Optimal. Leaf size=188 \[ \frac {5 a (6 A b+a B) (e x)^{3/2} \sqrt {a+b x^3}}{24 e^4}+\frac {5 (6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{36 e^4}+\frac {(6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 a e^4}-\frac {2 A \left (a+b x^3\right )^{7/2}}{3 a e (e x)^{3/2}}+\frac {5 a^2 (6 A b+a B) \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{24 \sqrt {b} e^{5/2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 188, 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 = {464, 285, 335,
281, 223, 212} \begin {gather*} \frac {5 a^2 (a B+6 A b) \tanh ^{-1}\left (\frac {\sqrt {b} (e x)^{3/2}}{e^{3/2} \sqrt {a+b x^3}}\right )}{24 \sqrt {b} e^{5/2}}+\frac {(e x)^{3/2} \left (a+b x^3\right )^{5/2} (a B+6 A b)}{9 a e^4}+\frac {5 (e x)^{3/2} \left (a+b x^3\right )^{3/2} (a B+6 A b)}{36 e^4}+\frac {5 a (e x)^{3/2} \sqrt {a+b x^3} (a B+6 A b)}{24 e^4}-\frac {2 A \left (a+b x^3\right )^{7/2}}{3 a e (e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 281
Rule 285
Rule 335
Rule 464
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{5/2} \left (A+B x^3\right )}{(e x)^{5/2}} \, dx &=-\frac {2 A \left (a+b x^3\right )^{7/2}}{3 a e (e x)^{3/2}}+\frac {(6 A b+a B) \int \sqrt {e x} \left (a+b x^3\right )^{5/2} \, dx}{a e^3}\\ &=\frac {(6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 a e^4}-\frac {2 A \left (a+b x^3\right )^{7/2}}{3 a e (e x)^{3/2}}+\frac {(5 (6 A b+a B)) \int \sqrt {e x} \left (a+b x^3\right )^{3/2} \, dx}{6 e^3}\\ &=\frac {5 (6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{36 e^4}+\frac {(6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 a e^4}-\frac {2 A \left (a+b x^3\right )^{7/2}}{3 a e (e x)^{3/2}}+\frac {(5 a (6 A b+a B)) \int \sqrt {e x} \sqrt {a+b x^3} \, dx}{8 e^3}\\ &=\frac {5 a (6 A b+a B) (e x)^{3/2} \sqrt {a+b x^3}}{24 e^4}+\frac {5 (6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{36 e^4}+\frac {(6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 a e^4}-\frac {2 A \left (a+b x^3\right )^{7/2}}{3 a e (e x)^{3/2}}+\frac {\left (5 a^2 (6 A b+a B)\right ) \int \frac {\sqrt {e x}}{\sqrt {a+b x^3}} \, dx}{16 e^3}\\ &=\frac {5 a (6 A b+a B) (e x)^{3/2} \sqrt {a+b x^3}}{24 e^4}+\frac {5 (6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{36 e^4}+\frac {(6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 a e^4}-\frac {2 A \left (a+b x^3\right )^{7/2}}{3 a e (e x)^{3/2}}+\frac {\left (5 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 e^4}\\ &=\frac {5 a (6 A b+a B) (e x)^{3/2} \sqrt {a+b x^3}}{24 e^4}+\frac {5 (6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{36 e^4}+\frac {(6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 a e^4}-\frac {2 A \left (a+b x^3\right )^{7/2}}{3 a e (e x)^{3/2}}+\frac {\left (5 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 e^4}\\ &=\frac {5 a (6 A b+a B) (e x)^{3/2} \sqrt {a+b x^3}}{24 e^4}+\frac {5 (6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{36 e^4}+\frac {(6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 a e^4}-\frac {2 A \left (a+b x^3\right )^{7/2}}{3 a e (e x)^{3/2}}+\frac {\left (5 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 e^4}\\ &=\frac {5 a (6 A b+a B) (e x)^{3/2} \sqrt {a+b x^3}}{24 e^4}+\frac {5 (6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{3/2}}{36 e^4}+\frac {(6 A b+a B) (e x)^{3/2} \left (a+b x^3\right )^{5/2}}{9 a e^4}-\frac {2 A \left (a+b x^3\right )^{7/2}}{3 a e (e x)^{3/2}}+\frac {5 a^2 (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 \sqrt {b} e^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.37, size = 126, normalized size = 0.67 \begin {gather*} \frac {x \left (\sqrt {b} \sqrt {a+b x^3} \left (4 b^2 x^6 \left (3 A+2 B x^3\right )+a^2 \left (-48 A+33 B x^3\right )+a \left (54 A b x^3+26 b B x^6\right )\right )+15 a^2 (6 A b+a B) x^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {b} x^{3/2}}\right )\right )}{72 \sqrt {b} (e x)^{5/2}} \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 = 0.41, size = 7544, normalized size = 40.13
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1093\) |
elliptic | \(\text {Expression too large to display}\) | \(1246\) |
default | \(\text {Expression too large to display}\) | \(7544\) |
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. 306 vs.
