3.10.32 \(\int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{12}} \, dx\)

Optimal. Leaf size=235 \[ \frac {b^3 (a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{9240 e (d+e x)^7 (b d-a e)^5}+\frac {b^2 (a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{1320 e (d+e x)^8 (b d-a e)^4}+\frac {b (a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{330 e (d+e x)^9 (b d-a e)^3}+\frac {(a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{110 e (d+e x)^{10} (b d-a e)^2}-\frac {(a+b x)^7 (B d-A e)}{11 e (d+e x)^{11} (b d-a e)} \]

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Rubi [A]  time = 0.11, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {78, 45, 37} \begin {gather*} \frac {b^3 (a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{9240 e (d+e x)^7 (b d-a e)^5}+\frac {b^2 (a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{1320 e (d+e x)^8 (b d-a e)^4}+\frac {b (a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{330 e (d+e x)^9 (b d-a e)^3}+\frac {(a+b x)^7 (-11 a B e+4 A b e+7 b B d)}{110 e (d+e x)^{10} (b d-a e)^2}-\frac {(a+b x)^7 (B d-A e)}{11 e (d+e x)^{11} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^12,x]

[Out]

-((B*d - A*e)*(a + b*x)^7)/(11*e*(b*d - a*e)*(d + e*x)^11) + ((7*b*B*d + 4*A*b*e - 11*a*B*e)*(a + b*x)^7)/(110
*e*(b*d - a*e)^2*(d + e*x)^10) + (b*(7*b*B*d + 4*A*b*e - 11*a*B*e)*(a + b*x)^7)/(330*e*(b*d - a*e)^3*(d + e*x)
^9) + (b^2*(7*b*B*d + 4*A*b*e - 11*a*B*e)*(a + b*x)^7)/(1320*e*(b*d - a*e)^4*(d + e*x)^8) + (b^3*(7*b*B*d + 4*
A*b*e - 11*a*B*e)*(a + b*x)^7)/(9240*e*(b*d - a*e)^5*(d + e*x)^7)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rubi steps

\begin {align*} \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{12}} \, dx &=-\frac {(B d-A e) (a+b x)^7}{11 e (b d-a e) (d+e x)^{11}}+\frac {(7 b B d+4 A b e-11 a B e) \int \frac {(a+b x)^6}{(d+e x)^{11}} \, dx}{11 e (b d-a e)}\\ &=-\frac {(B d-A e) (a+b x)^7}{11 e (b d-a e) (d+e x)^{11}}+\frac {(7 b B d+4 A b e-11 a B e) (a+b x)^7}{110 e (b d-a e)^2 (d+e x)^{10}}+\frac {(3 b (7 b B d+4 A b e-11 a B e)) \int \frac {(a+b x)^6}{(d+e x)^{10}} \, dx}{110 e (b d-a e)^2}\\ &=-\frac {(B d-A e) (a+b x)^7}{11 e (b d-a e) (d+e x)^{11}}+\frac {(7 b B d+4 A b e-11 a B e) (a+b x)^7}{110 e (b d-a e)^2 (d+e x)^{10}}+\frac {b (7 b B d+4 A b e-11 a B e) (a+b x)^7}{330 e (b d-a e)^3 (d+e x)^9}+\frac {\left (b^2 (7 b B d+4 A b e-11 a B e)\right ) \int \frac {(a+b x)^6}{(d+e x)^9} \, dx}{165 e (b d-a e)^3}\\ &=-\frac {(B d-A e) (a+b x)^7}{11 e (b d-a e) (d+e x)^{11}}+\frac {(7 b B d+4 A b e-11 a B e) (a+b x)^7}{110 e (b d-a e)^2 (d+e x)^{10}}+\frac {b (7 b B d+4 A b e-11 a B e) (a+b x)^7}{330 e (b d-a e)^3 (d+e x)^9}+\frac {b^2 (7 b B d+4 A b e-11 a B e) (a+b x)^7}{1320 e (b d-a e)^4 (d+e x)^8}+\frac {\left (b^3 (7 b B d+4 A b e-11 a B e)\right ) \int \frac {(a+b x)^6}{(d+e x)^8} \, dx}{1320 e (b d-a e)^4}\\ &=-\frac {(B d-A e) (a+b x)^7}{11 e (b d-a e) (d+e x)^{11}}+\frac {(7 b B d+4 A b e-11 a B e) (a+b x)^7}{110 e (b d-a e)^2 (d+e x)^{10}}+\frac {b (7 b B d+4 A b e-11 a B e) (a+b x)^7}{330 e (b d-a e)^3 (d+e x)^9}+\frac {b^2 (7 b B d+4 A b e-11 a B e) (a+b x)^7}{1320 e (b d-a e)^4 (d+e x)^8}+\frac {b^3 (7 b B d+4 A b e-11 a B e) (a+b x)^7}{9240 e (b d-a e)^5 (d+e x)^7}\\ \end {align*}

