3.13.5 \(\int \frac {(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{14}} \, dx\)

Optimal. Leaf size=171 \[ \frac {3 b^5 (b d-a e)}{4 e^7 (d+e x)^8}-\frac {5 b^4 (b d-a e)^2}{3 e^7 (d+e x)^9}+\frac {2 b^3 (b d-a e)^3}{e^7 (d+e x)^{10}}-\frac {15 b^2 (b d-a e)^4}{11 e^7 (d+e x)^{11}}+\frac {b (b d-a e)^5}{2 e^7 (d+e x)^{12}}-\frac {(b d-a e)^6}{13 e^7 (d+e x)^{13}}-\frac {b^6}{7 e^7 (d+e x)^7} \]

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Rubi [A]  time = 0.13, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 43} \begin {gather*} \frac {3 b^5 (b d-a e)}{4 e^7 (d+e x)^8}-\frac {5 b^4 (b d-a e)^2}{3 e^7 (d+e x)^9}+\frac {2 b^3 (b d-a e)^3}{e^7 (d+e x)^{10}}-\frac {15 b^2 (b d-a e)^4}{11 e^7 (d+e x)^{11}}+\frac {b (b d-a e)^5}{2 e^7 (d+e x)^{12}}-\frac {(b d-a e)^6}{13 e^7 (d+e x)^{13}}-\frac {b^6}{7 e^7 (d+e x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^14,x]

[Out]

-(b*d - a*e)^6/(13*e^7*(d + e*x)^13) + (b*(b*d - a*e)^5)/(2*e^7*(d + e*x)^12) - (15*b^2*(b*d - a*e)^4)/(11*e^7
*(d + e*x)^11) + (2*b^3*(b*d - a*e)^3)/(e^7*(d + e*x)^10) - (5*b^4*(b*d - a*e)^2)/(3*e^7*(d + e*x)^9) + (3*b^5
*(b*d - a*e))/(4*e^7*(d + e*x)^8) - b^6/(7*e^7*(d + e*x)^7)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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])

Rubi steps

\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{14}} \, dx &=\int \frac {(a+b x)^6}{(d+e x)^{14}} \, dx\\ &=\int \left (\frac {(-b d+a e)^6}{e^6 (d+e x)^{14}}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^{13}}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^{12}}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^{11}}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^{10}}-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)^9}+\frac {b^6}{e^6 (d+e x)^8}\right ) \, dx\\ &=-\frac {(b d-a e)^6}{13 e^7 (d+e x)^{13}}+\frac {b (b d-a e)^5}{2 e^7 (d+e x)^{12}}-\frac {15 b^2 (b d-a e)^4}{11 e^7 (d+e x)^{11}}+\frac {2 b^3 (b d-a e)^3}{e^7 (d+e x)^{10}}-\frac {5 b^4 (b d-a e)^2}{3 e^7 (d+e x)^9}+\frac {3 b^5 (b d-a e)}{4 e^7 (d+e x)^8}-\frac {b^6}{7 e^7 (d+e x)^7}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 277, normalized size = 1.62 \begin {gather*} -\frac {924 a^6 e^6+462 a^5 b e^5 (d+13 e x)+210 a^4 b^2 e^4 \left (d^2+13 d e x+78 e^2 x^2\right )+84 a^3 b^3 e^3 \left (d^3+13 d^2 e x+78 d e^2 x^2+286 e^3 x^3\right )+28 a^2 b^4 e^2 \left (d^4+13 d^3 e x+78 d^2 e^2 x^2+286 d e^3 x^3+715 e^4 x^4\right )+7 a b^5 e \left (d^5+13 d^4 e x+78 d^3 e^2 x^2+286 d^2 e^3 x^3+715 d e^4 x^4+1287 e^5 x^5\right )+b^6 \left (d^6+13 d^5 e x+78 d^4 e^2 x^2+286 d^3 e^3 x^3+715 d^2 e^4 x^4+1287 d e^5 x^5+1716 e^6 x^6\right )}{12012 e^7 (d+e x)^{13}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^14,x]

[Out]

-1/12012*(924*a^6*e^6 + 462*a^5*b*e^5*(d + 13*e*x) + 210*a^4*b^2*e^4*(d^2 + 13*d*e*x + 78*e^2*x^2) + 84*a^3*b^
3*e^3*(d^3 + 13*d^2*e*x + 78*d*e^2*x^2 + 286*e^3*x^3) + 28*a^2*b^4*e^2*(d^4 + 13*d^3*e*x + 78*d^2*e^2*x^2 + 28
6*d*e^3*x^3 + 715*e^4*x^4) + 7*a*b^5*e*(d^5 + 13*d^4*e*x + 78*d^3*e^2*x^2 + 286*d^2*e^3*x^3 + 715*d*e^4*x^4 +
1287*e^5*x^5) + b^6*(d^6 + 13*d^5*e*x + 78*d^4*e^2*x^2 + 286*d^3*e^3*x^3 + 715*d^2*e^4*x^4 + 1287*d*e^5*x^5 +
1716*e^6*x^6))/(e^7*(d + e*x)^13)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{14}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^14,x]

