3.82 \(\int \frac{1}{x \left (c+(a+b x)^2\right )} \, dx\)

Optimal. Leaf size=59 \[ -\frac{\log \left ((a+b x)^2+c\right )}{2 \left (a^2+c\right )}-\frac{a \tan ^{-1}\left (\frac{a+b x}{\sqrt{c}}\right )}{\sqrt{c} \left (a^2+c\right )}+\frac{\log (x)}{a^2+c} \]

[Out]

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

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Rubi [A]  time = 0.082148, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\log \left ((a+b x)^2+c\right )}{2 \left (a^2+c\right )}-\frac{a \tan ^{-1}\left (\frac{a+b x}{\sqrt{c}}\right )}{\sqrt{c} \left (a^2+c\right )}+\frac{\log (x)}{a^2+c} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(c + (a + b*x)^2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 10.9615, size = 53, normalized size = 0.9 \[ - \frac{a \operatorname{atan}{\left (\frac{a + b x}{\sqrt{c}} \right )}}{\sqrt{c} \left (a^{2} + c\right )} + \frac{\log{\left (- b x \right )}}{a^{2} + c} - \frac{\log{\left (c + \left (a + b x\right )^{2} \right )}}{2 \left (a^{2} + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0572488, size = 48, normalized size = 0.81 \[ -\frac{\log \left ((a+b x)^2+c\right )+\frac{2 a \tan ^{-1}\left (\frac{a+b x}{\sqrt{c}}\right )}{\sqrt{c}}-2 \log (b x)}{2 \left (a^2+c\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 72, normalized size = 1.2 \[ -{\frac{\ln \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2}+c \right ) }{2\,{a}^{2}+2\,c}}-{\frac{a}{{a}^{2}+c}\arctan \left ({\frac{2\,{b}^{2}x+2\,ab}{2\,b}{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}}+{\frac{\ln \left ( x \right ) }{{a}^{2}+c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(c+(b*x+a)^2),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.274189, size = 1, normalized size = 0.02 \[ \left [\frac{a \log \left (-\frac{2 \, b c x + 2 \, a c -{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - c\right )} \sqrt{-c}}{b^{2} x^{2} + 2 \, a b x + a^{2} + c}\right ) - \sqrt{-c}{\left (\log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right ) - 2 \, \log \left (x\right )\right )}}{2 \,{\left (a^{2} + c\right )} \sqrt{-c}}, -\frac{2 \, a \arctan \left (\frac{b x + a}{\sqrt{c}}\right ) + \sqrt{c}{\left (\log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right ) - 2 \, \log \left (x\right )\right )}}{2 \,{\left (a^{2} + c\right )} \sqrt{c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [A]  time = 5.9779, size = 738, normalized size = 12.51 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(c+(b*x+a)**2),x)

[Out]

(-a*sqrt(-c)/(2*c*(a**2 + c)) - 1/(2*(a**2 + c)))*log(x + (-4*a**6*c*(-a*sqrt(-c
)/(2*c*(a**2 + c)) - 1/(2*(a**2 + c)))**2 + 4*a**4*c**2*(-a*sqrt(-c)/(2*c*(a**2
+ c)) - 1/(2*(a**2 + c)))**2 - 6*a**4*c*(-a*sqrt(-c)/(2*c*(a**2 + c)) - 1/(2*(a*
*2 + c))) + 20*a**2*c**3*(-a*sqrt(-c)/(2*c*(a**2 + c)) - 1/(2*(a**2 + c)))**2 -
12*a**2*c**2*(-a*sqrt(-c)/(2*c*(a**2 + c)) - 1/(2*(a**2 + c))) + 10*a**2*c + 12*
c**4*(-a*sqrt(-c)/(2*c*(a**2 + c)) - 1/(2*(a**2 + c)))**2 - 6*c**3*(-a*sqrt(-c)/
(2*c*(a**2 + c)) - 1/(2*(a**2 + c))) - 6*c**2)/(a**3*b + 9*a*b*c)) + (a*sqrt(-c)
/(2*c*(a**2 + c)) - 1/(2*(a**2 + c)))*log(x + (-4*a**6*c*(a*sqrt(-c)/(2*c*(a**2
+ c)) - 1/(2*(a**2 + c)))**2 + 4*a**4*c**2*(a*sqrt(-c)/(2*c*(a**2 + c)) - 1/(2*(
a**2 + c)))**2 - 6*a**4*c*(a*sqrt(-c)/(2*c*(a**2 + c)) - 1/(2*(a**2 + c))) + 20*
a**2*c**3*(a*sqrt(-c)/(2*c*(a**2 + c)) - 1/(2*(a**2 + c)))**2 - 12*a**2*c**2*(a*
sqrt(-c)/(2*c*(a**2 + c)) - 1/(2*(a**2 + c))) + 10*a**2*c + 12*c**4*(a*sqrt(-c)/
(2*c*(a**2 + c)) - 1/(2*(a**2 + c)))**2 - 6*c**3*(a*sqrt(-c)/(2*c*(a**2 + c)) -
1/(2*(a**2 + c))) - 6*c**2)/(a**3*b + 9*a*b*c)) + log(x + (-4*a**6*c/(a**2 + c)*
*2 + 4*a**4*c**2/(a**2 + c)**2 - 6*a**4*c/(a**2 + c) + 20*a**2*c**3/(a**2 + c)**
2 - 12*a**2*c**2/(a**2 + c) + 10*a**2*c + 12*c**4/(a**2 + c)**2 - 6*c**3/(a**2 +
c) - 6*c**2)/(a**3*b + 9*a*b*c))/(a**2 + c)

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GIAC/XCAS [A]  time = 0.26141, size = 84, normalized size = 1.42 \[ -\frac{a \arctan \left (\frac{b x + a}{\sqrt{c}}\right )}{{\left (a^{2} + c\right )} \sqrt{c}} - \frac{{\rm ln}\left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{2 \,{\left (a^{2} + c\right )}} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a^{2} + c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((b*x + a)^2 + c)*x),x, algorithm="giac")

[Out]

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