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Mathematica [B]  time = 0.28, size = 605, normalized size = 2.57 \begin {gather*} -\frac {84 a^6 e^6 (10 A e+B (d+11 e x))+56 a^5 b e^5 \left (9 A e (d+11 e x)+2 B \left (d^2+11 d e x+55 e^2 x^2\right )\right )+35 a^4 b^2 e^4 \left (8 A e \left (d^2+11 d e x+55 e^2 x^2\right )+3 B \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )\right )+20 a^3 b^3 e^3 \left (7 A e \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+4 B \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 )\right )+10 a^2 b^4 e^2 \left (6 A e \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 B \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 )\right )+4 a b^5 e \left (5 A 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 )+6 B \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 )\right )+b^6 \left (4 A e \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 )+7 B \left (d^7+11 d^6 e x+55 d^5 e^2 x^2+165 d^4 e^3 x^3+330 d^3 e^4 x^4+462 d^2 e^5 x^5+462 d e^6 x^6+330 e^7 x^7\right )\right )}{9240 e^8 (d+e x)^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^12,x]

[Out]

-1/9240*(84*a^6*e^6*(10*A*e + B*(d + 11*e*x)) + 56*a^5*b*e^5*(9*A*e*(d + 11*e*x) + 2*B*(d^2 + 11*d*e*x + 55*e^
2*x^2)) + 35*a^4*b^2*e^4*(8*A*e*(d^2 + 11*d*e*x + 55*e^2*x^2) + 3*B*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3
*x^3)) + 20*a^3*b^3*e^3*(7*A*e*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3) + 4*B*(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)) + 10*a^2*b^4*e^2*(6*A*e*(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*B*(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
)) + 4*a*b^5*e*(5*A*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 + 462*e^5*x^5) + 6*
B*(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)) + b^6
*(4*A*e*(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)
+ 7*B*(d^7 + 11*d^6*e*x + 55*d^5*e^2*x^2 + 165*d^4*e^3*x^3 + 330*d^3*e^4*x^4 + 462*d^2*e^5*x^5 + 462*d*e^6*x^6
+ 330*e^7*x^7)))/(e^8*(d + e*x)^11)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{12}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)^6*(A + B*x))/(d + e*x)^12,x]

[Out]

IntegrateAlgebraic[((a + b*x)^6*(A + B*x))/(d + e*x)^12, x]

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fricas [B]  time = 1.01, size = 883, normalized size = 3.76 \begin {gather*} -\frac {2310 \, B b^{6} e^{7} x^{7} + 7 \, B b^{6} d^{7} + 840 \, A a^{6} e^{7} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} + 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 35 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} + 56 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + 84 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 462 \, {\left (7 \, B b^{6} d e^{6} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 462 \, {\left (7 \, B b^{6} d^{2} e^{5} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} + 330 \, {\left (7 \, B b^{6} d^{3} e^{4} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} + 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 165 \, {\left (7 \, B b^{6} d^{4} e^{3} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} + 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + 35 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} + 55 \, {\left (7 \, B b^{6} d^{5} e^{2} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} + 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + 35 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} + 56 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 11 \, {\left (7 \, B b^{6} d^{6} e + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} + 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 35 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} + 56 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + 84 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x}{9240 \, {\left (e^{19} x^{11} + 11 \, d e^{18} x^{10} + 55 \, d^{2} e^{17} x^{9} + 165 \, d^{3} e^{16} x^{8} + 330 \, d^{4} e^{15} x^{7} + 462 \, d^{5} e^{14} x^{6} + 462 \, d^{6} e^{13} x^{5} + 330 \, d^{7} e^{12} x^{4} + 165 \, d^{8} e^{11} x^{3} + 55 \, d^{9} e^{10} x^{2} + 11 \, d^{10} e^{9} x + d^{11} e^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^12,x, algorithm="fricas")

[Out]

