Optimal. Leaf size=117 \[ \frac{16 b^2 (a+b x)^{5/2} (6 A b-11 a B)}{3465 a^4 x^{5/2}}-\frac{8 b (a+b x)^{5/2} (6 A b-11 a B)}{693 a^3 x^{7/2}}+\frac{2 (a+b x)^{5/2} (6 A b-11 a B)}{99 a^2 x^{9/2}}-\frac{2 A (a+b x)^{5/2}}{11 a x^{11/2}} \]
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Rubi [A] time = 0.0408825, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {78, 45, 37} \[ \frac{16 b^2 (a+b x)^{5/2} (6 A b-11 a B)}{3465 a^4 x^{5/2}}-\frac{8 b (a+b x)^{5/2} (6 A b-11 a B)}{693 a^3 x^{7/2}}+\frac{2 (a+b x)^{5/2} (6 A b-11 a B)}{99 a^2 x^{9/2}}-\frac{2 A (a+b x)^{5/2}}{11 a x^{11/2}} \]
Antiderivative was successfully verified.
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Rule 78
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2} (A+B x)}{x^{13/2}} \, dx &=-\frac{2 A (a+b x)^{5/2}}{11 a x^{11/2}}+\frac{\left (2 \left (-3 A b+\frac{11 a B}{2}\right )\right ) \int \frac{(a+b x)^{3/2}}{x^{11/2}} \, dx}{11 a}\\ &=-\frac{2 A (a+b x)^{5/2}}{11 a x^{11/2}}+\frac{2 (6 A b-11 a B) (a+b x)^{5/2}}{99 a^2 x^{9/2}}+\frac{(4 b (6 A b-11 a B)) \int \frac{(a+b x)^{3/2}}{x^{9/2}} \, dx}{99 a^2}\\ &=-\frac{2 A (a+b x)^{5/2}}{11 a x^{11/2}}+\frac{2 (6 A b-11 a B) (a+b x)^{5/2}}{99 a^2 x^{9/2}}-\frac{8 b (6 A b-11 a B) (a+b x)^{5/2}}{693 a^3 x^{7/2}}-\frac{\left (8 b^2 (6 A b-11 a B)\right ) \int \frac{(a+b x)^{3/2}}{x^{7/2}} \, dx}{693 a^3}\\ &=-\frac{2 A (a+b x)^{5/2}}{11 a x^{11/2}}+\frac{2 (6 A b-11 a B) (a+b x)^{5/2}}{99 a^2 x^{9/2}}-\frac{8 b (6 A b-11 a B) (a+b x)^{5/2}}{693 a^3 x^{7/2}}+\frac{16 b^2 (6 A b-11 a B) (a+b x)^{5/2}}{3465 a^4 x^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0304975, size = 76, normalized size = 0.65 \[ -\frac{2 (a+b x)^{5/2} \left (-10 a^2 b x (21 A+22 B x)+35 a^3 (9 A+11 B x)+8 a b^2 x^2 (15 A+11 B x)-48 A b^3 x^3\right )}{3465 a^4 x^{11/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 77, normalized size = 0.7 \begin{align*} -{\frac{-96\,A{b}^{3}{x}^{3}+176\,B{x}^{3}a{b}^{2}+240\,aA{b}^{2}{x}^{2}-440\,B{x}^{2}{a}^{2}b-420\,{a}^{2}Abx+770\,{a}^{3}Bx+630\,A{a}^{3}}{3465\,{a}^{4}} \left ( bx+a \right ) ^{{\frac{5}{2}}}{x}^{-{\frac{11}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.55662, size = 297, normalized size = 2.54 \begin{align*} -\frac{2 \,{\left (315 \, A a^{5} + 8 \,{\left (11 \, B a b^{4} - 6 \, A b^{5}\right )} x^{5} - 4 \,{\left (11 \, B a^{2} b^{3} - 6 \, A a b^{4}\right )} x^{4} + 3 \,{\left (11 \, B a^{3} b^{2} - 6 \, A a^{2} b^{3}\right )} x^{3} + 5 \,{\left (110 \, B a^{4} b + 3 \, A a^{3} b^{2}\right )} x^{2} + 35 \,{\left (11 \, B a^{5} + 12 \, A a^{4} b\right )} x\right )} \sqrt{b x + a}}{3465 \, a^{4} x^{\frac{11}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 3.53016, size = 208, normalized size = 1.78 \begin{align*} \frac{{\left ({\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (11 \, B a^{2} b^{10} - 6 \, A a b^{11}\right )}{\left (b x + a\right )}}{a^{6} b^{18}} - \frac{11 \,{\left (11 \, B a^{3} b^{10} - 6 \, A a^{2} b^{11}\right )}}{a^{6} b^{18}}\right )} + \frac{99 \,{\left (11 \, B a^{4} b^{10} - 6 \, A a^{3} b^{11}\right )}}{a^{6} b^{18}}\right )} - \frac{693 \,{\left (B a^{5} b^{10} - A a^{4} b^{11}\right )}}{a^{6} b^{18}}\right )}{\left (b x + a\right )}^{\frac{5}{2}} b}{14192640 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{11}{2}}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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