### 3.504 $$\int \frac{1}{(d+e x)^3 (a+c x^2)} \, dx$$

Optimal. Leaf size=176 $\frac{c^{3/2} d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \left (a e^2+c d^2\right )^3}-\frac{c e \left (3 c d^2-a e^2\right ) \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^3}-\frac{2 c d e}{(d+e x) \left (a e^2+c d^2\right )^2}-\frac{e}{2 (d+e x)^2 \left (a e^2+c d^2\right )}+\frac{c e \left (3 c d^2-a e^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^3}$

[Out]

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

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Rubi [A]  time = 0.166154, antiderivative size = 176, 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 = {710, 801, 635, 205, 260} $\frac{c^{3/2} d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \left (a e^2+c d^2\right )^3}-\frac{c e \left (3 c d^2-a e^2\right ) \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^3}-\frac{2 c d e}{(d+e x) \left (a e^2+c d^2\right )^2}-\frac{e}{2 (d+e x)^2 \left (a e^2+c d^2\right )}+\frac{c e \left (3 c d^2-a e^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)^3*(a + c*x^2)),x]

[Out]

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

Rule 710

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

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

Mathematica [A]  time = 0.289447, size = 140, normalized size = 0.8 $\frac{\frac{2 c^{3/2} d \left (c d^2-3 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a}}+e \left (c \left (a e^2-3 c d^2\right ) \log \left (a+c x^2\right )-\frac{\left (a e^2+c d^2\right ) \left (a e^2+c d (5 d+4 e x)\right )}{(d+e x)^2}+2 c \left (3 c d^2-a e^2\right ) \log (d+e x)\right )}{2 \left (a e^2+c d^2\right )^3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((d + e*x)^3*(a + c*x^2)),x]

[Out]

((2*c^(3/2)*d*(c*d^2 - 3*a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/Sqrt[a] + e*(-(((c*d^2 + a*e^2)*(a*e^2 + c*d*(5*d
+ 4*e*x)))/(d + e*x)^2) + 2*c*(3*c*d^2 - a*e^2)*Log[d + e*x] + c*(-3*c*d^2 + a*e^2)*Log[a + c*x^2]))/(2*(c*d^
2 + a*e^2)^3)

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Maple [A]  time = 0.062, size = 233, normalized size = 1.3 \begin{align*}{\frac{c\ln \left ( c{x}^{2}+a \right ) a{e}^{3}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}}-{\frac{3\,{c}^{2}\ln \left ( c{x}^{2}+a \right ){d}^{2}e}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}}-3\,{\frac{a{c}^{2}d{e}^{2}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }+{\frac{{c}^{3}{d}^{3}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{e}{ \left ( 2\,a{e}^{2}+2\,c{d}^{2} \right ) \left ( ex+d \right ) ^{2}}}-2\,{\frac{cde}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( ex+d \right ) }}-{\frac{c{e}^{3}\ln \left ( ex+d \right ) a}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}}+3\,{\frac{{c}^{2}e\ln \left ( ex+d \right ){d}^{2}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^3/(c*x^2+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 8.88927, size = 1716, normalized size = 9.75 \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)^3/(c*x^2+a),x, algorithm="fricas")

[Out]