-1/9240*(2310*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 840*A*a^6*e^7 + 4*(6*B*a*b^5 + A*b^6)*d^6*e + 10*(5*B*a^2*b^4 + 2*
A*a*b^5)*d^5*e^2 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 35*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 56*(2*B*a
^5*b + 5*A*a^4*b^2)*d^2*e^5 + 84*(B*a^6 + 6*A*a^5*b)*d*e^6 + 462*(7*B*b^6*d*e^6 + 4*(6*B*a*b^5 + A*b^6)*e^7)*x
^6 + 462*(7*B*b^6*d^2*e^5 + 4*(6*B*a*b^5 + A*b^6)*d*e^6 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 330*(7*B*b^6
*d^3*e^4 + 4*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)
*e^7)*x^4 + 165*(7*B*b^6*d^4*e^3 + 4*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + 20*(
4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 35*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 55*(7*B*b^6*d^5*e^2 + 4*(6*B*a*b^
5 + A*b^6)*d^4*e^3 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 35*(3*B*a
^4*b^2 + 4*A*a^3*b^3)*d*e^6 + 56*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 11*(7*B*b^6*d^6*e + 4*(6*B*a*b^5 + A*b^6
)*d^5*e^2 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 35*(3*B*a^4*b^2 +
4*A*a^3*b^3)*d^2*e^5 + 56*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 84*(B*a^6 + 6*A*a^5*b)*e^7)*x)/(e^19*x^11 + 11*d*e
^18*x^10 + 55*d^2*e^17*x^9 + 165*d^3*e^16*x^8 + 330*d^4*e^15*x^7 + 462*d^5*e^14*x^6 + 462*d^6*e^13*x^5 + 330*d
^7*e^12*x^4 + 165*d^8*e^11*x^3 + 55*d^9*e^10*x^2 + 11*d^10*e^9*x + d^11*e^8)

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giac [B]  time = 1.18, size = 856, normalized size = 3.64 \begin {gather*} -\frac {{\left (2310 \, B b^{6} x^{7} e^{7} + 3234 \, B b^{6} d x^{6} e^{6} + 3234 \, B b^{6} d^{2} x^{5} e^{5} + 2310 \, B b^{6} d^{3} x^{4} e^{4} + 1155 \, B b^{6} d^{4} x^{3} e^{3} + 385 \, B b^{6} d^{5} x^{2} e^{2} + 77 \, B b^{6} d^{6} x e + 7 \, B b^{6} d^{7} + 11088 \, B a b^{5} x^{6} e^{7} + 1848 \, A b^{6} x^{6} e^{7} + 11088 \, B a b^{5} d x^{5} e^{6} + 1848 \, A b^{6} d x^{5} e^{6} + 7920 \, B a b^{5} d^{2} x^{4} e^{5} + 1320 \, A b^{6} d^{2} x^{4} e^{5} + 3960 \, B a b^{5} d^{3} x^{3} e^{4} + 660 \, A b^{6} d^{3} x^{3} e^{4} + 1320 \, B a b^{5} d^{4} x^{2} e^{3} + 220 \, A b^{6} d^{4} x^{2} e^{3} + 264 \, B a b^{5} d^{5} x e^{2} + 44 \, A b^{6} d^{5} x e^{2} + 24 \, B a b^{5} d^{6} e + 4 \, A b^{6} d^{6} e + 23100 \, B a^{2} b^{4} x^{5} e^{7} + 9240 \, A a b^{5} x^{5} e^{7} + 16500 \, B a^{2} b^{4} d x^{4} e^{6} + 6600 \, A a b^{5} d x^{4} e^{6} + 8250 \, B a^{2} b^{4} d^{2} x^{3} e^{5} + 3300 \, A a b^{5} d^{2} x^{3} e^{5} + 2750 \, B a^{2} b^{4} d^{3} x^{2} e^{4} + 1100 \, A a b^{5} d^{3} x^{2} e^{4} + 550 \, B a^{2} b^{4} d^{4} x e^{3} + 220 \, A a b^{5} d^{4} x e^{3} + 50 \, B a^{2} b^{4} d^{5} e^{2} + 20 \, A a b^{5} d^{5} e^{2} + 26400 \, B a^{3} b^{3} x^{4} e^{7} + 19800 \, A a^{2} b^{4} x^{4} e^{7} + 13200 \, B a^{3} b^{3} d x^{3} e^{6} + 9900 \, A a^{2} b^{4} d x^{3} e^{6} + 4400 \, B a^{3} b^{3} d^{2} x^{2} e^{5} + 3300 \, A a^{2} b^{4} d^{2} x^{2} e^{5} + 880 \, B a^{3} b^{3} d^{3} x e^{4} + 660 \, A a^{2} b^{4} d^{3} x e^{4} + 80 \, B a^{3} b^{3} d^{4} e^{3} + 60 \, A a^{2} b^{4} d^{4} e^{3} + 17325 \, B a^{4} b^{2} x^{3} e^{7} + 23100 \, A a^{3} b^{3} x^{3} e^{7} + 5775 \, B a^{4} b^{2} d x^{2} e^{6} + 7700 \, A a^{3} b^{3} d x^{2} e^{6} + 1155 \, B a^{4} b^{2} d^{2} x e^{5} + 1540 \, A a^{3} b^{3} d^{2} x e^{5} + 105 \, B a^{4} b^{2} d^{3} e^{4} + 140 \, A a^{3} b^{3} d^{3} e^{4} + 6160 \, B a^{5} b x^{2} e^{7} + 15400 \, A a^{4} b^{2} x^{2} e^{7} + 1232 \, B a^{5} b d x e^{6} + 3080 \, A a^{4} b^{2} d x e^{6} + 112 \, B a^{5} b d^{2} e^{5} + 280 \, A a^{4} b^{2} d^{2} e^{5} + 924 \, B a^{6} x e^{7} + 5544 \, A a^{5} b x e^{7} + 84 \, B a^{6} d e^{6} + 504 \, A a^{5} b d e^{6} + 840 \, A a^{6} e^{7}\right )} e^{\left (-8\right )}}{9240 \, {\left (x e + d\right )}^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^12,x, algorithm="giac")

