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

Optimal. Leaf size=2204 \[ \text{result too large to display} \]

[Out]

-e^11/(2*(c*d^4 + a*e^4)^3*(d + e*x)^2) - (12*c*d^3*e^11)/((c*d^4 + a*e^4)^4*(d
+ e*x)) + (c*x*(7*d*(c^2*d^8 - 12*a*c*d^4*e^4 + 3*a^2*e^8) - 6*e*(3*c^2*d^8 - 12
*a*c*d^4*e^4 + a^2*e^8)*x + 10*c*d^3*e^2*(3*c*d^4 - 5*a*e^4)*x^2))/(32*a^2*(c*d^
4 + a*e^4)^3*(a + c*x^4)) + (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)))/(8*a*(c*d^4 + a*e^4)^3*(a + c*x^4)^2) + (c*e
^4*(12*a*d^2*e^3*(3*c*d^4 - a*e^4) + x*(3*d*(5*c^2*d^8 - 10*a*c*d^4*e^4 + a^2*e^
8) - e*(21*c^2*d^8 - 26*a*c*d^4*e^4 + a^2*e^8)*x + 4*c*d^3*e^2*(7*c*d^4 - 5*a*e^
4)*x^2)))/(4*a*(c*d^4 + a*e^4)^4*(a + c*x^4)) - (Sqrt[c]*e^9*(55*c^2*d^8 - 40*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)^5
) - (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)/Sq
rt[a]])/(4*a^(3/2)*(c*d^4 + a*e^4)^4) - (3*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]])/(16*a^(5/2)*(c*d^4 + a*e^4)^3) - (3*c
^(3/4)*d*e^8*(15*c^2*d^8 - 16*a*c*d^4*e^4 + a^2*e^8 + 2*Sqrt[a]*Sqrt[c]*d^2*e^2*
(11*c*d^4 - 5*a*e^4))*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4
)*(c*d^4 + a*e^4)^5) - (c^(3/4)*d*e^4*(4*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4 - 5*a*
e^4) + 9*(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)])/(8*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^4) - (c^(3/4)*d*(10*Sqrt[a]*Sqrt[c]
*d^2*e^2*(3*c*d^4 - 5*a*e^4) + 21*(c^2*d^8 - 12*a*c*d^4*e^4 + 3*a^2*e^8))*ArcTan
[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*(c*d^4 + a*e^4)^3) + (3*
c^(3/4)*d*e^8*(15*c^2*d^8 - 16*a*c*d^4*e^4 + a^2*e^8 + 2*Sqrt[a]*Sqrt[c]*d^2*e^2
*(11*c*d^4 - 5*a*e^4))*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/
4)*(c*d^4 + a*e^4)^5) + (c^(3/4)*d*e^4*(4*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4 - 5*a
*e^4) + 9*(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)])/(8*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^4) + (c^(3/4)*d*(10*Sqrt[a]*Sqrt[c
]*d^2*e^2*(3*c*d^4 - 5*a*e^4) + 21*(c^2*d^8 - 12*a*c*d^4*e^4 + 3*a^2*e^8))*ArcTa
n[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*(c*d^4 + a*e^4)^3) + (6
*c*d^2*e^11*(13*c*d^4 - 3*a*e^4)*Log[d + e*x])/(c*d^4 + a*e^4)^5 - (3*c^(3/4)*d*
e^8*(15*c^2*d^8 - 16*a*c*d^4*e^4 + a^2*e^8 - 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(11*c*d^4
- 5*a*e^4))*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)^5) + (c^(3/4)*d*e^4*(4*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4
- 5*a*e^4) - 9*(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])/(16*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^4) + (c^(3/4)
*d*(10*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4) - 21*(c^2*d^8 - 12*a*c*d^4*e^
4 + 3*a^2*e^8))*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqr
t[2]*a^(11/4)*(c*d^4 + a*e^4)^3) + (3*c^(3/4)*d*e^8*(15*c^2*d^8 - 16*a*c*d^4*e^4
+ a^2*e^8 - 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(11*c*d^4 - 5*a*e^4))*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)^5) - (c^
(3/4)*d*e^4*(4*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4 - 5*a*e^4) - 9*(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])/(
16*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^4) - (c^(3/4)*d*(10*Sqrt[a]*Sqrt[c]*d^2*e^2*(
3*c*d^4 - 5*a*e^4) - 21*(c^2*d^8 - 12*a*c*d^4*e^4 + 3*a^2*e^8))*Log[Sqrt[a] + Sq
rt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(11/4)*(c*d^4 + a*e^4)^3)
- (3*c*d^2*e^11*(13*c*d^4 - 3*a*e^4)*Log[a + c*x^4])/(2*(c*d^4 + a*e^4)^5)

