3.524 $$\int \frac{1}{(d+e x)^2 (a+c x^2)^4} \, dx$$

Optimal. Leaf size=430 $-\frac{a e \left (5 c d^2-7 a e^2\right ) \left (5 a e^2+c d^2\right )-3 c d x \left (29 a^2 e^4+18 a c d^2 e^2+5 c^2 d^4\right )}{48 a^3 \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )^3}+\frac{e \left (47 a^2 c d^2 e^4-35 a^3 e^6+23 a c^2 d^4 e^2+5 c^3 d^6\right )}{16 a^3 (d+e x) \left (a e^2+c d^2\right )^4}+\frac{\sqrt{c} \left (70 a^2 c^2 d^4 e^4+140 a^3 c d^2 e^6-35 a^4 e^8+28 a c^3 d^6 e^2+5 c^4 d^8\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} \left (a e^2+c d^2\right )^5}-\frac{a e \left (c d^2-7 a e^2\right )-c d x \left (13 a e^2+5 c d^2\right )}{24 a^2 \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{6 a \left (a+c x^2\right )^3 (d+e x) \left (a e^2+c d^2\right )}-\frac{4 c d e^7 \log \left (a+c x^2\right )}{\left (a e^2+c d^2\right )^5}+\frac{8 c d e^7 \log (d+e x)}{\left (a e^2+c d^2\right )^5}$

[Out]

(e*(5*c^3*d^6 + 23*a*c^2*d^4*e^2 + 47*a^2*c*d^2*e^4 - 35*a^3*e^6))/(16*a^3*(c*d^2 + a*e^2)^4*(d + e*x)) + (a*e
+ c*d*x)/(6*a*(c*d^2 + a*e^2)*(d + e*x)*(a + c*x^2)^3) - (a*e*(c*d^2 - 7*a*e^2) - c*d*(5*c*d^2 + 13*a*e^2)*x)
/(24*a^2*(c*d^2 + a*e^2)^2*(d + e*x)*(a + c*x^2)^2) - (a*e*(5*c*d^2 - 7*a*e^2)*(c*d^2 + 5*a*e^2) - 3*c*d*(5*c^
2*d^4 + 18*a*c*d^2*e^2 + 29*a^2*e^4)*x)/(48*a^3*(c*d^2 + a*e^2)^3*(d + e*x)*(a + c*x^2)) + (Sqrt[c]*(5*c^4*d^8
+ 28*a*c^3*d^6*e^2 + 70*a^2*c^2*d^4*e^4 + 140*a^3*c*d^2*e^6 - 35*a^4*e^8)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(16*a^
(7/2)*(c*d^2 + a*e^2)^5) + (8*c*d*e^7*Log[d + e*x])/(c*d^2 + a*e^2)^5 - (4*c*d*e^7*Log[a + c*x^2])/(c*d^2 + a*
e^2)^5

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Rubi [A]  time = 0.533842, antiderivative size = 430, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.353, Rules used = {741, 823, 801, 635, 205, 260} $-\frac{a e \left (5 c d^2-7 a e^2\right ) \left (5 a e^2+c d^2\right )-3 c d x \left (29 a^2 e^4+18 a c d^2 e^2+5 c^2 d^4\right )}{48 a^3 \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )^3}+\frac{e \left (47 a^2 c d^2 e^4-35 a^3 e^6+23 a c^2 d^4 e^2+5 c^3 d^6\right )}{16 a^3 (d+e x) \left (a e^2+c d^2\right )^4}+\frac{\sqrt{c} \left (70 a^2 c^2 d^4 e^4+140 a^3 c d^2 e^6-35 a^4 e^8+28 a c^3 d^6 e^2+5 c^4 d^8\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} \left (a e^2+c d^2\right )^5}-\frac{a e \left (c d^2-7 a e^2\right )-c d x \left (13 a e^2+5 c d^2\right )}{24 a^2 \left (a+c x^2\right )^2 (d+e x) \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{6 a \left (a+c x^2\right )^3 (d+e x) \left (a e^2+c d^2\right )}-\frac{4 c d e^7 \log \left (a+c x^2\right )}{\left (a e^2+c d^2\right )^5}+\frac{8 c d e^7 \log (d+e x)}{\left (a e^2+c d^2\right )^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)^2*(a + c*x^2)^4),x]

[Out]

