3.16.60 \(\int \frac {(d+e x)^8}{(a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx\)

Optimal. Leaf size=219 \[ \frac {e^6 (a e+c d x)^5}{5 c^7 d^7}-\frac {\left (c d^2-a e^2\right )^6}{c^7 d^7 (a e+c d x)}+\frac {6 e \left (c d^2-a e^2\right )^5 \log (a e+c d x)}{c^7 d^7}+\frac {3 e^5 \left (c d^2-a e^2\right ) (a e+c d x)^4}{2 c^7 d^7}+\frac {5 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}{c^7 d^7}+\frac {10 e^3 \left (c d^2-a e^2\right )^3 (a e+c d x)^2}{c^7 d^7}+\frac {15 e^2 x \left (c d^2-a e^2\right )^4}{c^6 d^6} \]

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Rubi [A]  time = 0.28, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {626, 43} \begin {gather*} \frac {e^6 (a e+c d x)^5}{5 c^7 d^7}+\frac {3 e^5 \left (c d^2-a e^2\right ) (a e+c d x)^4}{2 c^7 d^7}+\frac {5 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}{c^7 d^7}+\frac {10 e^3 \left (c d^2-a e^2\right )^3 (a e+c d x)^2}{c^7 d^7}+\frac {15 e^2 x \left (c d^2-a e^2\right )^4}{c^6 d^6}-\frac {\left (c d^2-a e^2\right )^6}{c^7 d^7 (a e+c d x)}+\frac {6 e \left (c d^2-a e^2\right )^5 \log (a e+c d x)}{c^7 d^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^8/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

(15*e^2*(c*d^2 - a*e^2)^4*x)/(c^6*d^6) - (c*d^2 - a*e^2)^6/(c^7*d^7*(a*e + c*d*x)) + (10*e^3*(c*d^2 - a*e^2)^3
*(a*e + c*d*x)^2)/(c^7*d^7) + (5*e^4*(c*d^2 - a*e^2)^2*(a*e + c*d*x)^3)/(c^7*d^7) + (3*e^5*(c*d^2 - a*e^2)*(a*
e + c*d*x)^4)/(2*c^7*d^7) + (e^6*(a*e + c*d*x)^5)/(5*c^7*d^7) + (6*e*(c*d^2 - a*e^2)^5*Log[a*e + c*d*x])/(c^7*
d^7)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac {(d+e x)^6}{(a e+c d x)^2} \, dx\\ &=\int \left (\frac {15 e^2 \left (c d^2-a e^2\right )^4}{c^6 d^6}+\frac {\left (c d^2-a e^2\right )^6}{c^6 d^6 (a e+c d x)^2}+\frac {6 e \left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)}+\frac {20 \left (c d^2 e-a e^3\right )^3 (a e+c d x)}{c^6 d^6}+\frac {15 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{c^6 d^6}+\frac {6 \left (c d^2 e^5-a e^7\right ) (a e+c d x)^3}{c^6 d^6}+\frac {e^6 (a e+c d x)^4}{c^6 d^6}\right ) \, dx\\ &=\frac {15 e^2 \left (c d^2-a e^2\right )^4 x}{c^6 d^6}-\frac {\left (c d^2-a e^2\right )^6}{c^7 d^7 (a e+c d x)}+\frac {10 e^3 \left (c d^2-a e^2\right )^3 (a e+c d x)^2}{c^7 d^7}+\frac {5 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^3}{c^7 d^7}+\frac {3 e^5 \left (c d^2-a e^2\right ) (a e+c d x)^4}{2 c^7 d^7}+\frac {e^6 (a e+c d x)^5}{5 c^7 d^7}+\frac {6 e \left (c d^2-a e^2\right )^5 \log (a e+c d x)}{c^7 d^7}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 339, normalized size = 1.55 \begin {gather*} \frac {-10 a^6 e^{12}+10 a^5 c d e^{10} (6 d+5 e x)-30 a^4 c^2 d^2 e^8 \left (5 d^2+8 d e x-e^2 x^2\right )+10 a^3 c^3 d^3 e^6 \left (20 d^3+45 d^2 e x-15 d e^2 x^2-e^3 x^3\right )-5 a^2 c^4 d^4 e^4 \left (30 d^4+80 d^3 e x-60 d^2 e^2 x^2-10 d e^3 x^3-e^4 x^4\right )+a c^5 d^5 e^2 \left (60 d^5+150 d^4 e x-300 d^3 e^2 x^2-100 d^2 e^3 x^3-25 d e^4 x^4-3 e^5 x^5\right )-60 e \left (a e^2-c d^2\right )^5 (a e+c d x) \log (a e+c d x)+c^6 d^6 \left (-10 d^6+150 d^4 e^2 x^2+100 d^3 e^3 x^3+50 d^2 e^4 x^4+15 d e^5 x^5+2 e^6 x^6\right )}{10 c^7 d^7 (a e+c d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^8/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

