Optimal. Leaf size=119 \[ \frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}{4 a^2 x}-\frac{3 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{5/2} \sqrt{c}}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 a x^2} \]
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Rubi [A] time = 0.0475916, antiderivative size = 119, 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} \[ \frac{3 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}{4 a^2 x}-\frac{3 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{5/2} \sqrt{c}}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 a x^2} \]
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^3 \sqrt{a+b x}} \, dx &=-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 a x^2}-\frac{(3 (b c-a d)) \int \frac{\sqrt{c+d x}}{x^2 \sqrt{a+b x}} \, dx}{4 a}\\ &=\frac{3 (b c-a d) \sqrt{a+b x} \sqrt{c+d x}}{4 a^2 x}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 a x^2}+\frac{\left (3 (b c-a d)^2\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{8 a^2}\\ &=\frac{3 (b c-a d) \sqrt{a+b x} \sqrt{c+d x}}{4 a^2 x}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 a x^2}+\frac{\left (3 (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{4 a^2}\\ &=\frac{3 (b c-a d) \sqrt{a+b x} \sqrt{c+d x}}{4 a^2 x}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 a x^2}-\frac{3 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{5/2} \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.077777, size = 98, normalized size = 0.82 \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} (2 a c+5 a d x-3 b c x)}{4 a^2 x^2}-\frac{3 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{5/2} \sqrt{c}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.019, size = 255, normalized size = 2.1 \begin{align*} -{\frac{1}{8\,{a}^{2}{x}^{2}}\sqrt{bx+a}\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}{a}^{2}{d}^{2}-6\,\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}+10\,\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+4\,\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 ) }}}} \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 = 3.33864, size = 753, normalized size = 6.33 \begin{align*} \left [\frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{a c} x^{2} \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^{2} c^{2} -{\left (3 \, a b c^{2} - 5 \, a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{16 \, a^{3} c x^{2}}, \frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-a c} x^{2} \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^{2} c^{2} -{\left (3 \, a b c^{2} - 5 \, a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{8 \, a^{3} c x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c + d x\right )^{\frac{3}{2}}}{x^{3} \sqrt{a + b x}}\, dx \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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