3.10.30 \(\int \frac {(a+b x)^6 (A+B x)}{(d+e x)^{10}} \, dx\)

Optimal. Leaf size=135 \[ \frac {b (a+b x)^7 (-9 a B e+2 A b e+7 b B d)}{504 e (d+e x)^7 (b d-a e)^3}+\frac {(a+b x)^7 (-9 a B e+2 A b e+7 b B d)}{72 e (d+e x)^8 (b d-a e)^2}-\frac {(a+b x)^7 (B d-A e)}{9 e (d+e x)^9 (b d-a e)} \]

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Rubi [A]  time = 0.06, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 3, 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 (a+b x)^7 (-9 a B e+2 A b e+7 b B d)}{504 e (d+e x)^7 (b d-a e)^3}+\frac {(a+b x)^7 (-9 a B e+2 A b e+7 b B d)}{72 e (d+e x)^8 (b d-a e)^2}-\frac {(a+b x)^7 (B d-A e)}{9 e (d+e x)^9 (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^6*(A + B*x))/(d + e*x)^10,x]

[Out]

-((B*d - A*e)*(a + b*x)^7)/(9*e*(b*d - a*e)*(d + e*x)^9) + ((7*b*B*d + 2*A*b*e - 9*a*B*e)*(a + b*x)^7)/(72*e*(
b*d - a*e)^2*(d + e*x)^8) + (b*(7*b*B*d + 2*A*b*e - 9*a*B*e)*(a + b*x)^7)/(504*e*(b*d - a*e)^3*(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)^{10}} \, dx &=-\frac {(B d-A e) (a+b x)^7}{9 e (b d-a e) (d+e x)^9}+\frac {(7 b B d+2 A b e-9 a B e) \int \frac {(a+b x)^6}{(d+e x)^9} \, dx}{9 e (b d-a e)}\\ &=-\frac {(B d-A e) (a+b x)^7}{9 e (b d-a e) (d+e x)^9}+\frac {(7 b B d+2 A b e-9 a B e) (a+b x)^7}{72 e (b d-a e)^2 (d+e x)^8}+\frac {(b (7 b B d+2 A b e-9 a B e)) \int \frac {(a+b x)^6}{(d+e x)^8} \, dx}{72 e (b d-a e)^2}\\ &=-\frac {(B d-A e) (a+b x)^7}{9 e (b d-a e) (d+e x)^9}+\frac {(7 b B d+2 A b e-9 a B e) (a+b x)^7}{72 e (b d-a e)^2 (d+e x)^8}+\frac {b (7 b B d+2 A b e-9 a B e) (a+b x)^7}{504 e (b d-a e)^3 (d+e x)^7}\\ \end {align*}

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Mathematica [B]  time = 0.28, size = 603, normalized size = 4.47 \begin {gather*} -\frac {7 a^6 e^6 (8 A e+B (d+9 e x))+6 a^5 b e^5 \left (7 A e (d+9 e x)+2 B \left (d^2+9 d e x+36 e^2 x^2\right )\right )+15 a^4 b^2 e^4 \left (2 A e \left (d^2+9 d e x+36 e^2 x^2\right )+B \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )\right )+4 a^3 b^3 e^3 \left (5 A e \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+4 B \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )\right )+3 a^2 b^4 e^2 \left (4 A e \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+5 B \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )\right )+6 a b^5 e \left (A e \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )+2 B \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )\right )+b^6 \left (2 A e \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )+7 B \left (d^7+9 d^6 e x+36 d^5 e^2 x^2+84 d^4 e^3 x^3+126 d^3 e^4 x^4+126 d^2 e^5 x^5+84 d e^6 x^6+36 e^7 x^7\right )\right )}{504 e^8 (d+e x)^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^10,x]

[Out]