[Out]

-1/9240*(2310*B*b^6*x^7*e^7 + 3234*B*b^6*d*x^6*e^6 + 3234*B*b^6*d^2*x^5*e^5 + 2310*B*b^6*d^3*x^4*e^4 + 1155*B*
b^6*d^4*x^3*e^3 + 385*B*b^6*d^5*x^2*e^2 + 77*B*b^6*d^6*x*e + 7*B*b^6*d^7 + 11088*B*a*b^5*x^6*e^7 + 1848*A*b^6*
x^6*e^7 + 11088*B*a*b^5*d*x^5*e^6 + 1848*A*b^6*d*x^5*e^6 + 7920*B*a*b^5*d^2*x^4*e^5 + 1320*A*b^6*d^2*x^4*e^5 +
3960*B*a*b^5*d^3*x^3*e^4 + 660*A*b^6*d^3*x^3*e^4 + 1320*B*a*b^5*d^4*x^2*e^3 + 220*A*b^6*d^4*x^2*e^3 + 264*B*a
*b^5*d^5*x*e^2 + 44*A*b^6*d^5*x*e^2 + 24*B*a*b^5*d^6*e + 4*A*b^6*d^6*e + 23100*B*a^2*b^4*x^5*e^7 + 9240*A*a*b^
5*x^5*e^7 + 16500*B*a^2*b^4*d*x^4*e^6 + 6600*A*a*b^5*d*x^4*e^6 + 8250*B*a^2*b^4*d^2*x^3*e^5 + 3300*A*a*b^5*d^2
*x^3*e^5 + 2750*B*a^2*b^4*d^3*x^2*e^4 + 1100*A*a*b^5*d^3*x^2*e^4 + 550*B*a^2*b^4*d^4*x*e^3 + 220*A*a*b^5*d^4*x
*e^3 + 50*B*a^2*b^4*d^5*e^2 + 20*A*a*b^5*d^5*e^2 + 26400*B*a^3*b^3*x^4*e^7 + 19800*A*a^2*b^4*x^4*e^7 + 13200*B
*a^3*b^3*d*x^3*e^6 + 9900*A*a^2*b^4*d*x^3*e^6 + 4400*B*a^3*b^3*d^2*x^2*e^5 + 3300*A*a^2*b^4*d^2*x^2*e^5 + 880*
B*a^3*b^3*d^3*x*e^4 + 660*A*a^2*b^4*d^3*x*e^4 + 80*B*a^3*b^3*d^4*e^3 + 60*A*a^2*b^4*d^4*e^3 + 17325*B*a^4*b^2*
x^3*e^7 + 23100*A*a^3*b^3*x^3*e^7 + 5775*B*a^4*b^2*d*x^2*e^6 + 7700*A*a^3*b^3*d*x^2*e^6 + 1155*B*a^4*b^2*d^2*x
*e^5 + 1540*A*a^3*b^3*d^2*x*e^5 + 105*B*a^4*b^2*d^3*e^4 + 140*A*a^3*b^3*d^3*e^4 + 6160*B*a^5*b*x^2*e^7 + 15400
*A*a^4*b^2*x^2*e^7 + 1232*B*a^5*b*d*x*e^6 + 3080*A*a^4*b^2*d*x*e^6 + 112*B*a^5*b*d^2*e^5 + 280*A*a^4*b^2*d^2*e
^5 + 924*B*a^6*x*e^7 + 5544*A*a^5*b*x*e^7 + 84*B*a^6*d*e^6 + 504*A*a^5*b*d*e^6 + 840*A*a^6*e^7)*e^(-8)/(x*e +
d)^11

