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

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)*ArcTan[(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
)

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Rubi [A]  time = 0.125904, 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, Rules used = {741, 801, 635, 205, 260} \[ \frac{\sqrt{c} d \left (3 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}-\frac{e^3 \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^2}+\frac{e^3 \log (d+e x)}{\left (a e^2+c d^2\right )^2} \]

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)*ArcTan[(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
)

Rule 741

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(a*e + c*d*x)*(
a + c*x^2)^(p + 1))/(2*a*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*
Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a
, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{1}{(d+e x) \left (a+c x^2\right )^2} \, dx &=\frac{a e+c d x}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac{\int \frac{-c d^2-2 a e^2-c d e x}{(d+e x) \left (a+c x^2\right )} \, dx}{2 a \left (c d^2+a e^2\right )}\\ &=\frac{a e+c d x}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac{\int \left (-\frac{2 a e^4}{\left (c d^2+a e^2\right ) (d+e x)}-\frac{c \left (c d^3+3 a d e^2-2 a e^3 x\right )}{\left (c d^2+a e^2\right ) \left (a+c x^2\right )}\right ) \, dx}{2 a \left (c d^2+a e^2\right )}\\ &=\frac{a e+c d x}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac{e^3 \log (d+e x)}{\left (c d^2+a e^2\right )^2}+\frac{c \int \frac{c d^3+3 a d e^2-2 a e^3 x}{a+c x^2} \, dx}{2 a \left (c d^2+a e^2\right )^2}\\ &=\frac{a e+c d x}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac{e^3 \log (d+e x)}{\left (c d^2+a e^2\right )^2}-\frac{\left (c e^3\right ) \int \frac{x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^2}+\frac{\left (c d \left (c d^2+3 a e^2\right )\right ) \int \frac{1}{a+c x^2} \, dx}{2 a \left (c d^2+a e^2\right )^2}\\ &=\frac{a e+c d x}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac{\sqrt{c} d \left (c d^2+3 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \left (c d^2+a e^2\right )^2}+\frac{e^3 \log (d+e x)}{\left (c d^2+a e^2\right )^2}-\frac{e^3 \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.100066, size = 138, normalized size = 0.97 \[ \frac{\sqrt{a} \left (\left (a e^2+c d^2\right ) (a e+c d x)+2 a e^3 \left (a+c x^2\right ) \log (d+e x)-a e^3 \left (a+c x^2\right ) \log \left (a+c x^2\right )\right )+\sqrt{c} d \left (a+c x^2\right ) \left (3 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \left (a+c x^2\right ) \left (a e^2+c d^2\right )^2} \]

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))

