3.407 \(\int \frac{1}{(d+e x)^3 \left (a+c x^4\right )^2} \, dx\)

Optimal. Leaf size=1384 \[ \text{result too large to display} \]

[Out]

-e^7/(2*(c*d^4 + a*e^4)^2*(d + e*x)^2) - (8*c*d^3*e^7)/((c*d^4 + a*e^4)^3*(d + e
*x)) + (c*(2*a*d^2*e^3*(5*c*d^4 - 3*a*e^4) + x*(d*(c^2*d^8 - 12*a*c*d^4*e^4 + 3*
a^2*e^8) - e*(3*c^2*d^8 - 12*a*c*d^4*e^4 + a^2*e^8)*x + 2*c*d^3*e^2*(3*c*d^4 - 5
*a*e^4)*x^2)))/(4*a*(c*d^4 + a*e^4)^3*(a + c*x^4)) - (Sqrt[c]*e^5*(21*c^2*d^8 -
26*a*c*d^4*e^4 + a^2*e^8)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*Sqrt[a]*(c*d^4 + a*e
^4)^4) - (Sqrt[c]*e*(3*c^2*d^8 - 12*a*c*d^4*e^4 + a^2*e^8)*ArcTan[(Sqrt[c]*x^2)/
Sqrt[a]])/(4*a^(3/2)*(c*d^4 + a*e^4)^3) - (c^(3/4)*d*(3*c^2*d^8 - 36*a*c*d^4*e^4
+ 9*a^2*e^8 + 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4))*ArcTan[1 - (Sqrt[2
]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^3) - (c^(3/4)*d*e^4*(4
*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4 - 5*a*e^4) + 3*(5*c^2*d^8 - 10*a*c*d^4*e^4 + a
^2*e^8))*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^4 + a*
e^4)^4) + (c^(3/4)*d*(3*c^2*d^8 - 36*a*c*d^4*e^4 + 9*a^2*e^8 + 2*Sqrt[a]*Sqrt[c]
*d^2*e^2*(3*c*d^4 - 5*a*e^4))*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2
]*a^(7/4)*(c*d^4 + a*e^4)^3) + (c^(3/4)*d*e^4*(4*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^
4 - 5*a*e^4) + 3*(5*c^2*d^8 - 10*a*c*d^4*e^4 + a^2*e^8))*ArcTan[1 + (Sqrt[2]*c^(
1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^4) + (12*c*d^2*e^7*(3*c*d^4
- a*e^4)*Log[d + e*x])/(c*d^4 + a*e^4)^4 - (c^(3/4)*d*(3*c^2*d^8 - 36*a*c*d^4*e
^4 + 9*a^2*e^8 - 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4))*Log[Sqrt[a] - Sq
rt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^3) +
(c^(3/4)*d*e^4*(4*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4 - 5*a*e^4) - 3*(5*c^2*d^8 -
10*a*c*d^4*e^4 + a^2*e^8))*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2
])/(4*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^4) + (c^(3/4)*d*(3*c^2*d^8 - 36*a*c*d^4*e^
4 + 9*a^2*e^8 - 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4))*Log[Sqrt[a] + Sqr
t[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^3) -
(c^(3/4)*d*e^4*(4*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4 - 5*a*e^4) - 3*(5*c^2*d^8 - 1
0*a*c*d^4*e^4 + a^2*e^8))*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2]
)/(4*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^4) - (3*c*d^2*e^7*(3*c*d^4 - a*e^4)*Log[a +
c*x^4])/(c*d^4 + a*e^4)^4