[-1/2*(5*c^2*d^4*e + 6*a*c*d^2*e^3 + a^2*e^5 + (c^2*d^5 - 3*a*c*d^3*e^2 + (c^2*d^3*e^2 - 3*a*c*d*e^4)*x^2 + 2*
(c^2*d^4*e - 3*a*c*d^2*e^3)*x)*sqrt(-c/a)*log((c*x^2 - 2*a*x*sqrt(-c/a) - a)/(c*x^2 + a)) + 4*(c^2*d^3*e^2 + a
*c*d*e^4)*x + (3*c^2*d^4*e - a*c*d^2*e^3 + (3*c^2*d^2*e^3 - a*c*e^5)*x^2 + 2*(3*c^2*d^3*e^2 - a*c*d*e^4)*x)*lo
g(c*x^2 + a) - 2*(3*c^2*d^4*e - a*c*d^2*e^3 + (3*c^2*d^2*e^3 - a*c*e^5)*x^2 + 2*(3*c^2*d^3*e^2 - a*c*d*e^4)*x)
*log(e*x + d))/(c^3*d^8 + 3*a*c^2*d^6*e^2 + 3*a^2*c*d^4*e^4 + a^3*d^2*e^6 + (c^3*d^6*e^2 + 3*a*c^2*d^4*e^4 + 3
*a^2*c*d^2*e^6 + a^3*e^8)*x^2 + 2*(c^3*d^7*e + 3*a*c^2*d^5*e^3 + 3*a^2*c*d^3*e^5 + a^3*d*e^7)*x), -1/2*(5*c^2*
d^4*e + 6*a*c*d^2*e^3 + a^2*e^5 - 2*(c^2*d^5 - 3*a*c*d^3*e^2 + (c^2*d^3*e^2 - 3*a*c*d*e^4)*x^2 + 2*(c^2*d^4*e
- 3*a*c*d^2*e^3)*x)*sqrt(c/a)*arctan(x*sqrt(c/a)) + 4*(c^2*d^3*e^2 + a*c*d*e^4)*x + (3*c^2*d^4*e - a*c*d^2*e^3
+ (3*c^2*d^2*e^3 - a*c*e^5)*x^2 + 2*(3*c^2*d^3*e^2 - a*c*d*e^4)*x)*log(c*x^2 + a) - 2*(3*c^2*d^4*e - a*c*d^2*
e^3 + (3*c^2*d^2*e^3 - a*c*e^5)*x^2 + 2*(3*c^2*d^3*e^2 - a*c*d*e^4)*x)*log(e*x + d))/(c^3*d^8 + 3*a*c^2*d^6*e^
2 + 3*a^2*c*d^4*e^4 + a^3*d^2*e^6 + (c^3*d^6*e^2 + 3*a*c^2*d^4*e^4 + 3*a^2*c*d^2*e^6 + a^3*e^8)*x^2 + 2*(c^3*d
^7*e + 3*a*c^2*d^5*e^3 + 3*a^2*c*d^3*e^5 + a^3*d*e^7)*x)]

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Sympy [B]  time = 116.536, size = 4996, normalized size = 28.39 \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)**3/(c*x**2+a),x)

[Out]

