Optimal. Leaf size=77 \[ \frac {2 (A b-a B)}{9 a b \left (a+b x^3\right )^{3/2}}+\frac {2 A}{3 a^2 \sqrt {a+b x^3}}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{5/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {457, 79, 53, 65,
214} \begin {gather*} -\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{5/2}}+\frac {2 A}{3 a^2 \sqrt {a+b x^3}}+\frac {2 (A b-a B)}{9 a b \left (a+b x^3\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 79
Rule 214
Rule 457
Rubi steps
\begin {align*} \int \frac {A+B x^3}{x \left (a+b x^3\right )^{5/2}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {A+B x}{x (a+b x)^{5/2}} \, dx,x,x^3\right )\\ &=\frac {2 (A b-a B)}{9 a b \left (a+b x^3\right )^{3/2}}+\frac {A \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,x^3\right )}{3 a}\\ &=\frac {2 (A b-a B)}{9 a b \left (a+b x^3\right )^{3/2}}+\frac {2 A}{3 a^2 \sqrt {a+b x^3}}+\frac {A \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )}{3 a^2}\\ &=\frac {2 (A b-a B)}{9 a b \left (a+b x^3\right )^{3/2}}+\frac {2 A}{3 a^2 \sqrt {a+b x^3}}+\frac {(2 A) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )}{3 a^2 b}\\ &=\frac {2 (A b-a B)}{9 a b \left (a+b x^3\right )^{3/2}}+\frac {2 A}{3 a^2 \sqrt {a+b x^3}}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 70, normalized size = 0.91 \begin {gather*} -\frac {2 \left (-4 a A b+a^2 B-3 A b^2 x^3\right )}{9 a^2 b \left (a+b x^3\right )^{3/2}}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 85, normalized size = 1.10
method | result | size |
elliptic | \(\frac {2 \left (A b -B a \right ) \sqrt {b \,x^{3}+a}}{9 b^{3} a \left (x^{3}+\frac {a}{b}\right )^{2}}+\frac {2 A}{3 a^{2} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}-\frac {2 A \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 a^{\frac {5}{2}}}\) | \(77\) |
default | \(-\frac {2 B}{9 b \left (b \,x^{3}+a \right )^{\frac {3}{2}}}+A \left (\frac {2 \sqrt {b \,x^{3}+a}}{9 a \,b^{2} \left (x^{3}+\frac {a}{b}\right )^{2}}+\frac {2}{3 a^{2} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}-\frac {2 \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 a^{\frac {5}{2}}}\right )\) | \(85\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 81, normalized size = 1.05 \begin {gather*} \frac {1}{9} \, A {\left (\frac {3 \, \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, b x^{3} + 4 \, a\right )}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{2}}\right )} - \frac {2 \, B}{9 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.18, size = 243, normalized size = 3.16 \begin {gather*} \left [\frac {3 \, {\left (A b^{3} x^{6} + 2 \, A a b^{2} x^{3} + A a^{2} b\right )} \sqrt {a} \log \left (\frac {b x^{3} - 2 \, \sqrt {b x^{3} + a} \sqrt {a} + 2 \, a}{x^{3}}\right ) + 2 \, {\left (3 \, A a b^{2} x^{3} - B a^{3} + 4 \, A a^{2} b\right )} \sqrt {b x^{3} + a}}{9 \, {\left (a^{3} b^{3} x^{6} + 2 \, a^{4} b^{2} x^{3} + a^{5} b\right )}}, \frac {2 \, {\left (3 \, {\left (A b^{3} x^{6} + 2 \, A a b^{2} x^{3} + A a^{2} b\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{3} + a} \sqrt {-a}}{a}\right ) + {\left (3 \, A a b^{2} x^{3} - B a^{3} + 4 \, A a^{2} b\right )} \sqrt {b x^{3} + a}\right )}}{9 \, {\left (a^{3} b^{3} x^{6} + 2 \, a^{4} b^{2} x^{3} + a^{5} b\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 12.19, size = 76, normalized size = 0.99 \begin {gather*} \frac {2 A}{3 a^{2} \sqrt {a + b x^{3}}} + \frac {2 A \operatorname {atan}{\left (\frac {\sqrt {a + b x^{3}}}{\sqrt {- a}} \right )}}{3 a^{2} \sqrt {- a}} - \frac {2 \left (- A b + B a\right )}{9 a b \left (a + b x^{3}\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.99, size = 67, normalized size = 0.87 \begin {gather*} \frac {2 \, A \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{3 \, \sqrt {-a} a^{2}} - \frac {2 \, {\left (B a^{2} - 3 \, {\left (b x^{3} + a\right )} A b - A a b\right )}}{9 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.78, size = 80, normalized size = 1.04 \begin {gather*} \frac {\frac {2\,A}{9\,a}-\frac {2\,B}{9\,b}}{{\left (b\,x^3+a\right )}^{3/2}}+\frac {2\,A}{3\,a^2\,\sqrt {b\,x^3+a}}+\frac {A\,\ln \left (\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^3\,\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}{x^6}\right )}{3\,a^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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