Optimal. Leaf size=121 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2} c^{3/2}}+\frac{2 b}{a \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}+\frac{2 d \sqrt{a+b x} (a d+b c)}{a c \sqrt{c+d x} (b c-a d)^2} \]
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Rubi [A] time = 0.0691104, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {104, 152, 12, 93, 208} \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2} c^{3/2}}+\frac{2 b}{a \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}+\frac{2 d \sqrt{a+b x} (a d+b c)}{a c \sqrt{c+d x} (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 104
Rule 152
Rule 12
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x (a+b x)^{3/2} (c+d x)^{3/2}} \, dx &=\frac{2 b}{a (b c-a d) \sqrt{a+b x} \sqrt{c+d x}}+\frac{2 \int \frac{\frac{1}{2} (b c-a d)+b d x}{x \sqrt{a+b x} (c+d x)^{3/2}} \, dx}{a (b c-a d)}\\ &=\frac{2 b}{a (b c-a d) \sqrt{a+b x} \sqrt{c+d x}}+\frac{2 d (b c+a d) \sqrt{a+b x}}{a c (b c-a d)^2 \sqrt{c+d x}}-\frac{4 \int -\frac{(b c-a d)^2}{4 x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{a c (b c-a d)^2}\\ &=\frac{2 b}{a (b c-a d) \sqrt{a+b x} \sqrt{c+d x}}+\frac{2 d (b c+a d) \sqrt{a+b x}}{a c (b c-a d)^2 \sqrt{c+d x}}+\frac{\int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{a c}\\ &=\frac{2 b}{a (b c-a d) \sqrt{a+b x} \sqrt{c+d x}}+\frac{2 d (b c+a d) \sqrt{a+b x}}{a c (b c-a d)^2 \sqrt{c+d x}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{a c}\\ &=\frac{2 b}{a (b c-a d) \sqrt{a+b x} \sqrt{c+d x}}+\frac{2 d (b c+a d) \sqrt{a+b x}}{a c (b c-a d)^2 \sqrt{c+d x}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2} c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.127931, size = 104, normalized size = 0.86 \[ \frac{2 \left (a^2 d^2+a b d^2 x+b^2 c (c+d x)\right )}{a c \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2} c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.028, size = 638, normalized size = 5.3 \begin{align*}{\frac{1}{ac \left ( ad-bc \right ) ^{2}} \left ( -\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ){x}^{2}{a}^{2}b{d}^{3}+2\,\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{d}^{2}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ){x}^{2}{b}^{3}{c}^{2}d-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) x{a}^{3}{d}^{3}+\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) x{a}^{2}bc{d}^{2}+\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) xa{b}^{2}{c}^{2}d-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) x{b}^{3}{c}^{3}-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ){a}^{3}c{d}^{2}+2\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){a}^{2}b{c}^{2}d-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) a{b}^{2}{c}^{3}+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{ac}xab{d}^{2}+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{ac}x{b}^{2}cd+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{ac}{a}^{2}{d}^{2}+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }\sqrt{ac}{b}^{2}{c}^{2} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{dx+c}}}} \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 [B] time = 4.86171, size = 1476, normalized size = 12.2 \begin{align*} \left [\frac{{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x\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 (a b^{2} c^{3} + a^{3} c d^{2} +{\left (a b^{2} c^{2} d + a^{2} b c d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{2 \,{\left (a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2} +{\left (a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c^{2} d^{3}\right )} x^{2} +{\left (a^{2} b^{3} c^{5} - a^{3} b^{2} c^{4} d - a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}\right )} x\right )}}, \frac{{\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x\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 (a b^{2} c^{3} + a^{3} c d^{2} +{\left (a b^{2} c^{2} d + a^{2} b c d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + a^{5} c^{3} d^{2} +{\left (a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} + a^{4} b c^{2} d^{3}\right )} x^{2} +{\left (a^{2} b^{3} c^{5} - a^{3} b^{2} c^{4} d - a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3}\right )} x}\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{1}{x \left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}\, 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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