3.16.82 \(\int \frac {(d+e x)^{10}}{(a d e+(c d^2+a e^2) x+c d e x^2)^4} \, dx\)

Optimal. Leaf size=217 \[ \frac {e^4 x \left (10 a^2 e^4-24 a c d^2 e^2+15 c^2 d^4\right )}{c^6 d^6}-\frac {15 e^2 \left (c d^2-a e^2\right )^4}{c^7 d^7 (a e+c d x)}-\frac {3 e \left (c d^2-a e^2\right )^5}{c^7 d^7 (a e+c d x)^2}-\frac {\left (c d^2-a e^2\right )^6}{3 c^7 d^7 (a e+c d x)^3}+\frac {20 e^3 \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^7 d^7}+\frac {e^5 x^2 \left (3 c d^2-2 a e^2\right )}{c^5 d^5}+\frac {e^6 x^3}{3 c^4 d^4} \]

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Rubi [A]  time = 0.27, antiderivative size = 217, 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^4 x \left (10 a^2 e^4-24 a c d^2 e^2+15 c^2 d^4\right )}{c^6 d^6}+\frac {e^5 x^2 \left (3 c d^2-2 a e^2\right )}{c^5 d^5}-\frac {15 e^2 \left (c d^2-a e^2\right )^4}{c^7 d^7 (a e+c d x)}-\frac {3 e \left (c d^2-a e^2\right )^5}{c^7 d^7 (a e+c d x)^2}-\frac {\left (c d^2-a e^2\right )^6}{3 c^7 d^7 (a e+c d x)^3}+\frac {20 e^3 \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^7 d^7}+\frac {e^6 x^3}{3 c^4 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^10/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

(e^4*(15*c^2*d^4 - 24*a*c*d^2*e^2 + 10*a^2*e^4)*x)/(c^6*d^6) + (e^5*(3*c*d^2 - 2*a*e^2)*x^2)/(c^5*d^5) + (e^6*
x^3)/(3*c^4*d^4) - (c*d^2 - a*e^2)^6/(3*c^7*d^7*(a*e + c*d*x)^3) - (3*e*(c*d^2 - a*e^2)^5)/(c^7*d^7*(a*e + c*d
*x)^2) - (15*e^2*(c*d^2 - a*e^2)^4)/(c^7*d^7*(a*e + c*d*x)) + (20*e^3*(c*d^2 - a*e^2)^3*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)^{10}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx &=\int \frac {(d+e x)^6}{(a e+c d x)^4} \, dx\\ &=\int \left (\frac {15 c^2 d^4 e^4-24 a c d^2 e^6+10 a^2 e^8}{c^6 d^6}+\frac {2 e^5 \left (3 c d^2-2 a e^2\right ) x}{c^5 d^5}+\frac {e^6 x^2}{c^4 d^4}+\frac {\left (c d^2-a e^2\right )^6}{c^6 d^6 (a e+c d x)^4}+\frac {6 e \left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)^3}+\frac {15 e^2 \left (c d^2-a e^2\right )^4}{c^6 d^6 (a e+c d x)^2}+\frac {20 \left (c d^2 e-a e^3\right )^3}{c^6 d^6 (a e+c d x)}\right ) \, dx\\ &=\frac {e^4 \left (15 c^2 d^4-24 a c d^2 e^2+10 a^2 e^4\right ) x}{c^6 d^6}+\frac {e^5 \left (3 c d^2-2 a e^2\right ) x^2}{c^5 d^5}+\frac {e^6 x^3}{3 c^4 d^4}-\frac {\left (c d^2-a e^2\right )^6}{3 c^7 d^7 (a e+c d x)^3}-\frac {3 e \left (c d^2-a e^2\right )^5}{c^7 d^7 (a e+c d x)^2}-\frac {15 e^2 \left (c d^2-a e^2\right )^4}{c^7 d^7 (a e+c d x)}+\frac {20 e^3 \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^7 d^7}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 335, normalized size = 1.54 \begin {gather*} \frac {-37 a^6 e^{12}+3 a^5 c d e^{10} (47 d-17 e x)+3 a^4 c^2 d^2 e^8 \left (-65 d^2+81 d e x+13 e^2 x^2\right )+a^3 c^3 d^3 e^6 \left (110 d^3-405 d^2 e x-27 d e^2 x^2+73 e^3 x^3\right )-3 a^2 c^4 d^4 e^4 \left (5 d^4-90 d^3 e x+45 d^2 e^2 x^2+63 d e^3 x^3-5 e^4 x^4\right )-3 a c^5 d^5 e^2 \left (d^5+15 d^4 e x-60 d^3 e^2 x^2-45 d^2 e^3 x^3+15 d e^4 x^4+e^5 x^5\right )-60 e^3 \left (a e^2-c d^2\right )^3 (a e+c d x)^3 \log (a e+c d x)+c^6 d^6 \left (-d^6-9 d^5 e x-45 d^4 e^2 x^2+45 d^2 e^4 x^4+9 d e^5 x^5+e^6 x^6\right )}{3 c^7 d^7 (a e+c d x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^10/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