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maple [B]  time = 0.01, size = 814, normalized size = 3.46 \begin {gather*} -\frac {B \,b^{6}}{4 \left (e x +d \right )^{4} e^{8}}-\frac {\left (A b e +6 B a e -7 B b d \right ) b^{5}}{5 \left (e x +d \right )^{5} e^{8}}-\frac {\left (2 A a b \,e^{2}-2 A d \,b^{2} e +5 B \,a^{2} e^{2}-12 B a b d e +7 B \,b^{2} d^{2}\right ) b^{4}}{2 \left (e x +d \right )^{6} e^{8}}-\frac {5 \left (3 A \,a^{2} b \,e^{3}-6 A d a \,b^{2} e^{2}+3 A \,b^{3} d^{2} e +4 B \,a^{3} e^{3}-15 B d \,a^{2} b \,e^{2}+18 B a \,b^{2} d^{2} e -7 B \,b^{3} d^{3}\right ) b^{3}}{7 \left (e x +d \right )^{7} e^{8}}-\frac {5 \left (4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A a \,b^{3} d^{2} e^{2}-4 A \,b^{4} d^{3} e +3 B \,a^{4} e^{4}-16 B \,a^{3} b d \,e^{3}+30 B \,a^{2} b^{2} d^{2} e^{2}-24 B a \,b^{3} d^{3} e +7 B \,b^{4} d^{4}\right ) b^{2}}{8 \left (e x +d \right )^{8} e^{8}}-\frac {\left (5 A \,a^{4} b \,e^{5}-20 A \,a^{3} b^{2} d \,e^{4}+30 A \,a^{2} b^{3} d^{2} e^{3}-20 A a \,b^{4} d^{3} e^{2}+5 A \,b^{5} d^{4} e +2 B \,a^{5} e^{5}-15 B \,a^{4} b d \,e^{4}+40 B \,a^{3} b^{2} d^{2} e^{3}-50 B \,a^{2} b^{3} d^{3} e^{2}+30 B a \,b^{4} d^{4} e -7 B \,b^{5} d^{5}\right ) b}{3 \left (e x +d \right )^{9} e^{8}}-\frac {A \,a^{6} e^{7}-6 A d \,a^{5} b \,e^{6}+15 A \,d^{2} a^{4} b^{2} e^{5}-20 A \,d^{3} a^{3} b^{3} e^{4}+15 A \,d^{4} a^{2} b^{4} e^{3}-6 A \,d^{5} a \,b^{5} e^{2}+A \,d^{6} b^{6} e -B d \,a^{6} e^{6}+6 B \,d^{2} a^{5} b \,e^{5}-15 B \,d^{3} a^{4} b^{2} e^{4}+20 B \,d^{4} a^{3} b^{3} e^{3}-15 B \,d^{5} a^{2} b^{4} e^{2}+6 B \,d^{6} a \,b^{5} e -B \,b^{6} d^{7}}{11 \left (e x +d \right )^{11} e^{8}}-\frac {6 a^{5} b A \,e^{6}-30 A d \,a^{4} b^{2} e^{5}+60 A \,d^{2} a^{3} b^{3} e^{4}-60 A \,d^{3} a^{2} b^{4} e^{3}+30 A \,d^{4} a \,b^{5} e^{2}-6 A \,d^{5} b^{6} e +a^{6} B \,e^{6}-12 B d \,a^{5} b \,e^{5}+45 B \,d^{2} a^{4} b^{2} e^{4}-80 B \,d^{3} a^{3} b^{3} e^{3}+75 B \,d^{4} a^{2} b^{4} e^{2}-36 B \,d^{5} a \,b^{5} e +7 B \,b^{6} d^{6}}{10 \left (e x +d \right )^{10} e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^6*(B*x+A)/(e*x+d)^12,x)

