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

Optimal. Leaf size=370 \[ -\frac{\left (\sqrt{c} d \left (c d^2-3 a e^2\right )-\sqrt{a} e \left (3 c d^2-a e^2\right )\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} c^{7/4}}+\frac{\left (\sqrt{c} d \left (c d^2-3 a e^2\right )-\sqrt{a} e \left (3 c d^2-a e^2\right )\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} c^{7/4}}-\frac{\left (\sqrt{c} d \left (c d^2-3 a e^2\right )+\sqrt{a} e \left (3 c d^2-a e^2\right )\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} c^{7/4}}+\frac{\left (\sqrt{c} d \left (c d^2-3 a e^2\right )+\sqrt{a} e \left (3 c d^2-a e^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} c^{7/4}}+\frac{3 d e^2 x}{c}+\frac{e^3 x^3}{3 c} \]

[Out]

(3*d*e^2*x)/c + (e^3*x^3)/(3*c) - ((Sqrt[c]*d*(c*d^2 - 3*a*e^2) + Sqrt[a]*e*(3*c
*d^2 - a*e^2))*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*c^(7/
4)) + ((Sqrt[c]*d*(c*d^2 - 3*a*e^2) + Sqrt[a]*e*(3*c*d^2 - a*e^2))*ArcTan[1 + (S
qrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*c^(7/4)) - ((Sqrt[c]*d*(c*d^2 - 3
*a*e^2) - Sqrt[a]*e*(3*c*d^2 - a*e^2))*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x +
Sqrt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*c^(7/4)) + ((Sqrt[c]*d*(c*d^2 - 3*a*e^2) - Sqr
t[a]*e*(3*c*d^2 - a*e^2))*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2]
)/(4*Sqrt[2]*a^(3/4)*c^(7/4))

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Rubi [A]  time = 0.909209, antiderivative size = 370, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ -\frac{\left (\sqrt{c} d \left (c d^2-3 a e^2\right )-\sqrt{a} e \left (3 c d^2-a e^2\right )\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} c^{7/4}}+\frac{\left (\sqrt{c} d \left (c d^2-3 a e^2\right )-\sqrt{a} e \left (3 c d^2-a e^2\right )\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{4 \sqrt{2} a^{3/4} c^{7/4}}-\frac{\left (\sqrt{c} d \left (c d^2-3 a e^2\right )+\sqrt{a} e \left (3 c d^2-a e^2\right )\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} c^{7/4}}+\frac{\left (\sqrt{c} d \left (c d^2-3 a e^2\right )+\sqrt{a} e \left (3 c d^2-a e^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} c^{7/4}}+\frac{3 d e^2 x}{c}+\frac{e^3 x^3}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

(3*d*e^2*x)/c + (e^3*x^3)/(3*c) - ((Sqrt[c]*d*(c*d^2 - 3*a*e^2) + Sqrt[a]*e*(3*c
*d^2 - a*e^2))*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*c^(7/
4)) + ((Sqrt[c]*d*(c*d^2 - 3*a*e^2) + Sqrt[a]*e*(3*c*d^2 - a*e^2))*ArcTan[1 + (S
qrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*c^(7/4)) - ((Sqrt[c]*d*(c*d^2 - 3
*a*e^2) - Sqrt[a]*e*(3*c*d^2 - a*e^2))*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x +
Sqrt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*c^(7/4)) + ((Sqrt[c]*d*(c*d^2 - 3*a*e^2) - Sqr
t[a]*e*(3*c*d^2 - a*e^2))*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2]
)/(4*Sqrt[2]*a^(3/4)*c^(7/4))

