Optimal. Leaf size=266 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} (7 b c-9 a d)}{24 a^2 x^3}-\frac{\left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{9/2} c^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (3 a^2 d^2-46 a b c d+35 b^2 c^2\right )}{96 a^3 c x^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (9 a^3 d^3+15 a^2 b c d^2-145 a b^2 c^2 d+105 b^3 c^3\right )}{192 a^4 c^2 x}-\frac{c \sqrt{a+b x} \sqrt{c+d x}}{4 a x^4} \]
[Out]
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Rubi [A] time = 0.746812, 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 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} (7 b c-9 a d)}{24 a^2 x^3}-\frac{\left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) (b c-a d)^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{9/2} c^{5/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (3 a^2 d^2-46 a b c d+35 b^2 c^2\right )}{96 a^3 c x^2}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (9 a^3 d^3+15 a^2 b c d^2-145 a b^2 c^2 d+105 b^3 c^3\right )}{192 a^4 c^2 x}-\frac{c \sqrt{a+b x} \sqrt{c+d x}}{4 a x^4} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^(3/2)/(x^5*Sqrt[a + b*x]),x]
[Out]
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Rubi in Sympy [A] time = 109.864, size = 255, normalized size = 0.96 \[ - \frac{c \sqrt{a + b x} \sqrt{c + d x}}{4 a x^{4}} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (9 a d - 7 b c\right )}{24 a^{2} x^{3}} - \frac{\sqrt{a + b x} \sqrt{c + d x} \left (3 a^{2} d^{2} - 46 a b c d + 35 b^{2} c^{2}\right )}{96 a^{3} c x^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (9 a^{3} d^{3} + 15 a^{2} b c d^{2} - 145 a b^{2} c^{2} d + 105 b^{3} c^{3}\right )}{192 a^{4} c^{2} x} - \frac{\left (a d - b c\right )^{2} \left (3 a^{2} d^{2} + 10 a b c d + 35 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{64 a^{\frac{9}{2}} c^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(3/2)/x**5/(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.27528, size = 263, normalized size = 0.99 \[ \frac{3 x^4 \log (x) (b c-a d)^2 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right )-3 x^4 (b c-a d)^2 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x} \left (a^3 \left (48 c^3+72 c^2 d x+6 c d^2 x^2-9 d^3 x^3\right )-a^2 b c x \left (56 c^2+92 c d x+15 d^2 x^2\right )+5 a b^2 c^2 x^2 (14 c+29 d x)-105 b^3 c^3 x^3\right )}{384 a^{9/2} c^{5/2} x^4} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^(3/2)/(x^5*Sqrt[a + b*x]),x]
[Out]
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Maple [B] time = 0.04, size = 593, normalized size = 2.2 \[ -{\frac{1}{384\,{a}^{4}{c}^{2}{x}^{4}}\sqrt{bx+a}\sqrt{dx+c} \left ( 9\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{a}^{4}{d}^{4}+12\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{a}^{3}bc{d}^{3}+54\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}-180\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}a{b}^{3}{c}^{3}d+105\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{4}{b}^{4}{c}^{4}-18\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{d}^{3}{a}^{3}{x}^{3}\sqrt{ac}-30\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{d}^{2}bc{a}^{2}{x}^{3}\sqrt{ac}+290\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }d{b}^{2}{c}^{2}a{x}^{3}\sqrt{ac}-210\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{3}{c}^{3}{x}^{3}\sqrt{ac}+12\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{d}^{2}c{a}^{3}{x}^{2}\sqrt{ac}-184\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }db{c}^{2}{a}^{2}{x}^{2}\sqrt{ac}+140\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{b}^{2}{c}^{3}a{x}^{2}\sqrt{ac}+144\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }d{c}^{2}{a}^{3}x\sqrt{ac}-112\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }b{c}^{3}{a}^{2}x\sqrt{ac}+96\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }{c}^{3}{a}^{3}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(3/2)/x^5/(b*x+a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(3/2)/(sqrt(b*x + a)*x^5),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.755996, size = 1, normalized size = 0. \[ \left [\frac{3 \,{\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )} x^{4} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) - 4 \,{\left (48 \, a^{3} c^{3} -{\left (105 \, b^{3} c^{3} - 145 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} + 9 \, a^{3} d^{3}\right )} x^{3} + 2 \,{\left (35 \, a b^{2} c^{3} - 46 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} x^{2} - 8 \,{\left (7 \, a^{2} b c^{3} - 9 \, a^{3} c^{2} d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{768 \, \sqrt{a c} a^{4} c^{2} x^{4}}, -\frac{3 \,{\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )} x^{4} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) + 2 \,{\left (48 \, a^{3} c^{3} -{\left (105 \, b^{3} c^{3} - 145 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} + 9 \, a^{3} d^{3}\right )} x^{3} + 2 \,{\left (35 \, a b^{2} c^{3} - 46 \, a^{2} b c^{2} d + 3 \, a^{3} c d^{2}\right )} x^{2} - 8 \,{\left (7 \, a^{2} b c^{3} - 9 \, a^{3} c^{2} d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{384 \, \sqrt{-a c} a^{4} c^{2} x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(3/2)/(sqrt(b*x + a)*x^5),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**(3/2)/x**5/(b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^(3/2)/(sqrt(b*x + a)*x^5),x, algorithm="giac")
[Out]