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

Optimal. Leaf size=76 \[ \frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{a^{3/2} \sqrt{c}}-\frac{\sqrt{a+b x} \sqrt{c+d x}}{a x} \]

[Out]

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

_______________________________________________________________________________________

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]

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

_______________________________________________________________________________________

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]

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

_______________________________________________________________________________________

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]

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

_______________________________________________________________________________________

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]

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

_______________________________________________________________________________________

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]

Exception raised: ValueError

_______________________________________________________________________________________

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]

[-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)*s
qrt(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)
*a*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)*a*x)]

_______________________________________________________________________________________

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]

Timed out

_______________________________________________________________________________________

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]

Exception raised: TypeError