(-10*a^6*e^12 + 10*a^5*c*d*e^10*(6*d + 5*e*x) - 30*a^4*c^2*d^2*e^8*(5*d^2 + 8*d*e*x - e^2*x^2) + 10*a^3*c^3*d^
3*e^6*(20*d^3 + 45*d^2*e*x - 15*d*e^2*x^2 - e^3*x^3) - 5*a^2*c^4*d^4*e^4*(30*d^4 + 80*d^3*e*x - 60*d^2*e^2*x^2
- 10*d*e^3*x^3 - e^4*x^4) + a*c^5*d^5*e^2*(60*d^5 + 150*d^4*e*x - 300*d^3*e^2*x^2 - 100*d^2*e^3*x^3 - 25*d*e^
4*x^4 - 3*e^5*x^5) + c^6*d^6*(-10*d^6 + 150*d^4*e^2*x^2 + 100*d^3*e^3*x^3 + 50*d^2*e^4*x^4 + 15*d*e^5*x^5 + 2*
e^6*x^6) - 60*e*(-(c*d^2) + a*e^2)^5*(a*e + c*d*x)*Log[a*e + c*d*x])/(10*c^7*d^7*(a*e + c*d*x))

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^8/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

IntegrateAlgebraic[(d + e*x)^8/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2, x]

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fricas [B]  time = 0.40, size = 545, normalized size = 2.49 \begin {gather*} \frac {2 \, c^{6} d^{6} e^{6} x^{6} - 10 \, c^{6} d^{12} + 60 \, a c^{5} d^{10} e^{2} - 150 \, a^{2} c^{4} d^{8} e^{4} + 200 \, a^{3} c^{3} d^{6} e^{6} - 150 \, a^{4} c^{2} d^{4} e^{8} + 60 \, a^{5} c d^{2} e^{10} - 10 \, a^{6} e^{12} + 3 \, {\left (5 \, c^{6} d^{7} e^{5} - a c^{5} d^{5} e^{7}\right )} x^{5} + 5 \, {\left (10 \, c^{6} d^{8} e^{4} - 5 \, a c^{5} d^{6} e^{6} + a^{2} c^{4} d^{4} e^{8}\right )} x^{4} + 10 \, {\left (10 \, c^{6} d^{9} e^{3} - 10 \, a c^{5} d^{7} e^{5} + 5 \, a^{2} c^{4} d^{5} e^{7} - a^{3} c^{3} d^{3} e^{9}\right )} x^{3} + 30 \, {\left (5 \, c^{6} d^{10} e^{2} - 10 \, a c^{5} d^{8} e^{4} + 10 \, a^{2} c^{4} d^{6} e^{6} - 5 \, a^{3} c^{3} d^{4} e^{8} + a^{4} c^{2} d^{2} e^{10}\right )} x^{2} + 10 \, {\left (15 \, a c^{5} d^{9} e^{3} - 40 \, a^{2} c^{4} d^{7} e^{5} + 45 \, a^{3} c^{3} d^{5} e^{7} - 24 \, a^{4} c^{2} d^{3} e^{9} + 5 \, a^{5} c d e^{11}\right )} x + 60 \, {\left (a c^{5} d^{10} e^{2} - 5 \, a^{2} c^{4} d^{8} e^{4} + 10 \, a^{3} c^{3} d^{6} e^{6} - 10 \, a^{4} c^{2} d^{4} e^{8} + 5 \, a^{5} c d^{2} e^{10} - a^{6} e^{12} + {\left (c^{6} d^{11} e - 5 \, a c^{5} d^{9} e^{3} + 10 \, a^{2} c^{4} d^{7} e^{5} - 10 \, a^{3} c^{3} d^{5} e^{7} + 5 \, a^{4} c^{2} d^{3} e^{9} - a^{5} c d e^{11}\right )} x\right )} \log \left (c d x + a e\right )}{10 \, {\left (c^{8} d^{8} x + a c^{7} d^{7} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^8/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="fricas")

