Optimal. Leaf size=73 \[ -\frac{c (b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}}+\frac{c^2 x \sqrt{a+\frac{b}{x}}}{a}-\frac{2 d^2 \sqrt{a+\frac{b}{x}}}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0540239, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {375, 89, 80, 63, 208} \[ -\frac{c (b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}}+\frac{c^2 x \sqrt{a+\frac{b}{x}}}{a}-\frac{2 d^2 \sqrt{a+\frac{b}{x}}}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 375
Rule 89
Rule 80
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (c+\frac{d}{x}\right )^2}{\sqrt{a+\frac{b}{x}}} \, dx &=-\operatorname{Subst}\left (\int \frac{(c+d x)^2}{x^2 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{c^2 \sqrt{a+\frac{b}{x}} x}{a}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{1}{2} c (b c-4 a d)+a d^2 x}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{2 d^2 \sqrt{a+\frac{b}{x}}}{b}+\frac{c^2 \sqrt{a+\frac{b}{x}} x}{a}+\frac{(c (b c-4 a d)) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=-\frac{2 d^2 \sqrt{a+\frac{b}{x}}}{b}+\frac{c^2 \sqrt{a+\frac{b}{x}} x}{a}+\frac{(c (b c-4 a d)) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )}{a b}\\ &=-\frac{2 d^2 \sqrt{a+\frac{b}{x}}}{b}+\frac{c^2 \sqrt{a+\frac{b}{x}} x}{a}-\frac{c (b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0714159, size = 66, normalized size = 0.9 \[ \frac{c (4 a d-b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}}+\frac{\sqrt{a+\frac{b}{x}} \left (b c^2 x-2 a d^2\right )}{a b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.013, size = 348, normalized size = 4.8 \begin{align*} -{\frac{1}{2\,{b}^{2}x}\sqrt{{\frac{ax+b}{x}}} \left ( -2\,\sqrt{a{x}^{2}+bx}{a}^{5/2}{x}^{2}{d}^{2}-4\,\sqrt{a{x}^{2}+bx}{a}^{3/2}{x}^{2}bcd-\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ){x}^{2}{a}^{2}b{d}^{2}-2\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}a{b}^{2}cd-2\,{a}^{5/2}\sqrt{ \left ( ax+b \right ) x}{x}^{2}{d}^{2}+4\,{a}^{3/2}\sqrt{ \left ( ax+b \right ) x}{x}^{2}bcd-2\,\sqrt{a}\sqrt{ \left ( ax+b \right ) x}{x}^{2}{b}^{2}{c}^{2}+\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ){x}^{2}{a}^{2}b{d}^{2}-2\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}a{b}^{2}cd+\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ){x}^{2}{b}^{3}{c}^{2}+4\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{3/2}{d}^{2} \right ){\frac{1}{\sqrt{ \left ( ax+b \right ) x}}}{a}^{-{\frac{3}{2}}}} \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 = 1.36408, size = 359, normalized size = 4.92 \begin{align*} \left [-\frac{{\left (b^{2} c^{2} - 4 \, a b c d\right )} \sqrt{a} \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) - 2 \,{\left (a b c^{2} x - 2 \, a^{2} d^{2}\right )} \sqrt{\frac{a x + b}{x}}}{2 \, a^{2} b}, \frac{{\left (b^{2} c^{2} - 4 \, a b c d\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right ) +{\left (a b c^{2} x - 2 \, a^{2} d^{2}\right )} \sqrt{\frac{a x + b}{x}}}{a^{2} b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 25.404, size = 114, normalized size = 1.56 \begin{align*} d^{2} \left (\begin{cases} - \frac{1}{\sqrt{a} x} & \text{for}\: b = 0 \\- \frac{2 \sqrt{a + \frac{b}{x}}}{b} & \text{otherwise} \end{cases}\right ) + \frac{\sqrt{b} c^{2} \sqrt{x} \sqrt{\frac{a x}{b} + 1}}{a} - \frac{4 c d \operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{a}} \sqrt{a + \frac{b}{x}}} \right )}}{a \sqrt{- \frac{1}{a}}} - \frac{b c^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.20205, size = 131, normalized size = 1.79 \begin{align*} -b{\left (\frac{c^{2} \sqrt{\frac{a x + b}{x}}}{{\left (a - \frac{a x + b}{x}\right )} a} + \frac{2 \, d^{2} \sqrt{\frac{a x + b}{x}}}{b^{2}} - \frac{{\left (b c^{2} - 4 \, a c d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a b}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]