3.13.3 \(\int \frac {(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{12}} \, dx\)

Optimal. Leaf size=170 \[ \frac {b^5 (b d-a e)}{e^7 (d+e x)^6}-\frac {15 b^4 (b d-a e)^2}{7 e^7 (d+e x)^7}+\frac {5 b^3 (b d-a e)^3}{2 e^7 (d+e x)^8}-\frac {5 b^2 (b d-a e)^4}{3 e^7 (d+e x)^9}+\frac {3 b (b d-a e)^5}{5 e^7 (d+e x)^{10}}-\frac {(b d-a e)^6}{11 e^7 (d+e x)^{11}}-\frac {b^6}{5 e^7 (d+e x)^5} \]

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Rubi [A]  time = 0.13, antiderivative size = 170, 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 {b^5 (b d-a e)}{e^7 (d+e x)^6}-\frac {15 b^4 (b d-a e)^2}{7 e^7 (d+e x)^7}+\frac {5 b^3 (b d-a e)^3}{2 e^7 (d+e x)^8}-\frac {5 b^2 (b d-a e)^4}{3 e^7 (d+e x)^9}+\frac {3 b (b d-a e)^5}{5 e^7 (d+e x)^{10}}-\frac {(b d-a e)^6}{11 e^7 (d+e x)^{11}}-\frac {b^6}{5 e^7 (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^12,x]

[Out]

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

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

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Mathematica [A]  time = 0.09, size = 277, normalized size = 1.63 \begin {gather*} -\frac {210 a^6 e^6+126 a^5 b e^5 (d+11 e x)+70 a^4 b^2 e^4 \left (d^2+11 d e x+55 e^2 x^2\right )+35 a^3 b^3 e^3 \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+15 a^2 b^4 e^2 \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+5 a b^5 e \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )+b^6 \left (d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right )}{2310 e^7 (d+e x)^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^12,x]

[Out]

-1/2310*(210*a^6*e^6 + 126*a^5*b*e^5*(d + 11*e*x) + 70*a^4*b^2*e^4*(d^2 + 11*d*e*x + 55*e^2*x^2) + 35*a^3*b^3*
e^3*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3) + 15*a^2*b^4*e^2*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165*
d*e^3*x^3 + 330*e^4*x^4) + 5*a*b^5*e*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4*x^4 + 46
2*e^5*x^5) + b^6*(d^6 + 11*d^5*e*x + 55*d^4*e^2*x^2 + 165*d^3*e^3*x^3 + 330*d^2*e^4*x^4 + 462*d*e^5*x^5 + 462*
e^6*x^6))/(e^7*(d + e*x)^11)

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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)^{12}} \, 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)^12,x]

[Out]

IntegrateAlgebraic[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^12, x]

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fricas [B]  time = 0.39, size = 463, normalized size = 2.72 \begin {gather*} -\frac {462 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 5 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} + 35 \, a^{3} b^{3} d^{3} e^{3} + 70 \, a^{4} b^{2} d^{2} e^{4} + 126 \, a^{5} b d e^{5} + 210 \, a^{6} e^{6} + 462 \, {\left (b^{6} d e^{5} + 5 \, a b^{5} e^{6}\right )} x^{5} + 330 \, {\left (b^{6} d^{2} e^{4} + 5 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} + 165 \, {\left (b^{6} d^{3} e^{3} + 5 \, a b^{5} d^{2} e^{4} + 15 \, a^{2} b^{4} d e^{5} + 35 \, a^{3} b^{3} e^{6}\right )} x^{3} + 55 \, {\left (b^{6} d^{4} e^{2} + 5 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} + 35 \, a^{3} b^{3} d e^{5} + 70 \, a^{4} b^{2} e^{6}\right )} x^{2} + 11 \, {\left (b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 15 \, a^{2} b^{4} d^{3} e^{3} + 35 \, a^{3} b^{3} d^{2} e^{4} + 70 \, a^{4} b^{2} d e^{5} + 126 \, a^{5} b e^{6}\right )} x}{2310 \, {\left (e^{18} x^{11} + 11 \, d e^{17} x^{10} + 55 \, d^{2} e^{16} x^{9} + 165 \, d^{3} e^{15} x^{8} + 330 \, d^{4} e^{14} x^{7} + 462 \, d^{5} e^{13} x^{6} + 462 \, d^{6} e^{12} x^{5} + 330 \, d^{7} e^{11} x^{4} + 165 \, d^{8} e^{10} x^{3} + 55 \, d^{9} e^{9} x^{2} + 11 \, d^{10} e^{8} x + d^{11} 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)^12,x, algorithm="fricas")