(-37*a^6*e^12 + 3*a^5*c*d*e^10*(47*d - 17*e*x) + 3*a^4*c^2*d^2*e^8*(-65*d^2 + 81*d*e*x + 13*e^2*x^2) + a^3*c^3
*d^3*e^6*(110*d^3 - 405*d^2*e*x - 27*d*e^2*x^2 + 73*e^3*x^3) - 3*a^2*c^4*d^4*e^4*(5*d^4 - 90*d^3*e*x + 45*d^2*
e^2*x^2 + 63*d*e^3*x^3 - 5*e^4*x^4) - 3*a*c^5*d^5*e^2*(d^5 + 15*d^4*e*x - 60*d^3*e^2*x^2 - 45*d^2*e^3*x^3 + 15
*d*e^4*x^4 + e^5*x^5) + c^6*d^6*(-d^6 - 9*d^5*e*x - 45*d^4*e^2*x^2 + 45*d^2*e^4*x^4 + 9*d*e^5*x^5 + e^6*x^6) -
60*e^3*(-(c*d^2) + a*e^2)^3*(a*e + c*d*x)^3*Log[a*e + c*d*x])/(3*c^7*d^7*(a*e + c*d*x)^3)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^{10}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^10/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

IntegrateAlgebraic[(d + e*x)^10/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4, x]

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fricas [B]  time = 0.41, size = 644, normalized size = 2.97 \begin {gather*} \frac {c^{6} d^{6} e^{6} x^{6} - c^{6} d^{12} - 3 \, a c^{5} d^{10} e^{2} - 15 \, a^{2} c^{4} d^{8} e^{4} + 110 \, a^{3} c^{3} d^{6} e^{6} - 195 \, a^{4} c^{2} d^{4} e^{8} + 141 \, a^{5} c d^{2} e^{10} - 37 \, a^{6} e^{12} + 3 \, {\left (3 \, c^{6} d^{7} e^{5} - a c^{5} d^{5} e^{7}\right )} x^{5} + 15 \, {\left (3 \, c^{6} d^{8} e^{4} - 3 \, a c^{5} d^{6} e^{6} + a^{2} c^{4} d^{4} e^{8}\right )} x^{4} + {\left (135 \, a c^{5} d^{7} e^{5} - 189 \, a^{2} c^{4} d^{5} e^{7} + 73 \, a^{3} c^{3} d^{3} e^{9}\right )} x^{3} - 3 \, {\left (15 \, c^{6} d^{10} e^{2} - 60 \, a c^{5} d^{8} e^{4} + 45 \, a^{2} c^{4} d^{6} e^{6} + 9 \, a^{3} c^{3} d^{4} e^{8} - 13 \, a^{4} c^{2} d^{2} e^{10}\right )} x^{2} - 3 \, {\left (3 \, c^{6} d^{11} e + 15 \, a c^{5} d^{9} e^{3} - 90 \, a^{2} c^{4} d^{7} e^{5} + 135 \, a^{3} c^{3} d^{5} e^{7} - 81 \, a^{4} c^{2} d^{3} e^{9} + 17 \, a^{5} c d e^{11}\right )} x + 60 \, {\left (a^{3} c^{3} d^{6} e^{6} - 3 \, a^{4} c^{2} d^{4} e^{8} + 3 \, a^{5} c d^{2} e^{10} - a^{6} e^{12} + {\left (c^{6} d^{9} e^{3} - 3 \, a c^{5} d^{7} e^{5} + 3 \, a^{2} c^{4} d^{5} e^{7} - a^{3} c^{3} d^{3} e^{9}\right )} x^{3} + 3 \, {\left (a c^{5} d^{8} e^{4} - 3 \, a^{2} c^{4} d^{6} e^{6} + 3 \, a^{3} c^{3} d^{4} e^{8} - a^{4} c^{2} d^{2} e^{10}\right )} x^{2} + 3 \, {\left (a^{2} c^{4} d^{7} e^{5} - 3 \, a^{3} c^{3} d^{5} e^{7} + 3 \, 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 )}{3 \, {\left (c^{10} d^{10} x^{3} + 3 \, a c^{9} d^{9} e x^{2} + 3 \, a^{2} c^{8} d^{8} e^{2} x + a^{3} c^{7} d^{7} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^10/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="fricas")

