Optimal. Leaf size=136 \[ \frac{2 a^2 x^{3/2} (A b-a B)}{3 b^4}-\frac{2 a^3 \sqrt{x} (A b-a B)}{b^5}+\frac{2 a^{7/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{11/2}}+\frac{2 x^{7/2} (A b-a B)}{7 b^2}-\frac{2 a x^{5/2} (A b-a B)}{5 b^3}+\frac{2 B x^{9/2}}{9 b} \]
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Rubi [A] time = 0.0815894, antiderivative size = 136, 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 = {80, 50, 63, 205} \[ \frac{2 a^2 x^{3/2} (A b-a B)}{3 b^4}-\frac{2 a^3 \sqrt{x} (A b-a B)}{b^5}+\frac{2 a^{7/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{11/2}}+\frac{2 x^{7/2} (A b-a B)}{7 b^2}-\frac{2 a x^{5/2} (A b-a B)}{5 b^3}+\frac{2 B x^{9/2}}{9 b} \]
Antiderivative was successfully verified.
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Rule 80
Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{7/2} (A+B x)}{a+b x} \, dx &=\frac{2 B x^{9/2}}{9 b}+\frac{\left (2 \left (\frac{9 A b}{2}-\frac{9 a B}{2}\right )\right ) \int \frac{x^{7/2}}{a+b x} \, dx}{9 b}\\ &=\frac{2 (A b-a B) x^{7/2}}{7 b^2}+\frac{2 B x^{9/2}}{9 b}-\frac{(a (A b-a B)) \int \frac{x^{5/2}}{a+b x} \, dx}{b^2}\\ &=-\frac{2 a (A b-a B) x^{5/2}}{5 b^3}+\frac{2 (A b-a B) x^{7/2}}{7 b^2}+\frac{2 B x^{9/2}}{9 b}+\frac{\left (a^2 (A b-a B)\right ) \int \frac{x^{3/2}}{a+b x} \, dx}{b^3}\\ &=\frac{2 a^2 (A b-a B) x^{3/2}}{3 b^4}-\frac{2 a (A b-a B) x^{5/2}}{5 b^3}+\frac{2 (A b-a B) x^{7/2}}{7 b^2}+\frac{2 B x^{9/2}}{9 b}-\frac{\left (a^3 (A b-a B)\right ) \int \frac{\sqrt{x}}{a+b x} \, dx}{b^4}\\ &=-\frac{2 a^3 (A b-a B) \sqrt{x}}{b^5}+\frac{2 a^2 (A b-a B) x^{3/2}}{3 b^4}-\frac{2 a (A b-a B) x^{5/2}}{5 b^3}+\frac{2 (A b-a B) x^{7/2}}{7 b^2}+\frac{2 B x^{9/2}}{9 b}+\frac{\left (a^4 (A b-a B)\right ) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{b^5}\\ &=-\frac{2 a^3 (A b-a B) \sqrt{x}}{b^5}+\frac{2 a^2 (A b-a B) x^{3/2}}{3 b^4}-\frac{2 a (A b-a B) x^{5/2}}{5 b^3}+\frac{2 (A b-a B) x^{7/2}}{7 b^2}+\frac{2 B x^{9/2}}{9 b}+\frac{\left (2 a^4 (A b-a B)\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{b^5}\\ &=-\frac{2 a^3 (A b-a B) \sqrt{x}}{b^5}+\frac{2 a^2 (A b-a B) x^{3/2}}{3 b^4}-\frac{2 a (A b-a B) x^{5/2}}{5 b^3}+\frac{2 (A b-a B) x^{7/2}}{7 b^2}+\frac{2 B x^{9/2}}{9 b}+\frac{2 a^{7/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.0962008, size = 120, normalized size = 0.88 \[ \frac{2 \sqrt{x} \left (21 a^2 b^2 x (5 A+3 B x)-105 a^3 b (3 A+B x)+315 a^4 B-9 a b^3 x^2 (7 A+5 B x)+5 b^4 x^3 (9 A+7 B x)\right )}{315 b^5}-\frac{2 a^{7/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{11/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 150, normalized size = 1.1 \begin{align*}{\frac{2\,B}{9\,b}{x}^{{\frac{9}{2}}}}+{\frac{2\,A}{7\,b}{x}^{{\frac{7}{2}}}}-{\frac{2\,Ba}{7\,{b}^{2}}{x}^{{\frac{7}{2}}}}-{\frac{2\,Aa}{5\,{b}^{2}}{x}^{{\frac{5}{2}}}}+{\frac{2\,B{a}^{2}}{5\,{b}^{3}}{x}^{{\frac{5}{2}}}}+{\frac{2\,A{a}^{2}}{3\,{b}^{3}}{x}^{{\frac{3}{2}}}}-{\frac{2\,B{a}^{3}}{3\,{b}^{4}}{x}^{{\frac{3}{2}}}}-2\,{\frac{{a}^{3}A\sqrt{x}}{{b}^{4}}}+2\,{\frac{B{a}^{4}\sqrt{x}}{{b}^{5}}}+2\,{\frac{{a}^{4}A}{{b}^{4}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) }-2\,{\frac{B{a}^{5}}{{b}^{5}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\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 = 2.36002, size = 617, normalized size = 4.54 \begin{align*} \left [-\frac{315 \,{\left (B a^{4} - A a^{3} b\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (35 \, B b^{4} x^{4} + 315 \, B a^{4} - 315 \, A a^{3} b - 45 \,{\left (B a b^{3} - A b^{4}\right )} x^{3} + 63 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 105 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x\right )} \sqrt{x}}{315 \, b^{5}}, -\frac{2 \,{\left (315 \,{\left (B a^{4} - A a^{3} b\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{\frac{a}{b}}}{a}\right ) -{\left (35 \, B b^{4} x^{4} + 315 \, B a^{4} - 315 \, A a^{3} b - 45 \,{\left (B a b^{3} - A b^{4}\right )} x^{3} + 63 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 105 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x\right )} \sqrt{x}\right )}}{315 \, b^{5}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 150.866, size = 313, normalized size = 2.3 \begin{align*} \begin{cases} - \frac{i A a^{\frac{7}{2}} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{b^{5} \sqrt{\frac{1}{b}}} + \frac{i A a^{\frac{7}{2}} \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{b^{5} \sqrt{\frac{1}{b}}} - \frac{2 A a^{3} \sqrt{x}}{b^{4}} + \frac{2 A a^{2} x^{\frac{3}{2}}}{3 b^{3}} - \frac{2 A a x^{\frac{5}{2}}}{5 b^{2}} + \frac{2 A x^{\frac{7}{2}}}{7 b} + \frac{i B a^{\frac{9}{2}} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{b^{6} \sqrt{\frac{1}{b}}} - \frac{i B a^{\frac{9}{2}} \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{b^{6} \sqrt{\frac{1}{b}}} + \frac{2 B a^{4} \sqrt{x}}{b^{5}} - \frac{2 B a^{3} x^{\frac{3}{2}}}{3 b^{4}} + \frac{2 B a^{2} x^{\frac{5}{2}}}{5 b^{3}} - \frac{2 B a x^{\frac{7}{2}}}{7 b^{2}} + \frac{2 B x^{\frac{9}{2}}}{9 b} & \text{for}\: b \neq 0 \\\frac{\frac{2 A x^{\frac{9}{2}}}{9} + \frac{2 B x^{\frac{11}{2}}}{11}}{a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17552, size = 188, normalized size = 1.38 \begin{align*} -\frac{2 \,{\left (B a^{5} - A a^{4} b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{5}} + \frac{2 \,{\left (35 \, B b^{8} x^{\frac{9}{2}} - 45 \, B a b^{7} x^{\frac{7}{2}} + 45 \, A b^{8} x^{\frac{7}{2}} + 63 \, B a^{2} b^{6} x^{\frac{5}{2}} - 63 \, A a b^{7} x^{\frac{5}{2}} - 105 \, B a^{3} b^{5} x^{\frac{3}{2}} + 105 \, A a^{2} b^{6} x^{\frac{3}{2}} + 315 \, B a^{4} b^{4} \sqrt{x} - 315 \, A a^{3} b^{5} \sqrt{x}\right )}}{315 \, b^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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