3.10.31 \(\int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{11}} \, dx\)

Optimal. Leaf size=185 \[ \frac {b^2 (a+b x)^7 (-10 a B e+3 A b e+7 b B d)}{2520 e (d+e x)^7 (b d-a e)^4}+\frac {b (a+b x)^7 (-10 a B e+3 A b e+7 b B d)}{360 e (d+e x)^8 (b d-a e)^3}+\frac {(a+b x)^7 (-10 a B e+3 A b e+7 b B d)}{90 e (d+e x)^9 (b d-a e)^2}-\frac {(a+b x)^7 (B d-A e)}{10 e (d+e x)^{10} (b d-a e)} \]

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Rubi [A]  time = 0.08, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 4, 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^2 (a+b x)^7 (-10 a B e+3 A b e+7 b B d)}{2520 e (d+e x)^7 (b d-a e)^4}+\frac {b (a+b x)^7 (-10 a B e+3 A b e+7 b B d)}{360 e (d+e x)^8 (b d-a e)^3}+\frac {(a+b x)^7 (-10 a B e+3 A b e+7 b B d)}{90 e (d+e x)^9 (b d-a e)^2}-\frac {(a+b x)^7 (B d-A e)}{10 e (d+e x)^{10} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^11,x]

[Out]

-((B*d - A*e)*(a + b*x)^7)/(10*e*(b*d - a*e)*(d + e*x)^10) + ((7*b*B*d + 3*A*b*e - 10*a*B*e)*(a + b*x)^7)/(90*
e*(b*d - a*e)^2*(d + e*x)^9) + (b*(7*b*B*d + 3*A*b*e - 10*a*B*e)*(a + b*x)^7)/(360*e*(b*d - a*e)^3*(d + e*x)^8
) + (b^2*(7*b*B*d + 3*A*b*e - 10*a*B*e)*(a + b*x)^7)/(2520*e*(b*d - a*e)^4*(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)^{11}} \, dx &=-\frac {(B d-A e) (a+b x)^7}{10 e (b d-a e) (d+e x)^{10}}+\frac {(7 b B d+3 A b e-10 a B e) \int \frac {(a+b x)^6}{(d+e x)^{10}} \, dx}{10 e (b d-a e)}\\ &=-\frac {(B d-A e) (a+b x)^7}{10 e (b d-a e) (d+e x)^{10}}+\frac {(7 b B d+3 A b e-10 a B e) (a+b x)^7}{90 e (b d-a e)^2 (d+e x)^9}+\frac {(b (7 b B d+3 A b e-10 a B e)) \int \frac {(a+b x)^6}{(d+e x)^9} \, dx}{45 e (b d-a e)^2}\\ &=-\frac {(B d-A e) (a+b x)^7}{10 e (b d-a e) (d+e x)^{10}}+\frac {(7 b B d+3 A b e-10 a B e) (a+b x)^7}{90 e (b d-a e)^2 (d+e x)^9}+\frac {b (7 b B d+3 A b e-10 a B e) (a+b x)^7}{360 e (b d-a e)^3 (d+e x)^8}+\frac {\left (b^2 (7 b B d+3 A b e-10 a B e)\right ) \int \frac {(a+b x)^6}{(d+e x)^8} \, dx}{360 e (b d-a e)^3}\\ &=-\frac {(B d-A e) (a+b x)^7}{10 e (b d-a e) (d+e x)^{10}}+\frac {(7 b B d+3 A b e-10 a B e) (a+b x)^7}{90 e (b d-a e)^2 (d+e x)^9}+\frac {b (7 b B d+3 A b e-10 a B e) (a+b x)^7}{360 e (b d-a e)^3 (d+e x)^8}+\frac {b^2 (7 b B d+3 A b e-10 a B e) (a+b x)^7}{2520 e (b d-a e)^4 (d+e x)^7}\\ \end {align*}

