Optimal. Leaf size=161 \[ \frac {a (6 A b-a B) (e x)^{3/2} \sqrt {a+b x^3}}{24 b e}+\frac {(6 A b-a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{36 b e}+\frac {B (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 b e}+\frac {a^2 (6 A b-a B) \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{24 b^{3/2}} \]
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Rubi [A]
time = 0.08, 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 = {470, 285, 335,
281, 223, 212} \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 212
Rule 223
Rule 281
Rule 285
Rule 335
Rule 470
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 ) \text {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 ) \text {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 ) \text {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 = 0.29, size = 120, normalized size = 0.75 \begin {gather*} \frac {x \sqrt {e x} \sqrt {a+b x^3} \left (30 a A b+3 a^2 B+12 A b^2 x^3+14 a b B x^3+8 b^2 B x^6\right )}{72 b}-\frac {a^2 (-6 A b+a B) \sqrt {e x} \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {b} x^{3/2}}\right )}{24 b^{3/2} \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.52, size = 7290, normalized size = 45.28
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1080\) |
elliptic | \(\text {Expression too large to display}\) | \(1183\) |
default | \(\text {Expression too large to display}\) | \(7290\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 290 vs.
\(2 (111) = 222\).
time = 0.51, size = 290, normalized size = 1.80 \begin {gather*} -\frac {1}{144} \, {\left (6 \, {\left (\frac {3 \, a^{2} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}{\sqrt {b} + \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}\right )}{\sqrt {b}} + \frac {2 \, {\left (\frac {3 \, \sqrt {b x^{3} + a} a^{2} b}{x^{\frac {3}{2}}} - \frac {5 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{2}}{x^{\frac {9}{2}}}\right )}}{b^{2} - \frac {2 \, {\left (b x^{3} + a\right )} b}{x^{3}} + \frac {{\left (b x^{3} + a\right )}^{2}}{x^{6}}}\right )} A - {\left (\frac {3 \, a^{3} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}{\sqrt {b} + \frac {\sqrt {b x^{3} + a}}{x^{\frac {3}{2}}}}\right )}{b^{\frac {3}{2}}} + \frac {2 \, {\left (\frac {3 \, \sqrt {b x^{3} + a} a^{3} b^{2}}{x^{\frac {3}{2}}} - \frac {8 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{3} b}{x^{\frac {9}{2}}} - \frac {3 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a^{3}}{x^{\frac {15}{2}}}\right )}}{b^{4} - \frac {3 \, {\left (b x^{3} + a\right )} b^{3}}{x^{3}} + \frac {3 \, {\left (b x^{3} + a\right )}^{2} b^{2}}{x^{6}} - \frac {{\left (b x^{3} + a\right )}^{3} b}{x^{9}}}\right )} B\right )} e^{\frac {1}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.63, size = 258, normalized size = 1.60 \begin {gather*} \left [-\frac {3 \, {\left (B a^{3} - 6 \, A a^{2} b\right )} \sqrt {b} e^{\frac {1}{2}} \log \left (-8 \, b^{2} x^{6} - 8 \, a b x^{3} - 4 \, {\left (2 \, b x^{4} + a x\right )} \sqrt {b x^{3} + a} \sqrt {b} \sqrt {x} - a^{2}\right ) - 4 \, {\left (8 \, B b^{3} x^{7} + 2 \, {\left (7 \, B a b^{2} + 6 \, A b^{3}\right )} x^{4} + 3 \, {\left (B a^{2} b + 10 \, A a b^{2}\right )} x\right )} \sqrt {b x^{3} + a} \sqrt {x} e^{\frac {1}{2}}}{288 \, b^{2}}, \frac {3 \, {\left (B a^{3} - 6 \, A a^{2} b\right )} \sqrt {-b} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {-b} x^{\frac {3}{2}}}{2 \, b x^{3} + a}\right ) e^{\frac {1}{2}} + 2 \, {\left (8 \, B b^{3} x^{7} + 2 \, {\left (7 \, B a b^{2} + 6 \, A b^{3}\right )} x^{4} + 3 \, {\left (B a^{2} b + 10 \, A a b^{2}\right )} x\right )} \sqrt {b x^{3} + a} \sqrt {x} e^{\frac {1}{2}}}{144 \, b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 335 vs.
\(2 (138) = 276\).
time = 15.05, 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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 272 vs.
\(2 (111) = 222\).
time = 0.75, size = 272, normalized size = 1.69 \begin {gather*} \frac {1}{72} \, {\left (6 \, \sqrt {b x^{3} + a} {\left (2 \, x^{3} + \frac {a}{b}\right )} B a x^{\frac {3}{2}} + 6 \, \sqrt {b x^{3} + a} {\left (2 \, x^{3} + \frac {a}{b}\right )} A b x^{\frac {3}{2}} + {\left (2 \, {\left (4 \, x^{3} + \frac {a}{b}\right )} x^{3} - \frac {3 \, a^{2}}{b^{2}}\right )} \sqrt {b x^{3} + a} B b x^{\frac {3}{2}} + 24 \, {\left (\sqrt {b x^{3} + a} x^{\frac {3}{2}} - \frac {a \log \left ({\left | -\sqrt {b} x^{\frac {3}{2}} + \sqrt {b x^{3} + a} \right |}\right )}{\sqrt {b}}\right )} A a\right )} e^{\frac {1}{2}} - \frac {{\left (B^{2} a^{6} + 4 \, A B a^{5} b + 4 \, A^{2} a^{4} b^{2}\right )} e^{\frac {1}{2}} \log \left ({\left | {\left (B a^{3} x^{\frac {3}{2}} + 2 \, A a^{2} b x^{\frac {3}{2}}\right )} \sqrt {b} + \sqrt {B^{2} a^{7} + 4 \, A B a^{6} b + 4 \, A^{2} a^{5} b^{2} + {\left (B a^{3} x^{\frac {3}{2}} + 2 \, A a^{2} b x^{\frac {3}{2}}\right )}^{2} b} \right |}\right )}{24 \, b^{\frac {3}{2}} {\left | B a^{3} + 2 \, A a^{2} b \right |}} \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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