Optimal. Leaf size=241 \[ \frac {a^3 (10 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{384 b^2}+\frac {a^2 (10 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{192 b e}+\frac {a (10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{144 b e}+\frac {(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}-\frac {a^4 (10 A b-3 a B) e^{7/2} \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{384 b^{5/2}} \]
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Rubi [A]
time = 0.12, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps
used = 9, 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^4 e^{7/2} (10 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{384 b^{5/2}}+\frac {a^3 e^2 (e x)^{3/2} \sqrt {a+b x^3} (10 A b-3 a B)}{384 b^2}+\frac {a^2 (e x)^{9/2} \sqrt {a+b x^3} (10 A b-3 a B)}{192 b e}+\frac {(e x)^{9/2} \left (a+b x^3\right )^{5/2} (10 A b-3 a B)}{120 b e}+\frac {a (e x)^{9/2} \left (a+b x^3\right )^{3/2} (10 A b-3 a B)}{144 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 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 )^{5/2} \left (A+B x^3\right ) \, dx &=\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}-\frac {\left (-15 A b+\frac {9 a B}{2}\right ) \int (e x)^{7/2} \left (a+b x^3\right )^{5/2} \, dx}{15 b}\\ &=\frac {(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}+\frac {(a (10 A b-3 a B)) \int (e x)^{7/2} \left (a+b x^3\right )^{3/2} \, dx}{16 b}\\ &=\frac {a (10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{144 b e}+\frac {(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}+\frac {\left (a^2 (10 A b-3 a B)\right ) \int (e x)^{7/2} \sqrt {a+b x^3} \, dx}{32 b}\\ &=\frac {a^2 (10 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{192 b e}+\frac {a (10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{144 b e}+\frac {(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}+\frac {\left (a^3 (10 A b-3 a B)\right ) \int \frac {(e x)^{7/2}}{\sqrt {a+b x^3}} \, dx}{128 b}\\ &=\frac {a^3 (10 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{384 b^2}+\frac {a^2 (10 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{192 b e}+\frac {a (10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{144 b e}+\frac {(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}-\frac {\left (a^4 (10 A b-3 a B) e^3\right ) \int \frac {\sqrt {e x}}{\sqrt {a+b x^3}} \, dx}{256 b^2}\\ &=\frac {a^3 (10 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{384 b^2}+\frac {a^2 (10 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{192 b e}+\frac {a (10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{144 b e}+\frac {(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}-\frac {\left (a^4 (10 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 )}{128 b^2}\\ &=\frac {a^3 (10 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{384 b^2}+\frac {a^2 (10 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{192 b e}+\frac {a (10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{144 b e}+\frac {(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}-\frac {\left (a^4 (10 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 )}{384 b^2}\\ &=\frac {a^3 (10 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{384 b^2}+\frac {a^2 (10 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{192 b e}+\frac {a (10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{144 b e}+\frac {(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}-\frac {\left (a^4 (10 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 )}{384 b^2}\\ &=\frac {a^3 (10 A b-3 a B) e^2 (e x)^{3/2} \sqrt {a+b x^3}}{384 b^2}+\frac {a^2 (10 A b-3 a B) (e x)^{9/2} \sqrt {a+b x^3}}{192 b e}+\frac {a (10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{3/2}}{144 b e}+\frac {(10 A b-3 a B) (e x)^{9/2} \left (a+b x^3\right )^{5/2}}{120 b e}+\frac {B (e x)^{9/2} \left (a+b x^3\right )^{7/2}}{15 b e}-\frac {a^4 (10 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 )}{384 b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.42, size = 164, normalized size = 0.68 \begin {gather*} \frac {e^3 \sqrt {e x} \left (\sqrt {b} x^{3/2} \sqrt {a+b x^3} \left (-45 a^4 B+30 a^3 b \left (5 A+B x^3\right )+96 b^4 x^9 \left (5 A+4 B x^3\right )+16 a b^3 x^6 \left (85 A+63 B x^3\right )+4 a^2 b^2 x^3 \left (295 A+186 B x^3\right )\right )+15 a^4 (-10 A b+3 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {b} x^{3/2}}\right )\right )}{5760 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 = 0.35, size = 8117, normalized size = 33.68
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1135\) |
elliptic | \(\text {Expression too large to display}\) | \(1443\) |
default | \(\text {Expression too large to display}\) | \(8117\) |
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. 443 vs.
