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

Optimal. Leaf size=202 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a-b x) \sqrt{c+d x^2+e x}}{x (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (2 a d+b e) \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{2 \sqrt{d} (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a e+2 b c) \tanh ^{-1}\left (\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right )}{2 \sqrt{c} (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.500607, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a-b x) \sqrt{c+d x^2+e x}}{x (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (2 a d+b e) \tanh ^{-1}\left (\frac{2 d x+e}{2 \sqrt{d} \sqrt{c+d x^2+e x}}\right )}{2 \sqrt{d} (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a e+2 b c) \tanh ^{-1}\left (\frac{2 c+e x}{2 \sqrt{c} \sqrt{c+d x^2+e x}}\right )}{2 \sqrt{c} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 63.0626, size = 189, normalized size = 0.94 \[ - \frac{\left (2 a - 2 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \sqrt{c + d x^{2} + e x}}{2 x \left (a + b x\right )} + \frac{\left (2 a d + b e\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \operatorname{atanh}{\left (\frac{2 d x + e}{2 \sqrt{d} \sqrt{c + d x^{2} + e x}} \right )}}{2 \sqrt{d} \left (a + b x\right )} - \frac{\left (a e + 2 b c\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \operatorname{atanh}{\left (\frac{2 c + e x}{2 \sqrt{c} \sqrt{c + d x^{2} + e x}} \right )}}{2 \sqrt{c} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.389941, size = 168, normalized size = 0.83 \[ \frac{\sqrt{(a+b x)^2} \left (\sqrt{d} x \log (x) (a e+2 b c)-\sqrt{d} x (a e+2 b c) \log \left (2 \sqrt{c} \sqrt{c+x (d x+e)}+2 c+e x\right )+\sqrt{c} \left (2 \sqrt{d} (b x-a) \sqrt{c+x (d x+e)}+x (2 a d+b e) \log \left (2 \sqrt{d} \sqrt{c+x (d x+e)}+2 d x+e\right )\right )\right )}{2 \sqrt{c} \sqrt{d} x (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.018, size = 259, normalized size = 1.3 \[{\frac{{\it csgn} \left ( bx+a \right ) }{2\,x} \left ( 2\,ad\ln \left ( 1/2\,{\frac{2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e}{\sqrt{d}}} \right ){c}^{3/2}x-2\,b{c}^{2}\ln \left ({\frac{2\,c+ex+2\,\sqrt{c}\sqrt{d{x}^{2}+ex+c}}{x}} \right ) x\sqrt{d}+2\,a{d}^{3/2}\sqrt{d{x}^{2}+ex+c}{x}^{2}\sqrt{c}+be\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{d{x}^{2}+ex+c}\sqrt{d}+2\,dx+e \right ){\frac{1}{\sqrt{d}}}} \right ){c}^{{\frac{3}{2}}}x-2\,a \left ( d{x}^{2}+ex+c \right ) ^{3/2}\sqrt{c}\sqrt{d}+2\,ae\sqrt{d{x}^{2}+ex+c}x\sqrt{c}\sqrt{d}+2\,b\sqrt{d{x}^{2}+ex+c}{c}^{3/2}x\sqrt{d}-ae\ln \left ({\frac{1}{x} \left ( 2\,c+ex+2\,\sqrt{c}\sqrt{d{x}^{2}+ex+c} \right ) } \right ) cx\sqrt{d} \right ){c}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*csgn(b*x+a)*(2*a*d*ln(1/2*(2*(d*x^2+e*x+c)^(1/2)*d^(1/2)+2*d*x+e)/d^(1/2))*c