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Rubi [A]  time = 7.56783, antiderivative size = 2204, normalized size of antiderivative = 1., number of steps used = 46, number of rules used = 14, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.824 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

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

[Out]

-e^11/(2*(c*d^4 + a*e^4)^3*(d + e*x)^2) - (12*c*d^3*e^11)/((c*d^4 + a*e^4)^4*(d
+ e*x)) + (c*x*(7*d*(c^2*d^8 - 12*a*c*d^4*e^4 + 3*a^2*e^8) - 6*e*(3*c^2*d^8 - 12
*a*c*d^4*e^4 + a^2*e^8)*x + 10*c*d^3*e^2*(3*c*d^4 - 5*a*e^4)*x^2))/(32*a^2*(c*d^
4 + a*e^4)^3*(a + c*x^4)) + (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)))/(8*a*(c*d^4 + a*e^4)^3*(a + c*x^4)^2) + (c*e
^4*(12*a*d^2*e^3*(3*c*d^4 - a*e^4) + x*(3*d*(5*c^2*d^8 - 10*a*c*d^4*e^4 + a^2*e^
8) - e*(21*c^2*d^8 - 26*a*c*d^4*e^4 + a^2*e^8)*x + 4*c*d^3*e^2*(7*c*d^4 - 5*a*e^
4)*x^2)))/(4*a*(c*d^4 + a*e^4)^4*(a + c*x^4)) - (Sqrt[c]*e^9*(55*c^2*d^8 - 40*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)^5
) - (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)/Sq
rt[a]])/(4*a^(3/2)*(c*d^4 + a*e^4)^4) - (3*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]])/(16*a^(5/2)*(c*d^4 + a*e^4)^3) - (3*c
^(3/4)*d*e^8*(15*c^2*d^8 - 16*a*c*d^4*e^4 + a^2*e^8 + 2*Sqrt[a]*Sqrt[c]*d^2*e^2*
(11*c*d^4 - 5*a*e^4))*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4
)*(c*d^4 + a*e^4)^5) - (c^(3/4)*d*e^4*(4*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4 - 5*a*
e^4) + 9*(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)])/(8*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^4) - (c^(3/4)*d*(10*Sqrt[a]*Sqrt[c]
*d^2*e^2*(3*c*d^4 - 5*a*e^4) + 21*(c^2*d^8 - 12*a*c*d^4*e^4 + 3*a^2*e^8))*ArcTan
[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*(c*d^4 + a*e^4)^3) + (3*
c^(3/4)*d*e^8*(15*c^2*d^8 - 16*a*c*d^4*e^4 + a^2*e^8 + 2*Sqrt[a]*Sqrt[c]*d^2*e^2
*(11*c*d^4 - 5*a*e^4))*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/
4)*(c*d^4 + a*e^4)^5) + (c^(3/4)*d*e^4*(4*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4 - 5*a
*e^4) + 9*(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)])/(8*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^4) + (c^(3/4)*d*(10*Sqrt[a]*Sqrt[c
]*d^2*e^2*(3*c*d^4 - 5*a*e^4) + 21*(c^2*d^8 - 12*a*c*d^4*e^4 + 3*a^2*e^8))*ArcTa
n[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(11/4)*(c*d^4 + a*e^4)^3) + (6
*c*d^2*e^11*(13*c*d^4 - 3*a*e^4)*Log[d + e*x])/(c*d^4 + a*e^4)^5 - (3*c^(3/4)*d*
e^8*(15*c^2*d^8 - 16*a*c*d^4*e^4 + a^2*e^8 - 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(11*c*d^4
- 5*a*e^4))*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)^5) + (c^(3/4)*d*e^4*(4*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4
- 5*a*e^4) - 9*(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])/(16*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^4) + (c^(3/4)
*d*(10*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4) - 21*(c^2*d^8 - 12*a*c*d^4*e^
4 + 3*a^2*e^8))*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqr
t[2]*a^(11/4)*(c*d^4 + a*e^4)^3) + (3*c^(3/4)*d*e^8*(15*c^2*d^8 - 16*a*c*d^4*e^4
+ a^2*e^8 - 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(11*c*d^4 - 5*a*e^4))*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)^5) - (c^
(3/4)*d*e^4*(4*Sqrt[a]*Sqrt[c]*d^2*e^2*(7*c*d^4 - 5*a*e^4) - 9*(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])/(
16*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)^4) - (c^(3/4)*d*(10*Sqrt[a]*Sqrt[c]*d^2*e^2*(
3*c*d^4 - 5*a*e^4) - 21*(c^2*d^8 - 12*a*c*d^4*e^4 + 3*a^2*e^8))*Log[Sqrt[a] + Sq
rt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(11/4)*(c*d^4 + a*e^4)^3)
- (3*c*d^2*e^11*(13*c*d^4 - 3*a*e^4)*Log[a + c*x^4])/(2*(c*d^4 + a*e^4)^5)

