Optimal. Leaf size=163 \[ -\frac{(b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{5/2} c^{3/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}{8 a^2 c x}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (b c-a d)}{12 a c x^2}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{3 c x^3} \]
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Rubi [A] time = 0.0733824, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {94, 93, 208} \[ -\frac{(b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{5/2} c^{3/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2}{8 a^2 c x}-\frac{\sqrt{a+b x} (c+d x)^{3/2} (b c-a d)}{12 a c x^2}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{3 c x^3} \]
Antiderivative was successfully verified.
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Rule 94
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x} (c+d x)^{3/2}}{x^4} \, dx &=-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{3 c x^3}+\frac{(b c-a d) \int \frac{(c+d x)^{3/2}}{x^3 \sqrt{a+b x}} \, dx}{6 c}\\ &=-\frac{(b c-a d) \sqrt{a+b x} (c+d x)^{3/2}}{12 a c x^2}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{3 c x^3}-\frac{(b c-a d)^2 \int \frac{\sqrt{c+d x}}{x^2 \sqrt{a+b x}} \, dx}{8 a c}\\ &=\frac{(b c-a d)^2 \sqrt{a+b x} \sqrt{c+d x}}{8 a^2 c x}-\frac{(b c-a d) \sqrt{a+b x} (c+d x)^{3/2}}{12 a c x^2}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{3 c x^3}+\frac{(b c-a d)^3 \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{16 a^2 c}\\ &=\frac{(b c-a d)^2 \sqrt{a+b x} \sqrt{c+d x}}{8 a^2 c x}-\frac{(b c-a d) \sqrt{a+b x} (c+d x)^{3/2}}{12 a c x^2}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{3 c x^3}+\frac{(b c-a d)^3 \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{8 a^2 c}\\ &=\frac{(b c-a d)^2 \sqrt{a+b x} \sqrt{c+d x}}{8 a^2 c x}-\frac{(b c-a d) \sqrt{a+b x} (c+d x)^{3/2}}{12 a c x^2}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{3 c x^3}-\frac{(b c-a d)^3 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{5/2} c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.216033, size = 141, normalized size = 0.87 \[ -\frac{\frac{x (b c-a d) \left (3 x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} (2 a c+5 a d x-3 b c x)\right )}{a^{5/2} \sqrt{c}}+8 \sqrt{a+b x} (c+d x)^{5/2}}{24 c x^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.015, size = 485, normalized size = 3. \begin{align*}{\frac{1}{48\,{a}^{2}c{x}^{3}}\sqrt{bx+a}\sqrt{dx+c} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}{a}^{3}{d}^{3}-9\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}{a}^{2}bc{d}^{2}+9\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}a{b}^{2}{c}^{2}d-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}{b}^{3}{c}^{3}-6\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{ac}{x}^{2}{a}^{2}{d}^{2}-16\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}{x}^{2}abcd+6\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{ac}{x}^{2}{b}^{2}{c}^{2}-28\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}x{a}^{2}cd-4\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}xab{c}^{2}-16\,\sqrt{d{x}^{2}b+adx+bcx+ac}{a}^{2}{c}^{2}\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 = 6.93558, size = 961, normalized size = 5.9 \begin{align*} \left [-\frac{3 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{a c} x^{3} \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 (8 \, a^{3} c^{3} -{\left (3 \, a b^{2} c^{3} - 8 \, a^{2} b c^{2} d - 3 \, a^{3} c d^{2}\right )} x^{2} + 2 \,{\left (a^{2} b c^{3} + 7 \, a^{3} c^{2} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{96 \, a^{3} c^{2} x^{3}}, \frac{3 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{-a c} x^{3} \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 (8 \, a^{3} c^{3} -{\left (3 \, a b^{2} c^{3} - 8 \, a^{2} b c^{2} d - 3 \, a^{3} c d^{2}\right )} x^{2} + 2 \,{\left (a^{2} b c^{3} + 7 \, a^{3} c^{2} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{48 \, a^{3} c^{2} x^{3}}\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} \left (c + d x\right )^{\frac{3}{2}}}{x^{4}}\, 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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