[Out]

1/3*(c^6*d^6*e^6*x^6 - c^6*d^12 - 3*a*c^5*d^10*e^2 - 15*a^2*c^4*d^8*e^4 + 110*a^3*c^3*d^6*e^6 - 195*a^4*c^2*d^
4*e^8 + 141*a^5*c*d^2*e^10 - 37*a^6*e^12 + 3*(3*c^6*d^7*e^5 - a*c^5*d^5*e^7)*x^5 + 15*(3*c^6*d^8*e^4 - 3*a*c^5
*d^6*e^6 + a^2*c^4*d^4*e^8)*x^4 + (135*a*c^5*d^7*e^5 - 189*a^2*c^4*d^5*e^7 + 73*a^3*c^3*d^3*e^9)*x^3 - 3*(15*c
^6*d^10*e^2 - 60*a*c^5*d^8*e^4 + 45*a^2*c^4*d^6*e^6 + 9*a^3*c^3*d^4*e^8 - 13*a^4*c^2*d^2*e^10)*x^2 - 3*(3*c^6*
d^11*e + 15*a*c^5*d^9*e^3 - 90*a^2*c^4*d^7*e^5 + 135*a^3*c^3*d^5*e^7 - 81*a^4*c^2*d^3*e^9 + 17*a^5*c*d*e^11)*x
+ 60*(a^3*c^3*d^6*e^6 - 3*a^4*c^2*d^4*e^8 + 3*a^5*c*d^2*e^10 - a^6*e^12 + (c^6*d^9*e^3 - 3*a*c^5*d^7*e^5 + 3*
a^2*c^4*d^5*e^7 - a^3*c^3*d^3*e^9)*x^3 + 3*(a*c^5*d^8*e^4 - 3*a^2*c^4*d^6*e^6 + 3*a^3*c^3*d^4*e^8 - a^4*c^2*d^
2*e^10)*x^2 + 3*(a^2*c^4*d^7*e^5 - 3*a^3*c^3*d^5*e^7 + 3*a^4*c^2*d^3*e^9 - a^5*c*d*e^11)*x)*log(c*d*x + a*e))/
(c^10*d^10*x^3 + 3*a*c^9*d^9*e*x^2 + 3*a^2*c^8*d^8*e^2*x + a^3*c^7*d^7*e^3)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^10/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="giac")

[Out]

sage0*x

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maple [B]  time = 0.06, size = 578, normalized size = 2.66 \begin {gather*} -\frac {a^{6} e^{12}}{3 \left (c d x +a e \right )^{3} c^{7} d^{7}}+\frac {2 a^{5} e^{10}}{\left (c d x +a e \right )^{3} c^{6} d^{5}}-\frac {5 a^{4} e^{8}}{\left (c d x +a e \right )^{3} c^{5} d^{3}}+\frac {20 a^{3} e^{6}}{3 \left (c d x +a e \right )^{3} c^{4} d}-\frac {5 a^{2} d \,e^{4}}{\left (c d x +a e \right )^{3} c^{3}}+\frac {2 a \,d^{3} e^{2}}{\left (c d x +a e \right )^{3} c^{2}}-\frac {d^{5}}{3 \left (c d x +a e \right )^{3} c}+\frac {3 a^{5} e^{11}}{\left (c d x +a e \right )^{2} c^{7} d^{7}}-\frac {15 a^{4} e^{9}}{\left (c d x +a e \right )^{2} c^{6} d^{5}}+\frac {30 a^{3} e^{7}}{\left (c d x +a e \right )^{2} c^{5} d^{3}}-\frac {30 a^{2} e^{5}}{\left (c d x +a e \right )^{2} c^{4} d}+\frac {15 a d \,e^{3}}{\left (c d x +a e \right )^{2} c^{3}}-\frac {3 d^{3} e}{\left (c d x +a e \right )^{2} c^{2}}+\frac {e^{6} x^{3}}{3 c^{4} d^{4}}-\frac {15 a^{4} e^{10}}{\left (c d x +a e \right ) c^{7} d^{7}}+\frac {60 a^{3} e^{8}}{\left (c d x +a e \right ) c^{6} d^{5}}-\frac {90 a^{2} e^{6}}{\left (c d x +a e \right ) c^{5} d^{3}}+\frac {60 a \,e^{4}}{\left (c d x +a e \right ) c^{4} d}-\frac {2 a \,e^{7} x^{2}}{c^{5} d^{5}}-\frac {15 d \,e^{2}}{\left (c d x +a e \right ) c^{3}}+\frac {3 e^{5} x^{2}}{c^{4} d^{3}}-\frac {20 a^{3} e^{9} \ln \left (c d x +a e \right )}{c^{7} d^{7}}+\frac {60 a^{2} e^{7} \ln \left (c d x +a e \right )}{c^{6} d^{5}}+\frac {10 a^{2} e^{8} x}{c^{6} d^{6}}-\frac {60 a \,e^{5} \ln \left (c d x +a e \right )}{c^{5} d^{3}}-\frac {24 a \,e^{6} x}{c^{5} d^{4}}+\frac {20 e^{3} \ln \left (c d x +a e \right )}{c^{4} d}+\frac {15 e^{4} x}{c^{4} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^10/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x)