-c*e*(a*e**2 - 3*c*d**2)*log(x + (-12*a**9*c**2*e**18*(a*e**2 - 3*c*d**2)**2/(a*e**2 + c*d**2)**6 - 24*a**8*c*
*3*d**2*e**16*(a*e**2 - 3*c*d**2)**2/(a*e**2 + c*d**2)**6 + 104*a**7*c**4*d**4*e**14*(a*e**2 - 3*c*d**2)**2/(a
*e**2 + c*d**2)**6 + 6*a**7*c**2*e**14*(a*e**2 - 3*c*d**2)/(a*e**2 + c*d**2)**3 + 456*a**6*c**5*d**6*e**12*(a*
e**2 - 3*c*d**2)**2/(a*e**2 + c*d**2)**6 + 12*a**6*c**3*d**2*e**12*(a*e**2 - 3*c*d**2)/(a*e**2 + c*d**2)**3 +
720*a**5*c**6*d**8*e**10*(a*e**2 - 3*c*d**2)**2/(a*e**2 + c*d**2)**6 + 2*a**5*c**4*d**4*e**10*(a*e**2 - 3*c*d*
*2)/(a*e**2 + c*d**2)**3 + 6*a**5*c**2*e**10 + 568*a**4*c**7*d**10*e**8*(a*e**2 - 3*c*d**2)**2/(a*e**2 + c*d**
2)**6 + 8*a**4*c**5*d**6*e**8*(a*e**2 - 3*c*d**2)/(a*e**2 + c*d**2)**3 - 69*a**4*c**3*d**2*e**8 + 216*a**3*c**
8*d**12*e**6*(a*e**2 - 3*c*d**2)**2/(a*e**2 + c*d**2)**6 + 42*a**3*c**6*d**8*e**6*(a*e**2 - 3*c*d**2)/(a*e**2
+ c*d**2)**3 + 236*a**3*c**4*d**4*e**6 + 24*a**2*c**9*d**14*e**4*(a*e**2 - 3*c*d**2)**2/(a*e**2 + c*d**2)**6 +
44*a**2*c**7*d**10*e**4*(a*e**2 - 3*c*d**2)/(a*e**2 + c*d**2)**3 - 194*a**2*c**5*d**6*e**4 - 4*a*c**10*d**16*
e**2*(a*e**2 - 3*c*d**2)**2/(a*e**2 + c*d**2)**6 + 14*a*c**8*d**12*e**2*(a*e**2 - 3*c*d**2)/(a*e**2 + c*d**2)*
*3 + 6*a*c**6*d**8*e**2 - c**7*d**10)/(27*a**4*c**3*d*e**9 - 144*a**3*c**4*d**3*e**7 + 270*a**2*c**5*d**5*e**5
- 72*a*c**6*d**7*e**3 - c**7*d**9*e))/(a*e**2 + c*d**2)**3 + (c*e*(a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*d**2)**3
) - d*sqrt(-a*c**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6)
))*log(x + (-12*a**9*e**16*(c*e*(a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*d**2)**3) - d*sqrt(-a*c**3)*(3*a*e**2 - c*d
**2)/(2*a*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6)))**2 - 24*a**8*c*d**2*e**14*(c*e*(
a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*d**2)**3) - d*sqrt(-a*c**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3*e**6 + 3*a**2*c*d
**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6)))**2 + 104*a**7*c**2*d**4*e**12*(c*e*(a*e**2 - 3*c*d**2)/(2*(a*e**2
+ c*d**2)**3) - d*sqrt(-a*c**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2
+ c**3*d**6)))**2 - 6*a**7*c*e**13*(c*e*(a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*d**2)**3) - d*sqrt(-a*c**3)*(3*a*e
**2 - c*d**2)/(2*a*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6))) + 456*a**6*c**3*d**6*e*
*10*(c*e*(a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*d**2)**3) - d*sqrt(-a*c**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3*e**6 +
3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6)))**2 - 12*a**6*c**2*d**2*e**11*(c*e*(a*e**2 - 3*c*d**2)/(
2*(a*e**2 + c*d**2)**3) - d*sqrt(-a*c**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*
d**4*e**2 + c**3*d**6))) + 720*a**5*c**4*d**8*e**8*(c*e*(a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*d**2)**3) - d*sqrt(
-a*c**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6)))**2 - 2*a
**5*c**3*d**4*e**9*(c*e*(a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*d**2)**3) - d*sqrt(-a*c**3)*(3*a*e**2 - c*d**2)/(2*
a*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6))) + 6*a**5*c**2*e**10 + 568*a**4*c**5*d**1
0*e**6*(c*e*(a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*d**2)**3) - d*sqrt(-a*c**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3*e**6
+ 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6)))**2 - 8*a**4*c**4*d**6*e**7*(c*e*(a*e**2 - 3*c*d**2)/
(2*(a*e**2 + c*d**2)**3) - d*sqrt(-a*c**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2
*d**4*e**2 + c**3*d**6))) - 69*a**4*c**3*d**2*e**8 + 216*a**3*c**6*d**12*e**4*(c*e*(a*e**2 - 3*c*d**2)/(2*(a*e
**2 + c*d**2)**3) - d*sqrt(-a*c**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e
**2 + c**3*d**6)))**2 - 42*a**3*c**5*d**8*e**5*(c*e*(a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*d**2)**3) - d*sqrt(-a*c
**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6))) + 236*a**3*c
**4*d**4*e**6 + 24*a**2*c**7*d**14*e**2*(c*e*(a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*d**2)**3) - d*sqrt(-a*c**3)*(3
*a*e**2 - c*d**2)/(2*a*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6)))**2 - 44*a**2*c**6*d
**10*e**3*(c*e*(a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*d**2)**3) - d*sqrt(-a*c**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3*e
**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6))) - 194*a**2*c**5*d**6*e**4 - 4*a*c**8*d**16*(c*e*(
a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*d**2)**3) - d*sqrt(-a*c**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3*e**6 + 3*a**2*c*d
**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6)))**2 - 14*a*c**7*d**12*e*(c*e*(a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*d*
*2)**3) - d*sqrt(-a*c**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3
*d**6))) + 6*a*c**6*d**8*e**2 - c**7*d**10)/(27*a**4*c**3*d*e**9 - 144*a**3*c**4*d**3*e**7 + 270*a**2*c**5*d**
5*e**5 - 72*a*c**6*d**7*e**3 - c**7*d**9*e)) + (c*e*(a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*d**2)**3) + d*sqrt(-a*c
**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6)))*log(x + (-12
*a**9*e**16*(c*e*(a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*d**2)**3) + d*sqrt(-a*c**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3
*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6)))**2 - 24*a**8*c*d**2*e**14*(c*e*(a*e**2 - 3*c*d*
*2)/(2*(a*e**2 + c*d**2)**3) + d*sqrt(-a*c**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*
c**2*d**4*e**2 + c**3*d**6)))**2 + 104*a**7*c**2*d**4*e**12*(c*e*(a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*d**2)**3)
+ d*sqrt(-a*c**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6)))
**2 - 6*a**7*c*e**13*(c*e*(a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*d**2)**3) + d*sqrt(-a*c**3)*(3*a*e**2 - c*d**2)/(
2*a*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6))) + 456*a**6*c**3*d**6*e**10*(c*e*(a*e**
2 - 3*c*d**2)/(2*(a*e**2 + c*d**2)**3) + d*sqrt(-a*c**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3*e**6 + 3*a**2*c*d**2*e
**4 + 3*a*c**2*d**4*e**2 + c**3*d**6)))**2 - 12*a**6*c**2*d**2*e**11*(c*e*(a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*d
**2)**3) + d*sqrt(-a*c**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**
3*d**6))) + 720*a**5*c**4*d**8*e**8*(c*e*(a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*d**2)**3) + d*sqrt(-a*c**3)*(3*a*e
**2 - c*d**2)/(2*a*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6)))**2 - 2*a**5*c**3*d**4*e
**9*(c*e*(a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*d**2)**3) + d*sqrt(-a*c**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3*e**6 +
3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6))) + 6*a**5*c**2*e**10 + 568*a**4*c**5*d**10*e**6*(c*e*(a*
e**2 - 3*c*d**2)/(2*(a*e**2 + c*d**2)**3) + d*sqrt(-a*c**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3*e**6 + 3*a**2*c*d**
2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6)))**2 - 8*a**4*c**4*d**6*e**7*(c*e*(a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*
d**2)**3) + d*sqrt(-a*c**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c*
*3*d**6))) - 69*a**4*c**3*d**2*e**8 + 216*a**3*c**6*d**12*e**4*(c*e*(a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*d**2)**
3) + d*sqrt(-a*c**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6
)))**2 - 42*a**3*c**5*d**8*e**5*(c*e*(a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*d**2)**3) + d*sqrt(-a*c**3)*(3*a*e**2
- c*d**2)/(2*a*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6))) + 236*a**3*c**4*d**4*e**6 +
24*a**2*c**7*d**14*e**2*(c*e*(a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*d**2)**3) + d*sqrt(-a*c**3)*(3*a*e**2 - c*d**
2)/(2*a*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6)))**2 - 44*a**2*c**6*d**10*e**3*(c*e*
(a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*d**2)**3) + d*sqrt(-a*c**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3*e**6 + 3*a**2*c*
d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6))) - 194*a**2*c**5*d**6*e**4 - 4*a*c**8*d**16*(c*e*(a*e**2 - 3*c*d*
*2)/(2*(a*e**2 + c*d**2)**3) + d*sqrt(-a*c**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*
c**2*d**4*e**2 + c**3*d**6)))**2 - 14*a*c**7*d**12*e*(c*e*(a*e**2 - 3*c*d**2)/(2*(a*e**2 + c*d**2)**3) + d*sqr
t(-a*c**3)*(3*a*e**2 - c*d**2)/(2*a*(a**3*e**6 + 3*a**2*c*d**2*e**4 + 3*a*c**2*d**4*e**2 + c**3*d**6))) + 6*a*
c**6*d**8*e**2 - c**7*d**10)/(27*a**4*c**3*d*e**9 - 144*a**3*c**4*d**3*e**7 + 270*a**2*c**5*d**5*e**5 - 72*a*c
**6*d**7*e**3 - c**7*d**9*e)) - (a*e**3 + 5*c*d**2*e + 4*c*d*e**2*x)/(2*a**2*d**2*e**4 + 4*a*c*d**4*e**2 + 2*c
**2*d**6 + x**2*(2*a**2*e**6 + 4*a*c*d**2*e**4 + 2*c**2*d**4*e**2) + x*(4*a**2*d*e**5 + 8*a*c*d**3*e**3 + 4*c*
*2*d**5*e))