[Out]

-1/4*B*b^6/e^8/(e*x+d)^4-1/5*b^5*(A*b*e+6*B*a*e-7*B*b*d)/e^8/(e*x+d)^5-5/8*b^2*(4*A*a^3*b*e^4-12*A*a^2*b^2*d*e
^3+12*A*a*b^3*d^2*e^2-4*A*b^4*d^3*e+3*B*a^4*e^4-16*B*a^3*b*d*e^3+30*B*a^2*b^2*d^2*e^2-24*B*a*b^3*d^3*e+7*B*b^4
*d^4)/e^8/(e*x+d)^8-1/11*(A*a^6*e^7-6*A*a^5*b*d*e^6+15*A*a^4*b^2*d^2*e^5-20*A*a^3*b^3*d^3*e^4+15*A*a^2*b^4*d^4
*e^3-6*A*a*b^5*d^5*e^2+A*b^6*d^6*e-B*a^6*d*e^6+6*B*a^5*b*d^2*e^5-15*B*a^4*b^2*d^3*e^4+20*B*a^3*b^3*d^4*e^3-15*
B*a^2*b^4*d^5*e^2+6*B*a*b^5*d^6*e-B*b^6*d^7)/e^8/(e*x+d)^11-1/10*(6*A*a^5*b*e^6-30*A*a^4*b^2*d*e^5+60*A*a^3*b^
3*d^2*e^4-60*A*a^2*b^4*d^3*e^3+30*A*a*b^5*d^4*e^2-6*A*b^6*d^5*e+B*a^6*e^6-12*B*a^5*b*d*e^5+45*B*a^4*b^2*d^2*e^
4-80*B*a^3*b^3*d^3*e^3+75*B*a^2*b^4*d^4*e^2-36*B*a*b^5*d^5*e+7*B*b^6*d^6)/e^8/(e*x+d)^10-1/3*b*(5*A*a^4*b*e^5-
20*A*a^3*b^2*d*e^4+30*A*a^2*b^3*d^2*e^3-20*A*a*b^4*d^3*e^2+5*A*b^5*d^4*e+2*B*a^5*e^5-15*B*a^4*b*d*e^4+40*B*a^3
*b^2*d^2*e^3-50*B*a^2*b^3*d^3*e^2+30*B*a*b^4*d^4*e-7*B*b^5*d^5)/e^8/(e*x+d)^9-1/2*b^4*(2*A*a*b*e^2-2*A*b^2*d*e
+5*B*a^2*e^2-12*B*a*b*d*e+7*B*b^2*d^2)/e^8/(e*x+d)^6-5/7*b^3*(3*A*a^2*b*e^3-6*A*a*b^2*d*e^2+3*A*b^3*d^2*e+4*B*
a^3*e^3-15*B*a^2*b*d*e^2+18*B*a*b^2*d^2*e-7*B*b^3*d^3)/e^8/(e*x+d)^7

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maxima [B]  time = 0.82, size = 883, normalized size = 3.76 \begin {gather*} -\frac {2310 \, B b^{6} e^{7} x^{7} + 7 \, B b^{6} d^{7} + 840 \, A a^{6} e^{7} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} + 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 35 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} + 56 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + 84 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 462 \, {\left (7 \, B b^{6} d e^{6} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 462 \, {\left (7 \, B b^{6} d^{2} e^{5} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} + 330 \, {\left (7 \, B b^{6} d^{3} e^{4} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} + 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 165 \, {\left (7 \, B b^{6} d^{4} e^{3} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} + 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + 35 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} + 55 \, {\left (7 \, B b^{6} d^{5} e^{2} + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} + 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + 35 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} + 56 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 11 \, {\left (7 \, B b^{6} d^{6} e + 4 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 10 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} + 20 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 35 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} + 56 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + 84 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x}{9240 \, {\left (e^{19} x^{11} + 11 \, d e^{18} x^{10} + 55 \, d^{2} e^{17} x^{9} + 165 \, d^{3} e^{16} x^{8} + 330 \, d^{4} e^{15} x^{7} + 462 \, d^{5} e^{14} x^{6} + 462 \, d^{6} e^{13} x^{5} + 330 \, d^{7} e^{12} x^{4} + 165 \, d^{8} e^{11} x^{3} + 55 \, d^{9} e^{10} x^{2} + 11 \, d^{10} e^{9} x + d^{11} e^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^12,x, algorithm="maxima")

