3.1 \(\int \frac{d+e x+f x^2+g x^3+h x^4+j x^5+k x^6+l x^7+m x^8}{a+b x^3+c x^6} \, dx\)

Optimal. Leaf size=1668 \[ \text{result too large to display} \]

[Out]

(k*x)/c + (l*x^2)/(2*c) + (m*x^3)/(3*c) - ((g - (b*k)/c + (2*c^2*d + b^2*k - c*(
b*g + 2*a*k))/(c*Sqrt[b^2 - 4*a*c]))*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b - Sqrt
[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(2^(1/3)*Sqrt[3]*c^(1/3)*(b - Sqrt[b^2 - 4*a*c])
^(2/3)) - ((h - (b*l)/c + (2*c^2*e + b^2*l - c*(b*h + 2*a*l))/(c*Sqrt[b^2 - 4*a*
c]))*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b - Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/
(2^(2/3)*Sqrt[3]*c^(2/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3)) - ((g - (b*k)/c - (2*c^2
*d - b*c*g + b^2*k - 2*a*c*k)/(c*Sqrt[b^2 - 4*a*c]))*ArcTan[(1 - (2*2^(1/3)*c^(1
/3)*x)/(b + Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(2^(1/3)*Sqrt[3]*c^(1/3)*(b + Sq
rt[b^2 - 4*a*c])^(2/3)) - ((h - (b*l)/c - (2*c^2*e - b*c*h + b^2*l - 2*a*c*l)/(c
*Sqrt[b^2 - 4*a*c]))*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b + Sqrt[b^2 - 4*a*c])^(
1/3))/Sqrt[3]])/(2^(2/3)*Sqrt[3]*c^(2/3)*(b + Sqrt[b^2 - 4*a*c])^(1/3)) - ((2*c^
2*f - b*c*j + b^2*m - 2*a*c*m)*ArcTanh[(b + 2*c*x^3)/Sqrt[b^2 - 4*a*c]])/(3*c^2*
Sqrt[b^2 - 4*a*c]) + ((g - (b*k)/c + (2*c^2*d + b^2*k - c*(b*g + 2*a*k))/(c*Sqrt
[b^2 - 4*a*c]))*Log[(b - Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*2^(1/
3)*c^(1/3)*(b - Sqrt[b^2 - 4*a*c])^(2/3)) - ((h - (b*l)/c + (2*c^2*e + b^2*l - c
*(b*h + 2*a*l))/(c*Sqrt[b^2 - 4*a*c]))*Log[(b - Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/
3)*c^(1/3)*x])/(3*2^(2/3)*c^(2/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3)) + ((g - (b*k)/c
- (2*c^2*d - b*c*g + b^2*k - 2*a*c*k)/(c*Sqrt[b^2 - 4*a*c]))*Log[(b + Sqrt[b^2
- 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*2^(1/3)*c^(1/3)*(b + Sqrt[b^2 - 4*a*c])
^(2/3)) - ((h - (b*l)/c - (2*c^2*e - b*c*h + b^2*l - 2*a*c*l)/(c*Sqrt[b^2 - 4*a*
c]))*Log[(b + Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*2^(2/3)*c^(2/3)*
(b + Sqrt[b^2 - 4*a*c])^(1/3)) - ((g - (b*k)/c + (2*c^2*d + b^2*k - c*(b*g + 2*a
*k))/(c*Sqrt[b^2 - 4*a*c]))*Log[(b - Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*
(b - Sqrt[b^2 - 4*a*c])^(1/3)*x + 2^(2/3)*c^(2/3)*x^2])/(6*2^(1/3)*c^(1/3)*(b -
Sqrt[b^2 - 4*a*c])^(2/3)) + ((h - (b*l)/c + (2*c^2*e + b^2*l - c*(b*h + 2*a*l))/
(c*Sqrt[b^2 - 4*a*c]))*Log[(b - Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b -
Sqrt[b^2 - 4*a*c])^(1/3)*x + 2^(2/3)*c^(2/3)*x^2])/(6*2^(2/3)*c^(2/3)*(b - Sqrt[
b^2 - 4*a*c])^(1/3)) - ((g - (b*k)/c - (2*c^2*d - b*c*g + b^2*k - 2*a*c*k)/(c*Sq
rt[b^2 - 4*a*c]))*Log[(b + Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b + Sqrt[
b^2 - 4*a*c])^(1/3)*x + 2^(2/3)*c^(2/3)*x^2])/(6*2^(1/3)*c^(1/3)*(b + Sqrt[b^2 -
4*a*c])^(2/3)) + ((h - (b*l)/c - (2*c^2*e - b*c*h + b^2*l - 2*a*c*l)/(c*Sqrt[b^
2 - 4*a*c]))*Log[(b + Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b + Sqrt[b^2 -
4*a*c])^(1/3)*x + 2^(2/3)*c^(2/3)*x^2])/(6*2^(2/3)*c^(2/3)*(b + Sqrt[b^2 - 4*a*
c])^(1/3)) + ((c*j - b*m)*Log[a + b*x^3 + c*x^6])/(6*c^2)