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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)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 5.7129, size = 1338, normalized size = 0.61 \[ \frac{1536 c d^2 \left (13 c d^4-3 a e^4\right ) \log (d+e x) e^{11}-384 c d^2 \left (13 c d^4-3 a e^4\right ) \log \left (c x^4+a\right ) e^{11}-\frac{3072 c d^3 \left (c d^4+a e^4\right ) e^{11}}{d+e x}-\frac{128 \left (c d^4+a e^4\right )^2 e^{11}}{(d+e x)^2}+\frac{8 c \left (c d^4+a e^4\right ) \left (c^3 x \left (7 d^2-18 e x d+30 e^2 x^2\right ) d^{11}+a c^2 e^4 x \left (43 d^2-114 e x d+204 e^2 x^2\right ) d^7+a^2 c e^7 \left (288 d^3-303 e x d^2+274 e^2 x^2 d-210 e^3 x^3\right ) d^3+a^3 e^{11} \left (-96 d^2+45 e x d-14 e^2 x^2\right )\right )}{a^2 \left (c x^4+a\right )}+\frac{32 c \left (c d^4+a e^4\right )^2 \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 )^2}-\frac{6 \sqrt{c} \left (7 \sqrt{2} c^{17/4} d^{17}-24 \sqrt [4]{a} c^4 e d^{16}+10 \sqrt{2} \sqrt{a} c^{15/4} e^2 d^{15}+50 \sqrt{2} a c^{13/4} e^4 d^{13}-176 a^{5/4} c^3 e^5 d^{12}+78 \sqrt{2} a^{3/2} c^{11/4} e^6 d^{11}+220 \sqrt{2} a^2 c^{9/4} e^8 d^9-960 a^{9/4} c^2 e^9 d^8+702 \sqrt{2} a^{5/2} c^{7/4} e^{10} d^7-770 \sqrt{2} a^3 c^{5/4} e^{12} d^5+1200 a^{13/4} c e^{13} d^4-390 \sqrt{2} a^{7/2} c^{3/4} e^{14} d^3+77 \sqrt{2} a^4 \sqrt [4]{c} e^{16} d-40 a^{17/4} e^{17}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{11/4}}+\frac{6 \sqrt{c} \left (7 \sqrt{2} c^{17/4} d^{17}+24 \sqrt [4]{a} c^4 e d^{16}+10 \sqrt{2} \sqrt{a} c^{15/4} e^2 d^{15}+50 \sqrt{2} a c^{13/4} e^4 d^{13}+176 a^{5/4} c^3 e^5 d^{12}+78 \sqrt{2} a^{3/2} c^{11/4} e^6 d^{11}+220 \sqrt{2} a^2 c^{9/4} e^8 d^9+960 a^{9/4} c^2 e^9 d^8+702 \sqrt{2} a^{5/2} c^{7/4} e^{10} d^7-770 \sqrt{2} a^3 c^{5/4} e^{12} d^5-1200 a^{13/4} c e^{13} d^4-390 \sqrt{2} a^{7/2} c^{3/4} e^{14} d^3+77 \sqrt{2} a^4 \sqrt [4]{c} e^{16} d+40 a^{17/4} e^{17}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{11/4}}-\frac{3 \sqrt{2} c^{3/4} \left (7 c^4 d^{17}-10 \sqrt{a} c^{7/2} e^2 d^{15}+50 a c^3 e^4 d^{13}-78 a^{3/2} c^{5/2} e^6 d^{11}+220 a^2 c^2 e^8 d^9-702 a^{5/2} c^{3/2} e^{10} d^7-770 a^3 c e^{12} d^5+390 a^{7/2} \sqrt{c} e^{14} d^3+77 a^4 e^{16} d\right ) \log \left (\sqrt{c} x^2-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{a^{11/4}}+\frac{3 \sqrt{2} c^{3/4} \left (7 c^4 d^{17}-10 \sqrt{a} c^{7/2} e^2 d^{15}+50 a c^3 e^4 d^{13}-78 a^{3/2} c^{5/2} e^6 d^{11}+220 a^2 c^2 e^8 d^9-702 a^{5/2} c^{3/2} e^{10} d^7-770 a^3 c e^{12} d^5+390 a^{7/2} \sqrt{c} e^{14} d^3+77 a^4 e^{16} d\right ) \log \left (\sqrt{c} x^2+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}\right )}{a^{11/4}}}{256 \left (c d^4+a e^4\right )^5} \]