(e*(5*c^3*d^6 + 23*a*c^2*d^4*e^2 + 47*a^2*c*d^2*e^4 - 35*a^3*e^6))/(16*a^3*(c*d^2 + a*e^2)^4*(d + e*x)) + (a*e
+ c*d*x)/(6*a*(c*d^2 + a*e^2)*(d + e*x)*(a + c*x^2)^3) - (a*e*(c*d^2 - 7*a*e^2) - c*d*(5*c*d^2 + 13*a*e^2)*x)
/(24*a^2*(c*d^2 + a*e^2)^2*(d + e*x)*(a + c*x^2)^2) - (a*e*(5*c*d^2 - 7*a*e^2)*(c*d^2 + 5*a*e^2) - 3*c*d*(5*c^
2*d^4 + 18*a*c*d^2*e^2 + 29*a^2*e^4)*x)/(48*a^3*(c*d^2 + a*e^2)^3*(d + e*x)*(a + c*x^2)) + (Sqrt[c]*(5*c^4*d^8
+ 28*a*c^3*d^6*e^2 + 70*a^2*c^2*d^4*e^4 + 140*a^3*c*d^2*e^6 - 35*a^4*e^8)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(16*a^
(7/2)*(c*d^2 + a*e^2)^5) + (8*c*d*e^7*Log[d + e*x])/(c*d^2 + a*e^2)^5 - (4*c*d*e^7*Log[a + c*x^2])/(c*d^2 + a*
e^2)^5

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 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

