Optimal. Leaf size=183 \[ -\frac{128 b^3 \sqrt{a+b x} (10 A b-11 a B)}{3465 a^5 x^{3/2}}+\frac{32 b^2 \sqrt{a+b x} (10 A b-11 a B)}{1155 a^4 x^{5/2}}+\frac{256 b^4 \sqrt{a+b x} (10 A b-11 a B)}{3465 a^6 \sqrt{x}}-\frac{16 b \sqrt{a+b x} (10 A b-11 a B)}{693 a^3 x^{7/2}}+\frac{2 \sqrt{a+b x} (10 A b-11 a B)}{99 a^2 x^{9/2}}-\frac{2 A \sqrt{a+b x}}{11 a x^{11/2}} \]
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Rubi [A] time = 0.0711368, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {78, 45, 37} \[ -\frac{128 b^3 \sqrt{a+b x} (10 A b-11 a B)}{3465 a^5 x^{3/2}}+\frac{32 b^2 \sqrt{a+b x} (10 A b-11 a B)}{1155 a^4 x^{5/2}}+\frac{256 b^4 \sqrt{a+b x} (10 A b-11 a B)}{3465 a^6 \sqrt{x}}-\frac{16 b \sqrt{a+b x} (10 A b-11 a B)}{693 a^3 x^{7/2}}+\frac{2 \sqrt{a+b x} (10 A b-11 a B)}{99 a^2 x^{9/2}}-\frac{2 A \sqrt{a+b x}}{11 a x^{11/2}} \]
Antiderivative was successfully verified.
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Rule 78
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{A+B x}{x^{13/2} \sqrt{a+b x}} \, dx &=-\frac{2 A \sqrt{a+b x}}{11 a x^{11/2}}+\frac{\left (2 \left (-5 A b+\frac{11 a B}{2}\right )\right ) \int \frac{1}{x^{11/2} \sqrt{a+b x}} \, dx}{11 a}\\ &=-\frac{2 A \sqrt{a+b x}}{11 a x^{11/2}}+\frac{2 (10 A b-11 a B) \sqrt{a+b x}}{99 a^2 x^{9/2}}+\frac{(8 b (10 A b-11 a B)) \int \frac{1}{x^{9/2} \sqrt{a+b x}} \, dx}{99 a^2}\\ &=-\frac{2 A \sqrt{a+b x}}{11 a x^{11/2}}+\frac{2 (10 A b-11 a B) \sqrt{a+b x}}{99 a^2 x^{9/2}}-\frac{16 b (10 A b-11 a B) \sqrt{a+b x}}{693 a^3 x^{7/2}}-\frac{\left (16 b^2 (10 A b-11 a B)\right ) \int \frac{1}{x^{7/2} \sqrt{a+b x}} \, dx}{231 a^3}\\ &=-\frac{2 A \sqrt{a+b x}}{11 a x^{11/2}}+\frac{2 (10 A b-11 a B) \sqrt{a+b x}}{99 a^2 x^{9/2}}-\frac{16 b (10 A b-11 a B) \sqrt{a+b x}}{693 a^3 x^{7/2}}+\frac{32 b^2 (10 A b-11 a B) \sqrt{a+b x}}{1155 a^4 x^{5/2}}+\frac{\left (64 b^3 (10 A b-11 a B)\right ) \int \frac{1}{x^{5/2} \sqrt{a+b x}} \, dx}{1155 a^4}\\ &=-\frac{2 A \sqrt{a+b x}}{11 a x^{11/2}}+\frac{2 (10 A b-11 a B) \sqrt{a+b x}}{99 a^2 x^{9/2}}-\frac{16 b (10 A b-11 a B) \sqrt{a+b x}}{693 a^3 x^{7/2}}+\frac{32 b^2 (10 A b-11 a B) \sqrt{a+b x}}{1155 a^4 x^{5/2}}-\frac{128 b^3 (10 A b-11 a B) \sqrt{a+b x}}{3465 a^5 x^{3/2}}-\frac{\left (128 b^4 (10 A b-11 a B)\right ) \int \frac{1}{x^{3/2} \sqrt{a+b x}} \, dx}{3465 a^5}\\ &=-\frac{2 A \sqrt{a+b x}}{11 a x^{11/2}}+\frac{2 (10 A b-11 a B) \sqrt{a+b x}}{99 a^2 