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Rubi [A]  time = 4.66989, antiderivative size = 1384, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 13, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.765 \[ \frac{12 c d^2 \left (3 c d^4-a e^4\right ) \log (d+e x) e^7}{\left (c d^4+a e^4\right )^4}-\frac{3 c d^2 \left (3 c d^4-a e^4\right ) \log \left (c x^4+a\right ) e^7}{\left (c d^4+a e^4\right )^4}-\frac{8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}-\frac{e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac{\sqrt{c} \left (21 c^2 d^8-26 a c e^4 d^4+a^2 e^8\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right ) e^5}{2 \sqrt{a} \left (c d^4+a e^4\right )^4}-\frac{c^{3/4} d \left (4 \sqrt{a} \sqrt{c} d^2 \left (7 c d^4-5 a e^4\right ) e^2+3 \left (5 c^2 d^8-10 a c e^4 d^4+a^2 e^8\right )\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^4}{2 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac{c^{3/4} d \left (4 \sqrt{a} \sqrt{c} d^2 \left (7 c d^4-5 a e^4\right ) e^2+3 \left (5 c^2 d^8-10 a c e^4 d^4+a^2 e^8\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^4}{2 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac{c^{3/4} d \left (4 \sqrt{a} \sqrt{c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c e^4 d^4+a^2 e^8\right )\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right ) e^4}{4 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^4}-\frac{c^{3/4} d \left (4 \sqrt{a} \sqrt{c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c e^4 d^4+a^2 e^8\right )\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right ) e^4}{4 \sqrt{2} a^{3/4} \left (c d^4+a e^4\right )^4}-\frac{\sqrt{c} \left (3 c^2 d^8-12 a c e^4 d^4+a^2 e^8\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right ) e}{4 a^{3/2} \left (c d^4+a e^4\right )^3}+\frac{c \left (2 a d^2 \left (5 c d^4-3 a e^4\right ) e^3+x \left (2 c e^2 \left (3 c d^4-5 a e^4\right ) x^2 d^3+\left (c^2 d^8-12 a c e^4 d^4+3 a^2 e^8\right ) d-e \left (3 c^2 d^8-12 a c e^4 d^4+a^2 e^8\right ) x\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (c x^4+a\right )}-\frac{c^{3/4} d \left (3 c^2 d^8-36 a c e^4 d^4+2 \sqrt{a} \sqrt{c} e^2 \left (3 c d^4-5 a e^4\right ) d^2+9 a^2 e^8\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} \left (c d^4+a e^4\right )^3}+\frac{c^{3/4} d \left (3 c^2 d^8-36 a c e^4 d^4+2 \sqrt{a} \sqrt{c} e^2 \left (3 c d^4-5 a e^4\right ) d^2+9 a^2 e^8\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} \left (c d^4+a e^4\right )^3}-\frac{c^{3/4} d \left (3 c^2 d^8-36 a c e^4 d^4-2 \sqrt{a} \sqrt{c} e^2 \left (3 c d^4-5 a e^4\right ) d^2+9 a^2 e^8\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{16 \sqrt{2} a^{7/4} \left (c d^4+a e^4\right )^3}+\frac{c^{3/4} d \left (3 c^2 d^8-36 a c e^4 d^4-2 \sqrt{a} \sqrt{c} e^2 \left (3 c d^4-5 a e^4\right ) d^2+9 a^2 e^8\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{16 \sqrt{2} a^{7/4} \left (c d^4+a e^4\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^3*(a + c*x^4)^2),x]

[Out]