[Out]

IntegrateAlgebraic[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^14, x]

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fricas [B]  time = 0.38, size = 485, normalized size = 2.84 \begin {gather*} -\frac {1716 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 7 \, a b^{5} d^{5} e + 28 \, a^{2} b^{4} d^{4} e^{2} + 84 \, a^{3} b^{3} d^{3} e^{3} + 210 \, a^{4} b^{2} d^{2} e^{4} + 462 \, a^{5} b d e^{5} + 924 \, a^{6} e^{6} + 1287 \, {\left (b^{6} d e^{5} + 7 \, a b^{5} e^{6}\right )} x^{5} + 715 \, {\left (b^{6} d^{2} e^{4} + 7 \, a b^{5} d e^{5} + 28 \, a^{2} b^{4} e^{6}\right )} x^{4} + 286 \, {\left (b^{6} d^{3} e^{3} + 7 \, a b^{5} d^{2} e^{4} + 28 \, a^{2} b^{4} d e^{5} + 84 \, a^{3} b^{3} e^{6}\right )} x^{3} + 78 \, {\left (b^{6} d^{4} e^{2} + 7 \, a b^{5} d^{3} e^{3} + 28 \, a^{2} b^{4} d^{2} e^{4} + 84 \, a^{3} b^{3} d e^{5} + 210 \, a^{4} b^{2} e^{6}\right )} x^{2} + 13 \, {\left (b^{6} d^{5} e + 7 \, a b^{5} d^{4} e^{2} + 28 \, a^{2} b^{4} d^{3} e^{3} + 84 \, a^{3} b^{3} d^{2} e^{4} + 210 \, a^{4} b^{2} d e^{5} + 462 \, a^{5} b e^{6}\right )} x}{12012 \, {\left (e^{20} x^{13} + 13 \, d e^{19} x^{12} + 78 \, d^{2} e^{18} x^{11} + 286 \, d^{3} e^{17} x^{10} + 715 \, d^{4} e^{16} x^{9} + 1287 \, d^{5} e^{15} x^{8} + 1716 \, d^{6} e^{14} x^{7} + 1716 \, d^{7} e^{13} x^{6} + 1287 \, d^{8} e^{12} x^{5} + 715 \, d^{9} e^{11} x^{4} + 286 \, d^{10} e^{10} x^{3} + 78 \, d^{11} e^{9} x^{2} + 13 \, d^{12} e^{8} x + d^{13} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^14,x, algorithm="fricas")

[Out]

-1/12012*(1716*b^6*e^6*x^6 + b^6*d^6 + 7*a*b^5*d^5*e + 28*a^2*b^4*d^4*e^2 + 84*a^3*b^3*d^3*e^3 + 210*a^4*b^2*d
^2*e^4 + 462*a^5*b*d*e^5 + 924*a^6*e^6 + 1287*(b^6*d*e^5 + 7*a*b^5*e^6)*x^5 + 715*(b^6*d^2*e^4 + 7*a*b^5*d*e^5
+ 28*a^2*b^4*e^6)*x^4 + 286*(b^6*d^3*e^3 + 7*a*b^5*d^2*e^4 + 28*a^2*b^4*d*e^5 + 84*a^3*b^3*e^6)*x^3 + 78*(b^6
*d^4*e^2 + 7*a*b^5*d^3*e^3 + 28*a^2*b^4*d^2*e^4 + 84*a^3*b^3*d*e^5 + 210*a^4*b^2*e^6)*x^2 + 13*(b^6*d^5*e + 7*
a*b^5*d^4*e^2 + 28*a^2*b^4*d^3*e^3 + 84*a^3*b^3*d^2*e^4 + 210*a^4*b^2*d*e^5 + 462*a^5*b*e^6)*x)/(e^20*x^13 + 1
3*d*e^19*x^12 + 78*d^2*e^18*x^11 + 286*d^3*e^17*x^10 + 715*d^4*e^16*x^9 + 1287*d^5*e^15*x^8 + 1716*d^6*e^14*x^
7 + 1716*d^7*e^13*x^6 + 1287*d^8*e^12*x^5 + 715*d^9*e^11*x^4 + 286*d^10*e^10*x^3 + 78*d^11*e^9*x^2 + 13*d^12*e
^8*x + d^13*e^7)