x^{9/2}}-\frac{16 b (10 A b-11 a B) \sqrt{a+b x}}{693 a^3 x^{7/2}}+\frac{32 b^2 (10 A b-11 a B) \sqrt{a+b x}}{1155 a^4 x^{5/2}}-\frac{128 b^3 (10 A b-11 a B) \sqrt{a+b x}}{3465 a^5 x^{3/2}}+\frac{256 b^4 (10 A b-11 a B) \sqrt{a+b x}}{3465 a^6 \sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.0347568, size = 114, normalized size = 0.62 \[ -\frac{2 \sqrt{a+b x} \left (16 a^3 b^2 x^2 (25 A+33 B x)-32 a^2 b^3 x^3 (15 A+22 B x)-10 a^4 b x (35 A+44 B x)+35 a^5 (9 A+11 B x)+128 a b^4 x^4 (5 A+11 B x)-1280 A b^5 x^5\right )}{3465 a^6 x^{11/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 125, normalized size = 0.7 \begin{align*} -{\frac{-2560\,A{b}^{5}{x}^{5}+2816\,B{x}^{5}a{b}^{4}+1280\,aA{b}^{4}{x}^{4}-1408\,B{x}^{4}{a}^{2}{b}^{3}-960\,{a}^{2}A{b}^{3}{x}^{3}+1056\,B{x}^{3}{a}^{3}{b}^{2}+800\,{a}^{3}A{b}^{2}{x}^{2}-880\,B{x}^{2}{a}^{4}b-700\,{a}^{4}Abx+770\,{a}^{5}Bx+630\,A{a}^{5}}{3465\,{a}^{6}}\sqrt{bx+a}{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 = 2.51206, size = 308, normalized size = 1.68 \begin{align*} -\frac{2 \,{\left (315 \, A a^{5} + 128 \,{\left (11 \, B a b^{4} - 10 \, A b^{5}\right )} x^{5} - 64 \,{\left (11 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} x^{4} + 48 \,{\left (11 \, B a^{3} b^{2} - 10 \, A a^{2} b^{3}\right )} x^{3} - 40 \,{\left (11 \, B a^{4} b - 10 \, A a^{3} b^{2}\right )} x^{2} + 35 \,{\left (11 \, B a^{5} - 10 \, A a^{4} b\right )} x\right )} \sqrt{b x + a}}{3465 \, a^{6} 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 = 1.22825, size = 296, normalized size = 1.62 \begin{align*} \frac{{\left ({\left (8 \,{\left (2 \,{\left (b x + a\right )}{\left (4 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (11 \, B a b^{10} - 10 \, A b^{11}\right )}{\left (b x + a\right )}}{a^{6} b^{18}} - \frac{11 \,{\left (11 \, B a^{2} b^{10} - 10 \, A a b^{11}\right )}}{a^{6} b^{18}}\right )} + \frac{99 \,{\left (11 \, B a^{3} b^{10} - 10 \, A a^{2} b^{11}\right )}}{a^{6} b^{18}}\right )} - \frac{231 \,{\left (11 \, B a^{4} b^{10} - 10 \, A a^{3} b^{11}\right )}}{a^{6} b^{18}}\right )}{\left (b x + a\right )} + \frac{1155 \,{\left (11 \, B a^{5} b^{10} - 10 \, A a^{4} b^{11}\right )}}{a^{6} b^{18}}\right )}{\left (b x + a\right )} - \frac{3465 \,{\left (B a^{6} b^{10} - A a^{5} b^{11}\right )}}{a^{6} b^{18}}\right )} \sqrt{b x + a} 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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