[Out]

-1/2310*(462*b^6*e^6*x^6 + b^6*d^6 + 5*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 + 35*a^3*b^3*d^3*e^3 + 70*a^4*b^2*d^2*
e^4 + 126*a^5*b*d*e^5 + 210*a^6*e^6 + 462*(b^6*d*e^5 + 5*a*b^5*e^6)*x^5 + 330*(b^6*d^2*e^4 + 5*a*b^5*d*e^5 + 1
5*a^2*b^4*e^6)*x^4 + 165*(b^6*d^3*e^3 + 5*a*b^5*d^2*e^4 + 15*a^2*b^4*d*e^5 + 35*a^3*b^3*e^6)*x^3 + 55*(b^6*d^4
*e^2 + 5*a*b^5*d^3*e^3 + 15*a^2*b^4*d^2*e^4 + 35*a^3*b^3*d*e^5 + 70*a^4*b^2*e^6)*x^2 + 11*(b^6*d^5*e + 5*a*b^5
*d^4*e^2 + 15*a^2*b^4*d^3*e^3 + 35*a^3*b^3*d^2*e^4 + 70*a^4*b^2*d*e^5 + 126*a^5*b*e^6)*x)/(e^18*x^11 + 11*d*e^
17*x^10 + 55*d^2*e^16*x^9 + 165*d^3*e^15*x^8 + 330*d^4*e^14*x^7 + 462*d^5*e^13*x^6 + 462*d^6*e^12*x^5 + 330*d^
7*e^11*x^4 + 165*d^8*e^10*x^3 + 55*d^9*e^9*x^2 + 11*d^10*e^8*x + d^11*e^7)

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giac [B]  time = 0.16, size = 352, normalized size = 2.07 \begin {gather*} -\frac {{\left (462 \, b^{6} x^{6} e^{6} + 462 \, b^{6} d x^{5} e^{5} + 330 \, b^{6} d^{2} x^{4} e^{4} + 165 \, b^{6} d^{3} x^{3} e^{3} + 55 \, b^{6} d^{4} x^{2} e^{2} + 11 \, b^{6} d^{5} x e + b^{6} d^{6} + 2310 \, a b^{5} x^{5} e^{6} + 1650 \, a b^{5} d x^{4} e^{5} + 825 \, a b^{5} d^{2} x^{3} e^{4} + 275 \, a b^{5} d^{3} x^{2} e^{3} + 55 \, a b^{5} d^{4} x e^{2} + 5 \, a b^{5} d^{5} e + 4950 \, a^{2} b^{4} x^{4} e^{6} + 2475 \, a^{2} b^{4} d x^{3} e^{5} + 825 \, a^{2} b^{4} d^{2} x^{2} e^{4} + 165 \, a^{2} b^{4} d^{3} x e^{3} + 15 \, a^{2} b^{4} d^{4} e^{2} + 5775 \, a^{3} b^{3} x^{3} e^{6} + 1925 \, a^{3} b^{3} d x^{2} e^{5} + 385 \, a^{3} b^{3} d^{2} x e^{4} + 35 \, a^{3} b^{3} d^{3} e^{3} + 3850 \, a^{4} b^{2} x^{2} e^{6} + 770 \, a^{4} b^{2} d x e^{5} + 70 \, a^{4} b^{2} d^{2} e^{4} + 1386 \, a^{5} b x e^{6} + 126 \, a^{5} b d e^{5} + 210 \, a^{6} e^{6}\right )} e^{\left (-7\right )}}{2310 \, {\left (x e + d\right )}^{11}} \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)^12,x, algorithm="giac")

[Out]