[Out]

1/10*(2*c^6*d^6*e^6*x^6 - 10*c^6*d^12 + 60*a*c^5*d^10*e^2 - 150*a^2*c^4*d^8*e^4 + 200*a^3*c^3*d^6*e^6 - 150*a^
4*c^2*d^4*e^8 + 60*a^5*c*d^2*e^10 - 10*a^6*e^12 + 3*(5*c^6*d^7*e^5 - a*c^5*d^5*e^7)*x^5 + 5*(10*c^6*d^8*e^4 -
5*a*c^5*d^6*e^6 + a^2*c^4*d^4*e^8)*x^4 + 10*(10*c^6*d^9*e^3 - 10*a*c^5*d^7*e^5 + 5*a^2*c^4*d^5*e^7 - a^3*c^3*d
^3*e^9)*x^3 + 30*(5*c^6*d^10*e^2 - 10*a*c^5*d^8*e^4 + 10*a^2*c^4*d^6*e^6 - 5*a^3*c^3*d^4*e^8 + a^4*c^2*d^2*e^1
0)*x^2 + 10*(15*a*c^5*d^9*e^3 - 40*a^2*c^4*d^7*e^5 + 45*a^3*c^3*d^5*e^7 - 24*a^4*c^2*d^3*e^9 + 5*a^5*c*d*e^11)
*x + 60*(a*c^5*d^10*e^2 - 5*a^2*c^4*d^8*e^4 + 10*a^3*c^3*d^6*e^6 - 10*a^4*c^2*d^4*e^8 + 5*a^5*c*d^2*e^10 - a^6
*e^12 + (c^6*d^11*e - 5*a*c^5*d^9*e^3 + 10*a^2*c^4*d^7*e^5 - 10*a^3*c^3*d^5*e^7 + 5*a^4*c^2*d^3*e^9 - a^5*c*d*
e^11)*x)*log(c*d*x + a*e))/(c^8*d^8*x + a*c^7*d^7*e)

