3.16.73 \(\int \frac {(d+e x)^7}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\)

Optimal. Leaf size=142 \[ -\frac {4 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)}-\frac {\left (c d^2-a e^2\right )^4}{2 c^5 d^5 (a e+c d x)^2}+\frac {6 e^2 \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^5 d^5}+\frac {e^3 x \left (4 c d^2-3 a e^2\right )}{c^4 d^4}+\frac {e^4 x^2}{2 c^3 d^3} \]

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Rubi [A]  time = 0.14, antiderivative size = 142, 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^3 x \left (4 c d^2-3 a e^2\right )}{c^4 d^4}-\frac {4 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)}-\frac {\left (c d^2-a e^2\right )^4}{2 c^5 d^5 (a e+c d x)^2}+\frac {6 e^2 \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^5 d^5}+\frac {e^4 x^2}{2 c^3 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^7/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

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

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

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Mathematica [A]  time = 0.07, size = 191, normalized size = 1.35 \begin {gather*} \frac {7 a^4 e^8+2 a^3 c d e^6 (e x-10 d)+a^2 c^2 d^2 e^4 \left (18 d^2-16 d e x-11 e^2 x^2\right )-4 a c^3 d^3 e^2 \left (d^3-6 d^2 e x-4 d e^2 x^2+e^3 x^3\right )+12 e^2 \left (c d^2-a e^2\right )^2 (a e+c d x)^2 \log (a e+c d x)+c^4 d^4 \left (-d^4-8 d^3 e x+8 d e^3 x^3+e^4 x^4\right )}{2 c^5 d^5 (a e+c d x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^7/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(7*a^4*e^8 + 2*a^3*c*d*e^6*(-10*d + e*x) + a^2*c^2*d^2*e^4*(18*d^2 - 16*d*e*x - 11*e^2*x^2) - 4*a*c^3*d^3*e^2*
(d^3 - 6*d^2*e*x - 4*d*e^2*x^2 + e^3*x^3) + c^4*d^4*(-d^4 - 8*d^3*e*x + 8*d*e^3*x^3 + e^4*x^4) + 12*e^2*(c*d^2
- a*e^2)^2*(a*e + c*d*x)^2*Log[a*e + c*d*x])/(2*c^5*d^5*(a*e + c*d*x)^2)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^7/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

IntegrateAlgebraic[(d + e*x)^7/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3, x]

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fricas [B]  time = 0.40, size = 338, normalized size = 2.38 \begin {gather*} \frac {c^{4} d^{4} e^{4} x^{4} - c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 18 \, a^{2} c^{2} d^{4} e^{4} - 20 \, a^{3} c d^{2} e^{6} + 7 \, a^{4} e^{8} + 4 \, {\left (2 \, c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{3} + {\left (16 \, a c^{3} d^{4} e^{4} - 11 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{2} - 2 \, {\left (4 \, c^{4} d^{7} e - 12 \, a c^{3} d^{5} e^{3} + 8 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x + 12 \, {\left (a^{2} c^{2} d^{4} e^{4} - 2 \, a^{3} c d^{2} e^{6} + a^{4} e^{8} + {\left (c^{4} d^{6} e^{2} - 2 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \, {\left (a c^{3} d^{5} e^{3} - 2 \, a^{2} c^{2} d^{3} e^{5} + a^{3} c d e^{7}\right )} x\right )} \log \left (c d x + a e\right )}{2 \, {\left (c^{7} d^{7} x^{2} + 2 \, a c^{6} d^{6} e x + a^{2} c^{5} d^{5} e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^7/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="fricas")

[Out]

1/2*(c^4*d^4*e^4*x^4 - c^4*d^8 - 4*a*c^3*d^6*e^2 + 18*a^2*c^2*d^4*e^4 - 20*a^3*c*d^2*e^6 + 7*a^4*e^8 + 4*(2*c^
4*d^5*e^3 - a*c^3*d^3*e^5)*x^3 + (16*a*c^3*d^4*e^4 - 11*a^2*c^2*d^2*e^6)*x^2 - 2*(4*c^4*d^7*e - 12*a*c^3*d^5*e
^3 + 8*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x + 12*(a^2*c^2*d^4*e^4 - 2*a^3*c*d^2*e^6 + a^4*e^8 + (c^4*d^6*e^2 - 2*a
*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 + 2*(a*c^3*d^5*e^3 - 2*a^2*c^2*d^3*e^5 + a^3*c*d*e^7)*x)*log(c*d*x + a*e))
/(c^7*d^7*x^2 + 2*a*c^6*d^6*e*x + a^2*c^5*d^5*e^2)