-1/504*(7*a^6*e^6*(8*A*e + B*(d + 9*e*x)) + 6*a^5*b*e^5*(7*A*e*(d + 9*e*x) + 2*B*(d^2 + 9*d*e*x + 36*e^2*x^2))
+ 15*a^4*b^2*e^4*(2*A*e*(d^2 + 9*d*e*x + 36*e^2*x^2) + B*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3)) + 4*a
^3*b^3*e^3*(5*A*e*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 4*B*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d
*e^3*x^3 + 126*e^4*x^4)) + 3*a^2*b^4*e^2*(4*A*e*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4
) + 5*B*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5)) + 6*a*b^5*e*(A*e*(d
^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5) + 2*B*(d^6 + 9*d^5*e*x + 36*d^
4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6)) + b^6*(2*A*e*(d^6 + 9*d^5*e*x + 36
*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6) + 7*B*(d^7 + 9*d^6*e*x + 36*d^5*
e^2*x^2 + 84*d^4*e^3*x^3 + 126*d^3*e^4*x^4 + 126*d^2*e^5*x^5 + 84*d*e^6*x^6 + 36*e^7*x^7)))/(e^8*(d + e*x)^9)

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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)^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)^6*(A + B*x))/(d + e*x)^10,x]

[Out]

IntegrateAlgebraic[((a + b*x)^6*(A + B*x))/(d + e*x)^10, x]

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fricas [B]  time = 1.38, size = 861, normalized size = 6.38 \begin {gather*} -\frac {252 \, B b^{6} e^{7} x^{7} + 7 \, B b^{6} d^{7} + 56 \, A a^{6} e^{7} + 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} + 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} + 6 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + 7 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 84 \, {\left (7 \, B b^{6} d e^{6} + 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 126 \, {\left (7 \, B b^{6} d^{2} e^{5} + 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} + 126 \, {\left (7 \, B b^{6} d^{3} e^{4} + 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} + 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 84 \, {\left (7 \, B b^{6} d^{4} e^{3} + 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} + 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} + 36 \, {\left (7 \, B b^{6} d^{5} e^{2} + 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} + 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} + 6 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 9 \, {\left (7 \, B b^{6} d^{6} e + 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} + 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} + 6 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + 7 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x}{504 \, {\left (e^{17} x^{9} + 9 \, d e^{16} x^{8} + 36 \, d^{2} e^{15} x^{7} + 84 \, d^{3} e^{14} x^{6} + 126 \, d^{4} e^{13} x^{5} + 126 \, d^{5} e^{12} x^{4} + 84 \, d^{6} e^{11} x^{3} + 36 \, d^{7} e^{10} x^{2} + 9 \, d^{8} e^{9} x + d^{9} 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)^10,x, algorithm="fricas")

[Out]

-1/504*(252*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 56*A*a^6*e^7 + 2*(6*B*a*b^5 + A*b^6)*d^6*e + 3*(5*B*a^2*b^4 + 2*A*a*
b^5)*d^5*e^2 + 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 6*(2*B*a^5*b +
5*A*a^4*b^2)*d^2*e^5 + 7*(B*a^6 + 6*A*a^5*b)*d*e^6 + 84*(7*B*b^6*d*e^6 + 2*(6*B*a*b^5 + A*b^6)*e^7)*x^6 + 126*
(7*B*b^6*d^2*e^5 + 2*(6*B*a*b^5 + A*b^6)*d*e^6 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 126*(7*B*b^6*d^3*e^4 +
2*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 + 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 +
84*(7*B*b^6*d^4*e^3 + 2*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + 4*(4*B*a^3*b^3 + 3
*A*a^2*b^4)*d*e^6 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 36*(7*B*b^6*d^5*e^2 + 2*(6*B*a*b^5 + A*b^6)*d^4*e
^3 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 + 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^
3)*d*e^6 + 6*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 9*(7*B*b^6*d^6*e + 2*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 3*(5*B*a^
2*b^4 + 2*A*a*b^5)*d^4*e^3 + 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 + 6
*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 7*(B*a^6 + 6*A*a^5*b)*e^7)*x)/(e^17*x^9 + 9*d*e^16*x^8 + 36*d^2*e^15*x^7 +
84*d^3*e^14*x^6 + 126*d^4*e^13*x^5 + 126*d^5*e^12*x^4 + 84*d^6*e^11*x^3 + 36*d^7*e^10*x^2 + 9*d^8*e^9*x + d^9*
e^8)

