Optimal. Leaf size=76 \[ \frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2} \sqrt{c}}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{a x} \]
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Rubi [A] time = 0.024685, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {94, 93, 208} \[ \frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2} \sqrt{c}}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{a x} \]
Antiderivative was successfully verified.
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Rule 94
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{c+d x}}{x^2 \sqrt{a+b x}} \, dx &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{a x}-\frac{(b c-a d) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{2 a}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{a x}-\frac{(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}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{a x}+\frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2} \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0424579, size = 77, normalized size = 1.01 \[ -\frac{(a d-b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2} \sqrt{c}}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{a x} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 147, normalized size = 1.9 \begin{align*} -{\frac{1}{2\,ax}\sqrt{bx+a}\sqrt{dx+c} \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 ) xad-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) xbc+2\,\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 ) }}}} \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 = 2.57043, size = 587, normalized size = 7.72 \begin{align*} \left [-\frac{\sqrt{a c}{\left (b c - a d\right )} x \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 \, \sqrt{b x + a} \sqrt{d x + c} a c}{4 \, a^{2} c x}, -\frac{\sqrt{-a c}{\left (b c - a d\right )} x \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 \, \sqrt{b x + a} \sqrt{d x + c} a c}{2 \, a^{2} c 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{\sqrt{c + d x}}{x^{2} \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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