Optimal. Leaf size=100 \[ -\frac{\left (a+\frac{b}{x}\right )^{3/2} (2 a d+3 b c)}{3 a}-\sqrt{a+\frac{b}{x}} (2 a d+3 b c)+\sqrt{a} (2 a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )+\frac{c x \left (a+\frac{b}{x}\right )^{5/2}}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0627703, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {375, 78, 50, 63, 208} \[ -\frac{\left (a+\frac{b}{x}\right )^{3/2} (2 a d+3 b c)}{3 a}-\sqrt{a+\frac{b}{x}} (2 a d+3 b c)+\sqrt{a} (2 a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )+\frac{c x \left (a+\frac{b}{x}\right )^{5/2}}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 375
Rule 78
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right ) \, dx &=-\operatorname{Subst}\left (\int \frac{(a+b x)^{3/2} (c+d x)}{x^2} \, dx,x,\frac{1}{x}\right )\\ &=\frac{c \left (a+\frac{b}{x}\right )^{5/2} x}{a}-\frac{\left (\frac{3 b c}{2}+a d\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{(3 b c+2 a d) \left (a+\frac{b}{x}\right )^{3/2}}{3 a}+\frac{c \left (a+\frac{b}{x}\right )^{5/2} x}{a}-\frac{1}{2} (3 b c+2 a d) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\frac{1}{x}\right )\\ &=-(3 b c+2 a d) \sqrt{a+\frac{b}{x}}-\frac{(3 b c+2 a d) \left (a+\frac{b}{x}\right )^{3/2}}{3 a}+\frac{c \left (a+\frac{b}{x}\right )^{5/2} x}{a}-\frac{1}{2} (a (3 b c+2 a d)) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )\\ &=-(3 b c+2 a d) \sqrt{a+\frac{b}{x}}-\frac{(3 b c+2 a d) \left (a+\frac{b}{x}\right )^{3/2}}{3 a}+\frac{c \left (a+\frac{b}{x}\right )^{5/2} x}{a}-\frac{(a (3 b c+2 a d)) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )}{b}\\ &=-(3 b c+2 a d) \sqrt{a+\frac{b}{x}}-\frac{(3 b c+2 a d) \left (a+\frac{b}{x}\right )^{3/2}}{3 a}+\frac{c \left (a+\frac{b}{x}\right )^{5/2} x}{a}+\sqrt{a} (3 b c+2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0693263, size = 73, normalized size = 0.73 \[ \frac{\sqrt{a+\frac{b}{x}} (a x (3 c x-8 d)-2 b (3 c x+d))}{3 x}+\sqrt{a} (2 a d+3 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.009, size = 205, normalized size = 2.1 \begin{align*} -{\frac{1}{6\,b{x}^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( -12\,\sqrt{a{x}^{2}+bx}{a}^{5/2}{x}^{3}d-18\,\sqrt{a{x}^{2}+bx}{a}^{3/2}{x}^{3}bc-6\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{a}^{2}bd-9\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}a{b}^{2}c+12\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{3/2}xd+12\, \left ( a{x}^{2}+bx \right ) ^{3/2}\sqrt{a}xbc+4\,d \left ( a{x}^{2}+bx \right ) ^{3/2}\sqrt{a}b \right ){\frac{1}{\sqrt{ \left ( ax+b \right ) x}}}{\frac{1}{\sqrt{a}}}} \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 = 1.2941, size = 392, normalized size = 3.92 \begin{align*} \left [\frac{3 \,{\left (3 \, b c + 2 \, a d\right )} \sqrt{a} x \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) + 2 \,{\left (3 \, a c x^{2} - 2 \, b d - 2 \,{\left (3 \, b c + 4 \, a d\right )} x\right )} \sqrt{\frac{a x + b}{x}}}{6 \, x}, -\frac{3 \,{\left (3 \, b c + 2 \, a d\right )} \sqrt{-a} x \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right ) -{\left (3 \, a c x^{2} - 2 \, b d - 2 \,{\left (3 \, b c + 4 \, a d\right )} x\right )} \sqrt{\frac{a x + b}{x}}}{3 \, x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 27.8709, size = 163, normalized size = 1.63 \begin{align*} \sqrt{a} b c \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )} - \frac{2 a^{2} d \operatorname{atan}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{- a}} \right )}}{\sqrt{- a}} + a \sqrt{b} c \sqrt{x} \sqrt{\frac{a x}{b} + 1} - \frac{2 a b c \operatorname{atan}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{- a}} \right )}}{\sqrt{- a}} - 2 a d \sqrt{a + \frac{b}{x}} - 2 b c \sqrt{a + \frac{b}{x}} + b d \left (\begin{cases} - \frac{\sqrt{a}}{x} & \text{for}\: b = 0 \\- \frac{2 \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [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]