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giac [B]  time = 0.45, size = 796, normalized size = 3.63 \begin {gather*} \frac {6 \, {\left (c^{8} d^{16} e - 8 \, a c^{7} d^{14} e^{3} + 28 \, a^{2} c^{6} d^{12} e^{5} - 56 \, a^{3} c^{5} d^{10} e^{7} + 70 \, a^{4} c^{4} d^{8} e^{9} - 56 \, a^{5} c^{3} d^{6} e^{11} + 28 \, a^{6} c^{2} d^{4} e^{13} - 8 \, a^{7} c d^{2} e^{15} + a^{8} e^{17}\right )} \arctan \left (\frac {2 \, c d x e + c d^{2} + a e^{2}}{\sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{{\left (c^{9} d^{11} - 2 \, a c^{8} d^{9} e^{2} + a^{2} c^{7} d^{7} e^{4}\right )} \sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} + \frac {3 \, {\left (c^{5} d^{10} e - 5 \, a c^{4} d^{8} e^{3} + 10 \, a^{2} c^{3} d^{6} e^{5} - 10 \, a^{3} c^{2} d^{4} e^{7} + 5 \, a^{4} c d^{2} e^{9} - a^{5} e^{11}\right )} \log \left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{c^{7} d^{7}} - \frac {c^{8} d^{17} - 8 \, a c^{7} d^{15} e^{2} + 28 \, a^{2} c^{6} d^{13} e^{4} - 56 \, a^{3} c^{5} d^{11} e^{6} + 70 \, a^{4} c^{4} d^{9} e^{8} - 56 \, a^{5} c^{3} d^{7} e^{10} + 28 \, a^{6} c^{2} d^{5} e^{12} - 8 \, a^{7} c d^{3} e^{14} + a^{8} d e^{16} + {\left (c^{8} d^{16} e - 8 \, a c^{7} d^{14} e^{3} + 28 \, a^{2} c^{6} d^{12} e^{5} - 56 \, a^{3} c^{5} d^{10} e^{7} + 70 \, a^{4} c^{4} d^{8} e^{9} - 56 \, a^{5} c^{3} d^{6} e^{11} + 28 \, a^{6} c^{2} d^{4} e^{13} - 8 \, a^{7} c d^{2} e^{15} + a^{8} e^{17}\right )} x}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )} c^{7} d^{7}} + \frac {{\left (2 \, c^{8} d^{8} x^{5} e^{16} + 15 \, c^{8} d^{9} x^{4} e^{15} + 50 \, c^{8} d^{10} x^{3} e^{14} + 100 \, c^{8} d^{11} x^{2} e^{13} + 150 \, c^{8} d^{12} x e^{12} - 5 \, a c^{7} d^{7} x^{4} e^{17} - 40 \, a c^{7} d^{8} x^{3} e^{16} - 150 \, a c^{7} d^{9} x^{2} e^{15} - 400 \, a c^{7} d^{10} x e^{14} + 10 \, a^{2} c^{6} d^{6} x^{3} e^{18} + 90 \, a^{2} c^{6} d^{7} x^{2} e^{17} + 450 \, a^{2} c^{6} d^{8} x e^{16} - 20 \, a^{3} c^{5} d^{5} x^{2} e^{19} - 240 \, a^{3} c^{5} d^{6} x e^{18} + 50 \, a^{4} c^{4} d^{4} x e^{20}\right )} e^{\left (-10\right )}}{10 \, c^{10} d^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^8/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="giac")

[Out]

6*(c^8*d^16*e - 8*a*c^7*d^14*e^3 + 28*a^2*c^6*d^12*e^5 - 56*a^3*c^5*d^10*e^7 + 70*a^4*c^4*d^8*e^9 - 56*a^5*c^3
*d^6*e^11 + 28*a^6*c^2*d^4*e^13 - 8*a^7*c*d^2*e^15 + a^8*e^17)*arctan((2*c*d*x*e + c*d^2 + a*e^2)/sqrt(-c^2*d^
4 + 2*a*c*d^2*e^2 - a^2*e^4))/((c^9*d^11 - 2*a*c^8*d^9*e^2 + a^2*c^7*d^7*e^4)*sqrt(-c^2*d^4 + 2*a*c*d^2*e^2 -
a^2*e^4)) + 3*(c^5*d^10*e - 5*a*c^4*d^8*e^3 + 10*a^2*c^3*d^6*e^5 - 10*a^3*c^2*d^4*e^7 + 5*a^4*c*d^2*e^9 - a^5*
e^11)*log(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)/(c^7*d^7) - (c^8*d^17 - 8*a*c^7*d^15*e^2 + 28*a^2*c^6*d^13*e^
4 - 56*a^3*c^5*d^11*e^6 + 70*a^4*c^4*d^9*e^8 - 56*a^5*c^3*d^7*e^10 + 28*a^6*c^2*d^5*e^12 - 8*a^7*c*d^3*e^14 +
a^8*d*e^16 + (c^8*d^16*e - 8*a*c^7*d^14*e^3 + 28*a^2*c^6*d^12*e^5 - 56*a^3*c^5*d^10*e^7 + 70*a^4*c^4*d^8*e^9 -
56*a^5*c^3*d^6*e^11 + 28*a^6*c^2*d^4*e^13 - 8*a^7*c*d^2*e^15 + a^8*e^17)*x)/((c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e
^4)*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*c^7*d^7) + 1/10*(2*c^8*d^8*x^5*e^16 + 15*c^8*d^9*x^4*e^15 + 50*c^8
*d^10*x^3*e^14 + 100*c^8*d^11*x^2*e^13 + 150*c^8*d^12*x*e^12 - 5*a*c^7*d^7*x^4*e^17 - 40*a*c^7*d^8*x^3*e^16 -
150*a*c^7*d^9*x^2*e^15 - 400*a*c^7*d^10*x*e^14 + 10*a^2*c^6*d^6*x^3*e^18 + 90*a^2*c^6*d^7*x^2*e^17 + 450*a^2*c
^6*d^8*x*e^16 - 20*a^3*c^5*d^5*x^2*e^19 - 240*a^3*c^5*d^6*x*e^18 + 50*a^4*c^4*d^4*x*e^20)*e^(-10)/(c^10*d^10)

