Optimal. Leaf size=121 \[ \frac {a \sqrt {e} (4 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{12 b^{3/2}}+\frac {(e x)^{3/2} \sqrt {a+b x^3} (4 A b-a B)}{12 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{6 b e} \]
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Rubi [A] time = 0.09, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, 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 \sqrt {e} (4 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{12 b^{3/2}}+\frac {(e x)^{3/2} \sqrt {a+b x^3} (4 A b-a B)}{12 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{6 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} \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx &=\frac {B (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{6 b e}-\frac {\left (-6 A b+\frac {3 a B}{2}\right ) \int \sqrt {e x} \sqrt {a+b x^3} \, dx}{6 b}\\ &=\frac {(4 A b-a B) (e x)^{3/2} \sqrt {a+b x^3}}{12 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{6 b e}+\frac {(a (4 A b-a B)) \int \frac {\sqrt {e x}}{\sqrt {a+b x^3}} \, dx}{8 b}\\ &=\frac {(4 A b-a B) (e x)^{3/2} \sqrt {a+b x^3}}{12 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{6 b e}+\frac {(a (4 A b-a B)) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^6}{e^3}}} \, dx,x,\sqrt {e x}\right )}{4 b e}\\ &=\frac {(4 A b-a B) (e x)^{3/2} \sqrt {a+b x^3}}{12 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{6 b e}+\frac {(a (4 A b-a B)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^2}{e^3}}} \, dx,x,(e x)^{3/2}\right )}{12 b e}\\ &=\frac {(4 A b-a B) (e x)^{3/2} \sqrt {a+b x^3}}{12 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{6 b e}+\frac {(a (4 A b-a B)) \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 )}{12 b e}\\ &=\frac {(4 A b-a B) (e x)^{3/2} \sqrt {a+b x^3}}{12 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{6 b e}+\frac {a (4 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 )}{12 b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 119, normalized size = 0.98 \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 (B \left (a+2 b x^3\right )+4 A b\right )-\sqrt {a} (a B-4 A b) \sinh ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )\right )}{12 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 = 0.52, size = 123, normalized size = 1.02 \begin {gather*} \frac {\sqrt {a+b x^3} \left (a B e^3 (e x)^{3/2}+4 A b e^3 (e x)^{3/2}+2 b B (e x)^{9/2}\right )}{12 b e^4}-\frac {e^2 \sqrt {\frac {b}{e^3}} \left (4 a A b-a^2 B\right ) \log \left (\sqrt {a+b x^3}-\sqrt {\frac {b}{e^3}} (e x)^{3/2}\right )}{12 b^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.16, size = 221, normalized size = 1.83 \begin {gather*} \left [-\frac {{\left (B a^{2} - 4 \, A a 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 (2 \, B b x^{4} + {\left (B a + 4 \, A b\right )} x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{48 \, b}, \frac {{\left (B a^{2} - 4 \, A a 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 (2 \, B b x^{4} + {\left (B a + 4 \, A b\right )} x\right )} \sqrt {b x^{3} + a} \sqrt {e x}}{24 \, b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 137, normalized size = 1.13 \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 )} B x^{\frac {3}{2}} e^{\left (-\frac {1}{2}\right )} + \frac {B a^{2} e^{\frac {1}{2}} \log \left ({\left | -\sqrt {b} x^{\frac {3}{2}} e^{2} + \sqrt {b x^{3} e^{4} + a e^{4}} \right |}\right )}{12 \, b^{\frac {3}{2}}} + \frac {1}{3} \, {\left (\sqrt {b x^{3} e^{4} + a e^{4}} x^{\frac {3}{2}} e^{\frac {3}{2}} - \frac {a e^{\frac {7}{2}} \log \left ({\left | -\sqrt {b} x^{\frac {3}{2}} e^{2} + \sqrt {b x^{3} e^{4} + a e^{4}} \right |}\right )}{\sqrt {b}}\right )} A e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.09, size = 6858, normalized size = 56.68 \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 )} \sqrt {b x^{3} + a} \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}\,\sqrt {b\,x^3+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.67, size = 201, normalized size = 1.66 \begin {gather*} \frac {A \sqrt {a} \left (e x\right )^{\frac {3}{2}} \sqrt {1 + \frac {b x^{3}}{a}}}{3 e} + \frac {A a \sqrt {e} \operatorname {asinh}{\left (\frac {\sqrt {b} \left (e x\right )^{\frac {3}{2}}}{\sqrt {a} e^{\frac {3}{2}}} \right )}}{3 \sqrt {b}} + \frac {B a^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}}}{12 b e \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {B \sqrt {a} \left (e x\right )^{\frac {9}{2}}}{4 e^{4} \sqrt {1 + \frac {b x^{3}}{a}}} - \frac {B a^{2} \sqrt {e} \operatorname {asinh}{\left (\frac {\sqrt {b} \left (e x\right )^{\frac {3}{2}}}{\sqrt {a} e^{\frac {3}{2}}} \right )}}{12 b^{\frac {3}{2}}} + \frac {B b \left (e x\right )^{\frac {15}{2}}}{6 \sqrt {a} e^{7} \sqrt {1 + \frac {b x^{3}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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