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giac [B]  time = 1.22, size = 856, normalized size = 6.34 \begin {gather*} -\frac {{\left (252 \, B b^{6} x^{7} e^{7} + 588 \, B b^{6} d x^{6} e^{6} + 882 \, B b^{6} d^{2} x^{5} e^{5} + 882 \, B b^{6} d^{3} x^{4} e^{4} + 588 \, B b^{6} d^{4} x^{3} e^{3} + 252 \, B b^{6} d^{5} x^{2} e^{2} + 63 \, B b^{6} d^{6} x e + 7 \, B b^{6} d^{7} + 1008 \, B a b^{5} x^{6} e^{7} + 168 \, A b^{6} x^{6} e^{7} + 1512 \, B a b^{5} d x^{5} e^{6} + 252 \, A b^{6} d x^{5} e^{6} + 1512 \, B a b^{5} d^{2} x^{4} e^{5} + 252 \, A b^{6} d^{2} x^{4} e^{5} + 1008 \, B a b^{5} d^{3} x^{3} e^{4} + 168 \, A b^{6} d^{3} x^{3} e^{4} + 432 \, B a b^{5} d^{4} x^{2} e^{3} + 72 \, A b^{6} d^{4} x^{2} e^{3} + 108 \, B a b^{5} d^{5} x e^{2} + 18 \, A b^{6} d^{5} x e^{2} + 12 \, B a b^{5} d^{6} e + 2 \, A b^{6} d^{6} e + 1890 \, B a^{2} b^{4} x^{5} e^{7} + 756 \, A a b^{5} x^{5} e^{7} + 1890 \, B a^{2} b^{4} d x^{4} e^{6} + 756 \, A a b^{5} d x^{4} e^{6} + 1260 \, B a^{2} b^{4} d^{2} x^{3} e^{5} + 504 \, A a b^{5} d^{2} x^{3} e^{5} + 540 \, B a^{2} b^{4} d^{3} x^{2} e^{4} + 216 \, A a b^{5} d^{3} x^{2} e^{4} + 135 \, B a^{2} b^{4} d^{4} x e^{3} + 54 \, A a b^{5} d^{4} x e^{3} + 15 \, B a^{2} b^{4} d^{5} e^{2} + 6 \, A a b^{5} d^{5} e^{2} + 2016 \, B a^{3} b^{3} x^{4} e^{7} + 1512 \, A a^{2} b^{4} x^{4} e^{7} + 1344 \, B a^{3} b^{3} d x^{3} e^{6} + 1008 \, A a^{2} b^{4} d x^{3} e^{6} + 576 \, B a^{3} b^{3} d^{2} x^{2} e^{5} + 432 \, A a^{2} b^{4} d^{2} x^{2} e^{5} + 144 \, B a^{3} b^{3} d^{3} x e^{4} + 108 \, A a^{2} b^{4} d^{3} x e^{4} + 16 \, B a^{3} b^{3} d^{4} e^{3} + 12 \, A a^{2} b^{4} d^{4} e^{3} + 1260 \, B a^{4} b^{2} x^{3} e^{7} + 1680 \, A a^{3} b^{3} x^{3} e^{7} + 540 \, B a^{4} b^{2} d x^{2} e^{6} + 720 \, A a^{3} b^{3} d x^{2} e^{6} + 135 \, B a^{4} b^{2} d^{2} x e^{5} + 180 \, A a^{3} b^{3} d^{2} x e^{5} + 15 \, B a^{4} b^{2} d^{3} e^{4} + 20 \, A a^{3} b^{3} d^{3} e^{4} + 432 \, B a^{5} b x^{2} e^{7} + 1080 \, A a^{4} b^{2} x^{2} e^{7} + 108 \, B a^{5} b d x e^{6} + 270 \, A a^{4} b^{2} d x e^{6} + 12 \, B a^{5} b d^{2} e^{5} + 30 \, A a^{4} b^{2} d^{2} e^{5} + 63 \, B a^{6} x e^{7} + 378 \, A a^{5} b x e^{7} + 7 \, B a^{6} d e^{6} + 42 \, A a^{5} b d e^{6} + 56 \, A a^{6} e^{7}\right )} e^{\left (-8\right )}}{504 \, {\left (x e + d\right )}^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)/(e*x+d)^10,x, algorithm="giac")

[Out]