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Maple [A]  time = 0.087, size = 244, normalized size = 1.7 \begin{align*}{\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{a{e}^{3}}{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 ( c{x}^{2}+a \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}}}}+{\frac{{e}^{3}\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)/(c*x^2+a)^2,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.88093, size = 900, normalized size = 6.34 \begin{align*} \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 (x \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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [B]  time = 120.809, size = 3225, normalized size = 22.71 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e**3*log(x + (-96*a**9*e**18/(a*e**2 + c*d**2)**4 - 336*a**8*c*d**2*e**16/(a*e**2 + c*d**2)**4 - 368*a**7*c**2
*d**4*e**14/(a*e**2 + c*d**2)**4 + 48*a**7*e**14/(a*e**2 + c*d**2)**2 - 32*a**6*c**3*d**6*e**12/(a*e**2 + c*d*
*2)**4 + 180*a**6*c*d**2*e**12/(a*e**2 + c*d**2)**2 + 192*a**5*c**4*d**8*e**10/(a*e**2 + c*d**2)**4 + 256*a**5
*c**2*d**4*e**10/(a*e**2 + c*d**2)**2 + 48*a**5*e**10 + 112*a**4*c**5*d**10*e**8/(a*e**2 + c*d**2)**4 + 168*a*
*4*c**3*d**6*e**8/(a*e**2 + c*d**2)**2 - 24*a**4*c*d**2*e**8 + 16*a**3*c**6*d**12*e**6/(a*e**2 + c*d**2)**4 +
48*a**3*c**4*d**8*e**6/(a*e**2 + c*d**2)**2 + 7*a**3*c**2*d**4*e**6 + 4*a**2*c**5*d**10*e**4/(a*e**2 + c*d**2)
**2 + 23*a**2*c**3*d**6*e**4 + 9*a*c**4*d**8*e**2 + c**5*d**10)/(108*a**4*c*d*e**9 + 63*a**3*c**2*d**3*e**7 +
27*a**2*c**3*d**5*e**5 + 9*a*c**4*d**7*e**3 + c**5*d**9*e))/(a*e**2 + c*d**2)**2 + (a*e + c*d*x)/(2*a**3*e**2
+ 2*a**2*c*d**2 + x**2*(2*a**2*c*e**2 + 2*a*c**2*d**2)) + (-e**3/(2*(a*e**2 + c*d**2)**2) - d*sqrt(-a**3*c)*(3
*a*e**2 + c*d**2)/(4*a**3*(a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4)))*log(x + (-96*a**9*e**12*(-e**3/(2*(a*e**
2 + c*d**2)**2) - d*sqrt(-a**3*c)*(3*a*e**2 + c*d**2)/(4*a**3*(a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4)))**2 -
336*a**8*c*d**2*e**10*(-e**3/(2*(a*e**2 + c*d**2)**2) - d*sqrt(-a**3*c)*(3*a*e**2 + c*d**2)/(4*a**3*(a**2*e**
4 + 2*a*c*d**2*e**2 + c**2*d**4)))**2 - 368*a**7*c**2*d**4*e**8*(-e**3/(2*(a*e**2 + c*d**2)**2) - d*sqrt(-a**3
*c)*(3*a*e**2 + c*d**2)/(4*a**3*(a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4)))**2 + 48*a**7*e**11*(-e**3/(2*(a*e*
*2 + c*d**2)**2) - d*sqrt(-a**3*c)*(3*a*e**2 + c*d**2)/(4*a**3*(a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4))) - 3
2*a**6*c**3*d**6*e**6*(-e**3/(2*(a*e**2 + c*d**2)**2) - d*sqrt(-a**3*c)*(3*a*e**2 + c*d**2)/(4*a**3*(a**2*e**4
+ 2*a*c*d**2*e**2 + c**2*d**4)))**2 + 180*a**6*c*d**2*e**9*(-e**3/(2*(a*e**2 + c*d**2)**2) - d*sqrt(-a**3*c)*
(3*a*e**2 + c*d**2)/(4*a**3*(a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4))) + 192*a**5*c**4*d**8*e**4*(-e**3/(2*(a
*e**2 + c*d**2)**2) - d*sqrt(-a**3*c)*(3*a*e**2 + c*d**2)/(4*a**3*(a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4)))*
*2 + 256*a**5*c**2*d**4*e**7*(-e**3/(2*(a*e**2 + c*d**2)**2) - d*sqrt(-a**3*c)*(3*a*e**2 + c*d**2)/(4*a**3*(a*
*2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4))) + 48*a**5*e**10 + 112*a**4*c**5*d**10*e**2*(-e**3/(2*(a*e**2 + c*d**2
)**2) - d*sqrt(-a**3*c)*(3*a*e**2 + c*d**2)/(4*a**3*(a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4)))**2 + 168*a**4*
c**3*d**6*e**5*(-e**3/(2*(a*e**2 + c*d**2)**2) - d*sqrt(-a**3*c)*(3*a*e**2 + c*d**2)/(4*a**3*(a**2*e**4 + 2*a*