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giac [B]  time = 11.63, size = 803, normalized size = 5.65 \begin {gather*} \frac {6 \, {\left (c^{7} d^{14} e^{2} - 7 \, a c^{6} d^{12} e^{4} + 21 \, a^{2} c^{5} d^{10} e^{6} - 35 \, a^{3} c^{4} d^{8} e^{8} + 35 \, a^{4} c^{3} d^{6} e^{10} - 21 \, a^{5} c^{2} d^{4} e^{12} + 7 \, a^{6} c d^{2} e^{14} - a^{7} e^{16}\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^{13} - 4 \, a c^{8} d^{11} e^{2} + 6 \, a^{2} c^{7} d^{9} e^{4} - 4 \, a^{3} c^{6} d^{7} e^{6} + a^{4} c^{5} d^{5} e^{8}\right )} \sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} + \frac {3 \, {\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} \log \left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{c^{5} d^{5}} + \frac {{\left (c^{3} d^{3} x^{2} e^{10} + 8 \, c^{3} d^{4} x e^{9} - 6 \, a c^{2} d^{2} x e^{11}\right )} e^{\left (-6\right )}}{2 \, c^{6} d^{6}} - \frac {c^{8} d^{18} - 28 \, a^{2} c^{6} d^{14} e^{4} + 112 \, a^{3} c^{5} d^{12} e^{6} - 210 \, a^{4} c^{4} d^{10} e^{8} + 224 \, a^{5} c^{3} d^{8} e^{10} - 140 \, a^{6} c^{2} d^{6} e^{12} + 48 \, a^{7} c d^{4} e^{14} - 7 \, a^{8} d^{2} e^{16} + 8 \, {\left (c^{8} d^{15} e^{3} - 7 \, a c^{7} d^{13} e^{5} + 21 \, a^{2} c^{6} d^{11} e^{7} - 35 \, a^{3} c^{5} d^{9} e^{9} + 35 \, a^{4} c^{4} d^{7} e^{11} - 21 \, a^{5} c^{3} d^{5} e^{13} + 7 \, a^{6} c^{2} d^{3} e^{15} - a^{7} c d e^{17}\right )} x^{3} + {\left (17 \, c^{8} d^{16} e^{2} - 112 \, a c^{7} d^{14} e^{4} + 308 \, a^{2} c^{6} d^{12} e^{6} - 448 \, a^{3} c^{5} d^{10} e^{8} + 350 \, a^{4} c^{4} d^{8} e^{10} - 112 \, a^{5} c^{3} d^{6} e^{12} - 28 \, a^{6} c^{2} d^{4} e^{14} + 32 \, a^{7} c d^{2} e^{16} - 7 \, a^{8} e^{18}\right )} x^{2} + 2 \, {\left (5 \, c^{8} d^{17} e - 28 \, a c^{7} d^{15} e^{3} + 56 \, a^{2} c^{6} d^{13} e^{5} - 28 \, a^{3} c^{5} d^{11} e^{7} - 70 \, a^{4} c^{4} d^{9} e^{9} + 140 \, a^{5} c^{3} d^{7} e^{11} - 112 \, a^{6} c^{2} d^{5} e^{13} + 44 \, a^{7} c d^{3} e^{15} - 7 \, a^{8} d e^{17}\right )} x}{2 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}^{2} {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{2} c^{5} d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^7/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac")

[Out]