-1/2310*(462*b^6*x^6*e^6 + 462*b^6*d*x^5*e^5 + 330*b^6*d^2*x^4*e^4 + 165*b^6*d^3*x^3*e^3 + 55*b^6*d^4*x^2*e^2
+ 11*b^6*d^5*x*e + b^6*d^6 + 2310*a*b^5*x^5*e^6 + 1650*a*b^5*d*x^4*e^5 + 825*a*b^5*d^2*x^3*e^4 + 275*a*b^5*d^3
*x^2*e^3 + 55*a*b^5*d^4*x*e^2 + 5*a*b^5*d^5*e + 4950*a^2*b^4*x^4*e^6 + 2475*a^2*b^4*d*x^3*e^5 + 825*a^2*b^4*d^
2*x^2*e^4 + 165*a^2*b^4*d^3*x*e^3 + 15*a^2*b^4*d^4*e^2 + 5775*a^3*b^3*x^3*e^6 + 1925*a^3*b^3*d*x^2*e^5 + 385*a
^3*b^3*d^2*x*e^4 + 35*a^3*b^3*d^3*e^3 + 3850*a^4*b^2*x^2*e^6 + 770*a^4*b^2*d*x*e^5 + 70*a^4*b^2*d^2*e^4 + 1386
*a^5*b*x*e^6 + 126*a^5*b*d*e^5 + 210*a^6*e^6)*e^(-7)/(x*e + d)^11

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maple [B]  time = 0.05, size = 357, normalized size = 2.10 \begin {gather*} -\frac {b^{6}}{5 \left (e x +d \right )^{5} e^{7}}-\frac {\left (a e -b d \right ) b^{5}}{\left (e x +d \right )^{6} e^{7}}-\frac {15 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}}{7 \left (e x +d \right )^{7} e^{7}}-\frac {5 \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}}{2 \left (e x +d \right )^{8} e^{7}}-\frac {5 \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}}{3 \left (e x +d \right )^{9} e^{7}}-\frac {3 \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}{5 \left (e x +d \right )^{10} 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}}{11 \left (e x +d \right )^{11} 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)^12,x)

[Out]

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

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maxima [B]  time = 1.75, size = 463, normalized size = 2.72 \begin {gather*} -\frac {462 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 5 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} + 35 \, a^{3} b^{3} d^{3} e^{3} + 70 \, a^{4} b^{2} d^{2} e^{4} + 126 \, a^{5} b d e^{5} + 210 \, a^{6} e^{6} + 462 \, {\left (b^{6} d e^{5} + 5 \, a b^{5} e^{6}\right )} x^{5} + 330 \, {\left (b^{6} d^{2} e^{4} + 5 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} + 165 \, {\left (b^{6} d^{3} e^{3} + 5 \, a b^{5} d^{2} e^{4} + 15 \, a^{2} b^{4} d e^{5} + 35 \, a^{3} b^{3} e^{6}\right )} x^{3} + 55 \, {\left (b^{6} d^{4} e^{2} + 5 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} + 35 \, a^{3} b^{3} d e^{5} + 70 \, a^{4} b^{2} e^{6}\right )} x^{2} + 11 \, {\left (b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 15 \, a^{2} b^{4} d^{3} e^{3} + 35 \, a^{3} b^{3} d^{2} e^{4} + 70 \, a^{4} b^{2} d e^{5} + 126 \, a^{5} b e^{6}\right )} x}{2310 \, {\left (e^{18} x^{11} + 11 \, d e^{17} x^{10} + 55 \, d^{2} e^{16} x^{9} + 165 \, d^{3} e^{15} x^{8} + 330 \, d^{4} e^{14} x^{7} + 462 \, d^{5} e^{13} x^{6} + 462 \, d^{6} e^{12} x^{5} + 330 \, d^{7} e^{11} x^{4} + 165 \, d^{8} e^{10} x^{3} + 55 \, d^{9} e^{9} x^{2} + 11 \, d^{10} e^{8} x + d^{11} 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)^12,x, algorithm="maxima")

[Out]

