Optimal. Leaf size=238 \[ \frac{2 b^2 (a+b x) (A b-a B)}{a^4 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 b (a+b x) (A b-a B)}{3 a^3 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (A b-a B)}{5 a^2 x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 b^{5/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 A (a+b x)}{7 a x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.116351, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {770, 78, 51, 63, 205} \[ \frac{2 b^2 (a+b x) (A b-a B)}{a^4 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 b (a+b x) (A b-a B)}{3 a^3 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (a+b x) (A b-a B)}{5 a^2 x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 b^{5/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 A (a+b x)}{7 a x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 770
Rule 78
Rule 51
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B x}{x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}} \, dx &=\frac{\left (a b+b^2 x\right ) \int \frac{A+B x}{x^{9/2} \left (a b+b^2 x\right )} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{2 A (a+b x)}{7 a x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (2 \left (-\frac{7 A b^2}{2}+\frac{7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac{1}{x^{7/2} \left (a b+b^2 x\right )} \, dx}{7 a b \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{2 A (a+b x)}{7 a x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (A b-a B) (a+b x)}{5 a^2 x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (2 \left (-\frac{7 A b^2}{2}+\frac{7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac{1}{x^{5/2} \left (a b+b^2 x\right )} \, dx}{7 a^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{2 A (a+b x)}{7 a x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (A b-a B) (a+b x)}{5 a^2 x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 b (A b-a B) (a+b x)}{3 a^3 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (2 b \left (-\frac{7 A b^2}{2}+\frac{7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac{1}{x^{3/2} \left (a b+b^2 x\right )} \, dx}{7 a^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{2 A (a+b x)}{7 a x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (A b-a B) (a+b x)}{5 a^2 x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 b (A b-a B) (a+b x)}{3 a^3 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 b^2 (A b-a B) (a+b x)}{a^4 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (2 b^2 \left (-\frac{7 A b^2}{2}+\frac{7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac{1}{\sqrt{x} \left (a b+b^2 x\right )} \, dx}{7 a^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{2 A (a+b x)}{7 a x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (A b-a B) (a+b x)}{5 a^2 x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 b (A b-a B) (a+b x)}{3 a^3 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 b^2 (A b-a B) (a+b x)}{a^4 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (4 b^2 \left (-\frac{7 A b^2}{2}+\frac{7 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a b+b^2 x^2} \, dx,x,\sqrt{x}\right )}{7 a^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{2 A (a+b x)}{7 a x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (A b-a B) (a+b x)}{5 a^2 x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 b (A b-a B) (a+b x)}{3 a^3 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 b^2 (A b-a B) (a+b x)}{a^4 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 b^{5/2} (A b-a B) (a+b x) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [C] time = 0.0258333, size = 60, normalized size = 0.25 \[ -\frac{2 (a+b x) \left (\, _2F_1\left (-\frac{5}{2},1;-\frac{3}{2};-\frac{b x}{a}\right ) (7 a B x-7 A b x)+5 a A\right )}{35 a^2 x^{7/2} \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.013, size = 165, normalized size = 0.7 \begin{align*}{\frac{2\,bx+2\,a}{105\,{a}^{4}} \left ( 105\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{7/2}{b}^{4}-105\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{7/2}a{b}^{3}+105\,A\sqrt{ab}{x}^{3}{b}^{3}-105\,B\sqrt{ab}{x}^{3}a{b}^{2}-35\,A\sqrt{ab}{x}^{2}a{b}^{2}+35\,B\sqrt{ab}{x}^{2}{a}^{2}b+21\,A\sqrt{ab}x{a}^{2}b-21\,B\sqrt{ab}x{a}^{3}-15\,A{a}^{3}\sqrt{ab} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}{x}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{ab}}}} \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 = 1.3552, size = 548, normalized size = 2.3 \begin{align*} \left [-\frac{105 \,{\left (B a b^{2} - A b^{3}\right )} x^{4} \sqrt{-\frac{b}{a}} \log \left (\frac{b x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) + 2 \,{\left (15 \, A a^{3} + 105 \,{\left (B a b^{2} - A b^{3}\right )} x^{3} - 35 \,{\left (B a^{2} b - A a b^{2}\right )} x^{2} + 21 \,{\left (B a^{3} - A a^{2} b\right )} x\right )} \sqrt{x}}{105 \, a^{4} x^{4}}, \frac{2 \,{\left (105 \,{\left (B a b^{2} - A b^{3}\right )} x^{4} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) -{\left (15 \, A a^{3} + 105 \,{\left (B a b^{2} - A b^{3}\right )} x^{3} - 35 \,{\left (B a^{2} b - A a b^{2}\right )} x^{2} + 21 \,{\left (B a^{3} - A a^{2} b\right )} x\right )} \sqrt{x}\right )}}{105 \, a^{4} x^{4}}\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 = 1.13236, size = 213, normalized size = 0.89 \begin{align*} -\frac{2 \,{\left (B a b^{3} \mathrm{sgn}\left (b x + a\right ) - A b^{4} \mathrm{sgn}\left (b x + a\right )\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{4}} - \frac{2 \,{\left (105 \, B a b^{2} x^{3} \mathrm{sgn}\left (b x + a\right ) - 105 \, A b^{3} x^{3} \mathrm{sgn}\left (b x + a\right ) - 35 \, B a^{2} b x^{2} \mathrm{sgn}\left (b x + a\right ) + 35 \, A a b^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + 21 \, B a^{3} x \mathrm{sgn}\left (b x + a\right ) - 21 \, A a^{2} b x \mathrm{sgn}\left (b x + a\right ) + 15 \, A a^{3} \mathrm{sgn}\left (b x + a\right )\right )}}{105 \, a^{4} x^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]