[Out]

-5/c^5/d^3/(c*d*x+a*e)^3*a^4*e^8+20/3/c^4/d/(c*d*x+a*e)^3*a^3*e^6+20/c^4/d*e^3*ln(c*d*x+a*e)-15/c^3*d*e^2/(c*d
*x+a*e)+3*e^5/c^4/d^3*x^2+15*e^4/c^4/d^2*x-3*d^3/c^2*e/(c*d*x+a*e)^2-1/3/c^7/d^7/(c*d*x+a*e)^3*a^6*e^12+2/c^6/
d^5/(c*d*x+a*e)^3*a^5*e^10-2*e^7/c^5/d^5*x^2*a-1/3/c*d^5/(c*d*x+a*e)^3+60/c^4/d*e^4/(c*d*x+a*e)*a+60/c^6/d^5*e
^7*ln(c*d*x+a*e)*a^2-60/c^5/d^3*e^5*ln(c*d*x+a*e)*a-15/c^7/d^7*e^10/(c*d*x+a*e)*a^4+60/c^6/d^5*e^8/(c*d*x+a*e)
*a^3-90/c^5/d^3*e^6/(c*d*x+a*e)*a^2-5/c^3*d/(c*d*x+a*e)^3*a^2*e^4+2/c^2*d^3/(c*d*x+a*e)^3*a*e^2+3/d^7/c^7*e^11
/(c*d*x+a*e)^2*a^5-15/d^5/c^6*e^9/(c*d*x+a*e)^2*a^4+30/d^3/c^5*e^7/(c*d*x+a*e)^2*a^3-30/d/c^4*e^5/(c*d*x+a*e)^
2*a^2+15*d/c^3*e^3/(c*d*x+a*e)^2*a-24*e^6/c^5/d^4*a*x+10*e^8/c^6/d^6*a^2*x-20/c^7/d^7*e^9*ln(c*d*x+a*e)*a^3+1/
3*e^6*x^3/c^4/d^4

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maxima [A]  time = 1.18, size = 424, normalized size = 1.95 \begin {gather*} -\frac {c^{6} d^{12} + 3 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 110 \, a^{3} c^{3} d^{6} e^{6} + 195 \, a^{4} c^{2} d^{4} e^{8} - 141 \, a^{5} c d^{2} e^{10} + 37 \, a^{6} e^{12} + 45 \, {\left (c^{6} d^{10} e^{2} - 4 \, a c^{5} d^{8} e^{4} + 6 \, a^{2} c^{4} d^{6} e^{6} - 4 \, a^{3} c^{3} d^{4} e^{8} + a^{4} c^{2} d^{2} e^{10}\right )} x^{2} + 9 \, {\left (c^{6} d^{11} e + 5 \, a c^{5} d^{9} e^{3} - 30 \, a^{2} c^{4} d^{7} e^{5} + 50 \, a^{3} c^{3} d^{5} e^{7} - 35 \, a^{4} c^{2} d^{3} e^{9} + 9 \, a^{5} c d e^{11}\right )} x}{3 \, {\left (c^{10} d^{10} x^{3} + 3 \, a c^{9} d^{9} e x^{2} + 3 \, a^{2} c^{8} d^{8} e^{2} x + a^{3} c^{7} d^{7} e^{3}\right )}} + \frac {c^{2} d^{2} e^{6} x^{3} + 3 \, {\left (3 \, c^{2} d^{3} e^{5} - 2 \, a c d e^{7}\right )} x^{2} + 3 \, {\left (15 \, c^{2} d^{4} e^{4} - 24 \, a c d^{2} e^{6} + 10 \, a^{2} e^{8}\right )} x}{3 \, c^{6} d^{6}} + \frac {20 \, {\left (c^{3} d^{6} e^{3} - 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}\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)^10/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="maxima")

