Optimal. Leaf size=174 \[ \frac{35 b^3 (9 A b-8 a B)}{64 a^5 \sqrt{a+b x}}+\frac{35 b^2 (9 A b-8 a B)}{192 a^4 x \sqrt{a+b x}}-\frac{35 b^3 (9 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{64 a^{11/2}}-\frac{7 b (9 A b-8 a B)}{96 a^3 x^2 \sqrt{a+b x}}+\frac{9 A b-8 a B}{24 a^2 x^3 \sqrt{a+b x}}-\frac{A}{4 a x^4 \sqrt{a+b x}} \]
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Rubi [A] time = 0.0795249, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {78, 51, 63, 208} \[ \frac{35 b^2 \sqrt{a+b x} (9 A b-8 a B)}{64 a^5 x}-\frac{35 b^3 (9 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{64 a^{11/2}}-\frac{35 b \sqrt{a+b x} (9 A b-8 a B)}{96 a^4 x^2}+\frac{7 \sqrt{a+b x} (9 A b-8 a B)}{24 a^3 x^3}-\frac{9 A b-8 a B}{4 a^2 x^3 \sqrt{a+b x}}-\frac{A}{4 a x^4 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Rule 78
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B x}{x^5 (a+b x)^{3/2}} \, dx &=-\frac{A}{4 a x^4 \sqrt{a+b x}}+\frac{\left (-\frac{9 A b}{2}+4 a B\right ) \int \frac{1}{x^4 (a+b x)^{3/2}} \, dx}{4 a}\\ &=-\frac{A}{4 a x^4 \sqrt{a+b x}}-\frac{9 A b-8 a B}{4 a^2 x^3 \sqrt{a+b x}}-\frac{(7 (9 A b-8 a B)) \int \frac{1}{x^4 \sqrt{a+b x}} \, dx}{8 a^2}\\ &=-\frac{A}{4 a x^4 \sqrt{a+b x}}-\frac{9 A b-8 a B}{4 a^2 x^3 \sqrt{a+b x}}+\frac{7 (9 A b-8 a B) \sqrt{a+b x}}{24 a^3 x^3}+\frac{(35 b (9 A b-8 a B)) \int \frac{1}{x^3 \sqrt{a+b x}} \, dx}{48 a^3}\\ &=-\frac{A}{4 a x^4 \sqrt{a+b x}}-\frac{9 A b-8 a B}{4 a^2 x^3 \sqrt{a+b x}}+\frac{7 (9 A b-8 a B) \sqrt{a+b x}}{24 a^3 x^3}-\frac{35 b (9 A b-8 a B) \sqrt{a+b x}}{96 a^4 x^2}-\frac{\left (35 b^2 (9 A b-8 a B)\right ) \int \frac{1}{x^2 \sqrt{a+b x}} \, dx}{64 a^4}\\ &=-\frac{A}{4 a x^4 \sqrt{a+b x}}-\frac{9 A b-8 a B}{4 a^2 x^3 \sqrt{a+b x}}+\frac{7 (9 A b-8 a B) \sqrt{a+b x}}{24 a^3 x^3}-\frac{35 b (9 A b-8 a B) \sqrt{a+b x}}{96 a^4 x^2}+\frac{35 b^2 (9 A b-8 a B) \sqrt{a+b x}}{64 a^5 x}+\frac{\left (35 b^3 (9 A b-8 a B)\right ) \int \frac{1}{x \sqrt{a+b x}} \, dx}{128 a^5}\\ &=-\frac{A}{4 a x^4 \sqrt{a+b x}}-\frac{9 A b-8 a B}{4 a^2 x^3 \sqrt{a+b x}}+\frac{7 (9 A b-8 a B) \sqrt{a+b x}}{24 a^3 x^3}-\frac{35 b (9 A b-8 a B) \sqrt{a+b x}}{96 a^4 x^2}+\frac{35 b^2 (9 A b-8 a B) \sqrt{a+b x}}{64 a^5 x}+\frac{\left (35 b^2 (9 A b-8 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{64 a^5}\\ &=-\frac{A}{4 a x^4 \sqrt{a+b x}}-\frac{9 A b-8 a B}{4 a^2 x^3 \sqrt{a+b x}}+\frac{7 (9 A b-8 a B) \sqrt{a+b x}}{24 a^3 x^3}-\frac{35 b (9 A b-8 a B) \sqrt{a+b x}}{96 a^4 x^2}+\frac{35 b^2 (9 A b-8 a B) \sqrt{a+b x}}{64 a^5 x}-\frac{35 b^3 (9 A b-8 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{64 a^{11/2}}\\ \end{align*}