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Mathematica [B]  time = 0.27, size = 602, normalized size = 3.25 \begin {gather*} -\frac {28 a^6 e^6 (9 A e+B (d+10 e x))+42 a^5 b e^5 \left (4 A e (d+10 e x)+B \left (d^2+10 d e x+45 e^2 x^2\right )\right )+15 a^4 b^2 e^4 \left (7 A e \left (d^2+10 d e x+45 e^2 x^2\right )+3 B \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )\right )+20 a^3 b^3 e^3 \left (3 A e \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+2 B \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )\right )+30 a^2 b^4 e^2 \left (A e \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+B \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )\right )+6 a b^5 e \left (2 A e \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )+3 B \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )\right )+b^6 \left (3 A e \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )+7 B \left (d^7+10 d^6 e x+45 d^5 e^2 x^2+120 d^4 e^3 x^3+210 d^3 e^4 x^4+252 d^2 e^5 x^5+210 d e^6 x^6+120 e^7 x^7\right )\right )}{2520 e^8 (d+e x)^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^11,x]

[Out]

-1/2520*(28*a^6*e^6*(9*A*e + B*(d + 10*e*x)) + 42*a^5*b*e^5*(4*A*e*(d + 10*e*x) + B*(d^2 + 10*d*e*x + 45*e^2*x
^2)) + 15*a^4*b^2*e^4*(7*A*e*(d^2 + 10*d*e*x + 45*e^2*x^2) + 3*B*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^
3)) + 20*a^3*b^3*e^3*(3*A*e*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + 2*B*(d^4 + 10*d^3*e*x + 45*d^2*e
^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4)) + 30*a^2*b^4*e^2*(A*e*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^
3 + 210*e^4*x^4) + B*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x^4 + 252*e^5*x^5)) + 6*
a*b^5*e*(2*A*e*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x^4 + 252*e^5*x^5) + 3*B*(d^6
+ 10*d^5*e*x + 45*d^4*e^2*x^2 + 120*d^3*e^3*x^3 + 210*d^2*e^4*x^4 + 252*d*e^5*x^5 + 210*e^6*x^6)) + b^6*(3*A*e
*(d^6 + 10*d^5*e*x + 45*d^4*e^2*x^2 + 120*d^3*e^3*x^3 + 210*d^2*e^4*x^4 + 252*d*e^5*x^5 + 210*e^6*x^6) + 7*B*(
d^7 + 10*d^6*e*x + 45*d^5*e^2*x^2 + 120*d^4*e^3*x^3 + 210*d^3*e^4*x^4 + 252*d^2*e^5*x^5 + 210*d*e^6*x^6 + 120*
e^7*x^7)))/(e^8*(d + e*x)^10)

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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)^{11}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)^6*(A + B*x))/(d + e*x)^11,x]

[Out]

IntegrateAlgebraic[((a + b*x)^6*(A + B*x))/(d + e*x)^11, x]

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fricas [B]  time = 1.36, size = 872, normalized size = 4.71 \begin {gather*} -\frac {840 \, B b^{6} e^{7} x^{7} + 7 \, B b^{6} d^{7} + 252 \, A a^{6} e^{7} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} + 21 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + 28 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 210 \, {\left (7 \, B b^{6} d e^{6} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 252 \, {\left (7 \, B b^{6} d^{2} e^{5} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} + 210 \, {\left (7 \, B b^{6} d^{3} e^{4} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 120 \, {\left (7 \, B b^{6} d^{4} e^{3} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} + 45 \, {\left (7 \, B b^{6} d^{5} e^{2} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} + 21 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 10 \, {\left (7 \, B b^{6} d^{6} e + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} + 21 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + 28 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x}{2520 \, {\left (e^{18} x^{10} + 10 \, d e^{17} x^{9} + 45 \, d^{2} e^{16} x^{8} + 120 \, d^{3} e^{15} x^{7} + 210 \, d^{4} e^{14} x^{6} + 252 \, d^{5} e^{13} x^{5} + 210 \, d^{6} e^{12} x^{4} + 120 \, d^{7} e^{11} x^{3} + 45 \, d^{8} e^{10} x^{2} + 10 \, d^{9} e^{9} x + d^{10} 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)^11,x, algorithm="fricas")

[Out]

