3.569 \(\int \frac{\sqrt{a+b x}}{x^2 \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=77 \[ -\frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a} c^{3/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{c x} \]

[Out]

-((Sqrt[a + b*x]*Sqrt[c + d*x])/(c*x)) - ((b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a +
b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(Sqrt[a]*c^(3/2))

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Rubi [A]  time = 0.13005, antiderivative size = 77, 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 \[ -\frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a} c^{3/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{c x} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x]/(x^2*Sqrt[c + d*x]),x]

[Out]

-((Sqrt[a + b*x]*Sqrt[c + d*x])/(c*x)) - ((b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[a +
b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(Sqrt[a]*c^(3/2))

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Rubi in Sympy [A]  time = 10.1969, size = 65, normalized size = 0.84 \[ - \frac{\sqrt{a + b x} \sqrt{c + d x}}{c x} + \frac{\left (a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{\sqrt{a} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/2)/x**2/(d*x+c)**(1/2),x)

[Out]

-sqrt(a + b*x)*sqrt(c + d*x)/(c*x) + (a*d - b*c)*atanh(sqrt(c)*sqrt(a + b*x)/(sq
rt(a)*sqrt(c + d*x)))/(sqrt(a)*c**(3/2))

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Mathematica [A]  time = 0.0911497, size = 117, normalized size = 1.52 \[ -\frac{\log (x) (a d-b c)}{2 \sqrt{a} c^{3/2}}+\frac{(a d-b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{2 \sqrt{a} c^{3/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{c x} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x]/(x^2*Sqrt[c + d*x]),x]

[Out]

-((Sqrt[a + b*x]*Sqrt[c + d*x])/(c*x)) - ((-(b*c) + a*d)*Log[x])/(2*Sqrt[a]*c^(3
/2)) + ((-(b*c) + a*d)*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*
x]*Sqrt[c + d*x]])/(2*Sqrt[a]*c^(3/2))

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Maple [B]  time = 0.029, size = 147, normalized size = 1.9 \[{\frac{1}{2\,cx}\sqrt{bx+a}\sqrt{dx+c} \left ( \ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) xad-\ln \left ({\frac{1}{x} \left ( adx+bcx+2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }+2\,ac \right ) } \right ) xbc-2\,\sqrt{ac}\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) } \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( dx+c \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/2)/x^2/(d*x+c)^(1/2),x)

[Out]

1/2*(b*x+a)^(1/2)*(d*x+c)^(1/2)/c*(ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c
))^(1/2)+2*a*c)/x)*x*a*d-ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2
*a*c)/x)*x*b*c-2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/((b*x+a)*(d*x+c))^(1/2)/x/
(a*c)^(1/2)

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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^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.269105, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b c - a d\right )} x \log \left (\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}}{x^{2}}\right ) + 4 \, \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c}}{4 \, \sqrt{a c} c x}, -\frac{{\left (b c - a d\right )} x \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c}}{2 \, \sqrt{b x + a} \sqrt{d x + c} a c}\right ) + 2 \, \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{2 \, \sqrt{-a c} c x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)/(sqrt(d*x + c)*x^2),x, algorithm="fricas")

[Out]

[-1/4*((b*c - a*d)*x*log((4*(2*a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)*sqrt(b*x + a)*sq
rt(d*x + c) + (8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 + 8*(a*b*c^2 + a^
2*c*d)*x)*sqrt(a*c))/x^2) + 4*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*
c*x), -1/2*((b*c - a*d)*x*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)/(sqrt(b*
x + a)*sqrt(d*x + c)*a*c)) + 2*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-a*
c)*c*x)]

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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((b*x+a)**(1/2)/x**2/(d*x+c)**(1/2),x)

[Out]

Timed 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^2),x, algorithm="giac")

[Out]

Exception raised: TypeError