Optimal. Leaf size=122 \[ \frac{(b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{3/2} c^{3/2}}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 c x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)}{4 a c x} \]
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Rubi [A] time = 0.0478093, antiderivative size = 122, 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} \[ \frac{(b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{3/2} c^{3/2}}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 c x^2}-\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)}{4 a c x} \]
Antiderivative was successfully verified.
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Rule 94
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x} \sqrt{c+d x}}{x^3} \, dx &=-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 c x^2}+\frac{(b c-a d) \int \frac{\sqrt{c+d x}}{x^2 \sqrt{a+b x}} \, dx}{4 c}\\ &=-\frac{(b c-a d) \sqrt{a+b x} \sqrt{c+d x}}{4 a c x}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 c x^2}-\frac{(b c-a d)^2 \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{8 a c}\\ &=-\frac{(b c-a d) \sqrt{a+b x} \sqrt{c+d x}}{4 a c x}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 c x^2}-\frac{(b c-a d)^2 \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{4 a c}\\ &=-\frac{(b c-a d) \sqrt{a+b x} \sqrt{c+d x}}{4 a c x}-\frac{\sqrt{a+b x} (c+d x)^{3/2}}{2 c x^2}+\frac{(b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{3/2} c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.066321, size = 99, normalized size = 0.81 \[ \frac{(b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{3/2} c^{3/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (2 a c+a d x+b c x)}{4 a c x^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.015, size = 305, normalized size = 2.5 \begin{align*}{\frac{1}{8\,ac{x}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( \ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac \right ) } \right ){x}^{2}{a}^{2}{d}^{2}-2\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{2}abcd+\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac \right ) } \right ){x}^{2}{b}^{2}{c}^{2}-2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{ac}xad-2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{ac}xbc-4\,\sqrt{d{x}^{2}b+adx+bcx+ac}ac\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}} \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 = 4.28637, size = 744, normalized size = 6.1 \begin{align*} \left [\frac{{\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 (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{16 \, a^{2} c^{2} x^{2}}, -\frac{{\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 (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{8 \, a^{2} c^{2} 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{\sqrt{a + b x} \sqrt{c + d x}}{x^{3}}\, 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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