3.491 \(\int \frac{1}{(d+e x) \left (a+c x^2\right )} \, dx\)

Optimal. Leaf size=86 \[ -\frac{e \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )}+\frac{e \log (d+e x)}{a e^2+c d^2}+\frac{\sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \left (a e^2+c d^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0991698, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ -\frac{e \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )}+\frac{e \log (d+e x)}{a e^2+c d^2}+\frac{\sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.9293, size = 76, normalized size = 0.88 \[ - \frac{e \log{\left (a + c x^{2} \right )}}{2 \left (a e^{2} + c d^{2}\right )} + \frac{e \log{\left (d + e x \right )}}{a e^{2} + c d^{2}} + \frac{\sqrt{c} d \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{\sqrt{a} \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e*log(a + c*x**2)/(2*(a*e**2 + c*d**2)) + e*log(d + e*x)/(a*e**2 + c*d**2) + sq
rt(c)*d*atan(sqrt(c)*x/sqrt(a))/(sqrt(a)*(a*e**2 + c*d**2))

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Mathematica [A]  time = 0.067767, size = 63, normalized size = 0.73 \[ \frac{\frac{2 \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a}}-e \log \left (a+c x^2\right )+2 e \log (d+e x)}{2 a e^2+2 c d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 77, normalized size = 0.9 \[{\frac{e\ln \left ( ex+d \right ) }{a{e}^{2}+c{d}^{2}}}-{\frac{e\ln \left ( c{x}^{2}+a \right ) }{2\,a{e}^{2}+2\,c{d}^{2}}}+{\frac{cd}{a{e}^{2}+c{d}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)/(c*x^2+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229379, size = 1, normalized size = 0.01 \[ \left [\frac{d \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{2} + 2 \, a x \sqrt{-\frac{c}{a}} - a}{c x^{2} + a}\right ) - e \log \left (c x^{2} + a\right ) + 2 \, e \log \left (e x + d\right )}{2 \,{\left (c d^{2} + a e^{2}\right )}}, \frac{2 \, d \sqrt{\frac{c}{a}} \arctan \left (\frac{c x}{a \sqrt{\frac{c}{a}}}\right ) - e \log \left (c x^{2} + a\right ) + 2 \, e \log \left (e x + d\right )}{2 \,{\left (c d^{2} + a e^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)*(e*x + d)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 11.603, size = 1134, normalized size = 13.19 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*log(x + (-12*a**4*e**8/(a*e**2 + c*d**2)**2 - 20*a**3*c*d**2*e**6/(a*e**2 + c*
d**2)**2 + 6*a**3*e**6/(a*e**2 + c*d**2) - 4*a**2*c**2*d**4*e**4/(a*e**2 + c*d**
2)**2 + 12*a**2*c*d**2*e**4/(a*e**2 + c*d**2) + 6*a**2*e**4 + 4*a*c**3*d**6*e**2
/(a*e**2 + c*d**2)**2 + 6*a*c**2*d**4*e**2/(a*e**2 + c*d**2) - a*c*d**2*e**2 + c
**2*d**4)/(9*a*c*d*e**3 + c**2*d**3*e))/(a*e**2 + c*d**2) + (-e/(2*(a*e**2 + c*d
**2)) - d*sqrt(-a*c)/(2*a*(a*e**2 + c*d**2)))*log(x + (-12*a**4*e**6*(-e/(2*(a*e
**2 + c*d**2)) - d*sqrt(-a*c)/(2*a*(a*e**2 + c*d**2)))**2 - 20*a**3*c*d**2*e**4*
(-e/(2*(a*e**2 + c*d**2)) - d*sqrt(-a*c)/(2*a*(a*e**2 + c*d**2)))**2 + 6*a**3*e*
*5*(-e/(2*(a*e**2 + c*d**2)) - d*sqrt(-a*c)/(2*a*(a*e**2 + c*d**2))) - 4*a**2*c*
*2*d**4*e**2*(-e/(2*(a*e**2 + c*d**2)) - d*sqrt(-a*c)/(2*a*(a*e**2 + c*d**2)))**
2 + 12*a**2*c*d**2*e**3*(-e/(2*(a*e**2 + c*d**2)) - d*sqrt(-a*c)/(2*a*(a*e**2 +
c*d**2))) + 6*a**2*e**4 + 4*a*c**3*d**6*(-e/(2*(a*e**2 + c*d**2)) - d*sqrt(-a*c)
/(2*a*(a*e**2 + c*d**2)))**2 + 6*a*c**2*d**4*e*(-e/(2*(a*e**2 + c*d**2)) - d*sqr
t(-a*c)/(2*a*(a*e**2 + c*d**2))) - a*c*d**2*e**2 + c**2*d**4)/(9*a*c*d*e**3 + c*
*2*d**3*e)) + (-e/(2*(a*e**2 + c*d**2)) + d*sqrt(-a*c)/(2*a*(a*e**2 + c*d**2)))*
log(x + (-12*a**4*e**6*(-e/(2*(a*e**2 + c*d**2)) + d*sqrt(-a*c)/(2*a*(a*e**2 + c
*d**2)))**2 - 20*a**3*c*d**2*e**4*(-e/(2*(a*e**2 + c*d**2)) + d*sqrt(-a*c)/(2*a*
(a*e**2 + c*d**2)))**2 + 6*a**3*e**5*(-e/(2*(a*e**2 + c*d**2)) + d*sqrt(-a*c)/(2
*a*(a*e**2 + c*d**2))) - 4*a**2*c**2*d**4*e**2*(-e/(2*(a*e**2 + c*d**2)) + d*sqr
t(-a*c)/(2*a*(a*e**2 + c*d**2)))**2 + 12*a**2*c*d**2*e**3*(-e/(2*(a*e**2 + c*d**
2)) + d*sqrt(-a*c)/(2*a*(a*e**2 + c*d**2))) + 6*a**2*e**4 + 4*a*c**3*d**6*(-e/(2
*(a*e**2 + c*d**2)) + d*sqrt(-a*c)/(2*a*(a*e**2 + c*d**2)))**2 + 6*a*c**2*d**4*e
*(-e/(2*(a*e**2 + c*d**2)) + d*sqrt(-a*c)/(2*a*(a*e**2 + c*d**2))) - a*c*d**2*e*
*2 + c**2*d**4)/(9*a*c*d*e**3 + c**2*d**3*e))

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GIAC/XCAS [A]  time = 0.214597, size = 107, normalized size = 1.24 \[ \frac{c d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{{\left (c d^{2} + a e^{2}\right )} \sqrt{a c}} - \frac{e{\rm ln}\left (c x^{2} + a\right )}{2 \,{\left (c d^{2} + a e^{2}\right )}} + \frac{e^{2}{\rm ln}\left ({\left | x e + d \right |}\right )}{c d^{2} e + a e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*d*arctan(c*x/sqrt(a*c))/((c*d^2 + a*e^2)*sqrt(a*c)) - 1/2*e*ln(c*x^2 + a)/(c*d
^2 + a*e^2) + e^2*ln(abs(x*e + d))/(c*d^2*e + a*e^3)