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

Optimal. Leaf size=87 \[ -\frac{3 \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{e \log \left (a+c x^2\right )}{2 a^2}-\frac{3 d}{2 a^2 x}+\frac{e \log (x)}{a^2}+\frac{d+e x}{2 a x \left (a+c x^2\right )} \]

[Out]

(-3*d)/(2*a^2*x) + (d + e*x)/(2*a*x*(a + c*x^2)) - (3*Sqrt[c]*d*ArcTan[(Sqrt[c]*
x)/Sqrt[a]])/(2*a^(5/2)) + (e*Log[x])/a^2 - (e*Log[a + c*x^2])/(2*a^2)

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Rubi [A]  time = 0.167183, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ -\frac{3 \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{e \log \left (a+c x^2\right )}{2 a^2}-\frac{3 d}{2 a^2 x}+\frac{e \log (x)}{a^2}+\frac{d+e x}{2 a x \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-3*d)/(2*a^2*x) + (d + e*x)/(2*a*x*(a + c*x^2)) - (3*Sqrt[c]*d*ArcTan[(Sqrt[c]*
x)/Sqrt[a]])/(2*a^(5/2)) + (e*Log[x])/a^2 - (e*Log[a + c*x^2])/(2*a^2)

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Rubi in Sympy [A]  time = 26.2692, size = 78, normalized size = 0.9 \[ \frac{d + e x}{2 a x \left (a + c x^{2}\right )} - \frac{3 d}{2 a^{2} x} + \frac{e \log{\left (x \right )}}{a^{2}} - \frac{e \log{\left (a + c x^{2} \right )}}{2 a^{2}} - \frac{3 \sqrt{c} d \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d + e*x)/(2*a*x*(a + c*x**2)) - 3*d/(2*a**2*x) + e*log(x)/a**2 - e*log(a + c*x*
*2)/(2*a**2) - 3*sqrt(c)*d*atan(sqrt(c)*x/sqrt(a))/(2*a**(5/2))

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Mathematica [A]  time = 0.30859, size = 81, normalized size = 0.93 \[ -\frac{3 \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{\frac{2 a d-a e x+3 c d x^2}{a x+c x^3}+e \log \left (a+c x^2\right )-2 e \log (x)}{2 a^2} \]

Antiderivative was successfully verified.

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

[Out]

(-3*Sqrt[c]*d*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(5/2)) - ((2*a*d - a*e*x + 3*c*d
*x^2)/(a*x + c*x^3) - 2*e*Log[x] + e*Log[a + c*x^2])/(2*a^2)

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Maple [A]  time = 0.016, size = 85, normalized size = 1. \[ -{\frac{d}{{a}^{2}x}}+{\frac{e\ln \left ( x \right ) }{{a}^{2}}}-{\frac{cdx}{2\,{a}^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{e}{2\,a \left ( c{x}^{2}+a \right ) }}-{\frac{e\ln \left ( c{x}^{2}+a \right ) }{2\,{a}^{2}}}-{\frac{3\,cd}{2\,{a}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/x^2/(c*x^2+a)^2,x)

[Out]

-d/a^2/x+e*ln(x)/a^2-1/2/a^2*c/(c*x^2+a)*x*d+1/2/a/(c*x^2+a)*e-1/2*e*ln(c*x^2+a)
/a^2-3/2/a^2*c*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((e*x + d)/((c*x^2 + a)^2*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.306468, size = 1, normalized size = 0.01 \[ \left [-\frac{6 \, c d x^{2} - 2 \, a e x - 3 \,{\left (c d x^{3} + a d x\right )} \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{2} - 2 \, a x \sqrt{-\frac{c}{a}} - a}{c x^{2} + a}\right ) + 4 \, a d + 2 \,{\left (c e x^{3} + a e x\right )} \log \left (c x^{2} + a\right ) - 4 \,{\left (c e x^{3} + a e x\right )} \log \left (x\right )}{4 \,{\left (a^{2} c x^{3} + a^{3} x\right )}}, -\frac{3 \, c d x^{2} - a e x + 3 \,{\left (c d x^{3} + a d x\right )} \sqrt{\frac{c}{a}} \arctan \left (\frac{c x}{a \sqrt{\frac{c}{a}}}\right ) + 2 \, a d +{\left (c e x^{3} + a e x\right )} \log \left (c x^{2} + a\right ) - 2 \,{\left (c e x^{3} + a e x\right )} \log \left (x\right )}{2 \,{\left (a^{2} c x^{3} + a^{3} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/4*(6*c*d*x^2 - 2*a*e*x - 3*(c*d*x^3 + a*d*x)*sqrt(-c/a)*log((c*x^2 - 2*a*x*s
qrt(-c/a) - a)/(c*x^2 + a)) + 4*a*d + 2*(c*e*x^3 + a*e*x)*log(c*x^2 + a) - 4*(c*
e*x^3 + a*e*x)*log(x))/(a^2*c*x^3 + a^3*x), -1/2*(3*c*d*x^2 - a*e*x + 3*(c*d*x^3
+ a*d*x)*sqrt(c/a)*arctan(c*x/(a*sqrt(c/a))) + 2*a*d + (c*e*x^3 + a*e*x)*log(c*
x^2 + a) - 2*(c*e*x^3 + a*e*x)*log(x))/(a^2*c*x^3 + a^3*x)]