-1/504*(252*B*b^6*x^7*e^7 + 588*B*b^6*d*x^6*e^6 + 882*B*b^6*d^2*x^5*e^5 + 882*B*b^6*d^3*x^4*e^4 + 588*B*b^6*d^
4*x^3*e^3 + 252*B*b^6*d^5*x^2*e^2 + 63*B*b^6*d^6*x*e + 7*B*b^6*d^7 + 1008*B*a*b^5*x^6*e^7 + 168*A*b^6*x^6*e^7
+ 1512*B*a*b^5*d*x^5*e^6 + 252*A*b^6*d*x^5*e^6 + 1512*B*a*b^5*d^2*x^4*e^5 + 252*A*b^6*d^2*x^4*e^5 + 1008*B*a*b
^5*d^3*x^3*e^4 + 168*A*b^6*d^3*x^3*e^4 + 432*B*a*b^5*d^4*x^2*e^3 + 72*A*b^6*d^4*x^2*e^3 + 108*B*a*b^5*d^5*x*e^
2 + 18*A*b^6*d^5*x*e^2 + 12*B*a*b^5*d^6*e + 2*A*b^6*d^6*e + 1890*B*a^2*b^4*x^5*e^7 + 756*A*a*b^5*x^5*e^7 + 189
0*B*a^2*b^4*d*x^4*e^6 + 756*A*a*b^5*d*x^4*e^6 + 1260*B*a^2*b^4*d^2*x^3*e^5 + 504*A*a*b^5*d^2*x^3*e^5 + 540*B*a
^2*b^4*d^3*x^2*e^4 + 216*A*a*b^5*d^3*x^2*e^4 + 135*B*a^2*b^4*d^4*x*e^3 + 54*A*a*b^5*d^4*x*e^3 + 15*B*a^2*b^4*d
^5*e^2 + 6*A*a*b^5*d^5*e^2 + 2016*B*a^3*b^3*x^4*e^7 + 1512*A*a^2*b^4*x^4*e^7 + 1344*B*a^3*b^3*d*x^3*e^6 + 1008
*A*a^2*b^4*d*x^3*e^6 + 576*B*a^3*b^3*d^2*x^2*e^5 + 432*A*a^2*b^4*d^2*x^2*e^5 + 144*B*a^3*b^3*d^3*x*e^4 + 108*A
*a^2*b^4*d^3*x*e^4 + 16*B*a^3*b^3*d^4*e^3 + 12*A*a^2*b^4*d^4*e^3 + 1260*B*a^4*b^2*x^3*e^7 + 1680*A*a^3*b^3*x^3
*e^7 + 540*B*a^4*b^2*d*x^2*e^6 + 720*A*a^3*b^3*d*x^2*e^6 + 135*B*a^4*b^2*d^2*x*e^5 + 180*A*a^3*b^3*d^2*x*e^5 +
15*B*a^4*b^2*d^3*e^4 + 20*A*a^3*b^3*d^3*e^4 + 432*B*a^5*b*x^2*e^7 + 1080*A*a^4*b^2*x^2*e^7 + 108*B*a^5*b*d*x*
e^6 + 270*A*a^4*b^2*d*x*e^6 + 12*B*a^5*b*d^2*e^5 + 30*A*a^4*b^2*d^2*e^5 + 63*B*a^6*x*e^7 + 378*A*a^5*b*x*e^7 +
7*B*a^6*d*e^6 + 42*A*a^5*b*d*e^6 + 56*A*a^6*e^7)*e^(-8)/(x*e + d)^9

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maple [B]  time = 0.01, size = 814, normalized size = 6.03 \begin {gather*} -\frac {B \,b^{6}}{2 \left (e x +d \right )^{2} e^{8}}-\frac {\left (A b e +6 B a e -7 B b d \right ) b^{5}}{3 \left (e x +d \right )^{3} 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}}{4 \left (e x +d \right )^{4} e^{8}}-\frac {\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}}{\left (e x +d \right )^{5} 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}}{6 \left (e x +d \right )^{6} 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}{7 \left (e x +d \right )^{7} 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}}{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}}{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)^10,x)

[Out]

-1/3*b^5*(A*b*e+6*B*a*e-7*B*b*d)/e^8/(e*x+d)^3-3/4*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)^4-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)^5-1/2*B*b^6/e^8/(e*x+d)^2-1/8*(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^4-8
0*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)^8-1/9*(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)^9-5/6*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)^6-3/7*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)^7

