b-a B) (e x)^{3/2} \sqrt {a+b x^3}}{192 b e}+\frac {5 a (8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{288 b e}+\frac {(8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{72 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{7/2}}{12 b e}+\frac {\left (5 a^3 (8 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 )}{192 b e}\\ &=\frac {5 a^2 (8 A b-a B) (e x)^{3/2} \sqrt {a+b x^3}}{192 b e}+\frac {5 a (8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{288 b e}+\frac {(8 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{72 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{7/2}}{12 b e}+\frac {5 a^3 (8 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 )}{192 b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 146, normalized size = 0.73 \begin {gather*} \frac {x \sqrt {e x} \sqrt {a+b x^3} \left (\frac {(8 A b-a B) \left (15 a^{5/2} \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 (33 a^2+26 a b x^3+8 b^2 x^6\right )\right )}{48 \sqrt {b} x^{3/2} \sqrt {\frac {b x^3}{a}+1}}+B \left (a+b x^3\right )^3\right )}{12 b} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.77, size = 200, normalized size = 1.00 \begin {gather*} \frac {\sqrt {a+b x^3} \left (15 a^3 B e^9 (e x)^{3/2}+264 a^2 A b e^9 (e x)^{3/2}+118 a^2 b B e^6 (e x)^{9/2}+208 a A b^2 e^6 (e x)^{9/2}+136 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 e^{10}}-\frac {5 e^2 \sqrt {\frac {b}{e^3}} \left (8 a^3 A b-a^4 B\right ) \log \left (\sqrt {a+b x^3}-\sqrt {\frac {b}{e^3}} (e x)^{3/2}\right )}{192 b^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.39, size = 323, normalized size = 1.61 \begin {gather*} \left [-\frac {15 \, {\left (B a^{4} - 8 \, A a^{3} 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 (48 \, B b^{3} x^{10} + 8 \, {\left (17 \, B a b^{2} + 8 \, A b^{3}\right )} x^{7} + 2 \, {\left (59 \, B a^{2} b + 104 \, A a b^{2}\right )} x^{4} + 3 \, {\left (5 \, B a^{3} + 88 \, A a^{2} b\right )} x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{2304 \, b}, \frac {15 \, {\left (B a^{4} - 8 \, A a^{3} 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 (48 \, B b^{3} x^{10} + 8 \, {\left (17 \, B a b^{2} + 8 \, A b^{3}\right )} x^{7} + 2 \, {\left (59 \, B a^{2} b + 104 \, A a b^{2}\right )} x^{4} + 3 \, {\left (5 \, B a^{3} + 88 \, A a^{2} b\right )} x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{1152 \, 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 = 1.10, size = 7702, normalized size = 38.32 \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}} \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 = -0.00 \begin {gather*} \int \left (B\,x^3+A\right )\,\sqrt {e\,x}\,{\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 [B] time = 56.28, size = 413, normalized size = 2.05 \begin {gather*} \frac {A a^{\frac {5}{2}} \left (e x\right )^{\frac {3}{2}} \sqrt {1 + \frac {b x^{3}}{a}}}{3 e} + \frac {A a^{\frac {5}{2}} \left (e x\right )^{\frac {3}{2}}}{8 e \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {35 A a^{\frac {3}{2}} b \left (e x\right )^{\frac {9}{2}}}{72 e^{4} \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {17 A \sqrt {a} b^{2} \left (e x\right )^{\frac {15}{2}}}{36 e^{7} \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {5 A 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 \sqrt {b}} + \frac {A b^{3} \left (e x\right )^{\frac {21}{2}}}{9 \sqrt {a} e^{10} \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {5 B a^{\frac {7}{2}} \left (e x\right )^{\frac {3}{2}}}{192 b e \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {133 B a^{\frac {5}{2}} \left (e x\right )^{\frac {9}{2}}}{576 e^{4} \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {127 B a^{\frac {3}{2}} b \left (e x\right )^{\frac {15}{2}}}{288 e^{7} \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {23 B \sqrt {a} b^{2} \left (e x\right )^{\frac {21}{2}}}{72 e^{10} \sqrt {1 + \frac {b x^{3}}{a}}} - \frac {5 B a^{4} \sqrt {e} \operatorname {asinh}{\left (\frac {\sqrt {b} \left (e x\right )^{\frac {3}{2}}}{\sqrt {a} e^{\frac {3}{2}}} \right )}}{192 b^{\frac {3}{2}}} + \frac {B b^{3} \left (e x\right )^{\frac {27}{2}}}{12 \sqrt {a} e^{13} \sqrt {1 + \frac {b x^{3}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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