-1/2310*(462*b^6*e^6*x^6 + b^6*d^6 + 5*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 + 35*a^3*b^3*d^3*e^3 + 70*a^4*b^2*d^2*
e^4 + 126*a^5*b*d*e^5 + 210*a^6*e^6 + 462*(b^6*d*e^5 + 5*a*b^5*e^6)*x^5 + 330*(b^6*d^2*e^4 + 5*a*b^5*d*e^5 + 1
5*a^2*b^4*e^6)*x^4 + 165*(b^6*d^3*e^3 + 5*a*b^5*d^2*e^4 + 15*a^2*b^4*d*e^5 + 35*a^3*b^3*e^6)*x^3 + 55*(b^6*d^4
*e^2 + 5*a*b^5*d^3*e^3 + 15*a^2*b^4*d^2*e^4 + 35*a^3*b^3*d*e^5 + 70*a^4*b^2*e^6)*x^2 + 11*(b^6*d^5*e + 5*a*b^5
*d^4*e^2 + 15*a^2*b^4*d^3*e^3 + 35*a^3*b^3*d^2*e^4 + 70*a^4*b^2*d*e^5 + 126*a^5*b*e^6)*x)/(e^18*x^11 + 11*d*e^
17*x^10 + 55*d^2*e^16*x^9 + 165*d^3*e^15*x^8 + 330*d^4*e^14*x^7 + 462*d^5*e^13*x^6 + 462*d^6*e^12*x^5 + 330*d^
7*e^11*x^4 + 165*d^8*e^10*x^3 + 55*d^9*e^9*x^2 + 11*d^10*e^8*x + d^11*e^7)

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mupad [B]  time = 0.66, size = 445, normalized size = 2.62 \begin {gather*} -\frac {\frac {210\,a^6\,e^6+126\,a^5\,b\,d\,e^5+70\,a^4\,b^2\,d^2\,e^4+35\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2+5\,a\,b^5\,d^5\,e+b^6\,d^6}{2310\,e^7}+\frac {b^6\,x^6}{5\,e}+\frac {b^3\,x^3\,\left (35\,a^3\,e^3+15\,a^2\,b\,d\,e^2+5\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{14\,e^4}+\frac {b\,x\,\left (126\,a^5\,e^5+70\,a^4\,b\,d\,e^4+35\,a^3\,b^2\,d^2\,e^3+15\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e+b^5\,d^5\right )}{210\,e^6}+\frac {b^5\,x^5\,\left (5\,a\,e+b\,d\right )}{5\,e^2}+\frac {b^2\,x^2\,\left (70\,a^4\,e^4+35\,a^3\,b\,d\,e^3+15\,a^2\,b^2\,d^2\,e^2+5\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{42\,e^5}+\frac {b^4\,x^4\,\left (15\,a^2\,e^2+5\,a\,b\,d\,e+b^2\,d^2\right )}{7\,e^3}}{d^{11}+11\,d^{10}\,e\,x+55\,d^9\,e^2\,x^2+165\,d^8\,e^3\,x^3+330\,d^7\,e^4\,x^4+462\,d^6\,e^5\,x^5+462\,d^5\,e^6\,x^6+330\,d^4\,e^7\,x^7+165\,d^3\,e^8\,x^8+55\,d^2\,e^9\,x^9+11\,d\,e^{10}\,x^{10}+e^{11}\,x^{11}} \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)^12,x)

[Out]

-((210*a^6*e^6 + b^6*d^6 + 15*a^2*b^4*d^4*e^2 + 35*a^3*b^3*d^3*e^3 + 70*a^4*b^2*d^2*e^4 + 5*a*b^5*d^5*e + 126*
a^5*b*d*e^5)/(2310*e^7) + (b^6*x^6)/(5*e) + (b^3*x^3*(35*a^3*e^3 + b^3*d^3 + 5*a*b^2*d^2*e + 15*a^2*b*d*e^2))/
(14*e^4) + (b*x*(126*a^5*e^5 + b^5*d^5 + 15*a^2*b^3*d^3*e^2 + 35*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e + 70*a^4*b*d*
e^4))/(210*e^6) + (b^5*x^5*(5*a*e + b*d))/(5*e^2) + (b^2*x^2*(70*a^4*e^4 + b^4*d^4 + 15*a^2*b^2*d^2*e^2 + 5*a*
b^3*d^3*e + 35*a^3*b*d*e^3))/(42*e^5) + (b^4*x^4*(15*a^2*e^2 + b^2*d^2 + 5*a*b*d*e))/(7*e^3))/(d^11 + e^11*x^1
1 + 11*d*e^10*x^10 + 55*d^9*e^2*x^2 + 165*d^8*e^3*x^3 + 330*d^7*e^4*x^4 + 462*d^6*e^5*x^5 + 462*d^5*e^6*x^6 +
330*d^4*e^7*x^7 + 165*d^3*e^8*x^8 + 55*d^2*e^9*x^9 + 11*d^10*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)**12,x)

[Out]

Timed out

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