Optimal. Leaf size=150 \[ -\frac{32 b^3 (a+b x)^{5/2} (8 A b-13 a B)}{15015 a^5 x^{5/2}}+\frac{16 b^2 (a+b x)^{5/2} (8 A b-13 a B)}{3003 a^4 x^{7/2}}-\frac{4 b (a+b x)^{5/2} (8 A b-13 a B)}{429 a^3 x^{9/2}}+\frac{2 (a+b x)^{5/2} (8 A b-13 a B)}{143 a^2 x^{11/2}}-\frac{2 A (a+b x)^{5/2}}{13 a x^{13/2}} \]
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Rubi [A] time = 0.0563563, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {78, 45, 37} \[ -\frac{32 b^3 (a+b x)^{5/2} (8 A b-13 a B)}{15015 a^5 x^{5/2}}+\frac{16 b^2 (a+b x)^{5/2} (8 A b-13 a B)}{3003 a^4 x^{7/2}}-\frac{4 b (a+b x)^{5/2} (8 A b-13 a B)}{429 a^3 x^{9/2}}+\frac{2 (a+b x)^{5/2} (8 A b-13 a B)}{143 a^2 x^{11/2}}-\frac{2 A (a+b x)^{5/2}}{13 a x^{13/2}} \]
Antiderivative was successfully verified.
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Rule 78
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2} (A+B x)}{x^{15/2}} \, dx &=-\frac{2 A (a+b x)^{5/2}}{13 a x^{13/2}}+\frac{\left (2 \left (-4 A b+\frac{13 a B}{2}\right )\right ) \int \frac{(a+b x)^{3/2}}{x^{13/2}} \, dx}{13 a}\\ &=-\frac{2 A (a+b x)^{5/2}}{13 a x^{13/2}}+\frac{2 (8 A b-13 a B) (a+b x)^{5/2}}{143 a^2 x^{11/2}}+\frac{(6 b (8 A b-13 a B)) \int \frac{(a+b x)^{3/2}}{x^{11/2}} \, dx}{143 a^2}\\ &=-\frac{2 A (a+b x)^{5/2}}{13 a x^{13/2}}+\frac{2 (8 A b-13 a B) (a+b x)^{5/2}}{143 a^2 x^{11/2}}-\frac{4 b (8 A b-13 a B) (a+b x)^{5/2}}{429 a^3 x^{9/2}}-\frac{\left (8 b^2 (8 A b-13 a B)\right ) \int \frac{(a+b x)^{3/2}}{x^{9/2}} \, dx}{429 a^3}\\ &=-\frac{2 A (a+b x)^{5/2}}{13 a x^{13/2}}+\frac{2 (8 A b-13 a B) (a+b x)^{5/2}}{143 a^2 x^{11/2}}-\frac{4 b (8 A b-13 a B) (a+b x)^{5/2}}{429 a^3 x^{9/2}}+\frac{16 b^2 (8 A b-13 a B) (a+b x)^{5/2}}{3003 a^4 x^{7/2}}+\frac{\left (16 b^3 (8 A b-13 a B)\right ) \int \frac{(a+b x)^{3/2}}{x^{7/2}} \, dx}{3003 a^4}\\ &=-\frac{2 A (a+b x)^{5/2}}{13 a x^{13/2}}+\frac{2 (8 A b-13 a B) (a+b x)^{5/2}}{143 a^2 x^{11/2}}-\frac{4 b (8 A b-13 a B) (a+b x)^{5/2}}{429 a^3 x^{9/2}}+\frac{16 b^2 (8 A b-13 a B) (a+b x)^{5/2}}{3003 a^4 x^{7/2}}-\frac{32 b^3 (8 A b-13 a B) (a+b x)^{5/2}}{15015 a^5 x^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0360798, size = 95, normalized size = 0.63 \[ -\frac{2 (a+b x)^{5/2} \left (40 a^2 b^2 x^2 (14 A+13 B x)-70 a^3 b x (12 A+13 B x)+105 a^4 (11 A+13 B x)-16 a b^3 x^3 (20 A+13 B x)+128 A b^4 x^4\right )}{15015 a^5 x^{13/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 101, normalized size = 0.7 \begin{align*} -{\frac{256\,A{b}^{4}{x}^{4}-416\,Ba{b}^{3}{x}^{4}-640\,Aa{b}^{3}{x}^{3}+1040\,B{a}^{2}{b}^{2}{x}^{3}+1120\,A{a}^{2}{b}^{2}{x}^{2}-1820\,B{a}^{3}b{x}^{2}-1680\,A{a}^{3}bx+2730\,B{a}^{4}x+2310\,A{a}^{4}}{15015\,{a}^{5}} \left ( bx+a \right ) ^{{\frac{5}{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 = 2.60978, size = 350, normalized size = 2.33 \begin{align*} -\frac{2 \,{\left (1155 \, A a^{6} - 16 \,{\left (13 \, B a b^{5} - 8 \, A b^{6}\right )} x^{6} + 8 \,{\left (13 \, B a^{2} b^{4} - 8 \, A a b^{5}\right )} x^{5} - 6 \,{\left (13 \, B a^{3} b^{3} - 8 \, A a^{2} b^{4}\right )} x^{4} + 5 \,{\left (13 \, B a^{4} b^{2} - 8 \, A a^{3} b^{3}\right )} x^{3} + 35 \,{\left (52 \, B a^{5} b + A a^{4} b^{2}\right )} x^{2} + 105 \,{\left (13 \, B a^{6} + 14 \, A a^{5} b\right )} x\right )} \sqrt{b x + a}}{15015 \, a^{5} 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 = 2.53017, size = 255, normalized size = 1.7 \begin{align*} -\frac{{\left ({\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (13 \, B a^{2} b^{12} - 8 \, A a b^{13}\right )}{\left (b x + a\right )}}{a^{7} b^{21}} - \frac{13 \,{\left (13 \, B a^{3} b^{12} - 8 \, A a^{2} b^{13}\right )}}{a^{7} b^{21}}\right )} + \frac{143 \,{\left (13 \, B a^{4} b^{12} - 8 \, A a^{3} b^{13}\right )}}{a^{7} b^{21}}\right )} - \frac{429 \,{\left (13 \, B a^{5} b^{12} - 8 \, A a^{4} b^{13}\right )}}{a^{7} b^{21}}\right )}{\left (b x + a\right )} + \frac{3003 \,{\left (B a^{6} b^{12} - A a^{5} b^{13}\right )}}{a^{7} b^{21}}\right )}{\left (b x + a\right )}^{\frac{5}{2}} b}{11070259200 \,{\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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