Antiderivative was successfully verified.

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

[Out]

((-128*e^11*(c*d^4 + a*e^4)^2)/(d + e*x)^2 - (3072*c*d^3*e^11*(c*d^4 + a*e^4))/(
d + e*x) + (8*c*(c*d^4 + a*e^4)*(a^3*e^11*(-96*d^2 + 45*d*e*x - 14*e^2*x^2) + c^
3*d^11*x*(7*d^2 - 18*d*e*x + 30*e^2*x^2) + a*c^2*d^7*e^4*x*(43*d^2 - 114*d*e*x +
204*e^2*x^2) + a^2*c*d^3*e^7*(288*d^3 - 303*d^2*e*x + 274*d*e^2*x^2 - 210*e^3*x
^3)))/(a^2*(a + c*x^4)) + (32*c*(c*d^4 + a*e^4)^2*(-(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)^2) - (6*Sqrt[c]*(7*Sqrt[2]*c^(1
7/4)*d^17 - 24*a^(1/4)*c^4*d^16*e + 10*Sqrt[2]*Sqrt[a]*c^(15/4)*d^15*e^2 + 50*Sq
rt[2]*a*c^(13/4)*d^13*e^4 - 176*a^(5/4)*c^3*d^12*e^5 + 78*Sqrt[2]*a^(3/2)*c^(11/
4)*d^11*e^6 + 220*Sqrt[2]*a^2*c^(9/4)*d^9*e^8 - 960*a^(9/4)*c^2*d^8*e^9 + 702*Sq
rt[2]*a^(5/2)*c^(7/4)*d^7*e^10 - 770*Sqrt[2]*a^3*c^(5/4)*d^5*e^12 + 1200*a^(13/4
)*c*d^4*e^13 - 390*Sqrt[2]*a^(7/2)*c^(3/4)*d^3*e^14 + 77*Sqrt[2]*a^4*c^(1/4)*d*e
^16 - 40*a^(17/4)*e^17)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(11/4) + (6*S
qrt[c]*(7*Sqrt[2]*c^(17/4)*d^17 + 24*a^(1/4)*c^4*d^16*e + 10*Sqrt[2]*Sqrt[a]*c^(
15/4)*d^15*e^2 + 50*Sqrt[2]*a*c^(13/4)*d^13*e^4 + 176*a^(5/4)*c^3*d^12*e^5 + 78*
Sqrt[2]*a^(3/2)*c^(11/4)*d^11*e^6 + 220*Sqrt[2]*a^2*c^(9/4)*d^9*e^8 + 960*a^(9/4
)*c^2*d^8*e^9 + 702*Sqrt[2]*a^(5/2)*c^(7/4)*d^7*e^10 - 770*Sqrt[2]*a^3*c^(5/4)*d
^5*e^12 - 1200*a^(13/4)*c*d^4*e^13 - 390*Sqrt[2]*a^(7/2)*c^(3/4)*d^3*e^14 + 77*S
qrt[2]*a^4*c^(1/4)*d*e^16 + 40*a^(17/4)*e^17)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(
1/4)])/a^(11/4) + 1536*c*d^2*e^11*(13*c*d^4 - 3*a*e^4)*Log[d + e*x] - (3*Sqrt[2]
*c^(3/4)*(7*c^4*d^17 - 10*Sqrt[a]*c^(7/2)*d^15*e^2 + 50*a*c^3*d^13*e^4 - 78*a^(3
/2)*c^(5/2)*d^11*e^6 + 220*a^2*c^2*d^9*e^8 - 702*a^(5/2)*c^(3/2)*d^7*e^10 - 770*
a^3*c*d^5*e^12 + 390*a^(7/2)*Sqrt[c]*d^3*e^14 + 77*a^4*d*e^16)*Log[Sqrt[a] - Sqr
t[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(11/4) + (3*Sqrt[2]*c^(3/4)*(7*c^4*d^17
- 10*Sqrt[a]*c^(7/2)*d^15*e^2 + 50*a*c^3*d^13*e^4 - 78*a^(3/2)*c^(5/2)*d^11*e^6
+ 220*a^2*c^2*d^9*e^8 - 702*a^(5/2)*c^(3/2)*d^7*e^10 - 770*a^3*c*d^5*e^12 + 390
*a^(7/2)*Sqrt[c]*d^3*e^14 + 77*a^4*d*e^16)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)
*x + Sqrt[c]*x^2])/a^(11/4) - 384*c*d^2*e^11*(13*c*d^4 - 3*a*e^4)*Log[a + c*x^4]
)/(256*(c*d^4 + a*e^4)^5)

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Maple [A]  time = 0.048, size = 3352, normalized size = 1.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^3/(c*x^4+a)^3,x)

[Out]