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Giac [A]  time = 1.35745, size = 363, normalized size = 2.06 \begin{align*} -\frac{{\left (3 \, c^{2} d^{2} e - a c e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \,{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}} + \frac{{\left (3 \, c^{2} d^{2} e^{2} - a c e^{4}\right )} \log \left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e + 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} + a^{3} e^{7}} + \frac{{\left (c^{3} d^{3} - 3 \, a c^{2} d e^{2}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \sqrt{a c}} - \frac{5 \, c^{2} d^{4} e + 6 \, a c d^{2} e^{3} + a^{2} e^{5} + 4 \,{\left (c^{2} d^{3} e^{2} + a c d e^{4}\right )} x}{2 \,{\left (c d^{2} + a e^{2}\right )}^{3}{\left (x e + d\right )}^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(c*x^2+a),x, algorithm="giac")

[Out]

-1/2*(3*c^2*d^2*e - a*c*e^3)*log(c*x^2 + a)/(c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6) + (3*c^2*d
^2*e^2 - a*c*e^4)*log(abs(x*e + d))/(c^3*d^6*e + 3*a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 + a^3*e^7) + (c^3*d^3 - 3*a
*c^2*d*e^2)*arctan(c*x/sqrt(a*c))/((c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*sqrt(a*c)) - 1/2*(5
*c^2*d^4*e + 6*a*c*d^2*e^3 + a^2*e^5 + 4*(c^2*d^3*e^2 + a*c*d*e^4)*x)/((c*d^2 + a*e^2)^3*(x*e + d)^2)