Optimal. Leaf size=78 \[ \frac{(a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2} c^{3/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{a c x} \]
[Out]
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Rubi [A] time = 0.141238, antiderivative size = 78, 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 \[ \frac{(a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2} c^{3/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{a c x} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*Sqrt[a + b*x]*Sqrt[c + d*x]),x]
[Out]
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Rubi in Sympy [A] time = 10.612, size = 66, normalized size = 0.85 \[ - \frac{\sqrt{a + b x} \sqrt{c + d x}}{a c x} + \frac{\left (a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{a^{\frac{3}{2}} c^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(b*x+a)**(1/2)/(d*x+c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.115833, size = 113, normalized size = 1.45 \[ \frac{-2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+x \log (x) (-(a d+b c))+x (a d+b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{2 a^{3/2} c^{3/2} x} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*Sqrt[a + b*x]*Sqrt[c + d*x]),x]
[Out]
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Maple [B] time = 0.033, size = 149, normalized size = 1.9 \[{\frac{1}{2\,acx} \left ( \ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) xad+\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) xbc-2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) } \right ) \sqrt{dx+c}\sqrt{bx+a}{\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(b*x+a)^(1/2)/(d*x+c)^(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(1/(sqrt(b*x + a)*sqrt(d*x + c)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.278227, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (b c + a d\right )} x \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) - 4 \, \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{4 \, \sqrt{a c} a c x}, \frac{{\left (b c + a d\right )} x \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) - 2 \, \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{2 \, \sqrt{-a c} a c x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + a)*sqrt(d*x + c)*x^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \sqrt{a + b x} \sqrt{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(b*x+a)**(1/2)/(d*x+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.246683, size = 524, normalized size = 6.72 \[ \frac{\sqrt{b d} b^{4} d{\left (\frac{{\left (b c + a d\right )} \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{\sqrt{-a b c d} a b^{3} c d} - \frac{2 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2} -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} b c -{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a d\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \,{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{2} a b d +{\left (\sqrt{b d} \sqrt{b x + a} - \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d}\right )}^{4}\right )} a b^{2} c d}\right )}}{{\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(b*x + a)*sqrt(d*x + c)*x^2),x, algorithm="giac")
[Out]