Optimal. Leaf size=229 \[ -\frac{5 (b c-a d)^3 (a d+7 b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{9/2} c^{3/2}}+\frac{\sqrt{a+b x} (c+d x)^{5/2} (a d+7 b c)}{24 a^2 c x^3}-\frac{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d) (a d+7 b c)}{96 a^3 c x^2}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2 (a d+7 b c)}{64 a^4 c x}-\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 a c x^4} \]
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Rubi [A] time = 0.118983, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {96, 94, 93, 208} \[ -\frac{5 (b c-a d)^3 (a d+7 b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{9/2} c^{3/2}}+\frac{\sqrt{a+b x} (c+d x)^{5/2} (a d+7 b c)}{24 a^2 c x^3}-\frac{5 \sqrt{a+b x} (c+d x)^{3/2} (b c-a d) (a d+7 b c)}{96 a^3 c x^2}+\frac{5 \sqrt{a+b x} \sqrt{c+d x} (b c-a d)^2 (a d+7 b c)}{64 a^4 c x}-\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 a c x^4} \]
Antiderivative was successfully verified.
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Rule 96
Rule 94
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{(c+d x)^{5/2}}{x^5 \sqrt{a+b x}} \, dx &=-\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 a c x^4}-\frac{\left (\frac{7 b c}{2}+\frac{a d}{2}\right ) \int \frac{(c+d x)^{5/2}}{x^4 \sqrt{a+b x}} \, dx}{4 a c}\\ &=\frac{(7 b c+a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 a^2 c x^3}-\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 a c x^4}+\frac{(5 (b c-a d) (7 b c+a d)) \int \frac{(c+d x)^{3/2}}{x^3 \sqrt{a+b x}} \, dx}{48 a^2 c}\\ &=-\frac{5 (b c-a d) (7 b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{96 a^3 c x^2}+\frac{(7 b c+a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 a^2 c x^3}-\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 a c x^4}-\frac{\left (5 (b c-a d)^2 (7 b c+a d)\right ) \int \frac{\sqrt{c+d x}}{x^2 \sqrt{a+b x}} \, dx}{64 a^3 c}\\ &=\frac{5 (b c-a d)^2 (7 b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{64 a^4 c x}-\frac{5 (b c-a d) (7 b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{96 a^3 c x^2}+\frac{(7 b c+a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 a^2 c x^3}-\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 a c x^4}+\frac{\left (5 (b c-a d)^3 (7 b c+a d)\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{128 a^4 c}\\ &=\frac{5 (b c-a d)^2 (7 b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{64 a^4 c x}-\frac{5 (b c-a d) (7 b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{96 a^3 c x^2}+\frac{(7 b c+a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 a^2 c x^3}-\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 a c x^4}+\frac{\left (5 (b c-a d)^3 (7 b c+a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+b x}}{\sqrt{c+d x}}\right )}{64 a^4 c}\\ &=\frac{5 (b c-a d)^2 (7 b c+a d) \sqrt{a+b x} \sqrt{c+d x}}{64 a^4 c x}-\frac{5 (b c-a d) (7 b c+a d) \sqrt{a+b x} (c+d x)^{3/2}}{96 a^3 c x^2}+\frac{(7 b c+a d) \sqrt{a+b x} (c+d x)^{5/2}}{24 a^2 c x^3}-\frac{\sqrt{a+b x} (c+d x)^{7/2}}{4 a c x^4}-\frac{5 (b c-a d)^3 (7 b c+a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{9/2} c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.262957, size = 183, normalized size = 0.8 \[ -\frac{\frac{(a d+7 b c) \left (\frac{5 x (b c-a d) \left (3 x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} (2 a c+5 a d x-3 b c x)\right )}{a^{5/2} \sqrt{c}}-8 \sqrt{a+b x} (c+d x)^{5/2}\right )}{48 a x^3}+\frac{\sqrt{a+b x} (c+d x)^{7/2}}{x^4}}{4 a c} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 593, normalized size = 2.6 \begin{align*}{\frac{1}{384\,{a}^{4}c{x}^{4}}\sqrt{bx+a}\sqrt{dx+c} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{a}^{4}{d}^{4}+60\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{a}^{3}bc{d}^{3}-270\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}+300\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}a{b}^{3}{c}^{3}d-105\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{b}^{4}{c}^{4}-30\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{3}{a}^{3}{d}^{3}+382\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{3}{a}^{2}bc{d}^{2}-530\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{3}a{b}^{2}{c}^{2}d+210\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{3}{b}^{3}{c}^{3}-236\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{2}{a}^{3}c{d}^{2}+344\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{2}{a}^{2}b{c}^{2}d-140\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{2}a{b}^{2}{c}^{3}-272\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }x{a}^{3}{c}^{2}d+112\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }x{a}^{2}b{c}^{3}-96\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{a}^{3}{c}^{3} \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 = 21.0402, size = 1274, normalized size = 5.56 \begin{align*} \left [-\frac{15 \,{\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} \sqrt{a c} x^{4} \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 (48 \, a^{4} c^{4} -{\left (105 \, a b^{3} c^{4} - 265 \, a^{2} b^{2} c^{3} d + 191 \, a^{3} b c^{2} d^{2} - 15 \, a^{4} c d^{3}\right )} x^{3} + 2 \,{\left (35 \, a^{2} b^{2} c^{4} - 86 \, a^{3} b c^{3} d + 59 \, a^{4} c^{2} d^{2}\right )} x^{2} - 8 \,{\left (7 \, a^{3} b c^{4} - 17 \, a^{4} c^{3} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{768 \, a^{5} c^{2} x^{4}}, \frac{15 \,{\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} \sqrt{-a c} x^{4} \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 (48 \, a^{4} c^{4} -{\left (105 \, a b^{3} c^{4} - 265 \, a^{2} b^{2} c^{3} d + 191 \, a^{3} b c^{2} d^{2} - 15 \, a^{4} c d^{3}\right )} x^{3} + 2 \,{\left (35 \, a^{2} b^{2} c^{4} - 86 \, a^{3} b c^{3} d + 59 \, a^{4} c^{2} d^{2}\right )} x^{2} - 8 \,{\left (7 \, a^{3} b c^{4} - 17 \, a^{4} c^{3} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{384 \, a^{5} c^{2} x^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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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