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Rubi [A]  time = 10.4635, antiderivative size = 1668, normalized size of antiderivative = 1., number of steps used = 37, number of rules used = 16, integrand size = 55, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.291 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x + f*x^2 + g*x^3 + h*x^4 + j*x^5 + k*x^6 + l*x^7 + m*x^8)/(a + b*x^3 + c*x^6),x]

[Out]

(k*x)/c + (l*x^2)/(2*c) + (m*x^3)/(3*c) - ((g - (b*k)/c + (2*c^2*d + b^2*k - c*(
b*g + 2*a*k))/(c*Sqrt[b^2 - 4*a*c]))*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b - Sqrt
[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(2^(1/3)*Sqrt[3]*c^(1/3)*(b - Sqrt[b^2 - 4*a*c])
^(2/3)) - ((h - (b*l)/c + (2*c^2*e + b^2*l - c*(b*h + 2*a*l))/(c*Sqrt[b^2 - 4*a*
c]))*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b - Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/
(2^(2/3)*Sqrt[3]*c^(2/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3)) - ((g - (b*k)/c - (2*c^2
*d - b*c*g + b^2*k - 2*a*c*k)/(c*Sqrt[b^2 - 4*a*c]))*ArcTan[(1 - (2*2^(1/3)*c^(1
/3)*x)/(b + Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(2^(1/3)*Sqrt[3]*c^(1/3)*(b + Sq
rt[b^2 - 4*a*c])^(2/3)) - ((h - (b*l)/c - (2*c^2*e - b*c*h + b^2*l - 2*a*c*l)/(c
*Sqrt[b^2 - 4*a*c]))*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b + Sqrt[b^2 - 4*a*c])^(
1/3))/Sqrt[3]])/(2^(2/3)*Sqrt[3]*c^(2/3)*(b + Sqrt[b^2 - 4*a*c])^(1/3)) - ((2*c^
2*f - b*c*j + b^2*m - 2*a*c*m)*ArcTanh[(b + 2*c*x^3)/Sqrt[b^2 - 4*a*c]])/(3*c^2*
Sqrt[b^2 - 4*a*c]) + ((g - (b*k)/c + (2*c^2*d + b^2*k - c*(b*g + 2*a*k))/(c*Sqrt
[b^2 - 4*a*c]))*Log[(b - Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*2^(1/
3)*c^(1/3)*(b - Sqrt[b^2 - 4*a*c])^(2/3)) - ((h - (b*l)/c + (2*c^2*e + b^2*l - c
*(b*h + 2*a*l))/(c*Sqrt[b^2 - 4*a*c]))*Log[(b - Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/
3)*c^(1/3)*x])/(3*2^(2/3)*c^(2/3)*(b - Sqrt[b^2 - 4*a*c])^(1/3)) + ((g - (b*k)/c
- (2*c^2*d - b*c*g + b^2*k - 2*a*c*k)/(c*Sqrt[b^2 - 4*a*c]))*Log[(b + Sqrt[b^2
- 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*2^(1/3)*c^(1/3)*(b + Sqrt[b^2 - 4*a*c])
^(2/3)) - ((h - (b*l)/c - (2*c^2*e - b*c*h + b^2*l - 2*a*c*l)/(c*Sqrt[b^2 - 4*a*
c]))*Log[(b + Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*2^(2/3)*c^(2/3)*
(b + Sqrt[b^2 - 4*a*c])^(1/3)) - ((g - (b*k)/c + (2*c^2*d + b^2*k - c*(b*g + 2*a
*k))/(c*Sqrt[b^2 - 4*a*c]))*Log[(b - Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*
(b - Sqrt[b^2 - 4*a*c])^(1/3)*x + 2^(2/3)*c^(2/3)*x^2])/(6*2^(1/3)*c^(1/3)*(b -
Sqrt[b^2 - 4*a*c])^(2/3)) + ((h - (b*l)/c + (2*c^2*e + b^2*l - c*(b*h + 2*a*l))/
(c*Sqrt[b^2 - 4*a*c]))*Log[(b - Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b -
Sqrt[b^2 - 4*a*c])^(1/3)*x + 2^(2/3)*c^(2/3)*x^2])/(6*2^(2/3)*c^(2/3)*(b - Sqrt[
b^2 - 4*a*c])^(1/3)) - ((g - (b*k)/c - (2*c^2*d - b*c*g + b^2*k - 2*a*c*k)/(c*Sq
rt[b^2 - 4*a*c]))*Log[(b + Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b + Sqrt[
b^2 - 4*a*c])^(1/3)*x + 2^(2/3)*c^(2/3)*x^2])/(6*2^(1/3)*c^(1/3)*(b + Sqrt[b^2 -
4*a*c])^(2/3)) + ((h - (b*l)/c - (2*c^2*e - b*c*h + b^2*l - 2*a*c*l)/(c*Sqrt[b^
2 - 4*a*c]))*Log[(b + Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b + Sqrt[b^2 -
4*a*c])^(1/3)*x + 2^(2/3)*c^(2/3)*x^2])/(6*2^(2/3)*c^(2/3)*(b + Sqrt[b^2 - 4*a*
c])^(1/3)) + ((c*j - b*m)*Log[a + b*x^3 + c*x^6])/(6*c^2)