-1/2520*(840*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 252*A*a^6*e^7 + 3*(6*B*a*b^5 + A*b^6)*d^6*e + 6*(5*B*a^2*b^4 + 2*A*
a*b^5)*d^5*e^2 + 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 21*(2*B*a^5
*b + 5*A*a^4*b^2)*d^2*e^5 + 28*(B*a^6 + 6*A*a^5*b)*d*e^6 + 210*(7*B*b^6*d*e^6 + 3*(6*B*a*b^5 + A*b^6)*e^7)*x^6
+ 252*(7*B*b^6*d^2*e^5 + 3*(6*B*a*b^5 + A*b^6)*d*e^6 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 210*(7*B*b^6*d^
3*e^4 + 3*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 + 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7
)*x^4 + 120*(7*B*b^6*d^4*e^3 + 3*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + 10*(4*B*a
^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 45*(7*B*b^6*d^5*e^2 + 3*(6*B*a*b^5 + A
*b^6)*d^4*e^3 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 + 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 15*(3*B*a^4*b^2
+ 4*A*a^3*b^3)*d*e^6 + 21*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 10*(7*B*b^6*d^6*e + 3*(6*B*a*b^5 + A*b^6)*d^5*
e^2 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 + 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 15*(3*B*a^4*b^2 + 4*A*a^3
*b^3)*d^2*e^5 + 21*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 28*(B*a^6 + 6*A*a^5*b)*e^7)*x)/(e^18*x^10 + 10*d*e^17*x^9
+ 45*d^2*e^16*x^8 + 120*d^3*e^15*x^7 + 210*d^4*e^14*x^6 + 252*d^5*e^13*x^5 + 210*d^6*e^12*x^4 + 120*d^7*e^11*
x^3 + 45*d^8*e^10*x^2 + 10*d^9*e^9*x + d^10*e^8)

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giac [B]  time = 1.26, size = 856, normalized size = 4.63 \begin {gather*} -\frac {{\left (840 \, B b^{6} x^{7} e^{7} + 1470 \, B b^{6} d x^{6} e^{6} + 1764 \, B b^{6} d^{2} x^{5} e^{5} + 1470 \, B b^{6} d^{3} x^{4} e^{4} + 840 \, B b^{6} d^{4} x^{3} e^{3} + 315 \, B b^{6} d^{5} x^{2} e^{2} + 70 \, B b^{6} d^{6} x e + 7 \, B b^{6} d^{7} + 3780 \, B a b^{5} x^{6} e^{7} + 630 \, A b^{6} x^{6} e^{7} + 4536 \, B a b^{5} d x^{5} e^{6} + 756 \, A b^{6} d x^{5} e^{6} + 3780 \, B a b^{5} d^{2} x^{4} e^{5} + 630 \, A b^{6} d^{2} x^{4} e^{5} + 2160 \, B a b^{5} d^{3} x^{3} e^{4} + 360 \, A b^{6} d^{3} x^{3} e^{4} + 810 \, B a b^{5} d^{4} x^{2} e^{3} + 135 \, A b^{6} d^{4} x^{2} e^{3} + 180 \, B a b^{5} d^{5} x e^{2} + 30 \, A b^{6} d^{5} x e^{2} + 18 \, B a b^{5} d^{6} e + 3 \, A b^{6} d^{6} e + 7560 \, B a^{2} b^{4} x^{5} e^{7} + 3024 \, A a b^{5} x^{5} e^{7} + 6300 \, B a^{2} b^{4} d x^{4} e^{6} + 2520 \, A a b^{5} d x^{4} e^{6} + 3600 \, B a^{2} b^{4} d^{2} x^{3} e^{5} + 1440 \, A a b^{5} d^{2} x^{3} e^{5} + 1350 \, B a^{2} b^{4} d^{3} x^{2} e^{4} + 540 \, A a b^{5} d^{3} x^{2} e^{4} + 300 \, B a^{2} b^{4} d^{4} x e^{3} + 120 \, A a b^{5} d^{4} x e^{3} + 30 \, B a^{2} b^{4} d^{5} e^{2} + 12 \, A a b^{5} d^{5} e^{2} + 8400 \, B a^{3} b^{3} x^{4} e^{7} + 6300 \, A a^{2} b^{4} x^{4} e^{7} + 4800 \, B a^{3} b^{3} d x^{3} e^{6} + 3600 \, A a^{2} b^{4} d x^{3} e^{6} + 1800 \, B a^{3} b^{3} d^{2} x^{2} e^{5} + 1350 \, A a^{2} b^{4} d^{2} x^{2} e^{5} + 400 \, B a^{3} b^{3} d^{3} x e^{4} + 300 \, A a^{2} b^{4} d^{3} x e^{4} + 40 \, B a^{3} b^{3} d^{4} e^{3} + 30 \, A a^{2} b^{4} d^{4} e^{3} + 5400 \, B a^{4} b^{2} x^{3} e^{7} + 7200 \, A a^{3} b^{3} x^{3} e^{7} + 2025 \, B a^{4} b^{2} d x^{2} e^{6} + 2700 \, A a^{3} b^{3} d x^{2} e^{6} + 450 \, B a^{4} b^{2} d^{2} x e^{5} + 600 \, A a^{3} b^{3} d^{2} x e^{5} + 45 \, B a^{4} b^{2} d^{3} e^{4} + 60 \, A a^{3} b^{3} d^{3} e^{4} + 1890 \, B a^{5} b x^{2} e^{7} + 4725 \, A a^{4} b^{2} x^{2} e^{7} + 420 \, B a^{5} b d x e^{6} + 1050 \, A a^{4} b^{2} d x e^{6} + 42 \, B a^{5} b d^{2} e^{5} + 105 \, A a^{4} b^{2} d^{2} e^{5} + 280 \, B a^{6} x e^{7} + 1680 \, A a^{5} b x e^{7} + 28 \, B a^{6} d e^{6} + 168 \, A a^{5} b d e^{6} + 252 \, A a^{6} e^{7}\right )} e^{\left (-8\right )}}{2520 \, {\left (x e + d\right )}^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^11,x, algorithm="giac")