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maxima [B]  time = 0.82, size = 861, normalized size = 6.38 \begin {gather*} -\frac {252 \, B b^{6} e^{7} x^{7} + 7 \, B b^{6} d^{7} + 56 \, A a^{6} e^{7} + 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{6} e + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{5} e^{2} + 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{4} e^{3} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{3} e^{4} + 6 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} e^{5} + 7 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} d e^{6} + 84 \, {\left (7 \, B b^{6} d e^{6} + 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} e^{7}\right )} x^{6} + 126 \, {\left (7 \, B b^{6} d^{2} e^{5} + 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d e^{6} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{7}\right )} x^{5} + 126 \, {\left (7 \, B b^{6} d^{3} e^{4} + 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} e^{5} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e^{6} + 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{7}\right )} x^{4} + 84 \, {\left (7 \, B b^{6} d^{4} e^{3} + 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{3} e^{4} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} e^{5} + 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e^{6} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{7}\right )} x^{3} + 36 \, {\left (7 \, B b^{6} d^{5} e^{2} + 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{4} e^{3} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{3} e^{4} + 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} e^{5} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e^{6} + 6 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{7}\right )} x^{2} + 9 \, {\left (7 \, B b^{6} d^{6} e + 2 \, {\left (6 \, B a b^{5} + A b^{6}\right )} d^{5} e^{2} + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{4} e^{3} + 4 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{3} e^{4} + 5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} e^{5} + 6 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e^{6} + 7 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} e^{7}\right )} x}{504 \, {\left (e^{17} x^{9} + 9 \, d e^{16} x^{8} + 36 \, d^{2} e^{15} x^{7} + 84 \, d^{3} e^{14} x^{6} + 126 \, d^{4} e^{13} x^{5} + 126 \, d^{5} e^{12} x^{4} + 84 \, d^{6} e^{11} x^{3} + 36 \, d^{7} e^{10} x^{2} + 9 \, d^{8} e^{9} x + d^{9} 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)^10,x, algorithm="maxima")

[Out]

-1/504*(252*B*b^6*e^7*x^7 + 7*B*b^6*d^7 + 56*A*a^6*e^7 + 2*(6*B*a*b^5 + A*b^6)*d^6*e + 3*(5*B*a^2*b^4 + 2*A*a*
b^5)*d^5*e^2 + 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 6*(2*B*a^5*b +
5*A*a^4*b^2)*d^2*e^5 + 7*(B*a^6 + 6*A*a^5*b)*d*e^6 + 84*(7*B*b^6*d*e^6 + 2*(6*B*a*b^5 + A*b^6)*e^7)*x^6 + 126*
(7*B*b^6*d^2*e^5 + 2*(6*B*a*b^5 + A*b^6)*d*e^6 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*e^7)*x^5 + 126*(7*B*b^6*d^3*e^4 +
2*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 + 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 +
84*(7*B*b^6*d^4*e^3 + 2*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + 4*(4*B*a^3*b^3 + 3
*A*a^2*b^4)*d*e^6 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 + 36*(7*B*b^6*d^5*e^2 + 2*(6*B*a*b^5 + A*b^6)*d^4*e
^3 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 + 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^
3)*d*e^6 + 6*(2*B*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 9*(7*B*b^6*d^6*e + 2*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 3*(5*B*a^
2*b^4 + 2*A*a*b^5)*d^4*e^3 + 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 + 6
*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + 7*(B*a^6 + 6*A*a^5*b)*e^7)*x)/(e^17*x^9 + 9*d*e^16*x^8 + 36*d^2*e^15*x^7 +
84*d^3*e^14*x^6 + 126*d^4*e^13*x^5 + 126*d^5*e^12*x^4 + 84*d^6*e^11*x^3 + 36*d^7*e^10*x^2 + 9*d^8*e^9*x + d^9*
e^8)