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)^2 \left (a+c x^2\right )^4} \, dx &=\frac{a e+c d x}{6 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^3}-\frac{\int \frac{-5 c d^2-7 a e^2-6 c d e x}{(d+e x)^2 \left (a+c x^2\right )^3} \, dx}{6 a \left (c d^2+a e^2\right )}\\ &=\frac{a e+c d x}{6 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^3}-\frac{a e \left (c d^2-7 a e^2\right )-c d \left (5 c d^2+13 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )^2}+\frac{\int \frac{c \left (15 c^2 d^4+34 a c d^2 e^2+35 a^2 e^4\right )+4 c^2 d e \left (5 c d^2+13 a e^2\right ) x}{(d+e x)^2 \left (a+c x^2\right )^2} \, dx}{24 a^2 c \left (c d^2+a e^2\right )^2}\\ &=\frac{a e+c d x}{6 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^3}-\frac{a e \left (c d^2-7 a e^2\right )-c d \left (5 c d^2+13 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )^2}-\frac{a e \left (5 c d^2-7 a e^2\right ) \left (c d^2+5 a e^2\right )-3 c d \left (5 c^2 d^4+18 a c d^2 e^2+29 a^2 e^4\right ) x}{48 a^3 \left (c d^2+a e^2\right )^3 (d+e x) \left (a+c x^2\right )}-\frac{\int \frac{-3 c^2 \left (5 c^3 d^6+13 a c^2 d^4 e^2+11 a^2 c d^2 e^4+35 a^3 e^6\right )-6 c^3 d e \left (5 c^2 d^4+18 a c d^2 e^2+29 a^2 e^4\right ) x}{(d+e x)^2 \left (a+c x^2\right )} \, dx}{48 a^3 c^2 \left (c d^2+a e^2\right )^3}\\ &=\frac{a e+c d x}{6 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^3}-\frac{a e \left (c d^2-7 a e^2\right )-c d \left (5 c d^2+13 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )^2}-\frac{a e \left (5 c d^2-7 a e^2\right ) \left (c d^2+5 a e^2\right )-3 c d \left (5 c^2 d^4+18 a c d^2 e^2+29 a^2 e^4\right ) x}{48 a^3 \left (c d^2+a e^2\right )^3 (d+e x) \left (a+c x^2\right )}-\frac{\int \left (\frac{3 c^2 e^2 \left (5 c^3 d^6+23 a c^2 d^4 e^2+47 a^2 c d^2 e^4-35 a^3 e^6\right )}{\left (c d^2+a e^2\right ) (d+e x)^2}-\frac{384 a^3 c^3 d e^8}{\left (c d^2+a e^2\right )^2 (d+e x)}-\frac{3 c^3 \left (5 c^4 d^8+28 a c^3 d^6 e^2+70 a^2 c^2 d^4 e^4+140 a^3 c d^2 e^6-35 a^4 e^8-128 a^3 c d e^7 x\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}\right ) \, dx}{48 a^3 c^2 \left (c d^2+a e^2\right )^3}\\ &=\frac{e \left (5 c^3 d^6+23 a c^2 d^4 e^2+47 a^2 c d^2 e^4-35 a^3 e^6\right )}{16 a^3 \left (c d^2+a e^2\right )^4 (d+e x)}+\frac{a e+c d x}{6 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^3}-\frac{a e \left (c d^2-7 a e^2\right )-c d \left (5 c d^2+13 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )^2}-\frac{a e \left (5 c d^2-7 a e^2\right ) \left (c d^2+5 a e^2\right )-3 c d \left (5 c^2 d^4+18 a c d^2 e^2+29 a^2 e^4\right ) x}{48 a^3 \left (c d^2+a e^2\right )^3 (d+e x) \left (a+c x^2\right )}+\frac{8 c d e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^5}+\frac{c \int \frac{5 c^4 d^8+28 a c^3 d^6 e^2+70 a^2 c^2 d^4 e^4+140 a^3 c d^2 e^6-35 a^4 e^8-128 a^3 c d e^7 x}{a+c x^2} \, dx}{16 a^3 \left (c d^2+a e^2\right )^5}\\ &=\frac{e \left (5 c^3 d^6+23 a c^2 d^4 e^2+47 a^2 c d^2 e^4-35 a^3 e^6\right )}{16 a^3 \left (c d^2+a e^2\right )^4 (d+e x)}+\frac{a e+c d x}{6 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^3}-\frac{a e \left (c d^2-7 a e^2\right )-c d \left (5 c d^2+13 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )^2}-\frac{a e \left (5 c d^2-7 a e^2\right ) \left (c d^2+5 a e^2\right )-3 c d \left (5 c^2 d^4+18 a c d^2 e^2+29 a^2 e^4\right ) x}{48 a^3 \left (c d^2+a e^2\right )^3 (d+e x) \left (a+c x^2\right )}+\frac{8 c d e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^5}-\frac{\left (8 c^2 d e^7\right ) \int \frac{x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^5}+\frac{\left (c \left (5 c^4 d^8+28 a c^3 d^6 e^2+70 a^2 c^2 d^4 e^4+140 a^3 c d^2 e^6-35 a^4 e^8\right )\right ) \int \frac{1}{a+c x^2} \, dx}{16 a^3 \left (c d^2+a e^2\right )^5}\\ &=\frac{e \left (5 c^3 d^6+23 a c^2 d^4 e^2+47 a^2 c d^2 e^4-35 a^3 e^6\right )}{16 a^3 \left (c d^2+a e^2\right )^4 (d+e x)}+\frac{a e+c d x}{6 a \left (c d^2+a e^2\right ) (d+e x) \left (a+c x^2\right )^3}-\frac{a e \left (c d^2-7 a e^2\right )-c d \left (5 c d^2+13 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 (d+e x) \left (a+c x^2\right )^2}-\frac{a e \left (5 c d^2-7 a e^2\right ) \left (c d^2+5 a e^2\right )-3 c d \left (5 c^2 d^4+18 a c d^2 e^2+29 a^2 e^4\right ) x}{48 a^3 \left (c d^2+a e^2\right )^3 (d+e x) \left (a+c x^2\right )}+\frac{\sqrt{c} \left (5 c^4 d^8+28 a c^3 d^6 e^2+70 a^2 c^2 d^4 e^4+140 a^3 c d^2 e^6-35 a^4 e^8\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} \left (c d^2+a e^2\right )^5}+\frac{8 c d e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^5}-\frac{4 c d e^7 \log \left (a+c x^2\right )}{\left (c d^2+a e^2\right )^5}\\ \end{align*}

Mathematica [A]  time = 0.43234, size = 336, normalized size = 0.78 $\frac{\frac{3 c \left (a e^2+c d^2\right ) \left (47 a^2 c d^2 e^4 x+a^3 e^5 (48 d-19 e x)+23 a c^2 d^4 e^2 x+5 c^3 d^6 x\right )}{a^3 \left (a+c x^2\right )}+\frac{2 c \left (a e^2+c d^2\right )^2 \left (a^2 e^3 (24 d-11 e x)+18 a c d^2 e^2 x+5 c^2 d^4 x\right )}{a^2 \left (a+c x^2\right )^2}+\frac{3 \sqrt{c} \left (70 a^2 c^2 d^4 e^4+140 a^3 c d^2 e^6-35 a^4 e^8+28 a c^3 d^6 e^2+5 c^4 d^8\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{7/2}}+\frac{8 c \left (a e^2+c d^2\right )^3 \left (a e (2 d-e x)+c d^2 x\right )}{a \left (a+c x^2\right )^3}-\frac{48 e^7 \left (a e^2+c d^2\right )}{d+e x}-192 c d e^7 \log \left (a+c x^2\right )+384 c d e^7 \log (d+e x)}{48 \left (a e^2+c d^2\right )^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((d + e*x)^2*(a + c*x^2)^4),x]