[Out]

-1/2520*(840*B*b^6*x^7*e^7 + 1470*B*b^6*d*x^6*e^6 + 1764*B*b^6*d^2*x^5*e^5 + 1470*B*b^6*d^3*x^4*e^4 + 840*B*b^
6*d^4*x^3*e^3 + 315*B*b^6*d^5*x^2*e^2 + 70*B*b^6*d^6*x*e + 7*B*b^6*d^7 + 3780*B*a*b^5*x^6*e^7 + 630*A*b^6*x^6*
e^7 + 4536*B*a*b^5*d*x^5*e^6 + 756*A*b^6*d*x^5*e^6 + 3780*B*a*b^5*d^2*x^4*e^5 + 630*A*b^6*d^2*x^4*e^5 + 2160*B
*a*b^5*d^3*x^3*e^4 + 360*A*b^6*d^3*x^3*e^4 + 810*B*a*b^5*d^4*x^2*e^3 + 135*A*b^6*d^4*x^2*e^3 + 180*B*a*b^5*d^5
*x*e^2 + 30*A*b^6*d^5*x*e^2 + 18*B*a*b^5*d^6*e + 3*A*b^6*d^6*e + 7560*B*a^2*b^4*x^5*e^7 + 3024*A*a*b^5*x^5*e^7
+ 6300*B*a^2*b^4*d*x^4*e^6 + 2520*A*a*b^5*d*x^4*e^6 + 3600*B*a^2*b^4*d^2*x^3*e^5 + 1440*A*a*b^5*d^2*x^3*e^5 +
1350*B*a^2*b^4*d^3*x^2*e^4 + 540*A*a*b^5*d^3*x^2*e^4 + 300*B*a^2*b^4*d^4*x*e^3 + 120*A*a*b^5*d^4*x*e^3 + 30*B
*a^2*b^4*d^5*e^2 + 12*A*a*b^5*d^5*e^2 + 8400*B*a^3*b^3*x^4*e^7 + 6300*A*a^2*b^4*x^4*e^7 + 4800*B*a^3*b^3*d*x^3
*e^6 + 3600*A*a^2*b^4*d*x^3*e^6 + 1800*B*a^3*b^3*d^2*x^2*e^5 + 1350*A*a^2*b^4*d^2*x^2*e^5 + 400*B*a^3*b^3*d^3*
x*e^4 + 300*A*a^2*b^4*d^3*x*e^4 + 40*B*a^3*b^3*d^4*e^3 + 30*A*a^2*b^4*d^4*e^3 + 5400*B*a^4*b^2*x^3*e^7 + 7200*
A*a^3*b^3*x^3*e^7 + 2025*B*a^4*b^2*d*x^2*e^6 + 2700*A*a^3*b^3*d*x^2*e^6 + 450*B*a^4*b^2*d^2*x*e^5 + 600*A*a^3*
b^3*d^2*x*e^5 + 45*B*a^4*b^2*d^3*e^4 + 60*A*a^3*b^3*d^3*e^4 + 1890*B*a^5*b*x^2*e^7 + 4725*A*a^4*b^2*x^2*e^7 +
420*B*a^5*b*d*x*e^6 + 1050*A*a^4*b^2*d*x*e^6 + 42*B*a^5*b*d^2*e^5 + 105*A*a^4*b^2*d^2*e^5 + 280*B*a^6*x*e^7 +
1680*A*a^5*b*x*e^7 + 28*B*a^6*d*e^6 + 168*A*a^5*b*d*e^6 + 252*A*a^6*e^7)*e^(-8)/(x*e + d)^10

