Optimal. Leaf size=178 \[ \frac{(c+d x)^{3/2} (5 b c-a d)}{4 a^2 c x \sqrt{a+b x}}+\frac{3 \sqrt{c+d x} (b c-a d) (5 b c-a d)}{4 a^3 c \sqrt{a+b x}}-\frac{3 (b c-a d) (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{7/2} \sqrt{c}}-\frac{(c+d x)^{5/2}}{2 a c x^2 \sqrt{a+b x}} \]
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Rubi [A] time = 0.0763655, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {96, 94, 93, 208} \[ \frac{(c+d x)^{3/2} (5 b c-a d)}{4 a^2 c x \sqrt{a+b x}}+\frac{3 \sqrt{c+d x} (b c-a d) (5 b c-a d)}{4 a^3 c \sqrt{a+b x}}-\frac{3 (b c-a d) (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{7/2} \sqrt{c}}-\frac{(c+d x)^{5/2}}{2 a c x^2 \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Rule 96
Rule 94
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{(c+d x)^{3/2}}{x^3 (a+b x)^{3/2}} \, dx &=-\frac{(c+d x)^{5/2}}{2 a c x^2 \sqrt{a+b x}}-\frac{\left (\frac{5 b c}{2}-\frac{a d}{2}\right ) \int \frac{(c+d x)^{3/2}}{x^2 (a+b x)^{3/2}} \, dx}{2 a c}\\ &=\frac{(5 b c-a d) (c+d x)^{3/2}}{4 a^2 c x \sqrt{a+b x}}-\frac{(c+d x)^{5/2}}{2 a c x^2 \sqrt{a+b x}}+\frac{(3 (b c-a d) (5 b c-a d)) \int \frac{\sqrt{c+d x}}{x (a+b x)^{3/2}} \, dx}{8 a^2 c}\\ &=\frac{3 (b c-a d) (5 b c-a d) \sqrt{c+d x}}{4 a^3 c \sqrt{a+b x}}+\frac{(5 b c-a d) (c+d x)^{3/2}}{4 a^2 c x \sqrt{a+b x}}-\frac{(c+d x)^{5/2}}{2 a c x^2 \sqrt{a+b x}}+\frac{(3 (b c-a d) (5 b c-a d)) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{8 a^3}\\ &=\frac{3 (b c-a d) (5 b c-a d) \sqrt{c+d x}}{4 a^3 c \sqrt{a+b x}}+\frac{(5 b c-a d) (c+d x)^{3/2}}{4 a^2 c x \sqrt{a+b x}}-\frac{(c+d x)^{5/2}}{2 a c x^2 \sqrt{a+b x}}+\frac{(3 (b c-a d) (5 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 )}{4 a^3}\\ &=\frac{3 (b c-a d) (5 b c-a d) \sqrt{c+d x}}{4 a^3 c \sqrt{a+b x}}+\frac{(5 b c-a d) (c+d x)^{3/2}}{4 a^2 c x \sqrt{a+b x}}-\frac{(c+d x)^{5/2}}{2 a c x^2 \sqrt{a+b x}}-\frac{3 (b c-a d) (5 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{7/2} \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0968543, size = 130, normalized size = 0.73 \[ \frac{\sqrt{c+d x} \left (a^2 (-(2 c+5 d x))+a b x (5 c-13 d x)+15 b^2 c x^2\right )}{4 a^3 x^2 \sqrt{a+b x}}-\frac{3 \left (a^2 d^2-6 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{7/2} \sqrt{c}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.022, size = 464, normalized size = 2.6 \begin{align*} -{\frac{1}{8\,{a}^{3}{x}^{2}}\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}^{3}{a}^{2}b{d}^{2}-18\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}a{b}^{2}cd+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{3}{b}^{3}{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}^{2}{a}^{3}{d}^{2}-18\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{2}bcd+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}a{b}^{2}{c}^{2}+26\,{x}^{2}abd\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-30\,{x}^{2}{b}^{2}c\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+10\,x{a}^{2}d\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }-10\,xabc\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+4\,{a}^{2}c\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) } \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 = 6.58829, size = 1029, normalized size = 5.78 \begin{align*} \left [\frac{3 \,{\left ({\left (5 \, b^{3} c^{2} - 6 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} +{\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}\right )} \sqrt{a c} \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 c +{\left (b c + a d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \,{\left (2 \, a^{3} c^{2} -{\left (15 \, a b^{2} c^{2} - 13 \, a^{2} b c d\right )} x^{2} - 5 \,{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{16 \,{\left (a^{4} b c x^{3} + a^{5} c x^{2}\right )}}, \frac{3 \,{\left ({\left (5 \, b^{3} c^{2} - 6 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} +{\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}\right )} \sqrt{-a c} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{2 \,{\left (a b c d x^{2} + a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \,{\left (2 \, a^{3} c^{2} -{\left (15 \, a b^{2} c^{2} - 13 \, a^{2} b c d\right )} x^{2} - 5 \,{\left (a^{2} b c^{2} - a^{3} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{8 \,{\left (a^{4} b c x^{3} + a^{5} c x^{2}\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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