[Out]

-1/9240*(2310*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 840*A*a^6*e^7 + 4*(6*B*a*b^5 + A*b^6)*d^6*e + 10*(5*B*a^2*b^4 + 2*
A*a*b^5)*d^5*e^2 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 35*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 56*(2*B*a
^5*b + 5*A*a^4*b^2)*d^2*e^5 + 84*(B*a^6 + 6*A*a^5*b)*d*e^6 + 462*(7*B*b^6*d*e^6 + 4*(6*B*a*b^5 + A*b^6)*e^7)*x
^6 + 462*(7*B*b^6*d^2*e^5 + 4*(6*B*a*b^5 + A*b^6)*d*e^6 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 330*(7*B*b^6
*d^3*e^4 + 4*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)
*e^7)*x^4 + 165*(7*B*b^6*d^4*e^3 + 4*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + 20*(
4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 35*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 55*(7*B*b^6*d^5*e^2 + 4*(6*B*a*b^
5 + A*b^6)*d^4*e^3 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 35*(3*B*a
^4*b^2 + 4*A*a^3*b^3)*d*e^6 + 56*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 11*(7*B*b^6*d^6*e + 4*(6*B*a*b^5 + A*b^6
)*d^5*e^2 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 + 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 35*(3*B*a^4*b^2 +
4*A*a^3*b^3)*d^2*e^5 + 56*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 84*(B*a^6 + 6*A*a^5*b)*e^7)*x)/(e^19*x^11 + 11*d*e
^18*x^10 + 55*d^2*e^17*x^9 + 165*d^3*e^16*x^8 + 330*d^4*e^15*x^7 + 462*d^5*e^14*x^6 + 462*d^6*e^13*x^5 + 330*d
^7*e^12*x^4 + 165*d^8*e^11*x^3 + 55*d^9*e^10*x^2 + 11*d^10*e^9*x + d^11*e^8)

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mupad [B]  time = 1.30, size = 899, normalized size = 3.83 \begin {gather*} -\frac {\frac {84\,B\,a^6\,d\,e^6+840\,A\,a^6\,e^7+112\,B\,a^5\,b\,d^2\,e^5+504\,A\,a^5\,b\,d\,e^6+105\,B\,a^4\,b^2\,d^3\,e^4+280\,A\,a^4\,b^2\,d^2\,e^5+80\,B\,a^3\,b^3\,d^4\,e^3+140\,A\,a^3\,b^3\,d^3\,e^4+50\,B\,a^2\,b^4\,d^5\,e^2+60\,A\,a^2\,b^4\,d^4\,e^3+24\,B\,a\,b^5\,d^6\,e+20\,A\,a\,b^5\,d^5\,e^2+7\,B\,b^6\,d^7+4\,A\,b^6\,d^6\,e}{9240\,e^8}+\frac {x\,\left (84\,B\,a^6\,e^6+112\,B\,a^5\,b\,d\,e^5+504\,A\,a^5\,b\,e^6+105\,B\,a^4\,b^2\,d^2\,e^4+280\,A\,a^4\,b^2\,d\,e^5+80\,B\,a^3\,b^3\,d^3\,e^3+140\,A\,a^3\,b^3\,d^2\,e^4+50\,B\,a^2\,b^4\,d^4\,e^2+60\,A\,a^2\,b^4\,d^3\,e^3+24\,B\,a\,b^5\,d^5\,e+20\,A\,a\,b^5\,d^4\,e^2+7\,B\,b^6\,d^6+4\,A\,b^6\,d^5\,e\right )}{840\,e^7}+\frac {b^3\,x^4\,\left (80\,B\,a^3\,e^3+50\,B\,a^2\,b\,d\,e^2+60\,A\,a^2\,b\,e^3+24\,B\,a\,b^2\,d^2\,e+20\,A\,a\,b^2\,d\,e^2+7\,B\,b^3\,d^3+4\,A\,b^3\,d^2\,e\right )}{28\,e^4}+\frac {b^5\,x^6\,\left (4\,A\,b\,e+24\,B\,a\,e+7\,B\,b\,d\right )}{20\,e^2}+\frac {b\,x^2\,\left (112\,B\,a^5\,e^5+105\,B\,a^4\,b\,d\,e^4+280\,A\,a^4\,b\,e^5+80\,B\,a^3\,b^2\,d^2\,e^3+140\,A\,a^3\,b^2\,d\,e^4+50\,B\,a^2\,b^3\,d^3\,e^2+60\,A\,a^2\,b^3\,d^2\,e^3+24\,B\,a\,b^4\,d^4\,e+20\,A\,a\,b^4\,d^3\,e^2+7\,B\,b^5\,d^5+4\,A\,b^5\,d^4\,e\right )}{168\,e^6}+\frac {b^2\,x^3\,\left (105\,B\,a^4\,e^4+80\,B\,a^3\,b\,d\,e^3+140\,A\,a^3\,b\,e^4+50\,B\,a^2\,b^2\,d^2\,e^2+60\,A\,a^2\,b^2\,d\,e^3+24\,B\,a\,b^3\,d^3\,e+20\,A\,a\,b^3\,d^2\,e^2+7\,B\,b^4\,d^4+4\,A\,b^4\,d^3\,e\right )}{56\,e^5}+\frac {b^4\,x^5\,\left (50\,B\,a^2\,e^2+24\,B\,a\,b\,d\,e+20\,A\,a\,b\,e^2+7\,B\,b^2\,d^2+4\,A\,b^2\,d\,e\right )}{20\,e^3}+\frac {B\,b^6\,x^7}{4\,e}}{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 + B*x)*(a + b*x)^6)/(d + e*x)^12,x)