-e^7/(2*(c*d^4 + a*e^4)^2*(d + e*x)^2) - (8*c*d^3*e^7)/((c*d^4 + a*e^4)^3*(d + e
*x)) + (c*(2*a*d^2*e^3*(5*c*d^4 - 3*a*e^4) + x*(d*(c^2*d^8 - 12*a*c*d^4*e^4 + 3*
a^2*e^8) - e*(3*c^2*d^8 - 12*a*c*d^4*e^4 + a^2*e^8)*x + 2*c*d^3*e^2*(3*c*d^4 - 5
*a*e^4)*x^2)))/(4*a*(c*d^4 + a*e^4)^3*(a + c*x^4)) - (Sqrt[c]*e^5*(21*c^2*d^8 -
26*a*c*d^4*e^4 + a^2*e^8)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*Sqrt[a]*(c*d^4 + a*e
^4)^4) - (Sqrt[c]*e*(3*c^2*d^8 - 12*a*c*d^4*e^4 + a^2*e^8)*ArcTan[(Sqrt[c]*x^2)/
Sqrt[a]])/(4*a^(3/2)*(c*d^4 + a*e^4)^3) - (c^(3/4)*d*(3*c^2*d^8 - 36*a*c*d^4*e^4
+ 9*a^2*e^8 + 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4))*ArcTan[1 - (Sqrt[2
]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^3) - (c^(3/4)*d*e^4*(4
*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4 - 5*a*e^4) + 3*(5*c^2*d^8 - 10*a*c*d^4*e^4 + a
^2*e^8))*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^4 + a*
e^4)^4) + (c^(3/4)*d*(3*c^2*d^8 - 36*a*c*d^4*e^4 + 9*a^2*e^8 + 2*Sqrt[a]*Sqrt[c]
*d^2*e^2*(3*c*d^4 - 5*a*e^4))*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2
]*a^(7/4)*(c*d^4 + a*e^4)^3) + (c^(3/4)*d*e^4*(4*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^
4 - 5*a*e^4) + 3*(5*c^2*d^8 - 10*a*c*d^4*e^4 + a^2*e^8))*ArcTan[1 + (Sqrt[2]*c^(
1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^4) + (12*c*d^2*e^7*(3*c*d^4
- a*e^4)*Log[d + e*x])/(c*d^4 + a*e^4)^4 - (c^(3/4)*d*(3*c^2*d^8 - 36*a*c*d^4*e
^4 + 9*a^2*e^8 - 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4))*Log[Sqrt[a] - Sq
rt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^3) +
(c^(3/4)*d*e^4*(4*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4 - 5*a*e^4) - 3*(5*c^2*d^8 -
10*a*c*d^4*e^4 + a^2*e^8))*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2
])/(4*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^4) + (c^(3/4)*d*(3*c^2*d^8 - 36*a*c*d^4*e^
4 + 9*a^2*e^8 - 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4))*Log[Sqrt[a] + Sqr
t[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^3) -
(c^(3/4)*d*e^4*(4*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4 - 5*a*e^4) - 3*(5*c^2*d^8 - 1
0*a*c*d^4*e^4 + a^2*e^8))*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2]
)/(4*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^4) - (3*c*d^2*e^7*(3*c*d^4 - a*e^4)*Log[a +
c*x^4])/(c*d^4 + a*e^4)^4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**3/(c*x**4+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 2.43567, size = 996, normalized size = 0.72 \[ \frac{384 c d^2 \left (3 c d^4-a e^4\right ) \log (d+e x) e^7-96 c d^2 \left (3 c d^4-a e^4\right ) \log \left (c x^4+a\right ) e^7-\frac{256 c d^3 \left (c d^4+a e^4\right ) e^7}{d+e x}-\frac{16 \left (c d^4+a e^4\right )^2 e^7}{(d+e x)^2}+\frac{8 c \left (c d^4+a e^4\right ) \left (c^2 x \left (d^2-3 e x d+6 e^2 x^2\right ) d^7+2 a c e^3 \left (5 d^3-6 e x d^2+6 e^2 x^2 d-5 e^3 x^3\right ) d^3-a^2 e^7 \left (6 d^2-3 e x d+e^2 x^2\right )\right )}{a \left (c x^4+a\right )}-\frac{6 \sqrt{c} \left (\sqrt{2} c^{13/4} d^{13}-4 \sqrt [4]{a} c^3 e d^{12}+2 \sqrt{2} \sqrt{a} c^{11/4} e^2 d^{11}+9 \sqrt{2} a c^{9/4} e^4 d^9-44 a^{5/4} c^2 e^5 d^8+36 \sqrt{2} a^{3/2} c^{7/4} e^6 d^7-49 \sqrt{2} a^2 c^{5/4} e^8 d^5+84 a^{9/4} c e^9 d^4-30 \sqrt{2} a^{5/2} c^{3/4} e^{10} d^3+7 \sqrt{2} a^3 \sqrt [4]{c} e^{12} d-4 a^{13/4} e^{13}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac{6 \sqrt{c} \left (\sqrt{2} c^{13/4} d^{13}+4 \sqrt [4]{a} c^3 e d^{12}+2 \sqrt{2} \sqrt{a} c^{11/4} e^2 d^{11}+9 \sqrt{2} a c^{9/4} e^4 d^9+44 a^{5/4} c^2 e^5 d^8+36 \sqrt{2} a^{3/2} c^{7/4} e^6 d^7-49 \sqrt{2} a^2 c^{5/4} e^8 d^5-84 a^{9/4} c e^9 d^4-30 \sqrt{2} a^{5/2} c^{3/4} e^{10} d^3+7 \sqrt{2} a^3 \sqrt [4]{c} e^{12} d+4 a^{13/4} e^{13}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{7/4}}-\frac{3 \sqrt{2} c^{3/4} \left (c^3 d^{13}-2 \sqrt{a} c^{5/2} e^2 d^{11}+9 a c^2 e^4 d^9-36 a^{3/2} c^{3/2} e^6 d^7-49 a^2 c e^8 d^5+30 a^{5/2} \sqrt{c} e^{10} d^3+7 a^3 e^{12} d\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{a^{7/4}}+\frac{3 \sqrt{2} c^{3/4} \left (c^3 d^{13}-2 \sqrt{a} c^{5/2} e^2 d^{11}+9 a c^2 e^4 d^9-36 a^{3/2} c^{3/2} e^6 d^7-49 a^2 c e^8 d^5+30 a^{5/2} \sqrt{c} e^{10} d^3+7 a^3 e^{12} d\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{a^{7/4}}}{32 \left (c d^4+a e^4\right )^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^3*(a + c*x^4)^2),x]

