Optimal. Leaf size=103 \[ -\frac{6 a^2 A c}{5 x^{5/2}}-\frac{2 a^3 A}{9 x^{9/2}}-\frac{2 a^2 B c}{x^{3/2}}-\frac{2 a^3 B}{7 x^{7/2}}-\frac{6 a A c^2}{\sqrt{x}}+6 a B c^2 \sqrt{x}+\frac{2}{3} A c^3 x^{3/2}+\frac{2}{5} B c^3 x^{5/2} \]
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Rubi [A] time = 0.0390749, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {766} \[ -\frac{6 a^2 A c}{5 x^{5/2}}-\frac{2 a^3 A}{9 x^{9/2}}-\frac{2 a^2 B c}{x^{3/2}}-\frac{2 a^3 B}{7 x^{7/2}}-\frac{6 a A c^2}{\sqrt{x}}+6 a B c^2 \sqrt{x}+\frac{2}{3} A c^3 x^{3/2}+\frac{2}{5} B c^3 x^{5/2} \]
Antiderivative was successfully verified.
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Rule 766
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )^3}{x^{11/2}} \, dx &=\int \left (\frac{a^3 A}{x^{11/2}}+\frac{a^3 B}{x^{9/2}}+\frac{3 a^2 A c}{x^{7/2}}+\frac{3 a^2 B c}{x^{5/2}}+\frac{3 a A c^2}{x^{3/2}}+\frac{3 a B c^2}{\sqrt{x}}+A c^3 \sqrt{x}+B c^3 x^{3/2}\right ) \, dx\\ &=-\frac{2 a^3 A}{9 x^{9/2}}-\frac{2 a^3 B}{7 x^{7/2}}-\frac{6 a^2 A c}{5 x^{5/2}}-\frac{2 a^2 B c}{x^{3/2}}-\frac{6 a A c^2}{\sqrt{x}}+6 a B c^2 \sqrt{x}+\frac{2}{3} A c^3 x^{3/2}+\frac{2}{5} B c^3 x^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0274516, size = 71, normalized size = 0.69 \[ -\frac{2 \left (63 a^2 c x^2 (3 A+5 B x)+5 a^3 (7 A+9 B x)+945 a c^2 x^4 (A-B x)-21 c^3 x^6 (5 A+3 B x)\right )}{315 x^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 78, normalized size = 0.8 \begin{align*} -{\frac{-126\,B{c}^{3}{x}^{7}-210\,A{c}^{3}{x}^{6}-1890\,aB{c}^{2}{x}^{5}+1890\,aA{c}^{2}{x}^{4}+630\,{a}^{2}Bc{x}^{3}+378\,{a}^{2}Ac{x}^{2}+90\,{a}^{3}Bx+70\,A{a}^{3}}{315}{x}^{-{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00417, size = 105, normalized size = 1.02 \begin{align*} \frac{2}{5} \, B c^{3} x^{\frac{5}{2}} + \frac{2}{3} \, A c^{3} x^{\frac{3}{2}} + 6 \, B a c^{2} \sqrt{x} - \frac{2 \,{\left (945 \, A a c^{2} x^{4} + 315 \, B a^{2} c x^{3} + 189 \, A a^{2} c x^{2} + 45 \, B a^{3} x + 35 \, A a^{3}\right )}}{315 \, x^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.18719, size = 192, normalized size = 1.86 \begin{align*} \frac{2 \,{\left (63 \, B c^{3} x^{7} + 105 \, A c^{3} x^{6} + 945 \, B a c^{2} x^{5} - 945 \, A a c^{2} x^{4} - 315 \, B a^{2} c x^{3} - 189 \, A a^{2} c x^{2} - 45 \, B a^{3} x - 35 \, A a^{3}\right )}}{315 \, x^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.6632, size = 109, normalized size = 1.06 \begin{align*} - \frac{2 A a^{3}}{9 x^{\frac{9}{2}}} - \frac{6 A a^{2} c}{5 x^{\frac{5}{2}}} - \frac{6 A a c^{2}}{\sqrt{x}} + \frac{2 A c^{3} x^{\frac{3}{2}}}{3} - \frac{2 B a^{3}}{7 x^{\frac{7}{2}}} - \frac{2 B a^{2} c}{x^{\frac{3}{2}}} + 6 B a c^{2} \sqrt{x} + \frac{2 B c^{3} x^{\frac{5}{2}}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16012, size = 105, normalized size = 1.02 \begin{align*} \frac{2}{5} \, B c^{3} x^{\frac{5}{2}} + \frac{2}{3} \, A c^{3} x^{\frac{3}{2}} + 6 \, B a c^{2} \sqrt{x} - \frac{2 \,{\left (945 \, A a c^{2} x^{4} + 315 \, B a^{2} c x^{3} + 189 \, A a^{2} c x^{2} + 45 \, B a^{3} x + 35 \, A a^{3}\right )}}{315 \, x^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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