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maple [B]  time = 0.06, size = 502, normalized size = 2.29 \begin {gather*} \frac {e^{6} x^{5}}{5 c^{2} d^{2}}-\frac {a \,e^{7} x^{4}}{2 c^{3} d^{3}}+\frac {3 e^{5} x^{4}}{2 c^{2} d}+\frac {a^{2} e^{8} x^{3}}{c^{4} d^{4}}-\frac {4 a \,e^{6} x^{3}}{c^{3} d^{2}}+\frac {5 e^{4} x^{3}}{c^{2}}-\frac {a^{6} e^{12}}{\left (c d x +a e \right ) c^{7} d^{7}}+\frac {6 a^{5} e^{10}}{\left (c d x +a e \right ) c^{6} d^{5}}-\frac {15 a^{4} e^{8}}{\left (c d x +a e \right ) c^{5} d^{3}}+\frac {20 a^{3} e^{6}}{\left (c d x +a e \right ) c^{4} d}-\frac {2 a^{3} e^{9} x^{2}}{c^{5} d^{5}}-\frac {15 a^{2} d \,e^{4}}{\left (c d x +a e \right ) c^{3}}+\frac {9 a^{2} e^{7} x^{2}}{c^{4} d^{3}}+\frac {6 a \,d^{3} e^{2}}{\left (c d x +a e \right ) c^{2}}-\frac {15 a \,e^{5} x^{2}}{c^{3} d}-\frac {d^{5}}{\left (c d x +a e \right ) c}+\frac {10 d \,e^{3} x^{2}}{c^{2}}-\frac {6 a^{5} e^{11} \ln \left (c d x +a e \right )}{c^{7} d^{7}}+\frac {30 a^{4} e^{9} \ln \left (c d x +a e \right )}{c^{6} d^{5}}+\frac {5 a^{4} e^{10} x}{c^{6} d^{6}}-\frac {60 a^{3} e^{7} \ln \left (c d x +a e \right )}{c^{5} d^{3}}-\frac {24 a^{3} e^{8} x}{c^{5} d^{4}}+\frac {60 a^{2} e^{5} \ln \left (c d x +a e \right )}{c^{4} d}+\frac {45 a^{2} e^{6} x}{c^{4} d^{2}}-\frac {30 a d \,e^{3} \ln \left (c d x +a e \right )}{c^{3}}-\frac {40 a \,e^{4} x}{c^{3}}+\frac {6 d^{3} e \ln \left (c d x +a e \right )}{c^{2}}+\frac {15 d^{2} e^{2} x}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^8/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x)