21/128/(a*e^4+c*d^4)^5*c^5*(1/c*a)^(1/4)/a^3*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4
)*x-1)*d^17+65/8/(a*e^4+c*d^4)^5*c^3/(c*x^4+a)^2*e^13*a*x^6*d^4-33/8/(a*e^4+c*d^
4)^5*c^5/(c*x^4+a)^2*e^5/a*x^6*d^12+225/8/(a*e^4+c*d^4)^5*c^2/(a^5*c)^(1/2)*arct
an(x^2*(c/a)^(1/2))*a^3*d^4*e^13-45/2/(a*e^4+c*d^4)^5*c^3/(a^5*c)^(1/2)*arctan(x
^2*(c/a)^(1/2))*a^2*d^8*e^9-33/8/(a*e^4+c*d^4)^5*c^4/(a^5*c)^(1/2)*arctan(x^2*(c
/a)^(1/2))*a*d^12*e^5+21/128/(a*e^4+c*d^4)^5*c^5*(1/c*a)^(1/4)/a^3*2^(1/2)*arcta
n(2^(1/2)/(1/c*a)^(1/4)*x+1)*d^17+25/16/(a*e^4+c*d^4)^5*c^5/(c*x^4+a)^2*d^13/a*x
^5*e^4-125/16/(a*e^4+c*d^4)^5*c^2/(c*x^4+a)^2*d^3*e^14*a^2*x^3-31/16/(a*e^4+c*d^
4)^5*c^3/(c*x^4+a)^2*d^7*e^10*a*x^3+27/16/(a*e^4+c*d^4)^5*c^5/(c*x^4+a)^2*d^15*e
^2/a*x^3+75/8/(a*e^4+c*d^4)^5*c^2/(c*x^4+a)^2*e^13*a^2*x^2*d^4+15/2/(a*e^4+c*d^4
)^5*c^3/(c*x^4+a)^2*e^9*a*x^2*d^8-15/16/(a*e^4+c*d^4)^5*c^5/(c*x^4+a)^2*e/a*x^2*
d^16-141/16/(a*e^4+c*d^4)^5*c^2/(c*x^4+a)^2*d^5*a^2*x*e^12-85/8/(a*e^4+c*d^4)^5*
c^3/(c*x^4+a)^2*d^9*a*x*e^8-3/(a*e^4+c*d^4)^5*c^2/(c*x^4+a)^2*x^4*a^2*d^2*e^15-1
05/16/(a*e^4+c*d^4)^5*c^3/(c*x^4+a)^2*d^3*e^14*a*x^7+117/16/(a*e^4+c*d^4)^5*c^5/
(c*x^4+a)^2*d^11*e^6/a*x^7+15/16/(a*e^4+c*d^4)^5*c^6/(c*x^4+a)^2*d^15*e^2/a^2*x^
7-1155/128/(a*e^4+c*d^4)^5*c^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^5*e^12+21/256
/(a*e^4+c*d^4)^5*c^5*(1/c*a)^(1/4)/a^3*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^17+15/128/(a*e^4+c*
d^4)^5*c^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^15*e^2+15/64/(a*e^4+c*d^4)^5*
c^4/a^2/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)*d^15*e^2-585/64/
(a*e^4+c*d^4)^5*c*a/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)*d^3*
e^14+231/128/(a*e^4+c*d^4)^5*c*(1/c*a)^(1/4)*a*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1
