Optimal. Leaf size=131 \[ -\frac{b^{3/2} (7 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{9/2}}+\frac{7 A b-5 a B}{3 a^3 x^{3/2}}-\frac{7 A b-5 a B}{5 a^2 b x^{5/2}}-\frac{b (7 A b-5 a B)}{a^4 \sqrt{x}}+\frac{A b-a B}{a b x^{5/2} (a+b x)} \]
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Rubi [A] time = 0.0630578, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {27, 78, 51, 63, 205} \[ -\frac{b^{3/2} (7 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{9/2}}+\frac{7 A b-5 a B}{3 a^3 x^{3/2}}-\frac{7 A b-5 a B}{5 a^2 b x^{5/2}}-\frac{b (7 A b-5 a B)}{a^4 \sqrt{x}}+\frac{A b-a B}{a b x^{5/2} (a+b x)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 78
Rule 51
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B x}{x^{7/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac{A+B x}{x^{7/2} (a+b x)^2} \, dx\\ &=\frac{A b-a B}{a b x^{5/2} (a+b x)}-\frac{\left (-\frac{7 A b}{2}+\frac{5 a B}{2}\right ) \int \frac{1}{x^{7/2} (a+b x)} \, dx}{a b}\\ &=-\frac{7 A b-5 a B}{5 a^2 b x^{5/2}}+\frac{A b-a B}{a b x^{5/2} (a+b x)}-\frac{(7 A b-5 a B) \int \frac{1}{x^{5/2} (a+b x)} \, dx}{2 a^2}\\ &=-\frac{7 A b-5 a B}{5 a^2 b x^{5/2}}+\frac{7 A b-5 a B}{3 a^3 x^{3/2}}+\frac{A b-a B}{a b x^{5/2} (a+b x)}+\frac{(b (7 A b-5 a B)) \int \frac{1}{x^{3/2} (a+b x)} \, dx}{2 a^3}\\ &=-\frac{7 A b-5 a B}{5 a^2 b x^{5/2}}+\frac{7 A b-5 a B}{3 a^3 x^{3/2}}-\frac{b (7 A b-5 a B)}{a^4 \sqrt{x}}+\frac{A b-a B}{a b x^{5/2} (a+b x)}-\frac{\left (b^2 (7 A b-5 a B)\right ) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{2 a^4}\\ &=-\frac{7 A b-5 a B}{5 a^2 b x^{5/2}}+\frac{7 A b-5 a B}{3 a^3 x^{3/2}}-\frac{b (7 A b-5 a B)}{a^4 \sqrt{x}}+\frac{A b-a B}{a b x^{5/2} (a+b x)}-\frac{\left (b^2 (7 A b-5 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{a^4}\\ &=-\frac{7 A b-5 a B}{5 a^2 b x^{5/2}}+\frac{7 A b-5 a B}{3 a^3 x^{3/2}}-\frac{b (7 A b-5 a B)}{a^4 \sqrt{x}}+\frac{A b-a B}{a b x^{5/2} (a+b x)}-\frac{b^{3/2} (7 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0191568, size = 64, normalized size = 0.49 \[ \frac{(a+b x) (5 a B-7 A b) \, _2F_1\left (-\frac{5}{2},1;-\frac{3}{2};-\frac{b x}{a}\right )+5 a (A b-a B)}{5 a^2 b x^{5/2} (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 139, normalized size = 1.1 \begin{align*} -{\frac{2\,A}{5\,{a}^{2}}{x}^{-{\frac{5}{2}}}}+{\frac{4\,Ab}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}}-{\frac{2\,B}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}-6\,{\frac{A{b}^{2}}{{a}^{4}\sqrt{x}}}+4\,{\frac{bB}{{a}^{3}\sqrt{x}}}-{\frac{{b}^{3}A}{{a}^{4} \left ( bx+a \right ) }\sqrt{x}}+{\frac{{b}^{2}B}{{a}^{3} \left ( bx+a \right ) }\sqrt{x}}-7\,{\frac{{b}^{3}A}{{a}^{4}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) }+5\,{\frac{{b}^{2}B}{{a}^{3}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \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 = 1.67993, size = 698, normalized size = 5.33 \begin{align*} \left [-\frac{15 \,{\left ({\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{4} +{\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) + 2 \,{\left (6 \, A a^{3} - 15 \,{\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} - 10 \,{\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 2 \,{\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt{x}}{30 \,{\left (a^{4} b x^{4} + a^{5} x^{3}\right )}}, -\frac{15 \,{\left ({\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{4} +{\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) +{\left (6 \, A a^{3} - 15 \,{\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} - 10 \,{\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 2 \,{\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt{x}}{15 \,{\left (a^{4} b x^{4} + a^{5} x^{3}\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 = 1.14355, size = 149, normalized size = 1.14 \begin{align*} \frac{{\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{4}} + \frac{B a b^{2} \sqrt{x} - A b^{3} \sqrt{x}}{{\left (b x + a\right )} a^{4}} + \frac{2 \,{\left (30 \, B a b x^{2} - 45 \, A b^{2} x^{2} - 5 \, B a^{2} x + 10 \, A a b x - 3 \, A a^{2}\right )}}{15 \, a^{4} x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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