Optimal. Leaf size=117 \[ \frac{16 b^2 (a+b x)^{7/2} (6 A b-13 a B)}{9009 a^4 x^{7/2}}-\frac{8 b (a+b x)^{7/2} (6 A b-13 a B)}{1287 a^3 x^{9/2}}+\frac{2 (a+b x)^{7/2} (6 A b-13 a B)}{143 a^2 x^{11/2}}-\frac{2 A (a+b x)^{7/2}}{13 a x^{13/2}} \]
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Rubi [A] time = 0.0400097, 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} \[ \frac{16 b^2 (a+b x)^{7/2} (6 A b-13 a B)}{9009 a^4 x^{7/2}}-\frac{8 b (a+b x)^{7/2} (6 A b-13 a B)}{1287 a^3 x^{9/2}}+\frac{2 (a+b x)^{7/2} (6 A b-13 a B)}{143 a^2 x^{11/2}}-\frac{2 A (a+b x)^{7/2}}{13 a x^{13/2}} \]
Antiderivative was successfully verified.
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Rule 78
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{(a+b x)^{5/2} (A+B x)}{x^{15/2}} \, dx &=-\frac{2 A (a+b x)^{7/2}}{13 a x^{13/2}}+\frac{\left (2 \left (-3 A b+\frac{13 a B}{2}\right )\right ) \int \frac{(a+b x)^{5/2}}{x^{13/2}} \, dx}{13 a}\\ &=-\frac{2 A (a+b x)^{7/2}}{13 a x^{13/2}}+\frac{2 (6 A b-13 a B) (a+b x)^{7/2}}{143 a^2 x^{11/2}}+\frac{(4 b (6 A b-13 a B)) \int \frac{(a+b x)^{5/2}}{x^{11/2}} \, dx}{143 a^2}\\ &=-\frac{2 A (a+b x)^{7/2}}{13 a x^{13/2}}+\frac{2 (6 A b-13 a B) (a+b x)^{7/2}}{143 a^2 x^{11/2}}-\frac{8 b (6 A b-13 a B) (a+b x)^{7/2}}{1287 a^3 x^{9/2}}-\frac{\left (8 b^2 (6 A b-13 a B)\right ) \int \frac{(a+b x)^{5/2}}{x^{9/2}} \, dx}{1287 a^3}\\ &=-\frac{2 A (a+b x)^{7/2}}{13 a x^{13/2}}+\frac{2 (6 A b-13 a B) (a+b x)^{7/2}}{143 a^2 x^{11/2}}-\frac{8 b (6 A b-13 a B) (a+b x)^{7/2}}{1287 a^3 x^{9/2}}+\frac{16 b^2 (6 A b-13 a B) (a+b x)^{7/2}}{9009 a^4 x^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0346603, size = 76, normalized size = 0.65 \[ -\frac{2 (a+b x)^{7/2} \left (-14 a^2 b x (27 A+26 B x)+63 a^3 (11 A+13 B x)+8 a b^2 x^2 (21 A+13 B x)-48 A b^3 x^3\right )}{9009 a^4 x^{13/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 77, normalized size = 0.7 \begin{align*} -{\frac{-96\,A{b}^{3}{x}^{3}+208\,B{x}^{3}a{b}^{2}+336\,aA{b}^{2}{x}^{2}-728\,B{x}^{2}{a}^{2}b-756\,{a}^{2}Abx+1638\,{a}^{3}Bx+1386\,A{a}^{3}}{9009\,{a}^{4}} \left ( bx+a \right ) ^{{\frac{7}{2}}}{x}^{-{\frac{13}{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 = 2.64339, size = 351, normalized size = 3. \begin{align*} -\frac{2 \,{\left (693 \, A a^{6} + 8 \,{\left (13 \, B a b^{5} - 6 \, A b^{6}\right )} x^{6} - 4 \,{\left (13 \, B a^{2} b^{4} - 6 \, A a b^{5}\right )} x^{5} + 3 \,{\left (13 \, B a^{3} b^{3} - 6 \, A a^{2} b^{4}\right )} x^{4} +{\left (1469 \, B a^{4} b^{2} + 15 \, A a^{3} b^{3}\right )} x^{3} + 7 \,{\left (299 \, B a^{5} b + 159 \, A a^{4} b^{2}\right )} x^{2} + 63 \,{\left (13 \, B a^{6} + 27 \, A a^{5} b\right )} x\right )} \sqrt{b x + a}}{9009 \, a^{4} x^{\frac{13}{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 = 2.66771, size = 211, normalized size = 1.8 \begin{align*} \frac{{\left ({\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (13 \, B a^{3} b^{12} - 6 \, A a^{2} b^{13}\right )}{\left (b x + a\right )}}{a^{7} b^{21}} - \frac{13 \,{\left (13 \, B a^{4} b^{12} - 6 \, A a^{3} b^{13}\right )}}{a^{7} b^{21}}\right )} + \frac{143 \,{\left (13 \, B a^{5} b^{12} - 6 \, A a^{4} b^{13}\right )}}{a^{7} b^{21}}\right )} - \frac{1287 \,{\left (B a^{6} b^{12} - A a^{5} b^{13}\right )}}{a^{7} b^{21}}\right )}{\left (b x + a\right )}^{\frac{7}{2}} b}{6642155520 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{13}{2}}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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