[Out]

e^8/c^4/d^4*x^3*a^2-1/c*d^5/(c*d*x+a*e)+5*e^4/c^2*x^3+3/2*e^5/c^2/d*x^4+10*e^3/c^2*d*x^2-40*e^4/c^3*a*x+15*e^2
/c^2*d^2*x+6*d^3/c^2*e*ln(c*d*x+a*e)+1/5*e^6/c^2/d^2*x^5+6/c^6/d^5/(c*d*x+a*e)*a^5*e^10-15/c^5/d^3/(c*d*x+a*e)
*a^4*e^8+20/c^4/d/(c*d*x+a*e)*a^3*e^6-15/c^3*d/(c*d*x+a*e)*a^2*e^4+6/c^2*d^3/(c*d*x+a*e)*a*e^2-1/2*e^7/c^3/d^3
*x^4*a-6/d^7/c^7*e^11*ln(c*d*x+a*e)*a^5+30/d^5/c^6*e^9*ln(c*d*x+a*e)*a^4-60/d^3/c^5*e^7*ln(c*d*x+a*e)*a^3+60/d
/c^4*e^5*ln(c*d*x+a*e)*a^2-30*d/c^3*e^3*ln(c*d*x+a*e)*a-1/c^7/d^7/(c*d*x+a*e)*a^6*e^12-4*e^6/c^3/d^2*x^3*a-2*e
^9/c^5/d^5*x^2*a^3+9*e^7/c^4/d^3*x^2*a^2-15*e^5/c^3/d*x^2*a+5*e^10/c^6/d^6*a^4*x-24*e^8/c^5/d^4*a^3*x+45*e^6/c
^4/d^2*a^2*x

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maxima [A]  time = 1.12, size = 398, normalized size = 1.82 \begin {gather*} -\frac {c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}}{c^{8} d^{8} x + a c^{7} d^{7} e} + \frac {2 \, c^{4} d^{4} e^{6} x^{5} + 5 \, {\left (3 \, c^{4} d^{5} e^{5} - a c^{3} d^{3} e^{7}\right )} x^{4} + 10 \, {\left (5 \, c^{4} d^{6} e^{4} - 4 \, a c^{3} d^{4} e^{6} + a^{2} c^{2} d^{2} e^{8}\right )} x^{3} + 10 \, {\left (10 \, c^{4} d^{7} e^{3} - 15 \, a c^{3} d^{5} e^{5} + 9 \, a^{2} c^{2} d^{3} e^{7} - 2 \, a^{3} c d e^{9}\right )} x^{2} + 10 \, {\left (15 \, c^{4} d^{8} e^{2} - 40 \, a c^{3} d^{6} e^{4} + 45 \, a^{2} c^{2} d^{4} e^{6} - 24 \, a^{3} c d^{2} e^{8} + 5 \, a^{4} e^{10}\right )} x}{10 \, c^{6} d^{6}} + \frac {6 \, {\left (c^{5} d^{10} e - 5 \, a c^{4} d^{8} e^{3} + 10 \, a^{2} c^{3} d^{6} e^{5} - 10 \, a^{3} c^{2} d^{4} e^{7} + 5 \, a^{4} c d^{2} e^{9} - a^{5} e^{11}\right )} \log \left (c d x + a e\right )}{c^{7} d^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^8/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="maxima")

[Out]

-(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^1
0 + a^6*e^12)/(c^8*d^8*x + a*c^7*d^7*e) + 1/10*(2*c^4*d^4*e^6*x^5 + 5*(3*c^4*d^5*e^5 - a*c^3*d^3*e^7)*x^4 + 10
*(5*c^4*d^6*e^4 - 4*a*c^3*d^4*e^6 + a^2*c^2*d^2*e^8)*x^3 + 10*(10*c^4*d^7*e^3 - 15*a*c^3*d^5*e^5 + 9*a^2*c^2*d
^3*e^7 - 2*a^3*c*d*e^9)*x^2 + 10*(15*c^4*d^8*e^2 - 40*a*c^3*d^6*e^4 + 45*a^2*c^2*d^4*e^6 - 24*a^3*c*d^2*e^8 +
5*a^4*e^10)*x)/(c^6*d^6) + 6*(c^5*d^10*e - 5*a*c^4*d^8*e^3 + 10*a^2*c^3*d^6*e^5 - 10*a^3*c^2*d^4*e^7 + 5*a^4*c
*d^2*e^9 - a^5*e^11)*log(c*d*x + a*e)/(c^7*d^7)