[Out]

((-16*e^7*(c*d^4 + a*e^4)^2)/(d + e*x)^2 - (256*c*d^3*e^7*(c*d^4 + a*e^4))/(d +
e*x) + (8*c*(c*d^4 + a*e^4)*(-(a^2*e^7*(6*d^2 - 3*d*e*x + e^2*x^2)) + c^2*d^7*x*
(d^2 - 3*d*e*x + 6*e^2*x^2) + 2*a*c*d^3*e^3*(5*d^3 - 6*d^2*e*x + 6*d*e^2*x^2 - 5
*e^3*x^3)))/(a*(a + c*x^4)) - (6*Sqrt[c]*(Sqrt[2]*c^(13/4)*d^13 - 4*a^(1/4)*c^3*
d^12*e + 2*Sqrt[2]*Sqrt[a]*c^(11/4)*d^11*e^2 + 9*Sqrt[2]*a*c^(9/4)*d^9*e^4 - 44*
a^(5/4)*c^2*d^8*e^5 + 36*Sqrt[2]*a^(3/2)*c^(7/4)*d^7*e^6 - 49*Sqrt[2]*a^2*c^(5/4
)*d^5*e^8 + 84*a^(9/4)*c*d^4*e^9 - 30*Sqrt[2]*a^(5/2)*c^(3/4)*d^3*e^10 + 7*Sqrt[
2]*a^3*c^(1/4)*d*e^12 - 4*a^(13/4)*e^13)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)]
)/a^(7/4) + (6*Sqrt[c]*(Sqrt[2]*c^(13/4)*d^13 + 4*a^(1/4)*c^3*d^12*e + 2*Sqrt[2]
*Sqrt[a]*c^(11/4)*d^11*e^2 + 9*Sqrt[2]*a*c^(9/4)*d^9*e^4 + 44*a^(5/4)*c^2*d^8*e^
5 + 36*Sqrt[2]*a^(3/2)*c^(7/4)*d^7*e^6 - 49*Sqrt[2]*a^2*c^(5/4)*d^5*e^8 - 84*a^(
9/4)*c*d^4*e^9 - 30*Sqrt[2]*a^(5/2)*c^(3/4)*d^3*e^10 + 7*Sqrt[2]*a^3*c^(1/4)*d*e
^12 + 4*a^(13/4)*e^13)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(7/4) + 384*c*
d^2*e^7*(3*c*d^4 - a*e^4)*Log[d + e*x] - (3*Sqrt[2]*c^(3/4)*(c^3*d^13 - 2*Sqrt[a
]*c^(5/2)*d^11*e^2 + 9*a*c^2*d^9*e^4 - 36*a^(3/2)*c^(3/2)*d^7*e^6 - 49*a^2*c*d^5
*e^8 + 30*a^(5/2)*Sqrt[c]*d^3*e^10 + 7*a^3*d*e^12)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)
*c^(1/4)*x + Sqrt[c]*x^2])/a^(7/4) + (3*Sqrt[2]*c^(3/4)*(c^3*d^13 - 2*Sqrt[a]*c^
(5/2)*d^11*e^2 + 9*a*c^2*d^9*e^4 - 36*a^(3/2)*c^(3/2)*d^7*e^6 - 49*a^2*c*d^5*e^8
+ 30*a^(5/2)*Sqrt[c]*d^3*e^10 + 7*a^3*d*e^12)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(
1/4)*x + Sqrt[c]*x^2])/a^(7/4) - 96*c*d^2*e^7*(3*c*d^4 - a*e^4)*Log[a + c*x^4])/
(32*(c*d^4 + a*e^4)^4)

