Optimal. Leaf size=136 \[ \frac{2 b^2 (A b-a B)}{3 a^4 x^{3/2}}-\frac{2 b^3 (A b-a B)}{a^5 \sqrt{x}}-\frac{2 b^{7/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{11/2}}-\frac{2 b (A b-a B)}{5 a^3 x^{5/2}}+\frac{2 (A b-a B)}{7 a^2 x^{7/2}}-\frac{2 A}{9 a x^{9/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0721327, 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 = {78, 51, 63, 205} \[ \frac{2 b^2 (A b-a B)}{3 a^4 x^{3/2}}-\frac{2 b^3 (A b-a B)}{a^5 \sqrt{x}}-\frac{2 b^{7/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{11/2}}-\frac{2 b (A b-a B)}{5 a^3 x^{5/2}}+\frac{2 (A b-a B)}{7 a^2 x^{7/2}}-\frac{2 A}{9 a x^{9/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 78
Rule 51
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B x}{x^{11/2} (a+b x)} \, dx &=-\frac{2 A}{9 a x^{9/2}}+\frac{\left (2 \left (-\frac{9 A b}{2}+\frac{9 a B}{2}\right )\right ) \int \frac{1}{x^{9/2} (a+b x)} \, dx}{9 a}\\ &=-\frac{2 A}{9 a x^{9/2}}+\frac{2 (A b-a B)}{7 a^2 x^{7/2}}+\frac{(b (A b-a B)) \int \frac{1}{x^{7/2} (a+b x)} \, dx}{a^2}\\ &=-\frac{2 A}{9 a x^{9/2}}+\frac{2 (A b-a B)}{7 a^2 x^{7/2}}-\frac{2 b (A b-a B)}{5 a^3 x^{5/2}}-\frac{\left (b^2 (A b-a B)\right ) \int \frac{1}{x^{5/2} (a+b x)} \, dx}{a^3}\\ &=-\frac{2 A}{9 a x^{9/2}}+\frac{2 (A b-a B)}{7 a^2 x^{7/2}}-\frac{2 b (A b-a B)}{5 a^3 x^{5/2}}+\frac{2 b^2 (A b-a B)}{3 a^4 x^{3/2}}+\frac{\left (b^3 (A b-a B)\right ) \int \frac{1}{x^{3/2} (a+b x)} \, dx}{a^4}\\ &=-\frac{2 A}{9 a x^{9/2}}+\frac{2 (A b-a B)}{7 a^2 x^{7/2}}-\frac{2 b (A b-a B)}{5 a^3 x^{5/2}}+\frac{2 b^2 (A b-a B)}{3 a^4 x^{3/2}}-\frac{2 b^3 (A b-a B)}{a^5 \sqrt{x}}-\frac{\left (b^4 (A b-a B)\right ) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{a^5}\\ &=-\frac{2 A}{9 a x^{9/2}}+\frac{2 (A b-a B)}{7 a^2 x^{7/2}}-\frac{2 b (A b-a B)}{5 a^3 x^{5/2}}+\frac{2 b^2 (A b-a B)}{3 a^4 x^{3/2}}-\frac{2 b^3 (A b-a B)}{a^5 \sqrt{x}}-\frac{\left (2 b^4 (A b-a B)\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{a^5}\\ &=-\frac{2 A}{9 a x^{9/2}}+\frac{2 (A b-a B)}{7 a^2 x^{7/2}}-\frac{2 b (A b-a B)}{5 a^3 x^{5/2}}+\frac{2 b^2 (A b-a B)}{3 a^4 x^{3/2}}-\frac{2 b^3 (A b-a B)}{a^5 \sqrt{x}}-\frac{2 b^{7/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{11/2}}\\ \end{align*}
Mathematica [C] time = 0.0138916, size = 44, normalized size = 0.32 \[ -\frac{2 \left (\, _2F_1\left (-\frac{7}{2},1;-\frac{5}{2};-\frac{b x}{a}\right ) (9 a B x-9 A b x)+7 a A\right )}{63 a^2 x^{9/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 150, normalized size = 1.1 \begin{align*} -{\frac{2\,A}{9\,a}{x}^{-{\frac{9}{2}}}}+{\frac{2\,Ab}{7\,{a}^{2}}{x}^{-{\frac{7}{2}}}}-{\frac{2\,B}{7\,a}{x}^{-{\frac{7}{2}}}}-2\,{\frac{{b}^{4}A}{{a}^{5}\sqrt{x}}}+2\,{\frac{{b}^{3}B}{{a}^{4}\sqrt{x}}}-{\frac{2\,{b}^{2}A}{5\,{a}^{3}}{x}^{-{\frac{5}{2}}}}+{\frac{2\,Bb}{5\,{a}^{2}}{x}^{-{\frac{5}{2}}}}+{\frac{2\,{b}^{3}A}{3\,{a}^{4}}{x}^{-{\frac{3}{2}}}}-{\frac{2\,{b}^{2}B}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}}-2\,{\frac{A{b}^{5}}{{a}^{5}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) }+2\,{\frac{{b}^{4}B}{{a}^{4}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.37555, size = 641, normalized size = 4.71 \begin{align*} \left [-\frac{315 \,{\left (B a b^{3} - A b^{4}\right )} x^{5} \sqrt{-\frac{b}{a}} \log \left (\frac{b x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) + 2 \,{\left (35 \, A a^{4} - 315 \,{\left (B a b^{3} - A b^{4}\right )} x^{4} + 105 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{3} - 63 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x^{2} + 45 \,{\left (B a^{4} - A a^{3} b\right )} x\right )} \sqrt{x}}{315 \, a^{5} x^{5}}, -\frac{2 \,{\left (315 \,{\left (B a b^{3} - A b^{4}\right )} x^{5} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) +{\left (35 \, A a^{4} - 315 \,{\left (B a b^{3} - A b^{4}\right )} x^{4} + 105 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{3} - 63 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x^{2} + 45 \,{\left (B a^{4} - A a^{3} b\right )} x\right )} \sqrt{x}\right )}}{315 \, a^{5} x^{5}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1865, size = 173, normalized size = 1.27 \begin{align*} \frac{2 \,{\left (B a b^{4} - A b^{5}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{5}} + \frac{2 \,{\left (315 \, B a b^{3} x^{4} - 315 \, A b^{4} x^{4} - 105 \, B a^{2} b^{2} x^{3} + 105 \, A a b^{3} x^{3} + 63 \, B a^{3} b x^{2} - 63 \, A a^{2} b^{2} x^{2} - 45 \, B a^{4} x + 45 \, A a^{3} b x - 35 \, A a^{4}\right )}}{315 \, a^{5} x^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]