Optimal. Leaf size=108 \[ -\frac{3 \sqrt{c+d x} (b c-a d)}{a^2 \sqrt{a+b x}}+\frac{3 \sqrt{c} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{5/2}}-\frac{(c+d x)^{3/2}}{a x \sqrt{a+b x}} \]
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Rubi [A] time = 0.0402751, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {94, 93, 208} \[ -\frac{3 \sqrt{c+d x} (b c-a d)}{a^2 \sqrt{a+b x}}+\frac{3 \sqrt{c} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{5/2}}-\frac{(c+d x)^{3/2}}{a x \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Rule 94
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{(c+d x)^{3/2}}{x^2 (a+b x)^{3/2}} \, dx &=-\frac{(c+d x)^{3/2}}{a x \sqrt{a+b x}}-\frac{(3 (b c-a d)) \int \frac{\sqrt{c+d x}}{x (a+b x)^{3/2}} \, dx}{2 a}\\ &=-\frac{3 (b c-a d) \sqrt{c+d x}}{a^2 \sqrt{a+b x}}-\frac{(c+d x)^{3/2}}{a x \sqrt{a+b x}}-\frac{(3 c (b c-a d)) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{2 a^2}\\ &=-\frac{3 (b c-a d) \sqrt{c+d x}}{a^2 \sqrt{a+b x}}-\frac{(c+d x)^{3/2}}{a x \sqrt{a+b x}}-\frac{(3 c (b c-a d)) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{a^2}\\ &=-\frac{3 (b c-a d) \sqrt{c+d x}}{a^2 \sqrt{a+b x}}-\frac{(c+d x)^{3/2}}{a x \sqrt{a+b x}}+\frac{3 \sqrt{c} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0824342, size = 91, normalized size = 0.84 \[ \frac{\sqrt{c+d x} (-a c+2 a d x-3 b c x)}{a^2 x \sqrt{a+b x}}+\frac{3 \sqrt{c} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.021, size = 298, normalized size = 2.8 \begin{align*} -{\frac{1}{2\,{a}^{2}x}\sqrt{dx+c} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}abcd-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{b}^{2}{c}^{2}+3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) x{a}^{2}cd-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ) xab{c}^{2}-4\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{ac}xad+6\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{ac}xbc+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }ac\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bx+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A] time = 3.61694, size = 752, normalized size = 6.96 \begin{align*} \left [-\frac{3 \,{\left ({\left (b^{2} c - a b d\right )} x^{2} +{\left (a b c - a^{2} d\right )} x\right )} \sqrt{\frac{c}{a}} \log \left (\frac{8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \,{\left (2 \, a^{2} c +{\left (a b c + a^{2} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{c}{a}} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \,{\left (a c +{\left (3 \, b c - 2 \, a d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{4 \,{\left (a^{2} b x^{2} + a^{3} x\right )}}, -\frac{3 \,{\left ({\left (b^{2} c - a b d\right )} x^{2} +{\left (a b c - a^{2} d\right )} x\right )} \sqrt{-\frac{c}{a}} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{-\frac{c}{a}}}{2 \,{\left (b c d x^{2} + a c^{2} +{\left (b c^{2} + a c d\right )} x\right )}}\right ) + 2 \,{\left (a c +{\left (3 \, b c - 2 \, a d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{2 \,{\left (a^{2} b x^{2} + a^{3} x\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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