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mupad [B]  time = 0.11, size = 625, normalized size = 2.85 \begin {gather*} x^4\,\left (\frac {3\,e^5}{2\,c^2\,d}-\frac {a\,e^7}{2\,c^3\,d^3}\right )+x^2\,\left (\frac {10\,d\,e^3}{c^2}+\frac {a\,e\,\left (\frac {a^2\,e^8}{c^4\,d^4}-\frac {15\,e^4}{c^2}+\frac {2\,a\,e\,\left (\frac {6\,e^5}{c^2\,d}-\frac {2\,a\,e^7}{c^3\,d^3}\right )}{c\,d}\right )}{c\,d}-\frac {a^2\,e^2\,\left (\frac {6\,e^5}{c^2\,d}-\frac {2\,a\,e^7}{c^3\,d^3}\right )}{2\,c^2\,d^2}\right )-x^3\,\left (\frac {a^2\,e^8}{3\,c^4\,d^4}-\frac {5\,e^4}{c^2}+\frac {2\,a\,e\,\left (\frac {6\,e^5}{c^2\,d}-\frac {2\,a\,e^7}{c^3\,d^3}\right )}{3\,c\,d}\right )+x\,\left (\frac {15\,d^2\,e^2}{c^2}+\frac {a^2\,e^2\,\left (\frac {a^2\,e^8}{c^4\,d^4}-\frac {15\,e^4}{c^2}+\frac {2\,a\,e\,\left (\frac {6\,e^5}{c^2\,d}-\frac {2\,a\,e^7}{c^3\,d^3}\right )}{c\,d}\right )}{c^2\,d^2}-\frac {2\,a\,e\,\left (\frac {20\,d\,e^3}{c^2}+\frac {2\,a\,e\,\left (\frac {a^2\,e^8}{c^4\,d^4}-\frac {15\,e^4}{c^2}+\frac {2\,a\,e\,\left (\frac {6\,e^5}{c^2\,d}-\frac {2\,a\,e^7}{c^3\,d^3}\right )}{c\,d}\right )}{c\,d}-\frac {a^2\,e^2\,\left (\frac {6\,e^5}{c^2\,d}-\frac {2\,a\,e^7}{c^3\,d^3}\right )}{c^2\,d^2}\right )}{c\,d}\right )-\frac {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}}{c\,d\,\left (x\,c^7\,d^7+a\,e\,c^6\,d^6\right )}+\frac {e^6\,x^5}{5\,c^2\,d^2}-\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (6\,a^5\,e^{11}-30\,a^4\,c\,d^2\,e^9+60\,a^3\,c^2\,d^4\,e^7-60\,a^2\,c^3\,d^6\,e^5+30\,a\,c^4\,d^8\,e^3-6\,c^5\,d^{10}\,e\right )}{c^7\,d^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^8/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2,x)

[Out]