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Rubi in Sympy [A]  time = 88.3478, size = 345, normalized size = 0.93 \[ \frac{3 d e^{2} x}{c} + \frac{e^{3} x^{3}}{3 c} - \frac{\sqrt{2} \left (\sqrt{a} e \left (a e^{2} - 3 c d^{2}\right ) - \sqrt{c} d \left (3 a e^{2} - c d^{2}\right )\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} x + \sqrt{a} \sqrt{c} + c x^{2} \right )}}{8 a^{\frac{3}{4}} c^{\frac{7}{4}}} + \frac{\sqrt{2} \left (\sqrt{a} e \left (a e^{2} - 3 c d^{2}\right ) - \sqrt{c} d \left (3 a e^{2} - c d^{2}\right )\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} x + \sqrt{a} \sqrt{c} + c x^{2} \right )}}{8 a^{\frac{3}{4}} c^{\frac{7}{4}}} + \frac{\sqrt{2} \left (\sqrt{a} e \left (a e^{2} - 3 c d^{2}\right ) + \sqrt{c} d \left (3 a e^{2} - c d^{2}\right )\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} c^{\frac{7}{4}}} - \frac{\sqrt{2} \left (\sqrt{a} e \left (a e^{2} - 3 c d^{2}\right ) + \sqrt{c} d \left (3 a e^{2} - c d^{2}\right )\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{4 a^{\frac{3}{4}} c^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*d*e**2*x/c + e**3*x**3/(3*c) - sqrt(2)*(sqrt(a)*e*(a*e**2 - 3*c*d**2) - sqrt(c
)*d*(3*a*e**2 - c*d**2))*log(-sqrt(2)*a**(1/4)*c**(3/4)*x + sqrt(a)*sqrt(c) + c*
x**2)/(8*a**(3/4)*c**(7/4)) + sqrt(2)*(sqrt(a)*e*(a*e**2 - 3*c*d**2) - sqrt(c)*d
*(3*a*e**2 - c*d**2))*log(sqrt(2)*a**(1/4)*c**(3/4)*x + sqrt(a)*sqrt(c) + c*x**2
)/(8*a**(3/4)*c**(7/4)) + sqrt(2)*(sqrt(a)*e*(a*e**2 - 3*c*d**2) + sqrt(c)*d*(3*
a*e**2 - c*d**2))*atan(1 - sqrt(2)*c**(1/4)*x/a**(1/4))/(4*a**(3/4)*c**(7/4)) -
sqrt(2)*(sqrt(a)*e*(a*e**2 - 3*c*d**2) + sqrt(c)*d*(3*a*e**2 - c*d**2))*atan(1 +
sqrt(2)*c**(1/4)*x/a**(1/4))/(4*a**(3/4)*c**(7/4))

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Mathematica [A]  time = 0.51865, size = 360, normalized size = 0.97 \[ \frac{-3 \sqrt{2} \left (a^{3/2} e^3-3 \sqrt{a} c d^2 e-3 a \sqrt{c} d e^2+c^{3/2} d^3\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+3 \sqrt{2} \left (a^{3/2} e^3-3 \sqrt{a} c d^2 e-3 a \sqrt{c} d e^2+c^{3/2} d^3\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )+6 \sqrt{2} \left (a^{3/2} e^3-3 \sqrt{a} c d^2 e+3 a \sqrt{c} d e^2-c^{3/2} d^3\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+6 \sqrt{2} \left (-a^{3/2} e^3+3 \sqrt{a} c d^2 e-3 a \sqrt{c} d e^2+c^{3/2} d^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )+72 a^{3/4} c^{3/4} d e^2 x+8 a^{3/4} c^{3/4} e^3 x^3}{24 a^{3/4} c^{7/4}} \]

Antiderivative was successfully verified.

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

[Out]

(72*a^(3/4)*c^(3/4)*d*e^2*x + 8*a^(3/4)*c^(3/4)*e^3*x^3 + 6*Sqrt[2]*(-(c^(3/2)*d
^3) - 3*Sqrt[a]*c*d^2*e + 3*a*Sqrt[c]*d*e^2 + a^(3/2)*e^3)*ArcTan[1 - (Sqrt[2]*c
^(1/4)*x)/a^(1/4)] + 6*Sqrt[2]*(c^(3/2)*d^3 + 3*Sqrt[a]*c*d^2*e - 3*a*Sqrt[c]*d*
e^2 - a^(3/2)*e^3)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] - 3*Sqrt[2]*(c^(3/2)*
d^3 - 3*Sqrt[a]*c*d^2*e - 3*a*Sqrt[c]*d*e^2 + a^(3/2)*e^3)*Log[Sqrt[a] - Sqrt[2]
*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] + 3*Sqrt[2]*(c^(3/2)*d^3 - 3*Sqrt[a]*c*d^2*e -
3*a*Sqrt[c]*d*e^2 + a^(3/2)*e^3)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt
[c]*x^2])/(24*a^(3/4)*c^(7/4))

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Maple [A]  time = 0.006, size = 572, normalized size = 1.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+d)^3/(c*x^4+a),x)

[Out]

1/3*e^3*x^3/c+3*d*e^2*x/c-3/4/c*(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/
4)*x-1)*d*e^2+1/4*(1/c*a)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)*d^3-
3/8/c*(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^2+1/8*(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^3-3/4/c*(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)*d*e^2+1/
4*(1/c*a)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)*d^3-1/8/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)))*a*e^3+3/8/c/(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^2*e-1/
4/c^2/(1/c*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)*a*e^3+3/4/c/(1/c*a
)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x-1)*d^2*e-1/4/c^2/(1/c*a)^(1/4)*2^
(1/2)*arctan(2^(1/2)/(1/c*a)^(1/4)*x+1)*a*e^3+3/4/c/(1/c*a)^(1/4)*2^(1/2)*arctan
(2^(1/2)/(1/c*a)^(1/4)*x+1)*d^2*e