[Out]

-((840*A*a^6*e^7 + 7*B*b^6*d^7 + 4*A*b^6*d^6*e + 84*B*a^6*d*e^6 + 20*A*a*b^5*d^5*e^2 + 112*B*a^5*b*d^2*e^5 + 6
0*A*a^2*b^4*d^4*e^3 + 140*A*a^3*b^3*d^3*e^4 + 280*A*a^4*b^2*d^2*e^5 + 50*B*a^2*b^4*d^5*e^2 + 80*B*a^3*b^3*d^4*
e^3 + 105*B*a^4*b^2*d^3*e^4 + 504*A*a^5*b*d*e^6 + 24*B*a*b^5*d^6*e)/(9240*e^8) + (x*(84*B*a^6*e^6 + 7*B*b^6*d^
6 + 504*A*a^5*b*e^6 + 4*A*b^6*d^5*e + 20*A*a*b^5*d^4*e^2 + 280*A*a^4*b^2*d*e^5 + 60*A*a^2*b^4*d^3*e^3 + 140*A*
a^3*b^3*d^2*e^4 + 50*B*a^2*b^4*d^4*e^2 + 80*B*a^3*b^3*d^3*e^3 + 105*B*a^4*b^2*d^2*e^4 + 24*B*a*b^5*d^5*e + 112
*B*a^5*b*d*e^5))/(840*e^7) + (b^3*x^4*(80*B*a^3*e^3 + 7*B*b^3*d^3 + 60*A*a^2*b*e^3 + 4*A*b^3*d^2*e + 20*A*a*b^
2*d*e^2 + 24*B*a*b^2*d^2*e + 50*B*a^2*b*d*e^2))/(28*e^4) + (b^5*x^6*(4*A*b*e + 24*B*a*e + 7*B*b*d))/(20*e^2) +
(b*x^2*(112*B*a^5*e^5 + 7*B*b^5*d^5 + 280*A*a^4*b*e^5 + 4*A*b^5*d^4*e + 20*A*a*b^4*d^3*e^2 + 140*A*a^3*b^2*d*
e^4 + 60*A*a^2*b^3*d^2*e^3 + 50*B*a^2*b^3*d^3*e^2 + 80*B*a^3*b^2*d^2*e^3 + 24*B*a*b^4*d^4*e + 105*B*a^4*b*d*e^
4))/(168*e^6) + (b^2*x^3*(105*B*a^4*e^4 + 7*B*b^4*d^4 + 140*A*a^3*b*e^4 + 4*A*b^4*d^3*e + 20*A*a*b^3*d^2*e^2 +
60*A*a^2*b^2*d*e^3 + 50*B*a^2*b^2*d^2*e^2 + 24*B*a*b^3*d^3*e + 80*B*a^3*b*d*e^3))/(56*e^5) + (b^4*x^5*(50*B*a
^2*e^2 + 7*B*b^2*d^2 + 20*A*a*b*e^2 + 4*A*b^2*d*e + 24*B*a*b*d*e))/(20*e^3) + (B*b^6*x^7)/(4*e))/(d^11 + e^11*
x^11 + 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*x+a)**6*(B*x+A)/(e*x+d)**12,x)

[Out]

Timed out

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