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giac [B]  time = 0.17, size = 352, normalized size = 2.06 \begin {gather*} -\frac {{\left (1716 \, b^{6} x^{6} e^{6} + 1287 \, b^{6} d x^{5} e^{5} + 715 \, b^{6} d^{2} x^{4} e^{4} + 286 \, b^{6} d^{3} x^{3} e^{3} + 78 \, b^{6} d^{4} x^{2} e^{2} + 13 \, b^{6} d^{5} x e + b^{6} d^{6} + 9009 \, a b^{5} x^{5} e^{6} + 5005 \, a b^{5} d x^{4} e^{5} + 2002 \, a b^{5} d^{2} x^{3} e^{4} + 546 \, a b^{5} d^{3} x^{2} e^{3} + 91 \, a b^{5} d^{4} x e^{2} + 7 \, a b^{5} d^{5} e + 20020 \, a^{2} b^{4} x^{4} e^{6} + 8008 \, a^{2} b^{4} d x^{3} e^{5} + 2184 \, a^{2} b^{4} d^{2} x^{2} e^{4} + 364 \, a^{2} b^{4} d^{3} x e^{3} + 28 \, a^{2} b^{4} d^{4} e^{2} + 24024 \, a^{3} b^{3} x^{3} e^{6} + 6552 \, a^{3} b^{3} d x^{2} e^{5} + 1092 \, a^{3} b^{3} d^{2} x e^{4} + 84 \, a^{3} b^{3} d^{3} e^{3} + 16380 \, a^{4} b^{2} x^{2} e^{6} + 2730 \, a^{4} b^{2} d x e^{5} + 210 \, a^{4} b^{2} d^{2} e^{4} + 6006 \, a^{5} b x e^{6} + 462 \, a^{5} b d e^{5} + 924 \, a^{6} e^{6}\right )} e^{\left (-7\right )}}{12012 \, {\left (x e + d\right )}^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^14,x, algorithm="giac")

[Out]

-1/12012*(1716*b^6*x^6*e^6 + 1287*b^6*d*x^5*e^5 + 715*b^6*d^2*x^4*e^4 + 286*b^6*d^3*x^3*e^3 + 78*b^6*d^4*x^2*e
^2 + 13*b^6*d^5*x*e + b^6*d^6 + 9009*a*b^5*x^5*e^6 + 5005*a*b^5*d*x^4*e^5 + 2002*a*b^5*d^2*x^3*e^4 + 546*a*b^5
*d^3*x^2*e^3 + 91*a*b^5*d^4*x*e^2 + 7*a*b^5*d^5*e + 20020*a^2*b^4*x^4*e^6 + 8008*a^2*b^4*d*x^3*e^5 + 2184*a^2*
b^4*d^2*x^2*e^4 + 364*a^2*b^4*d^3*x*e^3 + 28*a^2*b^4*d^4*e^2 + 24024*a^3*b^3*x^3*e^6 + 6552*a^3*b^3*d*x^2*e^5
+ 1092*a^3*b^3*d^2*x*e^4 + 84*a^3*b^3*d^3*e^3 + 16380*a^4*b^2*x^2*e^6 + 2730*a^4*b^2*d*x*e^5 + 210*a^4*b^2*d^2
*e^4 + 6006*a^5*b*x*e^6 + 462*a^5*b*d*e^5 + 924*a^6*e^6)*e^(-7)/(x*e + d)^13

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maple [B]  time = 0.05, size = 357, normalized size = 2.09 \begin {gather*} -\frac {b^{6}}{7 \left (e x +d \right )^{7} e^{7}}-\frac {3 \left (a e -b d \right ) b^{5}}{4 \left (e x +d \right )^{8} e^{7}}-\frac {5 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}}{3 \left (e x +d \right )^{9} e^{7}}-\frac {2 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) b^{3}}{\left (e x +d \right )^{10} e^{7}}-\frac {15 \left (e^{4} a^{4}-4 d \,e^{3} a^{3} b +6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) b^{2}}{11 \left (e x +d \right )^{11} e^{7}}-\frac {\left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right ) b}{2 \left (e x +d \right )^{12} e^{7}}-\frac {a^{6} e^{6}-6 d \,e^{5} a^{5} b +15 d^{2} e^{4} a^{4} b^{2}-20 d^{3} e^{3} a^{3} b^{3}+15 d^{4} a^{2} b^{4} e^{2}-6 d^{5} e a \,b^{5}+b^{6} d^{6}}{13 \left (e x +d \right )^{13} e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^14,x)