c*d**2*e**2 + c**2*d**4))) - 24*a**4*c*d**2*e**8 + 16*a**3*c**6*d**12*(-e**3/(2*(a*e**2 + c*d**2)**2) - d*sqrt
(-a**3*c)*(3*a*e**2 + c*d**2)/(4*a**3*(a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4)))**2 + 48*a**3*c**4*d**8*e**3*
(-e**3/(2*(a*e**2 + c*d**2)**2) - d*sqrt(-a**3*c)*(3*a*e**2 + c*d**2)/(4*a**3*(a**2*e**4 + 2*a*c*d**2*e**2 + c
**2*d**4))) + 7*a**3*c**2*d**4*e**6 + 4*a**2*c**5*d**10*e*(-e**3/(2*(a*e**2 + c*d**2)**2) - d*sqrt(-a**3*c)*(3
*a*e**2 + c*d**2)/(4*a**3*(a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4))) + 23*a**2*c**3*d**6*e**4 + 9*a*c**4*d**8
*e**2 + c**5*d**10)/(108*a**4*c*d*e**9 + 63*a**3*c**2*d**3*e**7 + 27*a**2*c**3*d**5*e**5 + 9*a*c**4*d**7*e**3
+ c**5*d**9*e)) + (-e**3/(2*(a*e**2 + c*d**2)**2) + d*sqrt(-a**3*c)*(3*a*e**2 + c*d**2)/(4*a**3*(a**2*e**4 + 2
*a*c*d**2*e**2 + c**2*d**4)))*log(x + (-96*a**9*e**12*(-e**3/(2*(a*e**2 + c*d**2)**2) + d*sqrt(-a**3*c)*(3*a*e
**2 + c*d**2)/(4*a**3*(a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4)))**2 - 336*a**8*c*d**2*e**10*(-e**3/(2*(a*e**2
+ c*d**2)**2) + d*sqrt(-a**3*c)*(3*a*e**2 + c*d**2)/(4*a**3*(a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4)))**2 -
368*a**7*c**2*d**4*e**8*(-e**3/(2*(a*e**2 + c*d**2)**2) + d*sqrt(-a**3*c)*(3*a*e**2 + c*d**2)/(4*a**3*(a**2*e*
*4 + 2*a*c*d**2*e**2 + c**2*d**4)))**2 + 48*a**7*e**11*(-e**3/(2*(a*e**2 + c*d**2)**2) + d*sqrt(-a**3*c)*(3*a*
e**2 + c*d**2)/(4*a**3*(a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4))) - 32*a**6*c**3*d**6*e**6*(-e**3/(2*(a*e**2
+ c*d**2)**2) + d*sqrt(-a**3*c)*(3*a*e**2 + c*d**2)/(4*a**3*(a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4)))**2 + 1
80*a**6*c*d**2*e**9*(-e**3/(2*(a*e**2 + c*d**2)**2) + d*sqrt(-a**3*c)*(3*a*e**2 + c*d**2)/(4*a**3*(a**2*e**4 +
2*a*c*d**2*e**2 + c**2*d**4))) + 192*a**5*c**4*d**8*e**4*(-e**3/(2*(a*e**2 + c*d**2)**2) + d*sqrt(-a**3*c)*(3
*a*e**2 + c*d**2)/(4*a**3*(a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4)))**2 + 256*a**5*c**2*d**4*e**7*(-e**3/(2*(
a*e**2 + c*d**2)**2) + d*sqrt(-a**3*c)*(3*a*e**2 + c*d**2)/(4*a**3*(a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4)))
+ 48*a**5*e**10 + 112*a**4*c**5*d**10*e**2*(-e**3/(2*(a*e**2 + c*d**2)**2) + d*sqrt(-a**3*c)*(3*a*e**2 + c*d*
*2)/(4*a**3*(a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4)))**2 + 168*a**4*c**3*d**6*e**5*(-e**3/(2*(a*e**2 + c*d**
2)**2) + d*sqrt(-a**3*c)*(3*a*e**2 + c*d**2)/(4*a**3*(a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4))) - 24*a**4*c*d
**2*e**8 + 16*a**3*c**6*d**12*(-e**3/(2*(a*e**2 + c*d**2)**2) + d*sqrt(-a**3*c)*(3*a*e**2 + c*d**2)/(4*a**3*(a
**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4)))**2 + 48*a**3*c**4*d**8*e**3*(-e**3/(2*(a*e**2 + c*d**2)**2) + d*sqrt
(-a**3*c)*(3*a*e**2 + c*d**2)/(4*a**3*(a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4))) + 7*a**3*c**2*d**4*e**6 + 4*
a**2*c**5*d**10*e*(-e**3/(2*(a*e**2 + c*d**2)**2) + d*sqrt(-a**3*c)*(3*a*e**2 + c*d**2)/(4*a**3*(a**2*e**4 + 2
*a*c*d**2*e**2 + c**2*d**4))) + 23*a**2*c**3*d**6*e**4 + 9*a*c**4*d**8*e**2 + c**5*d**10)/(108*a**4*c*d*e**9 +
63*a**3*c**2*d**3*e**7 + 27*a**2*c**3*d**5*e**5 + 9*a*c**4*d**7*e**3 + c**5*d**9*e))

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Giac [A]  time = 1.32524, size = 259, normalized size = 1.82 \begin{align*} -\frac{e^{3} \log \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} \log \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*e^3*log(c*x^2 + a)/(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4) + e^4*log(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)