Optimal. Leaf size=100 \[ \frac{(3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{3/2} b^{5/2}}-\frac{\sqrt{x} (3 a B+A b)}{4 a b^2 (a+b x)}+\frac{x^{3/2} (A b-a B)}{2 a b (a+b x)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0381686, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {78, 47, 63, 205} \[ \frac{(3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{3/2} b^{5/2}}-\frac{\sqrt{x} (3 a B+A b)}{4 a b^2 (a+b x)}+\frac{x^{3/2} (A b-a B)}{2 a b (a+b x)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 78
Rule 47
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{\sqrt{x} (A+B x)}{(a+b x)^3} \, dx &=\frac{(A b-a B) x^{3/2}}{2 a b (a+b x)^2}+\frac{(A b+3 a B) \int \frac{\sqrt{x}}{(a+b x)^2} \, dx}{4 a b}\\ &=\frac{(A b-a B) x^{3/2}}{2 a b (a+b x)^2}-\frac{(A b+3 a B) \sqrt{x}}{4 a b^2 (a+b x)}+\frac{(A b+3 a B) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{8 a b^2}\\ &=\frac{(A b-a B) x^{3/2}}{2 a b (a+b x)^2}-\frac{(A b+3 a B) \sqrt{x}}{4 a b^2 (a+b x)}+\frac{(A b+3 a B) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{4 a b^2}\\ &=\frac{(A b-a B) x^{3/2}}{2 a b (a+b x)^2}-\frac{(A b+3 a B) \sqrt{x}}{4 a b^2 (a+b x)}+\frac{(A b+3 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{3/2} b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0645202, size = 85, normalized size = 0.85 \[ \frac{\sqrt{x} \left (-3 a^2 B-a b (A+5 B x)+A b^2 x\right )}{4 a b^2 (a+b x)^2}+\frac{(3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{3/2} b^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.012, size = 94, normalized size = 0.9 \begin{align*} 2\,{\frac{1}{ \left ( bx+a \right ) ^{2}} \left ( 1/8\,{\frac{ \left ( Ab-5\,Ba \right ){x}^{3/2}}{ab}}-1/8\,{\frac{ \left ( Ab+3\,Ba \right ) \sqrt{x}}{{b}^{2}}} \right ) }+{\frac{A}{4\,ab}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,B}{4\,{b}^{2}}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\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 = 2.44493, size = 633, normalized size = 6.33 \begin{align*} \left [-\frac{{\left (3 \, B a^{3} + A a^{2} b +{\left (3 \, B a b^{2} + A b^{3}\right )} x^{2} + 2 \,{\left (3 \, B a^{2} b + A a b^{2}\right )} x\right )} \sqrt{-a b} \log \left (\frac{b x - a - 2 \, \sqrt{-a b} \sqrt{x}}{b x + a}\right ) + 2 \,{\left (3 \, B a^{3} b + A a^{2} b^{2} +{\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} x\right )} \sqrt{x}}{8 \,{\left (a^{2} b^{5} x^{2} + 2 \, a^{3} b^{4} x + a^{4} b^{3}\right )}}, -\frac{{\left (3 \, B a^{3} + A a^{2} b +{\left (3 \, B a b^{2} + A b^{3}\right )} x^{2} + 2 \,{\left (3 \, B a^{2} b + A a b^{2}\right )} x\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b}}{b \sqrt{x}}\right ) +{\left (3 \, B a^{3} b + A a^{2} b^{2} +{\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} x\right )} \sqrt{x}}{4 \,{\left (a^{2} b^{5} x^{2} + 2 \, a^{3} b^{4} x + a^{4} b^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18574, size = 111, normalized size = 1.11 \begin{align*} \frac{{\left (3 \, B a + A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a b^{2}} - \frac{5 \, B a b x^{\frac{3}{2}} - A b^{2} x^{\frac{3}{2}} + 3 \, B a^{2} \sqrt{x} + A a b \sqrt{x}}{4 \,{\left (b x + a\right )}^{2} a b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]