6*(c^7*d^14*e^2 - 7*a*c^6*d^12*e^4 + 21*a^2*c^5*d^10*e^6 - 35*a^3*c^4*d^8*e^8 + 35*a^4*c^3*d^6*e^10 - 21*a^5*c
^2*d^4*e^12 + 7*a^6*c*d^2*e^14 - a^7*e^16)*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^13 - 4*a*c^8*d^11*e^2 + 6*a^2*c^7*d^9*e^4 - 4*a^3*c^6*d^7*e^6 + a^4*c^5*d^5*e^8)*sqrt(-c^2*d
^4 + 2*a*c*d^2*e^2 - a^2*e^4)) + 3*(c^2*d^4*e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*log(c*d*x^2*e + c*d^2*x + a*x*e^2 +
a*d*e)/(c^5*d^5) + 1/2*(c^3*d^3*x^2*e^10 + 8*c^3*d^4*x*e^9 - 6*a*c^2*d^2*x*e^11)*e^(-6)/(c^6*d^6) - 1/2*(c^8*
d^18 - 28*a^2*c^6*d^14*e^4 + 112*a^3*c^5*d^12*e^6 - 210*a^4*c^4*d^10*e^8 + 224*a^5*c^3*d^8*e^10 - 140*a^6*c^2*
d^6*e^12 + 48*a^7*c*d^4*e^14 - 7*a^8*d^2*e^16 + 8*(c^8*d^15*e^3 - 7*a*c^7*d^13*e^5 + 21*a^2*c^6*d^11*e^7 - 35*
a^3*c^5*d^9*e^9 + 35*a^4*c^4*d^7*e^11 - 21*a^5*c^3*d^5*e^13 + 7*a^6*c^2*d^3*e^15 - a^7*c*d*e^17)*x^3 + (17*c^8
*d^16*e^2 - 112*a*c^7*d^14*e^4 + 308*a^2*c^6*d^12*e^6 - 448*a^3*c^5*d^10*e^8 + 350*a^4*c^4*d^8*e^10 - 112*a^5*
c^3*d^6*e^12 - 28*a^6*c^2*d^4*e^14 + 32*a^7*c*d^2*e^16 - 7*a^8*e^18)*x^2 + 2*(5*c^8*d^17*e - 28*a*c^7*d^15*e^3
+ 56*a^2*c^6*d^13*e^5 - 28*a^3*c^5*d^11*e^7 - 70*a^4*c^4*d^9*e^9 + 140*a^5*c^3*d^7*e^11 - 112*a^6*c^2*d^5*e^1
3 + 44*a^7*c*d^3*e^15 - 7*a^8*d*e^17)*x)/((c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)^2*(c*d*x^2*e + c*d^2*x + a*x*e^2
+ a*d*e)^2*c^5*d^5)

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maple [B]  time = 0.05, size = 302, normalized size = 2.13 \begin {gather*} -\frac {a^{4} e^{8}}{2 \left (c d x +a e \right )^{2} c^{5} d^{5}}+\frac {2 a^{3} e^{6}}{\left (c d x +a e \right )^{2} c^{4} d^{3}}-\frac {3 a^{2} e^{4}}{\left (c d x +a e \right )^{2} c^{3} d}+\frac {2 a d \,e^{2}}{\left (c d x +a e \right )^{2} c^{2}}-\frac {d^{3}}{2 \left (c d x +a e \right )^{2} c}+\frac {4 a^{3} e^{7}}{\left (c d x +a e \right ) c^{5} d^{5}}-\frac {12 a^{2} e^{5}}{\left (c d x +a e \right ) c^{4} d^{3}}+\frac {12 a \,e^{3}}{\left (c d x +a e \right ) c^{3} d}-\frac {4 d e}{\left (c d x +a e \right ) c^{2}}+\frac {e^{4} x^{2}}{2 c^{3} d^{3}}+\frac {6 a^{2} e^{6} \ln \left (c d x +a e \right )}{c^{5} d^{5}}-\frac {12 a \,e^{4} \ln \left (c d x +a e \right )}{c^{4} d^{3}}-\frac {3 a \,e^{5} x}{c^{4} d^{4}}+\frac {6 e^{2} \ln \left (c d x +a e \right )}{c^{3} d}+\frac {4 e^{3} x}{c^{3} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^7/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)

[Out]

1/2*e^4*x^2/c^3/d^3-3*e^5/c^4/d^4*a*x+4*e^3/c^3/d^2*x-1/2/c^5/d^5/(c*d*x+a*e)^2*a^4*e^8+2/c^4/d^3/(c*d*x+a*e)^
2*a^3*e^6-3/c^3/d/(c*d*x+a*e)^2*a^2*e^4+2/c^2*d/(c*d*x+a*e)^2*a*e^2-1/2/c*d^3/(c*d*x+a*e)^2+6/c^5/d^5*e^6*ln(c
*d*x+a*e)*a^2-12/c^4/d^3*e^4*ln(c*d*x+a*e)*a+6/c^3/d*e^2*ln(c*d*x+a*e)+4/d^5/c^5*e^7/(c*d*x+a*e)*a^3-12/d^3/c^
4*e^5/(c*d*x+a*e)*a^2+12/d/c^3*e^3/(c*d*x+a*e)*a-4*d/c^2*e/(c*d*x+a*e)

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maxima [A]  time = 1.20, size = 224, normalized size = 1.58 \begin {gather*} -\frac {c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} - 18 \, a^{2} c^{2} d^{4} e^{4} + 20 \, a^{3} c d^{2} e^{6} - 7 \, a^{4} e^{8} + 8 \, {\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x}{2 \, {\left (c^{7} d^{7} x^{2} + 2 \, a c^{6} d^{6} e x + a^{2} c^{5} d^{5} e^{2}\right )}} + \frac {c d e^{4} x^{2} + 2 \, {\left (4 \, c d^{2} e^{3} - 3 \, a e^{5}\right )} x}{2 \, c^{4} d^{4}} + \frac {6 \, {\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} \log \left (c d x + a e\right )}{c^{5} d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^7/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="maxima")

