Optimal. Leaf size=137 \[ -\frac{\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{5/2} c^{5/2}}+\frac{3 \sqrt{a+b x} \sqrt{c+d x} (a d+b c)}{4 a^2 c^2 x}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{2 a c x^2} \]
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Rubi [A] time = 0.0679771, antiderivative size = 137, 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 = {103, 151, 12, 93, 208} \[ -\frac{\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{5/2} c^{5/2}}+\frac{3 \sqrt{a+b x} \sqrt{c+d x} (a d+b c)}{4 a^2 c^2 x}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{2 a c x^2} \]
Antiderivative was successfully verified.
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Rule 103
Rule 151
Rule 12
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^3 \sqrt{a+b x} \sqrt{c+d x}} \, dx &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{2 a c x^2}-\frac{\int \frac{\frac{3}{2} (b c+a d)+b d x}{x^2 \sqrt{a+b x} \sqrt{c+d x}} \, dx}{2 a c}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{2 a c x^2}+\frac{3 (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{4 a^2 c^2 x}+\frac{\int \frac{3 b^2 c^2+2 a b c d+3 a^2 d^2}{4 x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{2 a^2 c^2}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{2 a c x^2}+\frac{3 (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{4 a^2 c^2 x}+\frac{\left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{8 a^2 c^2}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{2 a c x^2}+\frac{3 (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{4 a^2 c^2 x}+\frac{\left (3 b^2 c^2+2 a b c d+3 a^2 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 c^2}\\ &=-\frac{\sqrt{a+b x} \sqrt{c+d x}}{2 a c x^2}+\frac{3 (b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{4 a^2 c^2 x}-\frac{\left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{5/2} c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0856526, size = 114, normalized size = 0.83 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} (-2 a c+3 a d x+3 b c x)}{4 a^2 c^2 x^2}-\frac{\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{5/2} c^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.026, size = 258, normalized size = 1.9 \begin{align*} -{\frac{1}{8\,{a}^{2}{c}^{2}{x}^{2}} \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}+2\,\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}-6\,\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 ) \sqrt{dx+c}\sqrt{bx+a}{\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 = 3.43028, size = 759, normalized size = 5.54 \begin{align*} \left [\frac{{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, 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} - 3 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{16 \, a^{3} c^{3} x^{2}}, \frac{{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, 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} - 3 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{8 \, a^{3} c^{3} 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{1}{x^{3} \sqrt{a + b x} \sqrt{c + d 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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