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

Optimal. Leaf size=142 \[ \frac{\sqrt{c} d \left (3 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}-\frac{e^3 \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^2}+\frac{e^3 \log (d+e x)}{\left (a e^2+c d^2\right )^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.319808, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{\sqrt{c} d \left (3 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}-\frac{e^3 \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^2}+\frac{e^3 \log (d+e x)}{\left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 50.1138, size = 126, normalized size = 0.89 \[ - \frac{e^{3} \log{\left (a + c x^{2} \right )}}{2 \left (a e^{2} + c d^{2}\right )^{2}} + \frac{e^{3} \log{\left (d + e x \right )}}{\left (a e^{2} + c d^{2}\right )^{2}} + \frac{a e + c d x}{2 a \left (a + c x^{2}\right ) \left (a e^{2} + c d^{2}\right )} + \frac{\sqrt{c} d \left (3 a e^{2} + c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.195253, size = 138, normalized size = 0.97 \[ \frac{\sqrt{c} d \left (a+c x^2\right ) \left (3 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )+\sqrt{a} \left (\left (a e^2+c d^2\right ) (a e+c d x)+2 a e^3 \left (a+c x^2\right ) \log (d+e x)-a e^3 \left (a+c x^2\right ) \log \left (a+c x^2\right )\right )}{2 a^{3/2} \left (a+c x^2\right ) \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.029, size = 246, normalized size = 1.7 \[{\frac{{e}^{3}\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}+{\frac{cdx{e}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{{c}^{2}{d}^{3}x}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) a}}+{\frac{{e}^{3}a}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{ce{d}^{2}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{{e}^{3}\ln \left ( a \left ( c{x}^{2}+a \right ) \right ) }{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}+{\frac{3\,d{e}^{2}c}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{{c}^{2}{d}^{3}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)/(c*x^2+a)^2,x)

[Out]

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

_______________________________________________________________________________________

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)^2*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.753943, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, a c d^{2} e + 2 \, a^{2} e^{3} +{\left (a c d^{3} + 3 \, a^{2} d e^{2} +{\left (c^{2} d^{3} + 3 \, a c d e^{2}\right )} x^{2}\right )} \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{2} + 2 \, a x \sqrt{-\frac{c}{a}} - a}{c x^{2} + a}\right ) + 2 \,{\left (c^{2} d^{3} + a c d e^{2}\right )} x - 2 \,{\left (a c e^{3} x^{2} + a^{2} e^{3}\right )} \log \left (c x^{2} + a\right ) + 4 \,{\left (a c e^{3} x^{2} + a^{2} e^{3}\right )} \log \left (e x + d\right )}{4 \,{\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4} +{\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}\right )} x^{2}\right )}}, \frac{a c d^{2} e + a^{2} e^{3} +{\left (a c d^{3} + 3 \, a^{2} d e^{2} +{\left (c^{2} d^{3} + 3 \, a c d e^{2}\right )} x^{2}\right )} \sqrt{\frac{c}{a}} \arctan \left (\frac{c x}{a \sqrt{\frac{c}{a}}}\right ) +{\left (c^{2} d^{3} + a c d e^{2}\right )} x -{\left (a c e^{3} x^{2} + a^{2} e^{3}\right )} \log \left (c x^{2} + a\right ) + 2 \,{\left (a c e^{3} x^{2} + a^{2} e^{3}\right )} \log \left (e x + d\right )}{2 \,{\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4} +{\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}\right )} x^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.214105, size = 259, normalized size = 1.82 \[ -\frac{e^{3}{\rm ln}\left (c x^{2} + a\right )}{2 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} + \frac{e^{4}{\rm ln}\left ({\left | x e + d \right |}\right )}{c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}} + \frac{{\left (c^{2} d^{3} + 3 \, a c d e^{2}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \,{\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt{a c}} + \frac{a c d^{2} e + a^{2} e^{3} +{\left (c^{2} d^{3} + a c d e^{2}\right )} x}{2 \,{\left (c d^{2} + a e^{2}\right )}^{2}{\left (c x^{2} + a\right )} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*e^3*ln(c*x^2 + a)/(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4) + e^4*ln(abs(x*e + d)
)/(c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5) + 1/2*(c^2*d^3 + 3*a*c*d*e^2)*arctan(c*x
/sqrt(a*c))/((a*c^2*d^4 + 2*a^2*c*d^2*e^2 + a^3*e^4)*sqrt(a*c)) + 1/2*(a*c*d^2*e
+ a^2*e^3 + (c^2*d^3 + a*c*d*e^2)*x)/((c*d^2 + a*e^2)^2*(c*x^2 + a)*a)