Optimal. Leaf size=83 \[ -\frac {2 a^3 A}{3 x^{3/2}}+2 a^2 \sqrt {x} (a B+3 A b)+\frac {2}{9} b^2 x^{9/2} (3 a B+A b)+\frac {6}{5} a b x^{5/2} (a B+A b)+\frac {2}{13} b^3 B x^{13/2} \]
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Rubi [A] time = 0.04, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {448} \begin {gather*} 2 a^2 \sqrt {x} (a B+3 A b)-\frac {2 a^3 A}{3 x^{3/2}}+\frac {2}{9} b^2 x^{9/2} (3 a B+A b)+\frac {6}{5} a b x^{5/2} (a B+A b)+\frac {2}{13} b^3 B x^{13/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^3 \left (A+B x^2\right )}{x^{5/2}} \, dx &=\int \left (\frac {a^3 A}{x^{5/2}}+\frac {a^2 (3 A b+a B)}{\sqrt {x}}+3 a b (A b+a B) x^{3/2}+b^2 (A b+3 a B) x^{7/2}+b^3 B x^{11/2}\right ) \, dx\\ &=-\frac {2 a^3 A}{3 x^{3/2}}+2 a^2 (3 A b+a B) \sqrt {x}+\frac {6}{5} a b (A b+a B) x^{5/2}+\frac {2}{9} b^2 (A b+3 a B) x^{9/2}+\frac {2}{13} b^3 B x^{13/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 78, normalized size = 0.94 \begin {gather*} \frac {-390 a^3 \left (A-3 B x^2\right )+702 a^2 b x^2 \left (5 A+B x^2\right )+78 a b^2 x^4 \left (9 A+5 B x^2\right )+10 b^3 x^6 \left (13 A+9 B x^2\right )}{585 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 83, normalized size = 1.00 \begin {gather*} \frac {2 \left (-195 a^3 A+585 a^3 B x^2+1755 a^2 A b x^2+351 a^2 b B x^4+351 a A b^2 x^4+195 a b^2 B x^6+65 A b^3 x^6+45 b^3 B x^8\right )}{585 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 75, normalized size = 0.90 \begin {gather*} \frac {2 \, {\left (45 \, B b^{3} x^{8} + 65 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + 351 \, {\left (B a^{2} b + A a b^{2}\right )} x^{4} - 195 \, A a^{3} + 585 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}\right )}}{585 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 77, normalized size = 0.93 \begin {gather*} \frac {2}{13} \, B b^{3} x^{\frac {13}{2}} + \frac {2}{3} \, B a b^{2} x^{\frac {9}{2}} + \frac {2}{9} \, A b^{3} x^{\frac {9}{2}} + \frac {6}{5} \, B a^{2} b x^{\frac {5}{2}} + \frac {6}{5} \, A a b^{2} x^{\frac {5}{2}} + 2 \, B a^{3} \sqrt {x} + 6 \, A a^{2} b \sqrt {x} - \frac {2 \, A a^{3}}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 80, normalized size = 0.96 \begin {gather*} -\frac {2 \left (-45 B \,b^{3} x^{8}-65 x^{6} A \,b^{3}-195 B a \,b^{2} x^{6}-351 x^{4} A a \,b^{2}-351 x^{4} B \,a^{2} b -1755 A \,a^{2} b \,x^{2}-585 B \,a^{3} x^{2}+195 A \,a^{3}\right )}{585 x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.15, size = 73, normalized size = 0.88 \begin {gather*} \frac {2}{13} \, B b^{3} x^{\frac {13}{2}} + \frac {2}{9} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {9}{2}} + \frac {6}{5} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {5}{2}} - \frac {2 \, A a^{3}}{3 \, x^{\frac {3}{2}}} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 69, normalized size = 0.83 \begin {gather*} \sqrt {x}\,\left (2\,B\,a^3+6\,A\,b\,a^2\right )+x^{9/2}\,\left (\frac {2\,A\,b^3}{9}+\frac {2\,B\,a\,b^2}{3}\right )-\frac {2\,A\,a^3}{3\,x^{3/2}}+\frac {2\,B\,b^3\,x^{13/2}}{13}+\frac {6\,a\,b\,x^{5/2}\,\left (A\,b+B\,a\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.06, size = 110, normalized size = 1.33 \begin {gather*} - \frac {2 A a^{3}}{3 x^{\frac {3}{2}}} + 6 A a^{2} b \sqrt {x} + \frac {6 A a b^{2} x^{\frac {5}{2}}}{5} + \frac {2 A b^{3} x^{\frac {9}{2}}}{9} + 2 B a^{3} \sqrt {x} + \frac {6 B a^{2} b x^{\frac {5}{2}}}{5} + \frac {2 B a b^{2} x^{\frac {9}{2}}}{3} + \frac {2 B b^{3} x^{\frac {13}{2}}}{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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