Optimal. Leaf size=73 \[ -\frac {2 \sqrt {a} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )}{3 b^{5/2}}+\frac {2 x^{3/2} (A b-a B)}{3 b^2}+\frac {2 B x^{9/2}}{9 b} \]
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Rubi [A] time = 0.05, antiderivative size = 73, 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 = {459, 321, 329, 275, 205} \[ \frac {2 x^{3/2} (A b-a B)}{3 b^2}-\frac {2 \sqrt {a} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )}{3 b^{5/2}}+\frac {2 B x^{9/2}}{9 b} \]
Antiderivative was successfully verified.
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Rule 205
Rule 275
Rule 321
Rule 329
Rule 459
Rubi steps
\begin {align*} \int \frac {x^{7/2} \left (A+B x^3\right )}{a+b x^3} \, dx &=\frac {2 B x^{9/2}}{9 b}-\frac {\left (2 \left (-\frac {9 A b}{2}+\frac {9 a B}{2}\right )\right ) \int \frac {x^{7/2}}{a+b x^3} \, dx}{9 b}\\ &=\frac {2 (A b-a B) x^{3/2}}{3 b^2}+\frac {2 B x^{9/2}}{9 b}-\frac {(a (A b-a B)) \int \frac {\sqrt {x}}{a+b x^3} \, dx}{b^2}\\ &=\frac {2 (A b-a B) x^{3/2}}{3 b^2}+\frac {2 B x^{9/2}}{9 b}-\frac {(2 a (A b-a B)) \operatorname {Subst}\left (\int \frac {x^2}{a+b x^6} \, dx,x,\sqrt {x}\right )}{b^2}\\ &=\frac {2 (A b-a B) x^{3/2}}{3 b^2}+\frac {2 B x^{9/2}}{9 b}-\frac {(2 a (A b-a B)) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^{3/2}\right )}{3 b^2}\\ &=\frac {2 (A b-a B) x^{3/2}}{3 b^2}+\frac {2 B x^{9/2}}{9 b}-\frac {2 \sqrt {a} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )}{3 b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 67, normalized size = 0.92 \[ \frac {2 \sqrt {a} (a B-A b) \tan ^{-1}\left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )}{3 b^{5/2}}+\frac {2 x^{3/2} \left (-3 a B+3 A b+b B x^3\right )}{9 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 143, normalized size = 1.96 \[ \left [-\frac {3 \, {\left (B a - A b\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{3} - 2 \, b x^{\frac {3}{2}} \sqrt {-\frac {a}{b}} - a}{b x^{3} + a}\right ) - 2 \, {\left (B b x^{4} - 3 \, {\left (B a - A b\right )} x\right )} \sqrt {x}}{9 \, b^{2}}, \frac {2 \, {\left (3 \, {\left (B a - A b\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x^{\frac {3}{2}} \sqrt {\frac {a}{b}}}{a}\right ) + {\left (B b x^{4} - 3 \, {\left (B a - A b\right )} x\right )} \sqrt {x}\right )}}{9 \, b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 64, normalized size = 0.88 \[ \frac {2 \, {\left (B a^{2} - A a b\right )} \arctan \left (\frac {b x^{\frac {3}{2}}}{\sqrt {a b}}\right )}{3 \, \sqrt {a b} b^{2}} + \frac {2 \, {\left (B b^{2} x^{\frac {9}{2}} - 3 \, B a b x^{\frac {3}{2}} + 3 \, A b^{2} x^{\frac {3}{2}}\right )}}{9 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 78, normalized size = 1.07 \[ \frac {2 B \,x^{\frac {9}{2}}}{9 b}-\frac {2 A a \arctan \left (\frac {b \,x^{\frac {3}{2}}}{\sqrt {a b}}\right )}{3 \sqrt {a b}\, b}+\frac {2 B \,a^{2} \arctan \left (\frac {b \,x^{\frac {3}{2}}}{\sqrt {a b}}\right )}{3 \sqrt {a b}\, b^{2}}+\frac {2 A \,x^{\frac {3}{2}}}{3 b}-\frac {2 B a \,x^{\frac {3}{2}}}{3 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 58, normalized size = 0.79 \[ \frac {2 \, {\left (B a^{2} - A a b\right )} \arctan \left (\frac {b x^{\frac {3}{2}}}{\sqrt {a b}}\right )}{3 \, \sqrt {a b} b^{2}} + \frac {2 \, {\left (B b x^{\frac {9}{2}} - 3 \, {\left (B a - A b\right )} x^{\frac {3}{2}}\right )}}{9 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.61, size = 111, normalized size = 1.52 \[ x^{3/2}\,\left (\frac {2\,A}{3\,b}-\frac {2\,B\,a}{3\,b^2}\right )+\frac {2\,B\,x^{9/2}}{9\,b}-\frac {2\,\sqrt {a}\,\mathrm {atan}\left (\frac {72\,b^{3/2}\,x^{3/2}\,\left (A^2\,a^2\,b^2-2\,A\,B\,a^3\,b+B^2\,a^4\right )}{\sqrt {a}\,\left (72\,A\,a^2\,b^2-72\,B\,a^3\,b\right )\,\left (A\,b-B\,a\right )}\right )\,\left (A\,b-B\,a\right )}{3\,b^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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