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

Optimal. Leaf size=41 \[ \frac{\log \left ((a+b x)^2+c\right )}{2 b^2}-\frac{a \tan ^{-1}\left (\frac{a+b x}{\sqrt{c}}\right )}{b^2 \sqrt{c}} \]

[Out]

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

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Rubi [A]  time = 0.0525262, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{\log \left ((a+b x)^2+c\right )}{2 b^2}-\frac{a \tan ^{-1}\left (\frac{a+b x}{\sqrt{c}}\right )}{b^2 \sqrt{c}} \]

Antiderivative was successfully verified.

[In]  Int[x/(c + (a + b*x)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.56939, size = 36, normalized size = 0.88 \[ - \frac{a \operatorname{atan}{\left (\frac{a + b x}{\sqrt{c}} \right )}}{b^{2} \sqrt{c}} + \frac{\log{\left (c + \left (a + b x\right )^{2} \right )}}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*atan((a + b*x)/sqrt(c))/(b**2*sqrt(c)) + log(c + (a + b*x)**2)/(2*b**2)

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Mathematica [A]  time = 0.0222516, size = 38, normalized size = 0.93 \[ \frac{\log \left ((a+b x)^2+c\right )-\frac{2 a \tan ^{-1}\left (\frac{a+b x}{\sqrt{c}}\right )}{\sqrt{c}}}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 54, normalized size = 1.3 \[{\frac{\ln \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2}+c \right ) }{2\,{b}^{2}}}-{\frac{a}{{b}^{2}}\arctan \left ({\frac{2\,{b}^{2}x+2\,ab}{2\,b}{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(c+(b*x+a)^2),x)

[Out]

1/2/b^2*ln(b^2*x^2+2*a*b*x+a^2+c)-a/b^2/c^(1/2)*arctan(1/2*(2*b^2*x+2*a*b)/b/c^(
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(x/((b*x + a)^2 + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.256959, size = 1, normalized size = 0.02 \[ \left [\frac{a \log \left (-\frac{2 \, b c x + 2 \, a c -{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - c\right )} \sqrt{-c}}{b^{2} x^{2} + 2 \, a b x + a^{2} + c}\right ) + \sqrt{-c} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{2 \, b^{2} \sqrt{-c}}, -\frac{2 \, a \arctan \left (\frac{b x + a}{\sqrt{c}}\right ) - \sqrt{c} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{2 \, b^{2} \sqrt{c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.64546, size = 124, normalized size = 3.02 \[ \left (- \frac{a \sqrt{- c}}{2 b^{2} c} + \frac{1}{2 b^{2}}\right ) \log{\left (x + \frac{a^{2} - 2 b^{2} c \left (- \frac{a \sqrt{- c}}{2 b^{2} c} + \frac{1}{2 b^{2}}\right ) + c}{a b} \right )} + \left (\frac{a \sqrt{- c}}{2 b^{2} c} + \frac{1}{2 b^{2}}\right ) \log{\left (x + \frac{a^{2} - 2 b^{2} c \left (\frac{a \sqrt{- c}}{2 b^{2} c} + \frac{1}{2 b^{2}}\right ) + c}{a b} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(c+(b*x+a)**2),x)

[Out]

(-a*sqrt(-c)/(2*b**2*c) + 1/(2*b**2))*log(x + (a**2 - 2*b**2*c*(-a*sqrt(-c)/(2*b
**2*c) + 1/(2*b**2)) + c)/(a*b)) + (a*sqrt(-c)/(2*b**2*c) + 1/(2*b**2))*log(x +
(a**2 - 2*b**2*c*(a*sqrt(-c)/(2*b**2*c) + 1/(2*b**2)) + c)/(a*b))

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GIAC/XCAS [A]  time = 0.267374, size = 58, normalized size = 1.41 \[ -\frac{a \arctan \left (\frac{b x + a}{\sqrt{c}}\right )}{b^{2} \sqrt{c}} + \frac{{\rm ln}\left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*arctan((b*x + a)/sqrt(c))/(b^2*sqrt(c)) + 1/2*ln(b^2*x^2 + 2*a*b*x + a^2 + c)
/b^2