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.21967, size = 2880, normalized size = 7.78 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(4*e^3*x^3 + 36*d*e^2*x - 3*c*sqrt(-(6*c^2*d^5*e - 20*a*c*d^3*e^3 + 6*a^2*d
*e^5 + a*c^3*sqrt(-(c^6*d^12 - 30*a*c^5*d^10*e^2 + 255*a^2*c^4*d^8*e^4 - 452*a^3
*c^3*d^6*e^6 + 255*a^4*c^2*d^4*e^8 - 30*a^5*c*d^2*e^10 + a^6*e^12)/(a^3*c^7)))/(
a*c^3))*log(-(c^6*d^12 - 12*a*c^5*d^10*e^2 - 27*a^2*c^4*d^8*e^4 + 27*a^4*c^2*d^4
*e^8 + 12*a^5*c*d^2*e^10 - a^6*e^12)*x + (a*c^6*d^9 - 18*a^2*c^5*d^7*e^2 + 60*a^
3*c^4*d^5*e^4 - 46*a^4*c^3*d^3*e^6 + 3*a^5*c^2*d*e^8 + (3*a^3*c^6*d^2*e - a^4*c^
5*e^3)*sqrt(-(c^6*d^12 - 30*a*c^5*d^10*e^2 + 255*a^2*c^4*d^8*e^4 - 452*a^3*c^3*d
^6*e^6 + 255*a^4*c^2*d^4*e^8 - 30*a^5*c*d^2*e^10 + a^6*e^12)/(a^3*c^7)))*sqrt(-(
6*c^2*d^5*e - 20*a*c*d^3*e^3 + 6*a^2*d*e^5 + a*c^3*sqrt(-(c^6*d^12 - 30*a*c^5*d^
10*e^2 + 255*a^2*c^4*d^8*e^4 - 452*a^3*c^3*d^6*e^6 + 255*a^4*c^2*d^4*e^8 - 30*a^
5*c*d^2*e^10 + a^6*e^12)/(a^3*c^7)))/(a*c^3))) + 3*c*sqrt(-(6*c^2*d^5*e - 20*a*c
*d^3*e^3 + 6*a^2*d*e^5 + a*c^3*sqrt(-(c^6*d^12 - 30*a*c^5*d^10*e^2 + 255*a^2*c^4
*d^8*e^4 - 452*a^3*c^3*d^6*e^6 + 255*a^4*c^2*d^4*e^8 - 30*a^5*c*d^2*e^10 + a^6*e
^12)/(a^3*c^7)))/(a*c^3))*log(-(c^6*d^12 - 12*a*c^5*d^10*e^2 - 27*a^2*c^4*d^8*e^
4 + 27*a^4*c^2*d^4*e^8 + 12*a^5*c*d^2*e^10 - a^6*e^12)*x - (a*c^6*d^9 - 18*a^2*c
^5*d^7*e^2 + 60*a^3*c^4*d^5*e^4 - 46*a^4*c^3*d^3*e^6 + 3*a^5*c^2*d*e^8 + (3*a^3*
c^6*d^2*e - a^4*c^5*e^3)*sqrt(-(c^6*d^12 - 30*a*c^5*d^10*e^2 + 255*a^2*c^4*d^8*e
^4 - 452*a^3*c^3*d^6*e^6 + 255*a^4*c^2*d^4*e^8 - 30*a^5*c*d^2*e^10 + a^6*e^12)/(
a^3*c^7)))*sqrt(-(6*c^2*d^5*e - 20*a*c*d^3*e^3 + 6*a^2*d*e^5 + a*c^3*sqrt(-(c^6*
d^12 - 30*a*c^5*d^10*e^2 + 255*a^2*c^4*d^8*e^4 - 452*a^3*c^3*d^6*e^6 + 255*a^4*c
^2*d^4*e^8 - 30*a^5*c*d^2*e^10 + a^6*e^12)/(a^3*c^7)))/(a*c^3))) - 3*c*sqrt(-(6*
c^2*d^5*e - 20*a*c*d^3*e^3 + 6*a^2*d*e^5 - a*c^3*sqrt(-(c^6*d^12 - 30*a*c^5*d^10
*e^2 + 255*a^2*c^4*d^8*e^4 - 452*a^3*c^3*d^6*e^6 + 255*a^4*c^2*d^4*e^8 - 30*a^5*
c*d^2*e^10 + a^6*e^12)/(a^3*c^7)))/(a*c^3))*log(-(c^6*d^12 - 12*a*c^5*d^10*e^2 -
27*a^2*c^4*d^8*e^4 + 27*a^4*c^2*d^4*e^8 + 12*a^5*c*d^2*e^10 - a^6*e^12)*x + (a*
c^6*d^9 - 18*a^2*c^5*d^7*e^2 + 60*a^3*c^4*d^5*e^4 - 46*a^4*c^3*d^3*e^6 + 3*a^5*c
^2*d*e^8 - (3*a^3*c^6*d^2*e - a^4*c^5*e^3)*sqrt(-(c^6*d^12 - 30*a*c^5*d^10*e^2 +
255*a^2*c^4*d^8*e^4 - 452*a^3*c^3*d^6*e^6 + 255*a^4*c^2*d^4*e^8 - 30*a^5*c*d^2*
e^10 + a^6*e^12)/(a^3*c^7)))*sqrt(-(6*c^2*d^5*e - 20*a*c*d^3*e^3 + 6*a^2*d*e^5 -
a*c^3*sqrt(-(c^6*d^12 - 30*a*c^5*d^10*e^2 + 255*a^2*c^4*d^8*e^4 - 452*a^3*c^3*d
^6*e^6 + 255*a^4*c^2*d^4*e^8 - 30*a^5*c*d^2*e^10 + a^6*e^12)/(a^3*c^7)))/(a*c^3)
)) + 3*c*sqrt(-(6*c^2*d^5*e - 20*a*c*d^3*e^3 + 6*a^2*d*e^5 - a*c^3*sqrt(-(c^6*d^
12 - 30*a*c^5*d^10*e^2 + 255*a^2*c^4*d^8*e^4 - 452*a^3*c^3*d^6*e^6 + 255*a^4*c^2
*d^4*e^8 - 30*a^5*c*d^2*e^10 + a^6*e^12)/(a^3*c^7)))/(a*c^3))*log(-(c^6*d^12 - 1
2*a*c^5*d^10*e^2 - 27*a^2*c^4*d^8*e^4 + 27*a^4*c^2*d^4*e^8 + 12*a^5*c*d^2*e^10 -
a^6*e^12)*x - (a*c^6*d^9 - 18*a^2*c^5*d^7*e^2 + 60*a^3*c^4*d^5*e^4 - 46*a^4*c^3
*d^3*e^6 + 3*a^5*c^2*d*e^8 - (3*a^3*c^6*d^2*e - a^4*c^5*e^3)*sqrt(-(c^6*d^12 - 3
0*a*c^5*d^10*e^2 + 255*a^2*c^4*d^8*e^4 - 452*a^3*c^3*d^6*e^6 + 255*a^4*c^2*d^4*e
^8 - 30*a^5*c*d^2*e^10 + a^6*e^12)/(a^3*c^7)))*sqrt(-(6*c^2*d^5*e - 20*a*c*d^3*e
^3 + 6*a^2*d*e^5 - a*c^3*sqrt(-(c^6*d^12 - 30*a*c^5*d^10*e^2 + 255*a^2*c^4*d^8*e
^4 - 452*a^3*c^3*d^6*e^6 + 255*a^4*c^2*d^4*e^8 - 30*a^5*c*d^2*e^10 + a^6*e^12)/(
a^3*c^7)))/(a*c^3))))/c

