3.45 \(\int \frac{\sqrt{a^2+2 a b x+b^2 x^2} \sqrt{c+d x^2}}{x^3} \, dx\)

Optimal. Leaf size=161 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+2 b x) \sqrt{c+d x^2}}{2 x^2 (a+b x)}+\frac{b \sqrt{d} \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{a+b x}-\frac{a d \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 \sqrt{c} (a+b x)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.380228, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+2 b x) \sqrt{c+d x^2}}{2 x^2 (a+b x)}+\frac{b \sqrt{d} \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{a+b x}-\frac{a d \sqrt{a^2+2 a b x+b^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{2 \sqrt{c} (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2])/x^3,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 41.5783, size = 151, normalized size = 0.94 \[ - \frac{a d \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{2 \sqrt{c} \left (a + b x\right )} + \frac{b \sqrt{d} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{a + b x} - \frac{\left (2 a + 4 b x\right ) \sqrt{c + d x^{2}} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{4 x^{2} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(((b*x+a)**2)**(1/2)*(d*x**2+c)**(1/2)/x**3,x)

[Out]

-a*d*sqrt(a**2 + 2*a*b*x + b**2*x**2)*atanh(sqrt(c + d*x**2)/sqrt(c))/(2*sqrt(c)
*(a + b*x)) + b*sqrt(d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)*atanh(sqrt(d)*x/sqrt(c
+ d*x**2))/(a + b*x) - (2*a + 4*b*x)*sqrt(c + d*x**2)*sqrt(a**2 + 2*a*b*x + b**2
*x**2)/(4*x**2*(a + b*x))

_______________________________________________________________________________________

Mathematica [A]  time = 0.179201, size = 125, normalized size = 0.78 \[ \frac{\sqrt{(a+b x)^2} \left (\sqrt{c} \left (2 b \sqrt{d} x^2 \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )-(a+2 b x) \sqrt{c+d x^2}\right )-a d x^2 \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )+a d x^2 \log (x)\right )}{2 \sqrt{c} x^2 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Sqrt[c + d*x^2])/x^3,x]

[Out]

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

_______________________________________________________________________________________

Maple [C]  time = 0.018, size = 141, normalized size = 0.9 \[{\frac{{\it csgn} \left ( bx+a \right ) }{2\,{x}^{2}} \left ( 2\,b\sqrt{d}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){x}^{2}{c}^{3/2}+2\,bd{x}^{3}\sqrt{d{x}^{2}+c}\sqrt{c}-ad\ln \left ( 2\,{\frac{\sqrt{c}\sqrt{d{x}^{2}+c}+c}{x}} \right ){x}^{2}c-2\,b \left ( d{x}^{2}+c \right ) ^{3/2}x\sqrt{c}+ad\sqrt{d{x}^{2}+c}{x}^{2}\sqrt{c}-a \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}\sqrt{c} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x+a)^2)^(1/2)*(d*x^2+c)^(1/2)/x^3,x)

[Out]

1/2*csgn(b*x+a)*(2*b*d^(1/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))*x^2*c^(3/2)+2*b*d*x^3
*(d*x^2+c)^(1/2)*c^(1/2)-a*d*ln(2*(c^(1/2)*(d*x^2+c)^(1/2)+c)/x)*x^2*c-2*b*(d*x^
2+c)^(3/2)*x*c^(1/2)+a*d*(d*x^2+c)^(1/2)*x^2*c^(1/2)-a*(d*x^2+c)^(3/2)*c^(1/2))/
x^2/c^(3/2)

_______________________________________________________________________________________

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^2 + c)*sqrt((b*x + a)^2)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.334976, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, b \sqrt{c} \sqrt{d} x^{2} \log \left (-2 \, d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{d} x - c\right ) + a d x^{2} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right ) - 2 \, \sqrt{d x^{2} + c}{\left (2 \, b x + a\right )} \sqrt{c}}{4 \, \sqrt{c} x^{2}}, \frac{4 \, b \sqrt{c} \sqrt{-d} x^{2} \arctan \left (\frac{d x}{\sqrt{d x^{2} + c} \sqrt{-d}}\right ) + a d x^{2} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right ) - 2 \, \sqrt{d x^{2} + c}{\left (2 \, b x + a\right )} \sqrt{c}}{4 \, \sqrt{c} x^{2}}, -\frac{a d x^{2} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) - b \sqrt{-c} \sqrt{d} x^{2} \log \left (-2 \, d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{d} x - c\right ) + \sqrt{d x^{2} + c}{\left (2 \, b x + a\right )} \sqrt{-c}}{2 \, \sqrt{-c} x^{2}}, \frac{2 \, b \sqrt{-c} \sqrt{-d} x^{2} \arctan \left (\frac{d x}{\sqrt{d x^{2} + c} \sqrt{-d}}\right ) - a d x^{2} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) - \sqrt{d x^{2} + c}{\left (2 \, b x + a\right )} \sqrt{-c}}{2 \, \sqrt{-c} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^2 + c)*sqrt((b*x + a)^2)/x^3,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c + d x^{2}} \sqrt{\left (a + b x\right )^{2}}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x+a)**2)**(1/2)*(d*x**2+c)**(1/2)/x**3,x)

[Out]

Integral(sqrt(c + d*x**2)*sqrt((a + b*x)**2)/x**3, x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.282307, size = 269, normalized size = 1.67 \[ \frac{a d \arctan \left (-\frac{\sqrt{d} x - \sqrt{d x^{2} + c}}{\sqrt{-c}}\right ){\rm sign}\left (b x + a\right )}{\sqrt{-c}} - b \sqrt{d}{\rm ln}\left ({\left | -\sqrt{d} x + \sqrt{d x^{2} + c} \right |}\right ){\rm sign}\left (b x + a\right ) + \frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{3} a d{\rm sign}\left (b x + a\right ) + 2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b c \sqrt{d}{\rm sign}\left (b x + a\right ) +{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )} a c d{\rm sign}\left (b x + a\right ) - 2 \, b c^{2} \sqrt{d}{\rm sign}\left (b x + a\right )}{{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^2 + c)*sqrt((b*x + a)^2)/x^3,x, algorithm="giac")

[Out]

a*d*arctan(-(sqrt(d)*x - sqrt(d*x^2 + c))/sqrt(-c))*sign(b*x + a)/sqrt(-c) - b*s
qrt(d)*ln(abs(-sqrt(d)*x + sqrt(d*x^2 + c)))*sign(b*x + a) + ((sqrt(d)*x - sqrt(
d*x^2 + c))^3*a*d*sign(b*x + a) + 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b*c*sqrt(d)*
sign(b*x + a) + (sqrt(d)*x - sqrt(d*x^2 + c))*a*c*d*sign(b*x + a) - 2*b*c^2*sqrt
(d)*sign(b*x + a))/((sqrt(d)*x - sqrt(d*x^2 + c))^2 - c)^2