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Maple [A]  time = 0.037, size = 2133, normalized size = 1.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^3/(c*x^4+a)^2,x)

[Out]

27/16*c^3/(a*e^4+c*d^4)^4/a*(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x
-1)*d^9*e^4-1/2*e^7/(a*e^4+c*d^4)^2/(e*x+d)^2-45/16*c/(a*e^4+c*d^4)^4*a/(1/c*a)^
(1/4)*2^(1/2)*ln((x^2-(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2))/(x^2+(1/c*a)^(1/4)*
x*2^(1/2)+(1/c*a)^(1/2)))*d^3*e^10+21/16*c/(a*e^4+c*d^4)^4*a*(1/c*a)^(1/4)*2^(1/
2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)*d*e^12-45/8*c/(a*e^4+c*d^4)^4*a/(1/c*a)^(1/
4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)*d^3*e^10+21/16*c/(a*e^4+c*d^4)^4*a*
(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)*d*e^12+27/32*c^3/(a*e^4+
c*d^4)^4/a*(1/c*a)^(1/4)*2^(1/2)*ln((x^2+(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2))/
(x^2-(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2)))*d^9*e^4+21/32*c/(a*e^4+c*d^4)^4*a*(
1/c*a)^(1/4)*2^(1/2)*ln((x^2+(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2))/(x^2-(1/c*a)
^(1/4)*x*2^(1/2)+(1/c*a)^(1/2)))*d*e^12+3/8*c^3/(a*e^4+c*d^4)^4/a/(1/c*a)^(1/4)*
2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)*d^11*e^2+3/8*c^3/(a*e^4+c*d^4)^4/a/(1/
c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)*d^11*e^2-45/8*c/(a*e^4+c*d^
4)^4*a/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)*d^3*e^10+3/16*c^3
/(a*e^4+c*d^4)^4/a/(1/c*a)^(1/4)*2^(1/2)*ln((x^2-(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)
^(1/2))/(x^2+(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2)))*d^11*e^2+3/2*c^4/(a*e^4+c*d
^4)^4/(c*x^4+a)*d^11*e^2/a*x^3+3/32*c^4/(a*e^4+c*d^4)^4/a^2*(1/c*a)^(1/4)*2^(1/2
)*ln((x^2+(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2))/(x^2-(1/c*a)^(1/4)*x*2^(1/2)+(1
/c*a)^(1/2)))*d^13-147/16*c^2/(a*e^4+c*d^4)^4*(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/
2)/(1/c*a)^(1/4)*x+1)*d^5*e^8-147/16*c^2/(a*e^4+c*d^4)^4*(1/c*a)^(1/4)*2^(1/2)*a
rctan(2^(1/2)/(1/c*a)^(1/4)*x-1)*d^5*e^8-147/32*c^2/(a*e^4+c*d^4)^4*(1/c*a)^(1/4
)*2^(1/2)*ln((x^2+(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2))/(x^2-(1/c*a)^(1/4)*x*2^
(1/2)+(1/c*a)^(1/2)))*d^5*e^8+27/4*c^2/(a*e^4+c*d^4)^4/(1/c*a)^(1/4)*2^(1/2)*arc