^(3/2)*x-2*b*c^2*ln((2*c+e*x+2*c^(1/2)*(d*x^2+e*x+c)^(1/2))/x)*x*d^(1/2)+2*a*d^(
3/2)*(d*x^2+e*x+c)^(1/2)*x^2*c^(1/2)+b*e*ln(1/2*(2*(d*x^2+e*x+c)^(1/2)*d^(1/2)+2
*d*x+e)/d^(1/2))*c^(3/2)*x-2*a*(d*x^2+e*x+c)^(3/2)*c^(1/2)*d^(1/2)+2*a*e*(d*x^2+
e*x+c)^(1/2)*x*c^(1/2)*d^(1/2)+2*b*(d*x^2+e*x+c)^(1/2)*c^(3/2)*x*d^(1/2)-a*e*ln(
(2*c+e*x+2*c^(1/2)*(d*x^2+e*x+c)^(1/2))/x)*c*x*d^(1/2))/c^(3/2)/x/d^(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(d*x^2 + e*x + c)*sqrt((b*x + a)^2)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.46141, size = 1, normalized size = 0. \[ \left [\frac{{\left (2 \, a d + b e\right )} \sqrt{c} x \log \left (4 \,{\left (2 \, d^{2} x + d e\right )} \sqrt{d x^{2} + e x + c} +{\left (8 \, d^{2} x^{2} + 8 \, d e x + 4 \, c d + e^{2}\right )} \sqrt{d}\right ) +{\left (2 \, b c + a e\right )} \sqrt{d} x \log \left (-\frac{4 \,{\left (c e x + 2 \, c^{2}\right )} \sqrt{d x^{2} + e x + c} -{\left (8 \, c e x +{\left (4 \, c d + e^{2}\right )} x^{2} + 8 \, c^{2}\right )} \sqrt{c}}{x^{2}}\right ) + 4 \, \sqrt{d x^{2} + e x + c}{\left (b x - a\right )} \sqrt{c} \sqrt{d}}{4 \, \sqrt{c} \sqrt{d} x}, \frac{2 \,{\left (2 \, a d + b e\right )} \sqrt{c} x \arctan \left (\frac{{\left (2 \, d x + e\right )} \sqrt{-d}}{2 \, \sqrt{d x^{2} + e x + c} d}\right ) +{\left (2 \, b c + a e\right )} \sqrt{-d} x \log \left (-\frac{4 \,{\left (c e x + 2 \, c^{2}\right )} \sqrt{d x^{2} + e x + c} -{\left (8 \, c e x +{\left (4 \, c d + e^{2}\right )} x^{2} + 8 \, c^{2}\right )} \sqrt{c}}{x^{2}}\right ) + 4 \, \sqrt{d x^{2} + e x + c}{\left (b x - a\right )} \sqrt{c} \sqrt{-d}}{4 \, \sqrt{c} \sqrt{-d} x}, -\frac{2 \,{\left (2 \, b c + a e\right )} \sqrt{d} x \arctan \left (\frac{{\left (e x + 2 \, c\right )} \sqrt{-c}}{2 \, \sqrt{d x^{2} + e x + c} c}\right ) -{\left (2 \, a d + b e\right )} \sqrt{-c} x \log \left (4 \,{\left (2 \, d^{2} x + d e\right )} \sqrt{d x^{2} + e x + c} +{\left (8 \, d^{2} x^{2} + 8 \, d e x + 4 \, c d + e^{2}\right )} \sqrt{d}\right ) - 4 \, \sqrt{d x^{2} + e x + c}{\left (b x - a\right )} \sqrt{-c} \sqrt{d}}{4 \, \sqrt{-c} \sqrt{d} x}, -\frac{{\left (2 \, b c + a e\right )} \sqrt{-d} x \arctan \left (\frac{{\left (e x + 2 \, c\right )} \sqrt{-c}}{2 \, \sqrt{d x^{2} + e x + c} c}\right ) -{\left (2 \, a d + b e\right )} \sqrt{-c} x \arctan \left (\frac{{\left (2 \, d x + e\right )} \sqrt{-d}}{2 \, \sqrt{d x^{2} + e x + c} d}\right ) - 2 \, \sqrt{d x^{2} + e x + c}{\left (b x - a\right )} \sqrt{-c} \sqrt{-d}}{2 \, \sqrt{-c} \sqrt{-d} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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(d*x^2 + e*x + c)*sqrt((b*x + a)^2)/x^2,x, algorithm="giac")

[Out]

Exception raised: TypeError