Mathematica [C] time = 0.0170232, size = 58, normalized size = 0.33 \[ \frac{b^3 x^4 (9 A b-8 a B) \, _2F_1\left (-\frac{1}{2},4;\frac{1}{2};\frac{b x}{a}+1\right )-a^4 A}{4 a^5 x^4 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 147, normalized size = 0.8 \begin{align*} 2\,{b}^{3} \left ({\frac{1}{{a}^{5}} \left ({\frac{1}{{b}^{4}{x}^{4}} \left ( \left ({\frac{187\,Ab}{128}}-{\frac{19\,Ba}{16}} \right ) \left ( bx+a \right ) ^{7/2}+ \left ( -{\frac{643\,Aba}{128}}+{\frac{193\,B{a}^{2}}{48}} \right ) \left ( bx+a \right ) ^{5/2}+ \left ({\frac{765\,Ab{a}^{2}}{128}}-{\frac{223\,B{a}^{3}}{48}} \right ) \left ( bx+a \right ) ^{3/2}+ \left ( -{\frac{325\,A{a}^{3}b}{128}}+{\frac{29\,B{a}^{4}}{16}} \right ) \sqrt{bx+a} \right ) }-{\frac{315\,Ab-280\,Ba}{128\,\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) }-{\frac{-Ab+Ba}{{a}^{5}\sqrt{bx+a}}} \right ) \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 = 2.53732, size = 836, normalized size = 4.8 \begin{align*} \left [-\frac{105 \,{\left ({\left (8 \, B a b^{4} - 9 \, A b^{5}\right )} x^{5} +{\left (8 \, B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{4}\right )} \sqrt{a} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (48 \, A a^{5} + 105 \,{\left (8 \, B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{4} + 35 \,{\left (8 \, B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{3} - 14 \,{\left (8 \, B a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{2} + 8 \,{\left (8 \, B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt{b x + a}}{384 \,{\left (a^{6} b x^{5} + a^{7} x^{4}\right )}}, -\frac{105 \,{\left ({\left (8 \, B a b^{4} - 9 \, A b^{5}\right )} x^{5} +{\left (8 \, B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{4}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (48 \, A a^{5} + 105 \,{\left (8 \, B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{4} + 35 \,{\left (8 \, B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{3} - 14 \,{\left (8 \, B a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{2} + 8 \,{\left (8 \, B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt{b x + a}}{192 \,{\left (a^{6} b x^{5} + a^{7} x^{4}\right )}}\right ] \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 = 1.18887, size = 266, normalized size = 1.53 \begin{align*} -\frac{35 \,{\left (8 \, B a b^{3} - 9 \, A b^{4}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{64 \, \sqrt{-a} a^{5}} - \frac{2 \,{\left (B a b^{3} - A b^{4}\right )}}{\sqrt{b x + a} a^{5}} - \frac{456 \,{\left (b x + a\right )}^{\frac{7}{2}} B a b^{3} - 1544 \,{\left (b x + a\right )}^{\frac{5}{2}} B a^{2} b^{3} + 1784 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{3} b^{3} - 696 \, \sqrt{b x + a} B a^{4} b^{3} - 561 \,{\left (b x + a\right )}^{\frac{7}{2}} A b^{4} + 1929 \,{\left (b x + a\right )}^{\frac{5}{2}} A a b^{4} - 2295 \,{\left (b x + a\right )}^{\frac{3}{2}} A a^{2} b^{4} + 975 \, \sqrt{b x + a} A a^{3} b^{4}}{192 \, a^{5} b^{4} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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