/4)*x-1)*d*e^16-585/128/(a*e^4+c*d^4)^5*c*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^14+117/128/(a*e^4+c*d^4)^5*c^3/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^6+
75/64/(a*e^4+c*d^4)^5*c^4*(1/c*a)^(1/4)/a^2*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)
*x-1)*d^13*e^4+165/64/(a*e^4+c*d^4)^5*c^3*(1/c*a)^(1/4)/a*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^8+75/128/(a*e^4+c*d^4)^5*c^4*(1/c*a)^(1/4)/a^2*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*e^4+
231/256/(a*e^4+c*d^4)^5*c*(1/c*a)^(1/4)*a*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^16-1/2*e^11/(a
*e^4+c*d^4)^3/(e*x+d)^2-9/16/(a*e^4+c*d^4)^5*c^6/(c*x^4+a)^2*e/a^2*x^6*d^16+45/3
2/(a*e^4+c*d^4)^5*c^2/(c*x^4+a)^2*d*a^2*x^5*e^16-129/16/(a*e^4+c*d^4)^5*c^3/(c*x
^4+a)^2*d^5*a*x^5*e^12+6/(a*e^4+c*d^4)^5*c^3/(c*x^4+a)^2*x^4*a*d^6*e^11+57/32/(a
*e^4+c*d^4)^5*c/(c*x^4+a)^2*d*a^3*x*e^16-65/8/(a*e^4+c*d^4)^5*c^4/(c*x^4+a)^2*d^
9*x^5*e^8+121/16/(a*e^4+c*d^4)^5*c^4/(c*x^4+a)^2*d^11*e^6*x^3-27/8/(a*e^4+c*d^4)
^5*c^4/(c*x^4+a)^2*e^5*x^2*d^12+5/16/(a*e^4+c*d^4)^5*c^4/(c*x^4+a)^2*d^13*x*e^4+
43/4/(a*e^4+c*d^4)^5*c^3/(c*x^4+a)^2*a*d^10*e^7+9/(a*e^4+c*d^4)^5*c^4/(c*x^4+a)^
2*x^4*d^10*e^7-7/16/(a*e^4+c*d^4)^5*c^2/(c*x^4+a)^2*e^17*a^2*x^6+7/32/(a*e^4+c*d
^4)^5*c^6/(c*x^4+a)^2*d^17/a^2*x^5+11/32/(a*e^4+c*d^4)^5*c^5/(c*x^4+a)^2*d^17/a*
x-3/16/(a*e^4+c*d^4)^5*c^4/(c*x^4+a)^2*d^7*e^10*x^7-18*e^15*c*d^2/(a*e^4+c*d^4)^
5*ln(e*x+d)*a+5/4/(a*e^4+c*d^4)^5*c^4/(c*x^4+a)^2*d^14*e^3-15/4/(a*e^4+c*d^4)^5*
c/(c*x^4+a)^2*a^3*d^2*e^15-15/16/(a*e^4+c*d^4)^5*c/(a^5*c)^(1/2)*arctan(x^2*(c/a
)^(1/2))*a^4*e^17-9/16/(a*e^4+c*d^4)^5*c^5/(a^5*c)^(1/2)*arctan(x^2*(c/a)^(1/2))
*d^16*e+9/2/(a*e^4+c*d^4)^5*c*a*ln(a^2*(c*x^4+a))*d^2*e^15-9/16/(a*e^4+c*d^4)^5*
c/(c*x^4+a)^2*e^17*a^3*x^2+23/4/(a*e^4+c*d^4)^5*c^2/(c*x^4+a)^2*a^2*d^6*e^11+5/(
