Optimal. Leaf size=266 \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} \left (35 a^2 d^2-46 a b c d+3 b^2 c^2\right )}{96 a c^3 x^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-145 a^2 b c d^2+105 a^3 d^3+15 a b^2 c^2 d+9 b^3 c^3\right )}{192 a^2 c^4 x}-\frac{\left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{5/2} c^{9/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (9 b c-7 a d)}{24 c^2 x^3}-\frac{a \sqrt{a+b x} \sqrt{c+d x}}{4 c x^4} \]
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Rubi [A] time = 0.213436, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {98, 151, 12, 93, 208} \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} \left (35 a^2 d^2-46 a b c d+3 b^2 c^2\right )}{96 a c^3 x^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-145 a^2 b c d^2+105 a^3 d^3+15 a b^2 c^2 d+9 b^3 c^3\right )}{192 a^2 c^4 x}-\frac{\left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{5/2} c^{9/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} (9 b c-7 a d)}{24 c^2 x^3}-\frac{a \sqrt{a+b x} \sqrt{c+d x}}{4 c x^4} \]
Antiderivative was successfully verified.
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Rule 98
Rule 151
Rule 12
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{(a+b x)^{3/2}}{x^5 \sqrt{c+d x}} \, dx &=-\frac{a \sqrt{a+b x} \sqrt{c+d x}}{4 c x^4}-\frac{\int \frac{-\frac{1}{2} a (9 b c-7 a d)-b (4 b c-3 a d) x}{x^4 \sqrt{a+b x} \sqrt{c+d x}} \, dx}{4 c}\\ &=-\frac{a \sqrt{a+b x} \sqrt{c+d x}}{4 c x^4}-\frac{(9 b c-7 a d) \sqrt{a+b x} \sqrt{c+d x}}{24 c^2 x^3}+\frac{\int \frac{\frac{1}{4} a \left (3 b^2 c^2-46 a b c d+35 a^2 d^2\right )-a b d (9 b c-7 a d) x}{x^3 \sqrt{a+b x} \sqrt{c+d x}} \, dx}{12 a c^2}\\ &=-\frac{a \sqrt{a+b x} \sqrt{c+d x}}{4 c x^4}-\frac{(9 b c-7 a d) \sqrt{a+b x} \sqrt{c+d x}}{24 c^2 x^3}-\frac{\left (3 b^2 c^2-46 a b c d+35 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{96 a c^3 x^2}-\frac{\int \frac{\frac{1}{8} a \left (9 b^3 c^3+15 a b^2 c^2 d-145 a^2 b c d^2+105 a^3 d^3\right )+\frac{1}{4} a b d \left (3 b^2 c^2-46 a b c d+35 a^2 d^2\right ) x}{x^2 \sqrt{a+b x} \sqrt{c+d x}} \, dx}{24 a^2 c^3}\\ &=-\frac{a \sqrt{a+b x} \sqrt{c+d x}}{4 c x^4}-\frac{(9 b c-7 a d) \sqrt{a+b x} \sqrt{c+d x}}{24 c^2 x^3}-\frac{\left (3 b^2 c^2-46 a b c d+35 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{96 a c^3 x^2}+\frac{\left (9 b^3 c^3+15 a b^2 c^2 d-145 a^2 b c d^2+105 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{192 a^2 c^4 x}+\frac{\int \frac{3 a (b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right )}{16 x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{24 a^3 c^4}\\ &=-\frac{a \sqrt{a+b x} \sqrt{c+d x}}{4 c x^4}-\frac{(9 b c-7 a d) \sqrt{a+b x} \sqrt{c+d x}}{24 c^2 x^3}-\frac{\left (3 b^2 c^2-46 a b c d+35 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{96 a c^3 x^2}+\frac{\left (9 b^3 c^3+15 a b^2 c^2 d-145 a^2 b c d^2+105 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{192 a^2 c^4 x}+\frac{\left ((b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right )\right ) \int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx}{128 a^2 c^4}\\ &=-\frac{a \sqrt{a+b x} \sqrt{c+d x}}{4 c x^4}-\frac{(9 b c-7 a d) \sqrt{a+b x} \sqrt{c+d x}}{24 c^2 x^3}-\frac{\left (3 b^2 c^2-46 a b c d+35 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{96 a c^3 x^2}+\frac{\left (9 b^3 c^3+15 a b^2 c^2 d-145 a^2 b c d^2+105 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{192 a^2 c^4 x}+\frac{\left ((b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right )\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^2 c^4}\\ &=-\frac{a \sqrt{a+b x} \sqrt{c+d x}}{4 c x^4}-\frac{(9 b c-7 a d) \sqrt{a+b x} \sqrt{c+d x}}{24 c^2 x^3}-\frac{\left (3 b^2 c^2-46 a b c d+35 a^2 d^2\right ) \sqrt{a+b x} \sqrt{c+d x}}{96 a c^3 x^2}+\frac{\left (9 b^3 c^3+15 a b^2 c^2 d-145 a^2 b c d^2+105 a^3 d^3\right ) \sqrt{a+b x} \sqrt{c+d x}}{192 a^2 c^4 x}-\frac{(b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{5/2} c^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.300535, size = 193, normalized size = 0.73 \[ -\frac{\frac{x^2 \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right ) \left (3 x^2 (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )+\sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} (2 a c-3 a d x+5 b c x)\right )}{\sqrt{a} c^{5/2}}+48 a c (a+b x)^{5/2} \sqrt{c+d x}-8 x (a+b x)^{5/2} \sqrt{c+d x} (7 a d+3 b c)}{192 a^2 c^2 x^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.024, size = 593, normalized size = 2.2 \begin{align*} -{\frac{1}{384\,{a}^{2}{c}^{4}{x}^{4}}\sqrt{bx+a}\sqrt{dx+c} \left ( 105\,\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}-180\,\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}+54\,\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}+12\,\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+9\,\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}-210\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{3}{a}^{3}{d}^{3}+290\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{3}{a}^{2}bc{d}^{2}-30\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{3}a{b}^{2}{c}^{2}d-18\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{3}{b}^{3}{c}^{3}+140\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{2}{a}^{3}c{d}^{2}-184\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{2}{a}^{2}b{c}^{2}d+12\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{x}^{2}a{b}^{2}{c}^{3}-112\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }x{a}^{3}{c}^{2}d+144\,\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 = 30.5592, size = 1268, normalized size = 4.77 \begin{align*} \left [\frac{3 \,{\left (3 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 60 \, a^{3} b c d^{3} + 35 \, 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 (9 \, a b^{3} c^{4} + 15 \, a^{2} b^{2} c^{3} d - 145 \, a^{3} b c^{2} d^{2} + 105 \, a^{4} c d^{3}\right )} x^{3} + 2 \,{\left (3 \, a^{2} b^{2} c^{4} - 46 \, a^{3} b c^{3} d + 35 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \,{\left (9 \, a^{3} b c^{4} - 7 \, a^{4} c^{3} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{768 \, a^{3} c^{5} x^{4}}, \frac{3 \,{\left (3 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 60 \, a^{3} b c d^{3} + 35 \, 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 (9 \, a b^{3} c^{4} + 15 \, a^{2} b^{2} c^{3} d - 145 \, a^{3} b c^{2} d^{2} + 105 \, a^{4} c d^{3}\right )} x^{3} + 2 \,{\left (3 \, a^{2} b^{2} c^{4} - 46 \, a^{3} b c^{3} d + 35 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \,{\left (9 \, a^{3} b c^{4} - 7 \, a^{4} c^{3} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{384 \, a^{3} c^{5} 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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