\(2 (176) = 352\).
time = 0.50, size = 443, normalized size = 1.84 \begin {gather*} \frac {1}{11520} \, {\left (10 \, {\left (\frac {15 \, 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 {3}{2}}} + \frac {2 \, {\left (\frac {15 \, \sqrt {b x^{3} + a} a^{4} b^{3}}{x^{\frac {3}{2}}} - \frac {55 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{4} b^{2}}{x^{\frac {9}{2}}} + \frac {73 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a^{4} b}{x^{\frac {15}{2}}} + \frac {15 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}} a^{4}}{x^{\frac {21}{2}}}\right )}}{b^{5} - \frac {4 \, {\left (b x^{3} + a\right )} b^{4}}{x^{3}} + \frac {6 \, {\left (b x^{3} + a\right )}^{2} b^{3}}{x^{6}} - \frac {4 \, {\left (b x^{3} + a\right )}^{3} b^{2}}{x^{9}} + \frac {{\left (b x^{3} + a\right )}^{4} b}{x^{12}}}\right )} A - 3 \, {\left (\frac {15 \, a^{5} \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 {15 \, \sqrt {b x^{3} + a} a^{5} b^{4}}{x^{\frac {3}{2}}} - \frac {70 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{5} b^{3}}{x^{\frac {9}{2}}} + \frac {128 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a^{5} b^{2}}{x^{\frac {15}{2}}} + \frac {70 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}} a^{5} b}{x^{\frac {21}{2}}} - \frac {15 \, {\left (b x^{3} + a\right )}^{\frac {9}{2}} a^{5}}{x^{\frac {27}{2}}}\right )}}{b^{7} - \frac {5 \, {\left (b x^{3} + a\right )} b^{6}}{x^{3}} + \frac {10 \, {\left (b x^{3} + a\right )}^{2} b^{5}}{x^{6}} - \frac {10 \, {\left (b x^{3} + a\right )}^{3} b^{4}}{x^{9}} + \frac {5 \, {\left (b x^{3} + a\right )}^{4} b^{3}}{x^{12}} - \frac {{\left (b x^{3} + a\right )}^{5} b^{2}}{x^{15}}}\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 = 2.17, size = 358, normalized size = 1.49 \begin {gather*} \left [-\frac {15 \, {\left (3 \, B a^{5} - 10 \, A a^{4} 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 (384 \, B b^{5} x^{13} + 48 \, {\left (21 \, B a b^{4} + 10 \, A b^{5}\right )} x^{10} + 8 \, {\left (93 \, B a^{2} b^{3} + 170 \, A a b^{4}\right )} x^{7} + 10 \, {\left (3 \, B a^{3} b^{2} + 118 \, A a^{2} b^{3}\right )} x^{4} - 15 \, {\left (3 \, B a^{4} b - 10 \, A a^{3} b^{2}\right )} x\right )} \sqrt {b x^{3} + a} \sqrt {x} e^{\frac {7}{2}}}{23040 \, b^{3}}, -\frac {15 \, {\left (3 \, B a^{5} - 10 \, A a^{4} 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 (384 \, B b^{5} x^{13} + 48 \, {\left (21 \, B a b^{4} + 10 \, A b^{5}\right )} x^{10} + 8 \, {\left (93 \, B a^{2} b^{3} + 170 \, A a b^{4}\right )} x^{7} + 10 \, {\left (3 \, B a^{3} b^{2} + 118 \, A a^{2} b^{3}\right )} x^{4} - 15 \, {\left (3 \, B a^{4} b - 10 \, A a^{3} b^{2}\right )} x\right )} \sqrt {b x^{3} + a} \sqrt {x} e^{\frac {7}{2}}}{11520 \, 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. 437 vs.
\(2 (176) = 352\).
time = 1.50, size = 437, normalized size = 1.81 \begin {gather*} \frac {1}{5760} \, {\left (480 \, \sqrt {b x^{3} + a} {\left (2 \, x^{3} + \frac {a}{b}\right )} A a^{2} x^{\frac {3}{2}} + 80 \, {\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^{2} x^{\frac {3}{2}} + 160 \, {\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 a b x^{\frac {3}{2}} + 20 \, {\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 a b x^{\frac {3}{2}} + 10 \, {\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} A b^{2} x^{\frac {3}{2}} + {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, x^{3} + \frac {a}{b}\right )} x^{3} - \frac {7 \, a^{2}}{b^{2}}\right )} x^{3} + \frac {35 \, a^{3}}{b^{3}}\right )} x^{3} - \frac {105 \, a^{4}}{b^{4}}\right )} \sqrt {b x^{3} + a} B b^{2} x^{\frac {3}{2}}\right )} e^{\frac {7}{2}} - \frac {{\left (9 \, B^{2} a^{10} - 60 \, A B a^{9} b + 100 \, A^{2} a^{8} b^{2}\right )} e^{\frac {7}{2}} \log \left ({\left | -{\left (3 \, B a^{5} x^{\frac {3}{2}} - 10 \, A a^{4} b x^{\frac {3}{2}}\right )} \sqrt {b} + \sqrt {9 \, B^{2} a^{11} - 60 \, A B a^{10} b + 100 \, A^{2} a^{9} b^{2} + {\left (3 \, B a^{5} x^{\frac {3}{2}} - 10 \, A a^{4} b x^{\frac {3}{2}}\right )}^{2} b} \right |}\right )}{384 \, b^{\frac {5}{2}} {\left | -3 \, B a^{5} + 10 \, A a^{4} 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 )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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