[Out]

-15/11*b^2*(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)/e^7/(e*x+d)^11-2*b^3*(a^3*e^3-3*a^2
*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)/e^7/(e*x+d)^10-1/7*b^6/e^7/(e*x+d)^7-1/2*b*(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d
^2*e^3-10*a^2*b^3*d^3*e^2+5*a*b^4*d^4*e-b^5*d^5)/e^7/(e*x+d)^12-5/3*b^4*(a^2*e^2-2*a*b*d*e+b^2*d^2)/e^7/(e*x+d
)^9-1/13*(a^6*e^6-6*a^5*b*d*e^5+15*a^4*b^2*d^2*e^4-20*a^3*b^3*d^3*e^3+15*a^2*b^4*d^4*e^2-6*a*b^5*d^5*e+b^6*d^6
)/e^7/(e*x+d)^13-3/4*b^5*(a*e-b*d)/e^7/(e*x+d)^8

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maxima [B]  time = 1.74, size = 485, normalized size = 2.84 \begin {gather*} -\frac {1716 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 7 \, a b^{5} d^{5} e + 28 \, a^{2} b^{4} d^{4} e^{2} + 84 \, a^{3} b^{3} d^{3} e^{3} + 210 \, a^{4} b^{2} d^{2} e^{4} + 462 \, a^{5} b d e^{5} + 924 \, a^{6} e^{6} + 1287 \, {\left (b^{6} d e^{5} + 7 \, a b^{5} e^{6}\right )} x^{5} + 715 \, {\left (b^{6} d^{2} e^{4} + 7 \, a b^{5} d e^{5} + 28 \, a^{2} b^{4} e^{6}\right )} x^{4} + 286 \, {\left (b^{6} d^{3} e^{3} + 7 \, a b^{5} d^{2} e^{4} + 28 \, a^{2} b^{4} d e^{5} + 84 \, a^{3} b^{3} e^{6}\right )} x^{3} + 78 \, {\left (b^{6} d^{4} e^{2} + 7 \, a b^{5} d^{3} e^{3} + 28 \, a^{2} b^{4} d^{2} e^{4} + 84 \, a^{3} b^{3} d e^{5} + 210 \, a^{4} b^{2} e^{6}\right )} x^{2} + 13 \, {\left (b^{6} d^{5} e + 7 \, a b^{5} d^{4} e^{2} + 28 \, a^{2} b^{4} d^{3} e^{3} + 84 \, a^{3} b^{3} d^{2} e^{4} + 210 \, a^{4} b^{2} d e^{5} + 462 \, a^{5} b e^{6}\right )} x}{12012 \, {\left (e^{20} x^{13} + 13 \, d e^{19} x^{12} + 78 \, d^{2} e^{18} x^{11} + 286 \, d^{3} e^{17} x^{10} + 715 \, d^{4} e^{16} x^{9} + 1287 \, d^{5} e^{15} x^{8} + 1716 \, d^{6} e^{14} x^{7} + 1716 \, d^{7} e^{13} x^{6} + 1287 \, d^{8} e^{12} x^{5} + 715 \, d^{9} e^{11} x^{4} + 286 \, d^{10} e^{10} x^{3} + 78 \, d^{11} e^{9} x^{2} + 13 \, d^{12} e^{8} x + d^{13} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^14,x, algorithm="maxima")

[Out]

-1/12012*(1716*b^6*e^6*x^6 + b^6*d^6 + 7*a*b^5*d^5*e + 28*a^2*b^4*d^4*e^2 + 84*a^3*b^3*d^3*e^3 + 210*a^4*b^2*d
^2*e^4 + 462*a^5*b*d*e^5 + 924*a^6*e^6 + 1287*(b^6*d*e^5 + 7*a*b^5*e^6)*x^5 + 715*(b^6*d^2*e^4 + 7*a*b^5*d*e^5
+ 28*a^2*b^4*e^6)*x^4 + 286*(b^6*d^3*e^3 + 7*a*b^5*d^2*e^4 + 28*a^2*b^4*d*e^5 + 84*a^3*b^3*e^6)*x^3 + 78*(b^6
*d^4*e^2 + 7*a*b^5*d^3*e^3 + 28*a^2*b^4*d^2*e^4 + 84*a^3*b^3*d*e^5 + 210*a^4*b^2*e^6)*x^2 + 13*(b^6*d^5*e + 7*
a*b^5*d^4*e^2 + 28*a^2*b^4*d^3*e^3 + 84*a^3*b^3*d^2*e^4 + 210*a^4*b^2*d*e^5 + 462*a^5*b*e^6)*x)/(e^20*x^13 + 1
3*d*e^19*x^12 + 78*d^2*e^18*x^11 + 286*d^3*e^17*x^10 + 715*d^4*e^16*x^9 + 1287*d^5*e^15*x^8 + 1716*d^6*e^14*x^
7 + 1716*d^7*e^13*x^6 + 1287*d^8*e^12*x^5 + 715*d^9*e^11*x^4 + 286*d^10*e^10*x^3 + 78*d^11*e^9*x^2 + 13*d^12*e
^8*x + d^13*e^7)

