Optimal. Leaf size=76 \[ \frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2} \sqrt{c}}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{a x} \]
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Rubi [A] time = 0.133349, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2} \sqrt{c}}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{a x} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c + d*x]/(x^2*Sqrt[a + b*x]),x]
[Out]
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Rubi in Sympy [A] time = 10.2325, size = 66, normalized size = 0.87 \[ - \frac{\sqrt{a + b x} \sqrt{c + d x}}{a x} - \frac{\left (a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{a^{\frac{3}{2}} \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**(1/2)/x**2/(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0925493, size = 117, normalized size = 1.54 \[ \frac{\log (x) (a d-b c)}{2 a^{3/2} \sqrt{c}}-\frac{(a d-b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{2 a^{3/2} \sqrt{c}}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{a x} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[c + d*x]/(x^2*Sqrt[a + b*x]),x]
[Out]
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Maple [B] time = 0.028, size = 147, normalized size = 1.9 \[ -{\frac{1}{2\,ax}\sqrt{bx+a}\sqrt{dx+c} \left ( \ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) xad-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) xbc+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) } \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^(1/2)/x^2/(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(sqrt(d*x + c)/(sqrt(b*x + a)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.277834, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b c - a d\right )} x \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) + 4 \, \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{4 \, \sqrt{a c} a x}, \frac{{\left (b c - a d\right )} x \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) - 2 \, \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{2 \, \sqrt{-a c} a x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x + c)/(sqrt(b*x + a)*x^2),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)**(1/2)/x**2/(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(sqrt(d*x + c)/(sqrt(b*x + a)*x^2),x, algorithm="giac")
[Out]