a*e^4+c*d^4)^5*c^4/(c*x^4+a)^2*e^9*x^6*d^8+1053/128/(a*e^4+c*d^4)^5*c^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^7*e^10+1053/64/(a*e^4+c*d^4)^5*c^2/(1/c*a)^(1/4)*2^(
1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)*d^7*e^10+1053/64/(a*e^4+c*d^4)^5*c^2/(1/c
*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)*d^7*e^10-1155/64/(a*e^4+c*d^
4)^5*c^2*(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)*d^5*e^12-1155/6
4/(a*e^4+c*d^4)^5*c^2*(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)*d^
5*e^12+117/64/(a*e^4+c*d^4)^5*c^3/a/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)
^(1/4)*x-1)*d^11*e^6+15/64/(a*e^4+c*d^4)^5*c^4/a^2/(1/c*a)^(1/4)*2^(1/2)*arctan(
2^(1/2)/(1/c*a)^(1/4)*x-1)*d^15*e^2+165/32/(a*e^4+c*d^4)^5*c^3*(1/c*a)^(1/4)/a*2
^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)*d^9*e^8+231/128/(a*e^4+c*d^4)^5*c*(1/c*
a)^(1/4)*a*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)*d*e^16+75/64/(a*e^4+c*d^4)^
5*c^4*(1/c*a)^(1/4)/a^2*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)*d^13*e^4+165/3
2/(a*e^4+c*d^4)^5*c^3*(1/c*a)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)*
d^9*e^8-585/64/(a*e^4+c*d^4)^5*c*a/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^
(1/4)*x+1)*d^3*e^14+117/64/(a*e^4+c*d^4)^5*c^3/a/(1/c*a)^(1/4)*2^(1/2)*arctan(2^
(1/2)/(1/c*a)^(1/4)*x+1)*d^11*e^6-39/2/(a*e^4+c*d^4)^5*c^2*ln(a^2*(c*x^4+a))*d^6
*e^11+78*e^11*c^2*d^6/(a*e^4+c*d^4)^5*ln(e*x+d)-12*c*d^3*e^11/(a*e^4+c*d^4)^4/(e
*x+d)

_______________________________________________________________________________________

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)^3*(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)^3*(e*x + d)^3),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

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)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.531328, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done