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Sympy [A]  time = 7.94747, size = 350, normalized size = 0.95 \[ \operatorname{RootSum}{\left (256 t^{4} a^{3} c^{7} + t^{2} \left (192 a^{4} c^{4} d e^{5} - 640 a^{3} c^{5} d^{3} e^{3} + 192 a^{2} c^{6} d^{5} e\right ) + a^{6} e^{12} + 6 a^{5} c d^{2} e^{10} + 15 a^{4} c^{2} d^{4} e^{8} + 20 a^{3} c^{3} d^{6} e^{6} + 15 a^{2} c^{4} d^{8} e^{4} + 6 a c^{5} d^{10} e^{2} + c^{6} d^{12}, \left ( t \mapsto t \log{\left (x + \frac{- 64 t^{3} a^{4} c^{5} e^{3} + 192 t^{3} a^{3} c^{6} d^{2} e - 36 t a^{5} c^{2} d e^{8} + 336 t a^{4} c^{3} d^{3} e^{6} - 504 t a^{3} c^{4} d^{5} e^{4} + 144 t a^{2} c^{5} d^{7} e^{2} - 4 t a c^{6} d^{9}}{a^{6} e^{12} - 12 a^{5} c d^{2} e^{10} - 27 a^{4} c^{2} d^{4} e^{8} + 27 a^{2} c^{4} d^{8} e^{4} + 12 a c^{5} d^{10} e^{2} - c^{6} d^{12}} \right )} \right )\right )} + \frac{3 d e^{2} x}{c} + \frac{e^{3} x^{3}}{3 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(256*_t**4*a**3*c**7 + _t**2*(192*a**4*c**4*d*e**5 - 640*a**3*c**5*d**3*e
**3 + 192*a**2*c**6*d**5*e) + a**6*e**12 + 6*a**5*c*d**2*e**10 + 15*a**4*c**2*d*
*4*e**8 + 20*a**3*c**3*d**6*e**6 + 15*a**2*c**4*d**8*e**4 + 6*a*c**5*d**10*e**2
+ c**6*d**12, Lambda(_t, _t*log(x + (-64*_t**3*a**4*c**5*e**3 + 192*_t**3*a**3*c
**6*d**2*e - 36*_t*a**5*c**2*d*e**8 + 336*_t*a**4*c**3*d**3*e**6 - 504*_t*a**3*c
**4*d**5*e**4 + 144*_t*a**2*c**5*d**7*e**2 - 4*_t*a*c**6*d**9)/(a**6*e**12 - 12*
a**5*c*d**2*e**10 - 27*a**4*c**2*d**4*e**8 + 27*a**2*c**4*d**8*e**4 + 12*a*c**5*
d**10*e**2 - c**6*d**12)))) + 3*d*e**2*x/c + e**3*x**3/(3*c)

