3.916 \(\int \frac{(A+B x) \sqrt{a+b x+c x^2}}{x^2} \, dx\)

Optimal. Leaf size=121 \[ -\frac{(A-B x) \sqrt{a+b x+c x^2}}{x}-\frac{(2 a B+A b) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{a}}+\frac{(2 A c+b B) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c}} \]

[Out]

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

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Rubi [A]  time = 0.263832, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ -\frac{(A-B x) \sqrt{a+b x+c x^2}}{x}-\frac{(2 a B+A b) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{a}}+\frac{(2 A c+b B) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 30.3041, size = 107, normalized size = 0.88 \[ - \frac{\left (A - B x\right ) \sqrt{a + b x + c x^{2}}}{x} + \frac{\left (2 A c + B b\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{2 \sqrt{c}} - \frac{\left (A b + 2 B a\right ) \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{2 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(A - B*x)*sqrt(a + b*x + c*x**2)/x + (2*A*c + B*b)*atanh((b + 2*c*x)/(2*sqrt(c)
*sqrt(a + b*x + c*x**2)))/(2*sqrt(c)) - (A*b + 2*B*a)*atanh((2*a + b*x)/(2*sqrt(
a)*sqrt(a + b*x + c*x**2)))/(2*sqrt(a))

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Mathematica [A]  time = 0.666203, size = 130, normalized size = 1.07 \[ \frac{1}{2} \left (\frac{2 (B x-A) \sqrt{a+x (b+c x)}}{x}-\frac{(2 a B+A b) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )}{\sqrt{a}}+\frac{(2 A c+b B) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{\sqrt{c}}+\frac{\log (x) (2 a B+A b)}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((2*(-A + B*x)*Sqrt[a + x*(b + c*x)])/x + ((A*b + 2*a*B)*Log[x])/Sqrt[a] - ((A*b
+ 2*a*B)*Log[2*a + b*x + 2*Sqrt[a]*Sqrt[a + x*(b + c*x)]])/Sqrt[a] + ((b*B + 2*
A*c)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/Sqrt[c])/2

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Maple [B]  time = 0.013, size = 207, normalized size = 1.7 \[ -{\frac{A}{ax} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{Ab}{a}\sqrt{c{x}^{2}+bx+a}}-{\frac{Ab}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{Acx}{a}\sqrt{c{x}^{2}+bx+a}}+A\sqrt{c}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ) +B\sqrt{c{x}^{2}+bx+a}+{\frac{Bb}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-B\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/a/x*(c*x^2+b*x+a)^(3/2)+A*b/a*(c*x^2+b*x+a)^(1/2)-1/2*A*b/a^(1/2)*ln((2*a+b*x
+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+A/a*c*(c*x^2+b*x+a)^(1/2)*x+A*c^(1/2)*ln((1/2
*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+B*(c*x^2+b*x+a)^(1/2)+1/2*B*b*ln((1/2*b+c*x
)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-B*a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*
x+a)^(1/2))/x)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.62405, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (B b + 2 \, A c\right )} \sqrt{a} x \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right ) +{\left (2 \, B a + A b\right )} \sqrt{c} x \log \left (\frac{4 \,{\left (a b x + 2 \, a^{2}\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, a b x +{\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{2}}\right ) + 4 \, \sqrt{c x^{2} + b x + a}{\left (B x - A\right )} \sqrt{a} \sqrt{c}}{4 \, \sqrt{a} \sqrt{c} x}, \frac{2 \,{\left (B b + 2 \, A c\right )} \sqrt{a} x \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right ) +{\left (2 \, B a + A b\right )} \sqrt{-c} x \log \left (\frac{4 \,{\left (a b x + 2 \, a^{2}\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, a b x +{\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{2}}\right ) + 4 \, \sqrt{c x^{2} + b x + a}{\left (B x - A\right )} \sqrt{a} \sqrt{-c}}{4 \, \sqrt{a} \sqrt{-c} x}, -\frac{2 \,{\left (2 \, B a + A b\right )} \sqrt{c} x \arctan \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{2} + b x + a} a}\right ) -{\left (B b + 2 \, A c\right )} \sqrt{-a} x \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right ) - 4 \, \sqrt{c x^{2} + b x + a}{\left (B x - A\right )} \sqrt{-a} \sqrt{c}}{4 \, \sqrt{-a} \sqrt{c} x}, -\frac{{\left (2 \, B a + A b\right )} \sqrt{-c} x \arctan \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{2} + b x + a} a}\right ) -{\left (B b + 2 \, A c\right )} \sqrt{-a} x \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right ) - 2 \, \sqrt{c x^{2} + b x + a}{\left (B x - A\right )} \sqrt{-a} \sqrt{-c}}{2 \, \sqrt{-a} \sqrt{-c} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)*sqrt(a + b*x + c*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(c*x^2 + b*x + a)*(B*x + A)/x^2,x, algorithm="giac")

[Out]

Exception raised: TypeError