[Out]

((-48*e^7*(c*d^2 + a*e^2))/(d + e*x) + (3*c*(c*d^2 + a*e^2)*(5*c^3*d^6*x + 23*a*c^2*d^4*e^2*x + 47*a^2*c*d^2*e
^4*x + a^3*e^5*(48*d - 19*e*x)))/(a^3*(a + c*x^2)) + (2*c*(c*d^2 + a*e^2)^2*(5*c^2*d^4*x + 18*a*c*d^2*e^2*x +
a^2*e^3*(24*d - 11*e*x)))/(a^2*(a + c*x^2)^2) + (8*c*(c*d^2 + a*e^2)^3*(c*d^2*x + a*e*(2*d - e*x)))/(a*(a + c*
x^2)^3) + (3*Sqrt[c]*(5*c^4*d^8 + 28*a*c^3*d^6*e^2 + 70*a^2*c^2*d^4*e^4 + 140*a^3*c*d^2*e^6 - 35*a^4*e^8)*ArcT
an[(Sqrt[c]*x)/Sqrt[a]])/a^(7/2) + 384*c*d*e^7*Log[d + e*x] - 192*c*d*e^7*Log[a + c*x^2])/(48*(c*d^2 + a*e^2)^
5)

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Maple [B]  time = 0.063, size = 1126, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^2/(c*x^2+a)^4,x)

[Out]

1/3*c^4/(a*e^2+c*d^2)^5/(c*x^2+a)^3*d^7*e-17/6*c^2/(a*e^2+c*d^2)^5/(c*x^2+a)^3*a^2*x^3*e^8+10*c^4/(a*e^2+c*d^2
)^5/(c*x^2+a)^3*x^3*d^4*e^4+5/6*c^6/(a*e^2+c*d^2)^5/(c*x^2+a)^3/a^2*x^3*d^8+5/16*c^7/(a*e^2+c*d^2)^5/(c*x^2+a)
^3/a^3*x^5*d^8+3*c^4/(a*e^2+c*d^2)^5/(c*x^2+a)^3*x^4*d^3*e^5+45/8*c^3/(a*e^2+c*d^2)^5/(c*x^2+a)^3*x*a*d^4*e^4+
35/8*c^3/(a*e^2+c*d^2)^5/a/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*d^4*e^4+7/4*c^4/(a*e^2+c*d^2)^5/a^2/(a*c)^(1/2)
*arctan(x*c/(a*c)^(1/2))*d^6*e^2+35/8*c^5/(a*e^2+c*d^2)^5/(c*x^2+a)^3/a*x^5*d^4*e^4+7/4*c^6/(a*e^2+c*d^2)^5/(c
*x^2+a)^3/a^2*x^5*d^6*e^2+3*c^3/(a*e^2+c*d^2)^5/(c*x^2+a)^3*x^4*a*d*e^7+10/3*c^3/(a*e^2+c*d^2)^5/(c*x^2+a)^3*a
*x^3*d^2*e^6+14/3*c^5/(a*e^2+c*d^2)^5/(c*x^2+a)^3/a*x^3*d^6*e^2+7*c^2/(a*e^2+c*d^2)^5/(c*x^2+a)^3*x^2*a^2*d*e^
7+8*c^3/(a*e^2+c*d^2)^5/(c*x^2+a)^3*x^2*a*d^3*e^5+5/4*c^2/(a*e^2+c*d^2)^5/(c*x^2+a)^3*x*a^2*d^2*e^6-e^7/(a*e^2
+c*d^2)^4/(e*x+d)+c^4/(a*e^2+c*d^2)^5/(c*x^2+a)^3*x^2*d^5*e^3+13/4*c^4/(a*e^2+c*d^2)^5/(c*x^2+a)^3*x*d^6*e^2+1
1/16*c^5/(a*e^2+c*d^2)^5/(c*x^2+a)^3*x/a*d^8-19/16*c^3/(a*e^2+c*d^2)^5/(c*x^2+a)^3*a*x^5*e^8+6*c^2/(a*e^2+c*d^
2)^5/(c*x^2+a)^3*a^2*d^3*e^5+2*c^3/(a*e^2+c*d^2)^5/(c*x^2+a)^3*a*d^5*e^3-29/16*c/(a*e^2+c*d^2)^5/(c*x^2+a)^3*x
*a^3*e^8+13/3*c/(a*e^2+c*d^2)^5/(c*x^2+a)^3*a^3*d*e^7+35/4*c^2/(a*e^2+c*d^2)^5/(a*c)^(1/2)*arctan(x*c/(a*c)^(1
/2))*d^2*e^6-35/16*c/(a*e^2+c*d^2)^5*a/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*e^8+5/16*c^5/(a*e^2+c*d^2)^5/a^3/(a
*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*d^8+7/4*c^4/(a*e^2+c*d^2)^5/(c*x^2+a)^3*x^5*d^2*e^6+8*c*d*e^7*ln(e*x+d)/(a*e
^2+c*d^2)^5-4*c*d*e^7*ln(c*x^2+a)/(a*e^2+c*d^2)^5