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mupad [B]  time = 1.17, size = 467, normalized size = 2.73 \begin {gather*} -\frac {\frac {924\,a^6\,e^6+462\,a^5\,b\,d\,e^5+210\,a^4\,b^2\,d^2\,e^4+84\,a^3\,b^3\,d^3\,e^3+28\,a^2\,b^4\,d^4\,e^2+7\,a\,b^5\,d^5\,e+b^6\,d^6}{12012\,e^7}+\frac {b^6\,x^6}{7\,e}+\frac {b^3\,x^3\,\left (84\,a^3\,e^3+28\,a^2\,b\,d\,e^2+7\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{42\,e^4}+\frac {b\,x\,\left (462\,a^5\,e^5+210\,a^4\,b\,d\,e^4+84\,a^3\,b^2\,d^2\,e^3+28\,a^2\,b^3\,d^3\,e^2+7\,a\,b^4\,d^4\,e+b^5\,d^5\right )}{924\,e^6}+\frac {3\,b^5\,x^5\,\left (7\,a\,e+b\,d\right )}{28\,e^2}+\frac {b^2\,x^2\,\left (210\,a^4\,e^4+84\,a^3\,b\,d\,e^3+28\,a^2\,b^2\,d^2\,e^2+7\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{154\,e^5}+\frac {5\,b^4\,x^4\,\left (28\,a^2\,e^2+7\,a\,b\,d\,e+b^2\,d^2\right )}{84\,e^3}}{d^{13}+13\,d^{12}\,e\,x+78\,d^{11}\,e^2\,x^2+286\,d^{10}\,e^3\,x^3+715\,d^9\,e^4\,x^4+1287\,d^8\,e^5\,x^5+1716\,d^7\,e^6\,x^6+1716\,d^6\,e^7\,x^7+1287\,d^5\,e^8\,x^8+715\,d^4\,e^9\,x^9+286\,d^3\,e^{10}\,x^{10}+78\,d^2\,e^{11}\,x^{11}+13\,d\,e^{12}\,x^{12}+e^{13}\,x^{13}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^2 + 2*a*b*x)^3/(d + e*x)^14,x)

[Out]

-((924*a^6*e^6 + b^6*d^6 + 28*a^2*b^4*d^4*e^2 + 84*a^3*b^3*d^3*e^3 + 210*a^4*b^2*d^2*e^4 + 7*a*b^5*d^5*e + 462
*a^5*b*d*e^5)/(12012*e^7) + (b^6*x^6)/(7*e) + (b^3*x^3*(84*a^3*e^3 + b^3*d^3 + 7*a*b^2*d^2*e + 28*a^2*b*d*e^2)
)/(42*e^4) + (b*x*(462*a^5*e^5 + b^5*d^5 + 28*a^2*b^3*d^3*e^2 + 84*a^3*b^2*d^2*e^3 + 7*a*b^4*d^4*e + 210*a^4*b
*d*e^4))/(924*e^6) + (3*b^5*x^5*(7*a*e + b*d))/(28*e^2) + (b^2*x^2*(210*a^4*e^4 + b^4*d^4 + 28*a^2*b^2*d^2*e^2
+ 7*a*b^3*d^3*e + 84*a^3*b*d*e^3))/(154*e^5) + (5*b^4*x^4*(28*a^2*e^2 + b^2*d^2 + 7*a*b*d*e))/(84*e^3))/(d^13
+ e^13*x^13 + 13*d*e^12*x^12 + 78*d^11*e^2*x^2 + 286*d^10*e^3*x^3 + 715*d^9*e^4*x^4 + 1287*d^8*e^5*x^5 + 1716
*d^7*e^6*x^6 + 1716*d^6*e^7*x^7 + 1287*d^5*e^8*x^8 + 715*d^4*e^9*x^9 + 286*d^3*e^10*x^10 + 78*d^2*e^11*x^11 +
13*d^12*e*x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**14,x)

[Out]

Timed out

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