[Out]

-1/2*(c^4*d^8 + 4*a*c^3*d^6*e^2 - 18*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 - 7*a^4*e^8 + 8*(c^4*d^7*e - 3*a*c^3*d
^5*e^3 + 3*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x)/(c^7*d^7*x^2 + 2*a*c^6*d^6*e*x + a^2*c^5*d^5*e^2) + 1/2*(c*d*e^4*
x^2 + 2*(4*c*d^2*e^3 - 3*a*e^5)*x)/(c^4*d^4) + 6*(c^2*d^4*e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*log(c*d*x + a*e)/(c^5
*d^5)

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mupad [B]  time = 0.11, size = 232, normalized size = 1.63 \begin {gather*} \frac {x\,\left (4\,a^3\,e^7-12\,a^2\,c\,d^2\,e^5+12\,a\,c^2\,d^4\,e^3-4\,c^3\,d^6\,e\right )-\frac {-7\,a^4\,e^8+20\,a^3\,c\,d^2\,e^6-18\,a^2\,c^2\,d^4\,e^4+4\,a\,c^3\,d^6\,e^2+c^4\,d^8}{2\,c\,d}}{a^2\,c^4\,d^4\,e^2+2\,a\,c^5\,d^5\,e\,x+c^6\,d^6\,x^2}+x\,\left (\frac {4\,e^3}{c^3\,d^2}-\frac {3\,a\,e^5}{c^4\,d^4}\right )+\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (6\,a^2\,e^6-12\,a\,c\,d^2\,e^4+6\,c^2\,d^4\,e^2\right )}{c^5\,d^5}+\frac {e^4\,x^2}{2\,c^3\,d^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^7/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3,x)

[Out]

(x*(4*a^3*e^7 - 4*c^3*d^6*e + 12*a*c^2*d^4*e^3 - 12*a^2*c*d^2*e^5) - (c^4*d^8 - 7*a^4*e^8 + 4*a*c^3*d^6*e^2 +
20*a^3*c*d^2*e^6 - 18*a^2*c^2*d^4*e^4)/(2*c*d))/(c^6*d^6*x^2 + a^2*c^4*d^4*e^2 + 2*a*c^5*d^5*e*x) + x*((4*e^3)
/(c^3*d^2) - (3*a*e^5)/(c^4*d^4)) + (log(a*e + c*d*x)*(6*a^2*e^6 + 6*c^2*d^4*e^2 - 12*a*c*d^2*e^4))/(c^5*d^5)
+ (e^4*x^2)/(2*c^3*d^3)

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sympy [A]  time = 2.16, size = 226, normalized size = 1.59 \begin {gather*} x \left (- \frac {3 a e^{5}}{c^{4} d^{4}} + \frac {4 e^{3}}{c^{3} d^{2}}\right ) + \frac {7 a^{4} e^{8} - 20 a^{3} c d^{2} e^{6} + 18 a^{2} c^{2} d^{4} e^{4} - 4 a c^{3} d^{6} e^{2} - c^{4} d^{8} + x \left (8 a^{3} c d e^{7} - 24 a^{2} c^{2} d^{3} e^{5} + 24 a c^{3} d^{5} e^{3} - 8 c^{4} d^{7} e\right )}{2 a^{2} c^{5} d^{5} e^{2} + 4 a c^{6} d^{6} e x + 2 c^{7} d^{7} x^{2}} + \frac {e^{4} x^{2}}{2 c^{3} d^{3}} + \frac {6 e^{2} \left (a e^{2} - c d^{2}\right )^{2} \log {\left (a e + c d x \right )}}{c^{5} d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**7/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

x*(-3*a*e**5/(c**4*d**4) + 4*e**3/(c**3*d**2)) + (7*a**4*e**8 - 20*a**3*c*d**2*e**6 + 18*a**2*c**2*d**4*e**4 -
4*a*c**3*d**6*e**2 - c**4*d**8 + x*(8*a**3*c*d*e**7 - 24*a**2*c**2*d**3*e**5 + 24*a*c**3*d**5*e**3 - 8*c**4*d
**7*e))/(2*a**2*c**5*d**5*e**2 + 4*a*c**6*d**6*e*x + 2*c**7*d**7*x**2) + e**4*x**2/(2*c**3*d**3) + 6*e**2*(a*e
**2 - c*d**2)**2*log(a*e + c*d*x)/(c**5*d**5)

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