tan(2^(1/2)/(1/c*a)^(1/4)*x-1)*d^7*e^6+1/4*c^4/(a*e^4+c*d^4)^4/(c*x^4+a)*d^13/a*
x+c^2/(a*e^4+c*d^4)^4/(c*x^4+a)*a*d^6*e^7+3*c/(a*e^4+c*d^4)^4*a*ln(a*(c*x^4+a))*
d^2*e^11-3/4*c/(a*e^4+c*d^4)^4/(a^3*c)^(1/2)*arctan(x^2*(c/a)^(1/2))*a^3*e^13-3/
4*c^4/(a*e^4+c*d^4)^4/(a^3*c)^(1/2)*arctan(x^2*(c/a)^(1/2))*d^12*e-12*e^11*c*d^2
/(a*e^4+c*d^4)^4*ln(e*x+d)*a+36*e^7*c^2*d^6/(a*e^4+c*d^4)^4*ln(e*x+d)+5/2*c^3/(a
*e^4+c*d^4)^4/(c*x^4+a)*d^10*e^3-9*c^2/(a*e^4+c*d^4)^4*ln(a*(c*x^4+a))*d^6*e^7+2
7/16*c^3/(a*e^4+c*d^4)^4/a*(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+
1)*d^9*e^4+27/4*c^2/(a*e^4+c*d^4)^4/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)
^(1/4)*x+1)*d^7*e^6+3/16*c^4/(a*e^4+c*d^4)^4/a^2*(1/c*a)^(1/4)*2^(1/2)*arctan(2^
(1/2)/(1/c*a)^(1/4)*x+1)*d^13+3/16*c^4/(a*e^4+c*d^4)^4/a^2*(1/c*a)^(1/4)*2^(1/2)
*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)*d^13+11/4*c^2/(a*e^4+c*d^4)^4/(c*x^4+a)*e^9*a
*x^2*d^4-3/4*c^4/(a*e^4+c*d^4)^4/(c*x^4+a)*e/a*x^2*d^12-9/4*c^2/(a*e^4+c*d^4)^4/
(c*x^4+a)*d^5*a*x*e^8-5/2*c^2/(a*e^4+c*d^4)^4/(c*x^4+a)*d^3*e^10*a*x^3-1/4*c/(a*
e^4+c*d^4)^4/(c*x^4+a)*e^13*a^2*x^2-3/2*c/(a*e^4+c*d^4)^4/(c*x^4+a)*a^2*d^2*e^11
-c^3/(a*e^4+c*d^4)^4/(c*x^4+a)*d^7*e^6*x^3+9/4*c^3/(a*e^4+c*d^4)^4/(c*x^4+a)*e^5
*x^2*d^8-11/4*c^3/(a*e^4+c*d^4)^4/(c*x^4+a)*d^9*x*e^4-8*c*d^3*e^7/(a*e^4+c*d^4)^
3/(e*x+d)-33/4*c^3/(a*e^4+c*d^4)^4/(a^3*c)^(1/2)*arctan(x^2*(c/a)^(1/2))*a*d^8*e
^5+27/8*c^2/(a*e^4+c*d^4)^4/(1/c*a)^(1/4)*2^(1/2)*ln((x^2-(1/c*a)^(1/4)*x*2^(1/2
)+(1/c*a)^(1/2))/(x^2+(1/c*a)^(1/4)*x*2^(1/2)+(1/c*a)^(1/2)))*d^7*e^6+3/4*c/(a*e
^4+c*d^4)^4/(c*x^4+a)*d*a^2*x*e^12+63/4*c^2/(a*e^4+c*d^4)^4/(a^3*c)^(1/2)*arctan
(x^2*(c/a)^(1/2))*a^2*d^4*e^9

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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(1/((c*x^4 + a)^2*(e*x + d)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [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/((c*x^4 + a)^2*(e*x + d)^3),x, algorithm="fricas")

[Out]

Timed out

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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)**3/(c*x**4+a)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.429836, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^2*(e*x + d)^3),x, algorithm="giac")

[Out]

Done