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maple [B]  time = 0.01, size = 814, normalized size = 4.40 \begin {gather*} -\frac {B \,b^{6}}{3 \left (e x +d \right )^{3} e^{8}}-\frac {\left (A b e +6 B a e -7 B b d \right ) b^{5}}{4 \left (e x +d \right )^{4} e^{8}}-\frac {3 \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}}{5 \left (e x +d \right )^{5} 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}}{6 \left (e x +d \right )^{6} 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}}{7 \left (e x +d \right )^{7} e^{8}}-\frac {3 \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}{8 \left (e x +d \right )^{8} 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}}{10 \left (e x +d \right )^{10} 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}}{9 \left (e x +d \right )^{9} 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)^11,x)

[Out]

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

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maxima [B]  time = 0.79, size = 872, normalized size = 4.71 \begin {gather*} -\frac {840 \, B b^{6} e^{7} x^{7} + 7 \, B b^{6} d^{7} + 252 \, A a^{6} e^{7} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} + 21 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + 28 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 210 \, {\left (7 \, B b^{6} d e^{6} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 252 \, {\left (7 \, B b^{6} d^{2} e^{5} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} + 210 \, {\left (7 \, B b^{6} d^{3} e^{4} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 120 \, {\left (7 \, B b^{6} d^{4} e^{3} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} + 45 \, {\left (7 \, B b^{6} d^{5} e^{2} + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} + 21 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 10 \, {\left (7 \, B b^{6} d^{6} e + 3 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} + 10 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 15 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} + 21 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + 28 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x}{2520 \, {\left (e^{18} x^{10} + 10 \, d e^{17} x^{9} + 45 \, d^{2} e^{16} x^{8} + 120 \, d^{3} e^{15} x^{7} + 210 \, d^{4} e^{14} x^{6} + 252 \, d^{5} e^{13} x^{5} + 210 \, d^{6} e^{12} x^{4} + 120 \, d^{7} e^{11} x^{3} + 45 \, d^{8} e^{10} x^{2} + 10 \, d^{9} e^{9} x + d^{10} 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)^11,x, algorithm="maxima")

[Out]

-1/2520*(840*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 252*A*a^6*e^7 + 3*(6*B*a*b^5 + A*b^6)*d^6*e + 6*(5*B*a^2*b^4 + 2*A*
a*b^5)*d^5*e^2 + 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 21*(2*B*a^5
*b + 5*A*a^4*b^2)*d^2*e^5 + 28*(B*a^6 + 6*A*a^5*b)*d*e^6 + 210*(7*B*b^6*d*e^6 + 3*(6*B*a*b^5 + A*b^6)*e^7)*x^6
+ 252*(7*B*b^6*d^2*e^5 + 3*(6*B*a*b^5 + A*b^6)*d*e^6 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 210*(7*B*b^6*d^
3*e^4 + 3*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 + 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7
)*x^4 + 120*(7*B*b^6*d^4*e^3 + 3*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + 10*(4*B*a
^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 45*(7*B*b^6*d^5*e^2 + 3*(6*B*a*b^5 + A
*b^6)*d^4*e^3 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 + 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 15*(3*B*a^4*b^2
+ 4*A*a^3*b^3)*d*e^6 + 21*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 10*(7*B*b^6*d^6*e + 3*(6*B*a*b^5 + A*b^6)*d^5*
e^2 + 6*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 + 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 15*(3*B*a^4*b^2 + 4*A*a^3
*b^3)*d^2*e^5 + 21*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 28*(B*a^6 + 6*A*a^5*b)*e^7)*x)/(e^18*x^10 + 10*d*e^17*x^9
+ 45*d^2*e^16*x^8 + 120*d^3*e^15*x^7 + 210*d^4*e^14*x^6 + 252*d^5*e^13*x^5 + 210*d^6*e^12*x^4 + 120*d^7*e^11*
x^3 + 45*d^8*e^10*x^2 + 10*d^9*e^9*x + d^10*e^8)

