Optimal. Leaf size=131 \[ \frac{(b c-a d) (3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{3/2} c^{5/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+b c)}{4 a c^2 x}-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{2 a c x^2} \]
[Out]
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Rubi [A] time = 0.223305, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{(b c-a d) (3 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{4 a^{3/2} c^{5/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} (3 a d+b c)}{4 a c^2 x}-\frac{(a+b x)^{3/2} \sqrt{c+d x}}{2 a c x^2} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*x]/(x^3*Sqrt[c + d*x]),x]
[Out]
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Rubi in Sympy [A] time = 17.0343, size = 114, normalized size = 0.87 \[ - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x}}{2 a c x^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (3 a d + b c\right )}{4 a c^{2} x} - \frac{\left (a d - b c\right ) \left (3 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{4 a^{\frac{3}{2}} c^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(1/2)/x**3/(d*x+c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.19229, size = 159, normalized size = 1.21 \[ -\frac{\log (x) (b c-a d) (3 a d+b c)}{8 a^{3/2} c^{5/2}}+\frac{(b c-a d) (3 a d+b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{8 a^{3/2} c^{5/2}}+\sqrt{a+b x} \sqrt{c+d x} \left (\frac{3 a d-b c}{4 a c^2 x}-\frac{1}{2 c x^2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*x]/(x^3*Sqrt[c + d*x]),x]
[Out]
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Maple [B] time = 0.031, size = 258, normalized size = 2. \[ -{\frac{1}{8\,{c}^{2}a{x}^{2}}\sqrt{bx+a}\sqrt{dx+c} \left ( 3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}{a}^{2}{d}^{2}-2\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac}{x}} \right ){x}^{2}abcd-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ){x}^{2}{b}^{2}{c}^{2}-6\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }dax\sqrt{ac}+2\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }bcx\sqrt{ac}+4\,\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }ca\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(1/2)/x^3/(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(sqrt(b*x + a)/(sqrt(d*x + c)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.293353, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} x^{2} \log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} -{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) + 4 \,{\left (2 \, a c +{\left (b c - 3 \, a d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{16 \, \sqrt{a c} a c^{2} x^{2}}, \frac{{\left (b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}\right )} x^{2} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) - 2 \,{\left (2 \, a c +{\left (b c - 3 \, a d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{8 \, \sqrt{-a c} a c^{2} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x + a)/(sqrt(d*x + c)*x^3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b x}}{x^{3} \sqrt{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**(1/2)/x**3/(d*x+c)**(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(b*x + a)/(sqrt(d*x + c)*x^3),x, algorithm="giac")
[Out]