Optimal. Leaf size=201 \[ \frac {a^2 (8 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{192 b^2}+\frac {a (8 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{96 b e}+\frac {(8 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{72 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}-\frac {a^3 (8 A b-3 a B) e^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{192 b^{5/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {470, 285, 327,
335, 281, 223, 212} \begin {gather*} -\frac {a^3 e^{7/2} (8 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{192 b^{5/2}}+\frac {a^2 e^2 (e x)^{3/2} \sqrt {a+b x^3} (8 A b-3 a B)}{192 b^2}+\frac {(e x)^{9/2} \left (a+b x^3\right )^{3/2} (8 A b-3 a B)}{72 b e}+\frac {a (e x)^{9/2} \sqrt {a+b x^3} (8 A b-3 a B)}{96 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 281
Rule 285
Rule 327
Rule 335
Rule 470
Rubi steps
\begin {align*} \int (e x)^{7/2} \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx &=\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}-\frac {\left (-12 A b+\frac {9 a B}{2}\right ) \int (e x)^{7/2} \left (a+b x^3\right )^{3/2} \, dx}{12 b}\\ &=\frac {(8 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{72 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}+\frac {(a (8 A b-3 a B)) \int (e x)^{7/2} \sqrt {a+b x^3} \, dx}{16 b}\\ &=\frac {a (8 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{96 b e}+\frac {(8 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{72 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}+\frac {\left (a^2 (8 A b-3 a B)\right ) \int \frac {(e x)^{7/2}}{\sqrt {a+b x^3}} \, dx}{64 b}\\ &=\frac {a^2 (8 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{192 b^2}+\frac {a (8 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{96 b e}+\frac {(8 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{72 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}-\frac {\left (a^3 (8 A b-3 a B) e^3\right ) \int \frac {\sqrt {e x}}{\sqrt {a+b x^3}} \, dx}{128 b^2}\\ &=\frac {a^2 (8 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{192 b^2}+\frac {a (8 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{96 b e}+\frac {(8 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{72 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}-\frac {\left (a^3 (8 A b-3 a B) e^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^6}{e^3}}} \, dx,x,\sqrt {e x}\right )}{64 b^2}\\ &=\frac {a^2 (8 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{192 b^2}+\frac {a (8 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{96 b e}+\frac {(8 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{72 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}-\frac {\left (a^3 (8 A b-3 a B) e^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^2}{e^3}}} \, dx,x,(e x)^{3/2}\right )}{192 b^2}\\ &=\frac {a^2 (8 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{192 b^2}+\frac {a (8 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{96 b e}+\frac {(8 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{72 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}-\frac {\left (a^3 (8 A b-3 a B) e^2\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 )}{192 b^2}\\ &=\frac {a^2 (8 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{192 b^2}+\frac {a (8 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{96 b e}+\frac {(8 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{72 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{12 b e}-\frac {a^3 (8 A b-3 a B) e^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{192 b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 143, normalized size = 0.71 \begin {gather*} \frac {e^3 \sqrt {e x} \left (\sqrt {b} x^{3/2} \sqrt {a+b x^3} \left (-9 a^3 B+6 a^2 b \left (4 A+B x^3\right )+16 b^3 x^6 \left (4 A+3 B x^3\right )+8 a b^2 x^3 \left (14 A+9 B x^3\right )\right )+3 a^3 (-8 A b+3 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {b} x^{3/2}}\right )\right )}{576 b^{5/2} \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 = 0.39, size = 7705, normalized size = 38.33
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1111\) |
elliptic | \(\text {Expression too large to display}\) | \(1285\) |
default | \(\text {Expression too large to display}\) | \(7705\) |
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. 369 vs.