[Out]

-1/3*(c^6*d^12 + 3*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 110*a^3*c^3*d^6*e^6 + 195*a^4*c^2*d^4*e^8 - 141*a^5*c
*d^2*e^10 + 37*a^6*e^12 + 45*(c^6*d^10*e^2 - 4*a*c^5*d^8*e^4 + 6*a^2*c^4*d^6*e^6 - 4*a^3*c^3*d^4*e^8 + a^4*c^2
*d^2*e^10)*x^2 + 9*(c^6*d^11*e + 5*a*c^5*d^9*e^3 - 30*a^2*c^4*d^7*e^5 + 50*a^3*c^3*d^5*e^7 - 35*a^4*c^2*d^3*e^
9 + 9*a^5*c*d*e^11)*x)/(c^10*d^10*x^3 + 3*a*c^9*d^9*e*x^2 + 3*a^2*c^8*d^8*e^2*x + a^3*c^7*d^7*e^3) + 1/3*(c^2*
d^2*e^6*x^3 + 3*(3*c^2*d^3*e^5 - 2*a*c*d*e^7)*x^2 + 3*(15*c^2*d^4*e^4 - 24*a*c*d^2*e^6 + 10*a^2*e^8)*x)/(c^6*d
^6) + 20*(c^3*d^6*e^3 - 3*a*c^2*d^4*e^5 + 3*a^2*c*d^2*e^7 - a^3*e^9)*log(c*d*x + a*e)/(c^7*d^7)

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mupad [B]  time = 0.15, size = 452, normalized size = 2.08 \begin {gather*} x^2\,\left (\frac {3\,e^5}{c^4\,d^3}-\frac {2\,a\,e^7}{c^5\,d^5}\right )-x\,\left (\frac {6\,a^2\,e^8}{c^6\,d^6}-\frac {15\,e^4}{c^4\,d^2}+\frac {4\,a\,e\,\left (\frac {6\,e^5}{c^4\,d^3}-\frac {4\,a\,e^7}{c^5\,d^5}\right )}{c\,d}\right )-\frac {x\,\left (27\,a^5\,e^{11}-105\,a^4\,c\,d^2\,e^9+150\,a^3\,c^2\,d^4\,e^7-90\,a^2\,c^3\,d^6\,e^5+15\,a\,c^4\,d^8\,e^3+3\,c^5\,d^{10}\,e\right )+x^2\,\left (15\,a^4\,c\,d\,e^{10}-60\,a^3\,c^2\,d^3\,e^8+90\,a^2\,c^3\,d^5\,e^6-60\,a\,c^4\,d^7\,e^4+15\,c^5\,d^9\,e^2\right )+\frac {37\,a^6\,e^{12}-141\,a^5\,c\,d^2\,e^{10}+195\,a^4\,c^2\,d^4\,e^8-110\,a^3\,c^3\,d^6\,e^6+15\,a^2\,c^4\,d^8\,e^4+3\,a\,c^5\,d^{10}\,e^2+c^6\,d^{12}}{3\,c\,d}}{a^3\,c^6\,d^6\,e^3+3\,a^2\,c^7\,d^7\,e^2\,x+3\,a\,c^8\,d^8\,e\,x^2+c^9\,d^9\,x^3}-\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (20\,a^3\,e^9-60\,a^2\,c\,d^2\,e^7+60\,a\,c^2\,d^4\,e^5-20\,c^3\,d^6\,e^3\right )}{c^7\,d^7}+\frac {e^6\,x^3}{3\,c^4\,d^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^10/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^4,x)