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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((m*x**8+l*x**7+k*x**6+j*x**5+h*x**4+g*x**3+f*x**2+e*x+d)/(c*x**6+b*x**3+a),x)

[Out]

Timed out

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Mathematica [C]  time = 3.07917, size = 223, normalized size = 0.13 \[ \frac{-2 \text{RootSum}\left [\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\text{$\#$1}^5 b m \log (x-\text{$\#$1})+\text{$\#$1}^5 (-c) j \log (x-\text{$\#$1})+\text{$\#$1}^4 b l \log (x-\text{$\#$1})-\text{$\#$1}^4 c h \log (x-\text{$\#$1})+\text{$\#$1}^3 b k \log (x-\text{$\#$1})-\text{$\#$1}^3 c g \log (x-\text{$\#$1})+\text{$\#$1}^2 a m \log (x-\text{$\#$1})-\text{$\#$1}^2 c f \log (x-\text{$\#$1})+a k \log (x-\text{$\#$1})+\text{$\#$1} a l \log (x-\text{$\#$1})-c d \log (x-\text{$\#$1})-\text{$\#$1} c e \log (x-\text{$\#$1})}{2 \text{$\#$1}^5 c+\text{$\#$1}^2 b}\&\right ]+6 k x+3 l x^2+2 m x^3}{6 c} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + j*x^5 + k*x^6 + l*x^7 + m*x^8)/(a + b*x^3 + c*x^6),x]

[Out]