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mupad [B]  time = 1.39, size = 888, normalized size = 4.80 \begin {gather*} -\frac {\frac {28\,B\,a^6\,d\,e^6+252\,A\,a^6\,e^7+42\,B\,a^5\,b\,d^2\,e^5+168\,A\,a^5\,b\,d\,e^6+45\,B\,a^4\,b^2\,d^3\,e^4+105\,A\,a^4\,b^2\,d^2\,e^5+40\,B\,a^3\,b^3\,d^4\,e^3+60\,A\,a^3\,b^3\,d^3\,e^4+30\,B\,a^2\,b^4\,d^5\,e^2+30\,A\,a^2\,b^4\,d^4\,e^3+18\,B\,a\,b^5\,d^6\,e+12\,A\,a\,b^5\,d^5\,e^2+7\,B\,b^6\,d^7+3\,A\,b^6\,d^6\,e}{2520\,e^8}+\frac {x\,\left (28\,B\,a^6\,e^6+42\,B\,a^5\,b\,d\,e^5+168\,A\,a^5\,b\,e^6+45\,B\,a^4\,b^2\,d^2\,e^4+105\,A\,a^4\,b^2\,d\,e^5+40\,B\,a^3\,b^3\,d^3\,e^3+60\,A\,a^3\,b^3\,d^2\,e^4+30\,B\,a^2\,b^4\,d^4\,e^2+30\,A\,a^2\,b^4\,d^3\,e^3+18\,B\,a\,b^5\,d^5\,e+12\,A\,a\,b^5\,d^4\,e^2+7\,B\,b^6\,d^6+3\,A\,b^6\,d^5\,e\right )}{252\,e^7}+\frac {b^3\,x^4\,\left (40\,B\,a^3\,e^3+30\,B\,a^2\,b\,d\,e^2+30\,A\,a^2\,b\,e^3+18\,B\,a\,b^2\,d^2\,e+12\,A\,a\,b^2\,d\,e^2+7\,B\,b^3\,d^3+3\,A\,b^3\,d^2\,e\right )}{12\,e^4}+\frac {b^5\,x^6\,\left (3\,A\,b\,e+18\,B\,a\,e+7\,B\,b\,d\right )}{12\,e^2}+\frac {b\,x^2\,\left (42\,B\,a^5\,e^5+45\,B\,a^4\,b\,d\,e^4+105\,A\,a^4\,b\,e^5+40\,B\,a^3\,b^2\,d^2\,e^3+60\,A\,a^3\,b^2\,d\,e^4+30\,B\,a^2\,b^3\,d^3\,e^2+30\,A\,a^2\,b^3\,d^2\,e^3+18\,B\,a\,b^4\,d^4\,e+12\,A\,a\,b^4\,d^3\,e^2+7\,B\,b^5\,d^5+3\,A\,b^5\,d^4\,e\right )}{56\,e^6}+\frac {b^2\,x^3\,\left (45\,B\,a^4\,e^4+40\,B\,a^3\,b\,d\,e^3+60\,A\,a^3\,b\,e^4+30\,B\,a^2\,b^2\,d^2\,e^2+30\,A\,a^2\,b^2\,d\,e^3+18\,B\,a\,b^3\,d^3\,e+12\,A\,a\,b^3\,d^2\,e^2+7\,B\,b^4\,d^4+3\,A\,b^4\,d^3\,e\right )}{21\,e^5}+\frac {b^4\,x^5\,\left (30\,B\,a^2\,e^2+18\,B\,a\,b\,d\,e+12\,A\,a\,b\,e^2+7\,B\,b^2\,d^2+3\,A\,b^2\,d\,e\right )}{10\,e^3}+\frac {B\,b^6\,x^7}{3\,e}}{d^{10}+10\,d^9\,e\,x+45\,d^8\,e^2\,x^2+120\,d^7\,e^3\,x^3+210\,d^6\,e^4\,x^4+252\,d^5\,e^5\,x^5+210\,d^4\,e^6\,x^6+120\,d^3\,e^7\,x^7+45\,d^2\,e^8\,x^8+10\,d\,e^9\,x^9+e^{10}\,x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^6)/(d + e*x)^11,x)