\(2 (132) = 264\).
time = 0.49, size = 306, normalized size = 1.63 \begin {gather*} -\frac {1}{144} \, {\left (6 \, {\left (15 \, a^{2} \sqrt {b} \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 ) + \frac {16 \, \sqrt {b x^{3} + a} a^{2}}{x^{\frac {3}{2}}} + \frac {2 \, {\left (\frac {7 \, \sqrt {b x^{3} + a} a^{2} b^{2}}{x^{\frac {3}{2}}} - \frac {9 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{2} b}{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 {15 \, 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 )}{\sqrt {b}} + \frac {2 \, {\left (\frac {15 \, \sqrt {b x^{3} + a} a^{3} b^{2}}{x^{\frac {3}{2}}} - \frac {40 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{3} b}{x^{\frac {9}{2}}} + \frac {33 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a^{3}}{x^{\frac {15}{2}}}\right )}}{b^{3} - \frac {3 \, {\left (b x^{3} + a\right )} b^{2}}{x^{3}} + \frac {3 \, {\left (b x^{3} + a\right )}^{2} b}{x^{6}} - \frac {{\left (b x^{3} + a\right )}^{3}}{x^{9}}}\right )} B\right )} e^{\left (-\frac {5}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.83, size = 286, normalized size = 1.52 \begin {gather*} \left [\frac {{\left (15 \, {\left (B a^{3} + 6 \, A a^{2} b\right )} \sqrt {b} x^{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^{9} + 2 \, {\left (13 \, B a b^{2} + 6 \, A b^{3}\right )} x^{6} - 48 \, A a^{2} b + 3 \, {\left (11 \, B a^{2} b + 18 \, A a b^{2}\right )} x^{3}\right )} \sqrt {b x^{3} + a} \sqrt {x}\right )} e^{\left (-\frac {5}{2}\right )}}{288 \, b x^{2}}, -\frac {{\left (15 \, {\left (B a^{3} + 6 \, A a^{2} b\right )} \sqrt {-b} x^{2} \arctan \left (\frac {2 \, \sqrt {b x^{3} + a} \sqrt {-b} x^{\frac {3}{2}}}{2 \, b x^{3} + a}\right ) - 2 \, {\left (8 \, B b^{3} x^{9} + 2 \, {\left (13 \, B a b^{2} + 6 \, A b^{3}\right )} x^{6} - 48 \, A a^{2} b + 3 \, {\left (11 \, B a^{2} b + 18 \, A a b^{2}\right )} x^{3}\right )} \sqrt {b x^{3} + a} \sqrt {x}\right )} e^{\left (-\frac {5}{2}\right )}}{144 \, b x^{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. 403 vs.
\(2 (180) = 360\).
time = 37.41, size = 403, normalized size = 2.14 \begin {gather*} - \frac {2 A a^{\frac {5}{2}}}{3 e^{\frac {5}{2}} x^{\frac {3}{2}} \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {2 A a^{\frac {3}{2}} b x^{\frac {3}{2}} \sqrt {1 + \frac {b x^{3}}{a}}}{3 e^{\frac {5}{2}}} - \frac {7 A a^{\frac {3}{2}} b x^{\frac {3}{2}}}{12 e^{\frac {5}{2}} \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {A \sqrt {a} b^{2} x^{\frac {9}{2}}}{4 e^{\frac {5}{2}} \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {5 A a^{2} \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} x^{\frac {3}{2}}}{\sqrt {a}} \right )}}{4 e^{\frac {5}{2}}} + \frac {A b^{3} x^{\frac {15}{2}}}{6 \sqrt {a} e^{\frac {5}{2}} \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {B a^{\frac {5}{2}} x^{\frac {3}{2}} \sqrt {1 + \frac {b x^{3}}{a}}}{3 e^{\frac {5}{2}}} + \frac {B a^{\frac {5}{2}} x^{\frac {3}{2}}}{8 e^{\frac {5}{2}} \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {35 B a^{\frac {3}{2}} b x^{\frac {9}{2}}}{72 e^{\frac {5}{2}} \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {17 B \sqrt {a} b^{2} x^{\frac {15}{2}}}{36 e^{\frac {5}{2}} \sqrt {1 + \frac {b x^{3}}{a}}} + \frac {5 B a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b} x^{\frac {3}{2}}}{\sqrt {a}} \right )}}{24 \sqrt {b} e^{\frac {5}{2}}} + \frac {B b^{3} x^{\frac {21}{2}}}{9 \sqrt {a} e^{\frac {5}{2}} \sqrt {1 + \frac {b x^{3}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 \frac {\left (B\,x^3+A\right )\,{\left (b\,x^3+a\right )}^{5/2}}{{\left (e\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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