[Out]

x^2*((3*e^5)/(c^4*d^3) - (2*a*e^7)/(c^5*d^5)) - x*((6*a^2*e^8)/(c^6*d^6) - (15*e^4)/(c^4*d^2) + (4*a*e*((6*e^5
)/(c^4*d^3) - (4*a*e^7)/(c^5*d^5)))/(c*d)) - (x*(27*a^5*e^11 + 3*c^5*d^10*e + 15*a*c^4*d^8*e^3 - 105*a^4*c*d^2
*e^9 - 90*a^2*c^3*d^6*e^5 + 150*a^3*c^2*d^4*e^7) + x^2*(15*c^5*d^9*e^2 - 60*a*c^4*d^7*e^4 + 90*a^2*c^3*d^5*e^6
- 60*a^3*c^2*d^3*e^8 + 15*a^4*c*d*e^10) + (37*a^6*e^12 + c^6*d^12 + 3*a*c^5*d^10*e^2 - 141*a^5*c*d^2*e^10 + 1
5*a^2*c^4*d^8*e^4 - 110*a^3*c^3*d^6*e^6 + 195*a^4*c^2*d^4*e^8)/(3*c*d))/(c^9*d^9*x^3 + a^3*c^6*d^6*e^3 + 3*a*c
^8*d^8*e*x^2 + 3*a^2*c^7*d^7*e^2*x) - (log(a*e + c*d*x)*(20*a^3*e^9 - 20*c^3*d^6*e^3 + 60*a*c^2*d^4*e^5 - 60*a
^2*c*d^2*e^7))/(c^7*d^7) + (e^6*x^3)/(3*c^4*d^4)

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sympy [B]  time = 102.95, size = 425, normalized size = 1.96 \begin {gather*} x^{2} \left (- \frac {2 a e^{7}}{c^{5} d^{5}} + \frac {3 e^{5}}{c^{4} d^{3}}\right ) + x \left (\frac {10 a^{2} e^{8}}{c^{6} d^{6}} - \frac {24 a e^{6}}{c^{5} d^{4}} + \frac {15 e^{4}}{c^{4} d^{2}}\right ) + \frac {- 37 a^{6} e^{12} + 141 a^{5} c d^{2} e^{10} - 195 a^{4} c^{2} d^{4} e^{8} + 110 a^{3} c^{3} d^{6} e^{6} - 15 a^{2} c^{4} d^{8} e^{4} - 3 a c^{5} d^{10} e^{2} - c^{6} d^{12} + x^{2} \left (- 45 a^{4} c^{2} d^{2} e^{10} + 180 a^{3} c^{3} d^{4} e^{8} - 270 a^{2} c^{4} d^{6} e^{6} + 180 a c^{5} d^{8} e^{4} - 45 c^{6} d^{10} e^{2}\right ) + x \left (- 81 a^{5} c d e^{11} + 315 a^{4} c^{2} d^{3} e^{9} - 450 a^{3} c^{3} d^{5} e^{7} + 270 a^{2} c^{4} d^{7} e^{5} - 45 a c^{5} d^{9} e^{3} - 9 c^{6} d^{11} e\right )}{3 a^{3} c^{7} d^{7} e^{3} + 9 a^{2} c^{8} d^{8} e^{2} x + 9 a c^{9} d^{9} e x^{2} + 3 c^{10} d^{10} x^{3}} + \frac {e^{6} x^{3}}{3 c^{4} d^{4}} - \frac {20 e^{3} \left (a e^{2} - c d^{2}\right )^{3} \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)**10/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)

[Out]

x**2*(-2*a*e**7/(c**5*d**5) + 3*e**5/(c**4*d**3)) + x*(10*a**2*e**8/(c**6*d**6) - 24*a*e**6/(c**5*d**4) + 15*e
**4/(c**4*d**2)) + (-37*a**6*e**12 + 141*a**5*c*d**2*e**10 - 195*a**4*c**2*d**4*e**8 + 110*a**3*c**3*d**6*e**6
- 15*a**2*c**4*d**8*e**4 - 3*a*c**5*d**10*e**2 - c**6*d**12 + x**2*(-45*a**4*c**2*d**2*e**10 + 180*a**3*c**3*
d**4*e**8 - 270*a**2*c**4*d**6*e**6 + 180*a*c**5*d**8*e**4 - 45*c**6*d**10*e**2) + x*(-81*a**5*c*d*e**11 + 315
*a**4*c**2*d**3*e**9 - 450*a**3*c**3*d**5*e**7 + 270*a**2*c**4*d**7*e**5 - 45*a*c**5*d**9*e**3 - 9*c**6*d**11*
e))/(3*a**3*c**7*d**7*e**3 + 9*a**2*c**8*d**8*e**2*x + 9*a*c**9*d**9*e*x**2 + 3*c**10*d**10*x**3) + e**6*x**3/
(3*c**4*d**4) - 20*e**3*(a*e**2 - c*d**2)**3*log(a*e + c*d*x)/(c**7*d**7)

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