[Out]

-((252*A*a^6*e^7 + 7*B*b^6*d^7 + 3*A*b^6*d^6*e + 28*B*a^6*d*e^6 + 12*A*a*b^5*d^5*e^2 + 42*B*a^5*b*d^2*e^5 + 30
*A*a^2*b^4*d^4*e^3 + 60*A*a^3*b^3*d^3*e^4 + 105*A*a^4*b^2*d^2*e^5 + 30*B*a^2*b^4*d^5*e^2 + 40*B*a^3*b^3*d^4*e^
3 + 45*B*a^4*b^2*d^3*e^4 + 168*A*a^5*b*d*e^6 + 18*B*a*b^5*d^6*e)/(2520*e^8) + (x*(28*B*a^6*e^6 + 7*B*b^6*d^6 +
168*A*a^5*b*e^6 + 3*A*b^6*d^5*e + 12*A*a*b^5*d^4*e^2 + 105*A*a^4*b^2*d*e^5 + 30*A*a^2*b^4*d^3*e^3 + 60*A*a^3*
b^3*d^2*e^4 + 30*B*a^2*b^4*d^4*e^2 + 40*B*a^3*b^3*d^3*e^3 + 45*B*a^4*b^2*d^2*e^4 + 18*B*a*b^5*d^5*e + 42*B*a^5
*b*d*e^5))/(252*e^7) + (b^3*x^4*(40*B*a^3*e^3 + 7*B*b^3*d^3 + 30*A*a^2*b*e^3 + 3*A*b^3*d^2*e + 12*A*a*b^2*d*e^
2 + 18*B*a*b^2*d^2*e + 30*B*a^2*b*d*e^2))/(12*e^4) + (b^5*x^6*(3*A*b*e + 18*B*a*e + 7*B*b*d))/(12*e^2) + (b*x^
2*(42*B*a^5*e^5 + 7*B*b^5*d^5 + 105*A*a^4*b*e^5 + 3*A*b^5*d^4*e + 12*A*a*b^4*d^3*e^2 + 60*A*a^3*b^2*d*e^4 + 30
*A*a^2*b^3*d^2*e^3 + 30*B*a^2*b^3*d^3*e^2 + 40*B*a^3*b^2*d^2*e^3 + 18*B*a*b^4*d^4*e + 45*B*a^4*b*d*e^4))/(56*e
^6) + (b^2*x^3*(45*B*a^4*e^4 + 7*B*b^4*d^4 + 60*A*a^3*b*e^4 + 3*A*b^4*d^3*e + 12*A*a*b^3*d^2*e^2 + 30*A*a^2*b^
2*d*e^3 + 30*B*a^2*b^2*d^2*e^2 + 18*B*a*b^3*d^3*e + 40*B*a^3*b*d*e^3))/(21*e^5) + (b^4*x^5*(30*B*a^2*e^2 + 7*B
*b^2*d^2 + 12*A*a*b*e^2 + 3*A*b^2*d*e + 18*B*a*b*d*e))/(10*e^3) + (B*b^6*x^7)/(3*e))/(d^10 + e^10*x^10 + 10*d*
e^9*x^9 + 45*d^8*e^2*x^2 + 120*d^7*e^3*x^3 + 210*d^6*e^4*x^4 + 252*d^5*e^5*x^5 + 210*d^4*e^6*x^6 + 120*d^3*e^7
*x^7 + 45*d^2*e^8*x^8 + 10*d^9*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)**11,x)

[Out]

Timed out

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