\(2 (145) = 290\).
time = 0.52, size = 369, normalized size = 1.84 \begin {gather*} \frac {1}{1152} \, {\left (8 \, {\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 )} A - 3 \, {\left (\frac {3 \, a^{4} \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 {5}{2}}} + \frac {2 \, {\left (\frac {3 \, \sqrt {b x^{3} + a} a^{4} b^{3}}{x^{\frac {3}{2}}} - \frac {11 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{4} b^{2}}{x^{\frac {9}{2}}} - \frac {11 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a^{4} b}{x^{\frac {15}{2}}} + \frac {3 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}} a^{4}}{x^{\frac {21}{2}}}\right )}}{b^{6} - \frac {4 \, {\left (b x^{3} + a\right )} b^{5}}{x^{3}} + \frac {6 \, {\left (b x^{3} + a\right )}^{2} b^{4}}{x^{6}} - \frac {4 \, {\left (b x^{3} + a\right )}^{3} b^{3}}{x^{9}} + \frac {{\left (b x^{3} + a\right )}^{4} b^{2}}{x^{12}}}\right )} B\right )} e^{\frac {7}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.37, size = 310, normalized size = 1.54 \begin {gather*} \left [-\frac {3 \, {\left (3 \, B a^{4} - 8 \, A a^{3} b\right )} \sqrt {b} e^{\frac {7}{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 (48 \, B b^{4} x^{10} + 8 \, {\left (9 \, B a b^{3} + 8 \, A b^{4}\right )} x^{7} + 2 \, {\left (3 \, B a^{2} b^{2} + 56 \, A a b^{3}\right )} x^{4} - 3 \, {\left (3 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x\right )} \sqrt {b x^{3} + a} \sqrt {x} e^{\frac {7}{2}}}{2304 \, b^{3}}, -\frac {3 \, {\left (3 \, B a^{4} - 8 \, A a^{3} 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 {7}{2}} - 2 \, {\left (48 \, B b^{4} x^{10} + 8 \, {\left (9 \, B a b^{3} + 8 \, A b^{4}\right )} x^{7} + 2 \, {\left (3 \, B a^{2} b^{2} + 56 \, A a b^{3}\right )} x^{4} - 3 \, {\left (3 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x\right )} \sqrt {b x^{3} + a} \sqrt {x} e^{\frac {7}{2}}}{1152 \, b^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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. 303 vs.
\(2 (145) = 290\).
time = 0.93, size = 303, normalized size = 1.51 \begin {gather*} \frac {1}{576} \, {\left (48 \, \sqrt {b x^{3} + a} {\left (2 \, x^{3} + \frac {a}{b}\right )} A a x^{\frac {3}{2}} + 8 \, {\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 a x^{\frac {3}{2}} + 8 \, {\left (2 \, {\left (4 \, x^{3} + \frac {a}{b}\right )} x^{3} - \frac {3 \, a^{2}}{b^{2}}\right )} \sqrt {b x^{3} + a} A b x^{\frac {3}{2}} + {\left (2 \, {\left (4 \, {\left (6 \, x^{3} + \frac {a}{b}\right )} x^{3} - \frac {5 \, a^{2}}{b^{2}}\right )} x^{3} + \frac {15 \, a^{3}}{b^{3}}\right )} \sqrt {b x^{3} + a} B b x^{\frac {3}{2}}\right )} e^{\frac {7}{2}} - \frac {{\left (9 \, B^{2} a^{8} - 48 \, A B a^{7} b + 64 \, A^{2} a^{6} b^{2}\right )} e^{\frac {7}{2}} \log \left ({\left | -{\left (3 \, B a^{4} x^{\frac {3}{2}} - 8 \, A a^{3} b x^{\frac {3}{2}}\right )} \sqrt {b} + \sqrt {9 \, B^{2} a^{9} - 48 \, A B a^{8} b + 64 \, A^{2} a^{7} b^{2} + {\left (3 \, B a^{4} x^{\frac {3}{2}} - 8 \, A a^{3} b x^{\frac {3}{2}}\right )}^{2} b} \right |}\right )}{192 \, b^{\frac {5}{2}} {\left | -3 \, B a^{4} + 8 \, A a^{3} 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 = -0.00 \begin {gather*} \int \left (B\,x^3+A\right )\,{\left (e\,x\right )}^{7/2}\,{\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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