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GIAC/XCAS [A]  time = 0.28042, size = 547, normalized size = 1.48 \[ \frac{c^{2} x^{3} e^{3} + 9 \, c^{2} d x e^{2}}{3 \, c^{3}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{3} d^{3} - 3 \, \left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d e^{2} + 3 \, \left (a c^{3}\right )^{\frac{3}{4}} c d^{2} e - \left (a c^{3}\right )^{\frac{3}{4}} a e^{3}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{4 \, a c^{4}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{3} d^{3} - 3 \, \left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d e^{2} + 3 \, \left (a c^{3}\right )^{\frac{3}{4}} c d^{2} e - \left (a c^{3}\right )^{\frac{3}{4}} a e^{3}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{4 \, a c^{4}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{3} d^{3} - 3 \, \left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d e^{2} - 3 \, \left (a c^{3}\right )^{\frac{3}{4}} c d^{2} e + \left (a c^{3}\right )^{\frac{3}{4}} a e^{3}\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, a c^{4}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} c^{3} d^{3} - 3 \, \left (a c^{3}\right )^{\frac{1}{4}} a c^{2} d e^{2} - 3 \, \left (a c^{3}\right )^{\frac{3}{4}} c d^{2} e + \left (a c^{3}\right )^{\frac{3}{4}} a e^{3}\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{8 \, a c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(c^2*x^3*e^3 + 9*c^2*d*x*e^2)/c^3 + 1/4*sqrt(2)*((a*c^3)^(1/4)*c^3*d^3 - 3*(
a*c^3)^(1/4)*a*c^2*d*e^2 + 3*(a*c^3)^(3/4)*c*d^2*e - (a*c^3)^(3/4)*a*e^3)*arctan
(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a*c^4) + 1/4*sqrt(2)*((a*
c^3)^(1/4)*c^3*d^3 - 3*(a*c^3)^(1/4)*a*c^2*d*e^2 + 3*(a*c^3)^(3/4)*c*d^2*e - (a*
c^3)^(3/4)*a*e^3)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a
*c^4) + 1/8*sqrt(2)*((a*c^3)^(1/4)*c^3*d^3 - 3*(a*c^3)^(1/4)*a*c^2*d*e^2 - 3*(a*
c^3)^(3/4)*c*d^2*e + (a*c^3)^(3/4)*a*e^3)*ln(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(
a/c))/(a*c^4) - 1/8*sqrt(2)*((a*c^3)^(1/4)*c^3*d^3 - 3*(a*c^3)^(1/4)*a*c^2*d*e^2
- 3*(a*c^3)^(3/4)*c*d^2*e + (a*c^3)^(3/4)*a*e^3)*ln(x^2 - sqrt(2)*x*(a/c)^(1/4)
+ sqrt(a/c))/(a*c^4)