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Sympy [A]  time = 5.78469, size = 389, normalized size = 4.47 \[ \left (- \frac{e}{2 a^{2}} - \frac{3 d \sqrt{- a^{5} c}}{4 a^{5}}\right ) \log{\left (x + \frac{32 a^{6} e \left (- \frac{e}{2 a^{2}} - \frac{3 d \sqrt{- a^{5} c}}{4 a^{5}}\right )^{2} - 16 a^{4} e^{2} \left (- \frac{e}{2 a^{2}} - \frac{3 d \sqrt{- a^{5} c}}{4 a^{5}}\right ) - 12 a^{3} c d^{2} \left (- \frac{e}{2 a^{2}} - \frac{3 d \sqrt{- a^{5} c}}{4 a^{5}}\right ) - 16 a^{2} e^{3} + 12 a c d^{2} e}{36 a c d e^{2} + 9 c^{2} d^{3}} \right )} + \left (- \frac{e}{2 a^{2}} + \frac{3 d \sqrt{- a^{5} c}}{4 a^{5}}\right ) \log{\left (x + \frac{32 a^{6} e \left (- \frac{e}{2 a^{2}} + \frac{3 d \sqrt{- a^{5} c}}{4 a^{5}}\right )^{2} - 16 a^{4} e^{2} \left (- \frac{e}{2 a^{2}} + \frac{3 d \sqrt{- a^{5} c}}{4 a^{5}}\right ) - 12 a^{3} c d^{2} \left (- \frac{e}{2 a^{2}} + \frac{3 d \sqrt{- a^{5} c}}{4 a^{5}}\right ) - 16 a^{2} e^{3} + 12 a c d^{2} e}{36 a c d e^{2} + 9 c^{2} d^{3}} \right )} - \frac{2 a d - a e x + 3 c d x^{2}}{2 a^{3} x + 2 a^{2} c x^{3}} + \frac{e \log{\left (x \right )}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-e/(2*a**2) - 3*d*sqrt(-a**5*c)/(4*a**5))*log(x + (32*a**6*e*(-e/(2*a**2) - 3*d
*sqrt(-a**5*c)/(4*a**5))**2 - 16*a**4*e**2*(-e/(2*a**2) - 3*d*sqrt(-a**5*c)/(4*a
**5)) - 12*a**3*c*d**2*(-e/(2*a**2) - 3*d*sqrt(-a**5*c)/(4*a**5)) - 16*a**2*e**3
+ 12*a*c*d**2*e)/(36*a*c*d*e**2 + 9*c**2*d**3)) + (-e/(2*a**2) + 3*d*sqrt(-a**5
*c)/(4*a**5))*log(x + (32*a**6*e*(-e/(2*a**2) + 3*d*sqrt(-a**5*c)/(4*a**5))**2 -
16*a**4*e**2*(-e/(2*a**2) + 3*d*sqrt(-a**5*c)/(4*a**5)) - 12*a**3*c*d**2*(-e/(2
*a**2) + 3*d*sqrt(-a**5*c)/(4*a**5)) - 16*a**2*e**3 + 12*a*c*d**2*e)/(36*a*c*d*e
**2 + 9*c**2*d**3)) - (2*a*d - a*e*x + 3*c*d*x**2)/(2*a**3*x + 2*a**2*c*x**3) +
e*log(x)/a**2

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GIAC/XCAS [A]  time = 0.270914, size = 108, normalized size = 1.24 \[ -\frac{3 \, c d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} a^{2}} - \frac{e{\rm ln}\left (c x^{2} + a\right )}{2 \, a^{2}} + \frac{e{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} - \frac{3 \, c d x^{2} - a x e + 2 \, a d}{2 \,{\left (c x^{3} + a x\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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