x^4*((3*e^5)/(2*c^2*d) - (a*e^7)/(2*c^3*d^3)) + x^2*((10*d*e^3)/c^2 + (a*e*((a^2*e^8)/(c^4*d^4) - (15*e^4)/c^2
+ (2*a*e*((6*e^5)/(c^2*d) - (2*a*e^7)/(c^3*d^3)))/(c*d)))/(c*d) - (a^2*e^2*((6*e^5)/(c^2*d) - (2*a*e^7)/(c^3*
d^3)))/(2*c^2*d^2)) - x^3*((a^2*e^8)/(3*c^4*d^4) - (5*e^4)/c^2 + (2*a*e*((6*e^5)/(c^2*d) - (2*a*e^7)/(c^3*d^3)
))/(3*c*d)) + x*((15*d^2*e^2)/c^2 + (a^2*e^2*((a^2*e^8)/(c^4*d^4) - (15*e^4)/c^2 + (2*a*e*((6*e^5)/(c^2*d) - (
2*a*e^7)/(c^3*d^3)))/(c*d)))/(c^2*d^2) - (2*a*e*((20*d*e^3)/c^2 + (2*a*e*((a^2*e^8)/(c^4*d^4) - (15*e^4)/c^2 +
(2*a*e*((6*e^5)/(c^2*d) - (2*a*e^7)/(c^3*d^3)))/(c*d)))/(c*d) - (a^2*e^2*((6*e^5)/(c^2*d) - (2*a*e^7)/(c^3*d^
3)))/(c^2*d^2)))/(c*d)) - (a^6*e^12 + c^6*d^12 - 6*a*c^5*d^10*e^2 - 6*a^5*c*d^2*e^10 + 15*a^2*c^4*d^8*e^4 - 20
*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8)/(c*d*(c^7*d^7*x + a*c^6*d^6*e)) + (e^6*x^5)/(5*c^2*d^2) - (log(a*e + c*
d*x)*(6*a^5*e^11 - 6*c^5*d^10*e + 30*a*c^4*d^8*e^3 - 30*a^4*c*d^2*e^9 - 60*a^2*c^3*d^6*e^5 + 60*a^3*c^2*d^4*e^
7))/(c^7*d^7)

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sympy [A]  time = 1.73, size = 345, normalized size = 1.58 \begin {gather*} x^{4} \left (- \frac {a e^{7}}{2 c^{3} d^{3}} + \frac {3 e^{5}}{2 c^{2} d}\right ) + x^{3} \left (\frac {a^{2} e^{8}}{c^{4} d^{4}} - \frac {4 a e^{6}}{c^{3} d^{2}} + \frac {5 e^{4}}{c^{2}}\right ) + x^{2} \left (- \frac {2 a^{3} e^{9}}{c^{5} d^{5}} + \frac {9 a^{2} e^{7}}{c^{4} d^{3}} - \frac {15 a e^{5}}{c^{3} d} + \frac {10 d e^{3}}{c^{2}}\right ) + x \left (\frac {5 a^{4} e^{10}}{c^{6} d^{6}} - \frac {24 a^{3} e^{8}}{c^{5} d^{4}} + \frac {45 a^{2} e^{6}}{c^{4} d^{2}} - \frac {40 a e^{4}}{c^{3}} + \frac {15 d^{2} e^{2}}{c^{2}}\right ) + \frac {- 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}}{a c^{7} d^{7} e + c^{8} d^{8} x} + \frac {e^{6} x^{5}}{5 c^{2} d^{2}} - \frac {6 e \left (a e^{2} - c d^{2}\right )^{5} \log {\left (a e + c d x \right )}}{c^{7} d^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**8/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)

[Out]

x**4*(-a*e**7/(2*c**3*d**3) + 3*e**5/(2*c**2*d)) + x**3*(a**2*e**8/(c**4*d**4) - 4*a*e**6/(c**3*d**2) + 5*e**4
/c**2) + x**2*(-2*a**3*e**9/(c**5*d**5) + 9*a**2*e**7/(c**4*d**3) - 15*a*e**5/(c**3*d) + 10*d*e**3/c**2) + x*(
5*a**4*e**10/(c**6*d**6) - 24*a**3*e**8/(c**5*d**4) + 45*a**2*e**6/(c**4*d**2) - 40*a*e**4/c**3 + 15*d**2*e**2
/c**2) + (-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)/(a*c**7*d**7*e + c**8*d**8*x) + e**6*x**5/(5*c**2*d**2) - 6*e*(a*
e**2 - c*d**2)**5*log(a*e + c*d*x)/(c**7*d**7)

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