(6*k*x + 3*l*x^2 + 2*m*x^3 - 2*RootSum[a + b*#1^3 + c*#1^6 & , (-(c*d*Log[x - #1
]) + a*k*Log[x - #1] - c*e*Log[x - #1]*#1 + a*l*Log[x - #1]*#1 - c*f*Log[x - #1]
*#1^2 + a*m*Log[x - #1]*#1^2 - c*g*Log[x - #1]*#1^3 + b*k*Log[x - #1]*#1^3 - c*h
*Log[x - #1]*#1^4 + b*l*Log[x - #1]*#1^4 - c*j*Log[x - #1]*#1^5 + b*m*Log[x - #1
]*#1^5)/(b*#1^2 + 2*c*#1^5) & ])/(6*c)

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Maple [C]  time = 0.032, size = 134, normalized size = 0.1 \[{\frac{m{x}^{3}}{3\,c}}+{\frac{l{x}^{2}}{2\,c}}+{\frac{kx}{c}}+{\frac{1}{3\,c}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}c+{{\it \_Z}}^{3}b+a \right ) }{\frac{ \left ( \left ( -bm+cj \right ){{\it \_R}}^{5}+ \left ( -bl+ch \right ){{\it \_R}}^{4}+ \left ( -bk+cg \right ){{\it \_R}}^{3}+ \left ( -am+cf \right ){{\it \_R}}^{2}+ \left ( -al+ce \right ){\it \_R}-ak+cd \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}c+{{\it \_R}}^{2}b}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((m*x^8+l*x^7+k*x^6+j*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^6+b*x^3+a),x)

[Out]

1/3*m*x^3/c+1/2*l*x^2/c+k*x/c+1/3/c*sum(((-b*m+c*j)*_R^5+(-b*l+c*h)*_R^4+(-b*k+c
*g)*_R^3+(-a*m+c*f)*_R^2+(-a*l+c*e)*_R-a*k+c*d)/(2*_R^5*c+_R^2*b)*ln(x-_R),_R=Ro
otOf(_Z^6*c+_Z^3*b+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 \, m x^{3} + 3 \, l x^{2} + 6 \, k x}{6 \, c} - \frac{-\int \frac{{\left (c j - b m\right )} x^{5} +{\left (c h - b l\right )} x^{4} +{\left (c g - b k\right )} x^{3} +{\left (c f - a m\right )} x^{2} + c d - a k +{\left (c e - a l\right )} x}{c x^{6} + b x^{3} + a}\,{d x}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((m*x^8 + l*x^7 + k*x^6 + j*x^5 + h*x^4 + g*x^3 + f*x^2 + e*x + d)/(c*x^6 + b*x^3 + a),x, algorithm="maxima")

[Out]

1/6*(2*m*x^3 + 3*l*x^2 + 6*k*x)/c - integrate(-((c*j - b*m)*x^5 + (c*h - b*l)*x^
4 + (c*g - b*k)*x^3 + (c*f - a*m)*x^2 + c*d - a*k + (c*e - a*l)*x)/(c*x^6 + b*x^
3 + a), x)/c

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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((m*x^8 + l*x^7 + k*x^6 + j*x^5 + h*x^4 + g*x^3 + f*x^2 + e*x + d)/(c*x^6 + b*x^3 + a),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((m*x**8+l*x**7+k*x**6+j*x**5+h*x**4+g*x**3+f*x**2+e*x+d)/(c*x**6+b*x**3+a),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{m x^{8} + l x^{7} + k x^{6} + j x^{5} + h x^{4} + g x^{3} + f x^{2} + e x + d}{c x^{6} + b x^{3} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((m*x^8 + l*x^7 + k*x^6 + j*x^5 + h*x^4 + g*x^3 + f*x^2 + e*x + d)/(c*x^6 + b*x^3 + a),x, algorithm="giac")

[Out]

integrate((m*x^8 + l*x^7 + k*x^6 + j*x^5 + h*x^4 + g*x^3 + f*x^2 + e*x + d)/(c*x
^6 + b*x^3 + a), x)