Optimal. Leaf size=190 \[ \frac{2 x^{3/2} (a+b x) (A b-a B)}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a \sqrt{x} (a+b x) (A b-a B)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^{3/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{5/2} (a+b x)}{5 b \sqrt{a^2+2 a b x+b^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0860083, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {770, 80, 50, 63, 205} \[ \frac{2 x^{3/2} (a+b x) (A b-a B)}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 a \sqrt{x} (a+b x) (A b-a B)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^{3/2} (a+b x) (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{5/2} (a+b x)}{5 b \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 770
Rule 80
Rule 50
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{3/2} (A+B x)}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx &=\frac{\left (a b+b^2 x\right ) \int \frac{x^{3/2} (A+B x)}{a b+b^2 x} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 B x^{5/2} (a+b x)}{5 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (2 \left (\frac{5 A b^2}{2}-\frac{5 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac{x^{3/2}}{a b+b^2 x} \, dx}{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 (A b-a B) x^{3/2} (a+b x)}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{5/2} (a+b x)}{5 b \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (2 a \left (\frac{5 A b^2}{2}-\frac{5 a b B}{2}\right ) \left (a b+b^2 x\right )\right ) \int \frac{\sqrt{x}}{a b+b^2 x} \, dx}{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{2 a (A b-a B) \sqrt{x} (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (A b-a B) x^{3/2} (a+b x)}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{5/2} (a+b x)}{5 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (2 a^2 \left (\frac{5 A b^2}{2}-\frac{5 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}{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{2 a (A b-a B) \sqrt{x} (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (A b-a B) x^{3/2} (a+b x)}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{5/2} (a+b x)}{5 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (4 a^2 \left (\frac{5 A b^2}{2}-\frac{5 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 )}{5 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{2 a (A b-a B) \sqrt{x} (a+b x)}{b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 (A b-a B) x^{3/2} (a+b x)}{3 b^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 B x^{5/2} (a+b x)}{5 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^{3/2} (A b-a B) (a+b x) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.05261, size = 100, normalized size = 0.53 \[ \frac{2 (a+b x) \left (\sqrt{b} \sqrt{x} \left (15 a^2 B-5 a b (3 A+B x)+b^2 x (5 A+3 B x)\right )-15 a^{3/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )\right )}{15 b^{7/2} \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 129, normalized size = 0.7 \begin{align*}{\frac{2\,bx+2\,a}{15\,{b}^{3}} \left ( 3\,B\sqrt{ab}{x}^{5/2}{b}^{2}+5\,A\sqrt{ab}{x}^{3/2}{b}^{2}-5\,B\sqrt{ab}{x}^{3/2}ab-15\,A\sqrt{ab}\sqrt{x}ab+15\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){a}^{2}b+15\,B\sqrt{ab}\sqrt{x}{a}^{2}-15\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){a}^{3} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) ^{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 = 1.37697, size = 414, normalized size = 2.18 \begin{align*} \left [-\frac{15 \,{\left (B a^{2} - A a 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 (3 \, B b^{2} x^{2} + 15 \, B a^{2} - 15 \, A a b - 5 \,{\left (B a b - A b^{2}\right )} x\right )} \sqrt{x}}{15 \, b^{3}}, -\frac{2 \,{\left (15 \,{\left (B a^{2} - A a b\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{\frac{a}{b}}}{a}\right ) -{\left (3 \, B b^{2} x^{2} + 15 \, B a^{2} - 15 \, A a b - 5 \,{\left (B a b - A b^{2}\right )} x\right )} \sqrt{x}\right )}}{15 \, b^{3}}\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 = 1.14601, size = 180, normalized size = 0.95 \begin{align*} -\frac{2 \,{\left (B a^{3} \mathrm{sgn}\left (b x + a\right ) - A a^{2} b \mathrm{sgn}\left (b x + a\right )\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} + \frac{2 \,{\left (3 \, B b^{4} x^{\frac{5}{2}} \mathrm{sgn}\left (b x + a\right ) - 5 \, B a b^{3} x^{\frac{3}{2}} \mathrm{sgn}\left (b x + a\right ) + 5 \, A b^{4} x^{\frac{3}{2}} \mathrm{sgn}\left (b x + a\right ) + 15 \, B a^{2} b^{2} \sqrt{x} \mathrm{sgn}\left (b x + a\right ) - 15 \, A a b^{3} \sqrt{x} \mathrm{sgn}\left (b x + a\right )\right )}}{15 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]