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

Optimal. Leaf size=72 \[ \frac{(A b-2 a B) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 a^{3/2}}-\frac{A \sqrt{a+b x+c x^2}}{a x} \]

[Out]

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

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Rubi [A]  time = 0.126628, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{(A b-2 a B) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 a^{3/2}}-\frac{A \sqrt{a+b x+c x^2}}{a x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.735, size = 61, normalized size = 0.85 \[ - \frac{A \sqrt{a + b x + c x^{2}}}{a x} + \frac{\left (A b - 2 B a\right ) \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{2 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*sqrt(a + b*x + c*x**2)/(a*x) + (A*b - 2*B*a)*atanh((2*a + b*x)/(2*sqrt(a)*sqr
t(a + b*x + c*x**2)))/(2*a**(3/2))

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Mathematica [A]  time = 0.254457, size = 86, normalized size = 1.19 \[ \frac{x (A b-2 a B) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )+\log (x) (2 a B x-A b x)-2 \sqrt{a} A \sqrt{a+x (b+c x)}}{2 a^{3/2} x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 94, normalized size = 1.3 \[ -{\frac{A}{ax}\sqrt{c{x}^{2}+bx+a}}+{\frac{Ab}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{B\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.32781, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (2 \, B a - A b\right )} x \log \left (-\frac{4 \,{\left (a b x + 2 \, a^{2}\right )} \sqrt{c x^{2} + b x + a} +{\left (8 \, a b x +{\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{2}}\right ) + 4 \, \sqrt{c x^{2} + b x + a} A \sqrt{a}}{4 \, a^{\frac{3}{2}} x}, -\frac{{\left (2 \, B a - A b\right )} x \arctan \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{2} + b x + a} a}\right ) + 2 \, \sqrt{c x^{2} + b x + a} A \sqrt{-a}}{2 \, \sqrt{-a} a x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/4*((2*B*a - A*b)*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)*A*sqrt(a))/
(a^(3/2)*x), -1/2*((2*B*a - A*b)*x*arctan(1/2*(b*x + 2*a)*sqrt(-a)/(sqrt(c*x^2 +
b*x + a)*a)) + 2*sqrt(c*x^2 + b*x + a)*A*sqrt(-a))/(sqrt(-a)*a*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)/(x**2*sqrt(a + b*x + c*x**2)), x)

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GIAC/XCAS [A]  time = 0.28044, size = 149, normalized size = 2.07 \[ \frac{{\left (2 \, B a - A b\right )} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} A b + 2 \, A a \sqrt{c}}{{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} - a\right )} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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