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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)^2/(c*x^2+a)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(c*x^2+a)^4,x, algorithm="fricas")

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**2/(c*x**2+a)**4,x)

[Out]

Timed out

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Giac [B]  time = 1.35452, size = 1156, normalized size = 2.69 \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)^2/(c*x^2+a)^4,x, algorithm="giac")

[Out]

-4*c*d*e^7*log(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + a*e^2/(x*e + d)^2)/(c^5*d^10 + 5*a*c^4*d^8*e^2 + 10*a
^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 + a^5*e^10) + 1/16*(5*c^5*d^8*e^2 + 28*a*c^4*d^6*e^4 + 7
0*a^2*c^3*d^4*e^6 + 140*a^3*c^2*d^2*e^8 - 35*a^4*c*e^10)*arctan((c*d - c*d^2/(x*e + d) - a*e^2/(x*e + d))*e^(-
1)/sqrt(a*c))*e^(-2)/((a^3*c^5*d^10 + 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 + 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^
2*e^8 + a^8*e^10)*sqrt(a*c)) - e^15/((c^4*d^8*e^8 + 4*a*c^3*d^6*e^10 + 6*a^2*c^2*d^4*e^12 + 4*a^3*c*d^2*e^14 +
a^4*e^16)*(x*e + d)) + 1/48*(15*c^7*d^7*e + 79*a*c^6*d^5*e^3 + 185*a^2*c^5*d^3*e^5 - 295*a^3*c^4*d*e^7 - 3*(2
5*c^7*d^8*e^2 + 130*a*c^6*d^6*e^4 + 300*a^2*c^5*d^4*e^6 - 618*a^3*c^4*d^2*e^8 + 19*a^4*c^3*e^10)*e^(-1)/(x*e +
d) + 6*(25*c^7*d^9*e^3 + 135*a*c^6*d^7*e^5 + 327*a^2*c^5*d^5*e^7 - 691*a^3*c^4*d^3*e^9 - 76*a^4*c^3*d*e^11)*e
^(-2)/(x*e + d)^2 - 2*(75*c^7*d^10*e^4 + 440*a*c^6*d^8*e^6 + 1162*a^2*c^5*d^6*e^8 - 2212*a^3*c^4*d^4*e^10 - 12
77*a^4*c^3*d^2*e^12 + 68*a^5*c^2*e^14)*e^(-3)/(x*e + d)^3 + 3*(25*c^7*d^11*e^5 + 165*a*c^6*d^9*e^7 + 490*a^2*c
^5*d^7*e^9 - 742*a^3*c^4*d^5*e^11 - 1139*a^4*c^3*d^3*e^13 - 47*a^5*c^2*d*e^15)*e^(-4)/(x*e + d)^4 - 3*(5*c^7*d
^12*e^6 + 38*a*c^6*d^10*e^8 + 131*a^2*c^5*d^8*e^10 - 140*a^3*c^4*d^6*e^12 - 517*a^4*c^3*d^4*e^14 - 250*a^5*c^2
*d^2*e^16 + 29*a^6*c*e^18)*e^(-5)/(x*e + d)^5)/((c*d^2 + a*e^2)^5*a^3*(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2
+ a*e^2/(x*e + d)^2)^3)