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mupad [B]  time = 0.24, size = 877, normalized size = 6.50 \begin {gather*} -\frac {\frac {7\,B\,a^6\,d\,e^6+56\,A\,a^6\,e^7+12\,B\,a^5\,b\,d^2\,e^5+42\,A\,a^5\,b\,d\,e^6+15\,B\,a^4\,b^2\,d^3\,e^4+30\,A\,a^4\,b^2\,d^2\,e^5+16\,B\,a^3\,b^3\,d^4\,e^3+20\,A\,a^3\,b^3\,d^3\,e^4+15\,B\,a^2\,b^4\,d^5\,e^2+12\,A\,a^2\,b^4\,d^4\,e^3+12\,B\,a\,b^5\,d^6\,e+6\,A\,a\,b^5\,d^5\,e^2+7\,B\,b^6\,d^7+2\,A\,b^6\,d^6\,e}{504\,e^8}+\frac {x\,\left (7\,B\,a^6\,e^6+12\,B\,a^5\,b\,d\,e^5+42\,A\,a^5\,b\,e^6+15\,B\,a^4\,b^2\,d^2\,e^4+30\,A\,a^4\,b^2\,d\,e^5+16\,B\,a^3\,b^3\,d^3\,e^3+20\,A\,a^3\,b^3\,d^2\,e^4+15\,B\,a^2\,b^4\,d^4\,e^2+12\,A\,a^2\,b^4\,d^3\,e^3+12\,B\,a\,b^5\,d^5\,e+6\,A\,a\,b^5\,d^4\,e^2+7\,B\,b^6\,d^6+2\,A\,b^6\,d^5\,e\right )}{56\,e^7}+\frac {b^3\,x^4\,\left (16\,B\,a^3\,e^3+15\,B\,a^2\,b\,d\,e^2+12\,A\,a^2\,b\,e^3+12\,B\,a\,b^2\,d^2\,e+6\,A\,a\,b^2\,d\,e^2+7\,B\,b^3\,d^3+2\,A\,b^3\,d^2\,e\right )}{4\,e^4}+\frac {b^5\,x^6\,\left (2\,A\,b\,e+12\,B\,a\,e+7\,B\,b\,d\right )}{6\,e^2}+\frac {b\,x^2\,\left (12\,B\,a^5\,e^5+15\,B\,a^4\,b\,d\,e^4+30\,A\,a^4\,b\,e^5+16\,B\,a^3\,b^2\,d^2\,e^3+20\,A\,a^3\,b^2\,d\,e^4+15\,B\,a^2\,b^3\,d^3\,e^2+12\,A\,a^2\,b^3\,d^2\,e^3+12\,B\,a\,b^4\,d^4\,e+6\,A\,a\,b^4\,d^3\,e^2+7\,B\,b^5\,d^5+2\,A\,b^5\,d^4\,e\right )}{14\,e^6}+\frac {b^2\,x^3\,\left (15\,B\,a^4\,e^4+16\,B\,a^3\,b\,d\,e^3+20\,A\,a^3\,b\,e^4+15\,B\,a^2\,b^2\,d^2\,e^2+12\,A\,a^2\,b^2\,d\,e^3+12\,B\,a\,b^3\,d^3\,e+6\,A\,a\,b^3\,d^2\,e^2+7\,B\,b^4\,d^4+2\,A\,b^4\,d^3\,e\right )}{6\,e^5}+\frac {b^4\,x^5\,\left (15\,B\,a^2\,e^2+12\,B\,a\,b\,d\,e+6\,A\,a\,b\,e^2+7\,B\,b^2\,d^2+2\,A\,b^2\,d\,e\right )}{4\,e^3}+\frac {B\,b^6\,x^7}{2\,e}}{d^9+9\,d^8\,e\,x+36\,d^7\,e^2\,x^2+84\,d^6\,e^3\,x^3+126\,d^5\,e^4\,x^4+126\,d^4\,e^5\,x^5+84\,d^3\,e^6\,x^6+36\,d^2\,e^7\,x^7+9\,d\,e^8\,x^8+e^9\,x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^6)/(d + e*x)^10,x)

[Out]

-((56*A*a^6*e^7 + 7*B*b^6*d^7 + 2*A*b^6*d^6*e + 7*B*a^6*d*e^6 + 6*A*a*b^5*d^5*e^2 + 12*B*a^5*b*d^2*e^5 + 12*A*
a^2*b^4*d^4*e^3 + 20*A*a^3*b^3*d^3*e^4 + 30*A*a^4*b^2*d^2*e^5 + 15*B*a^2*b^4*d^5*e^2 + 16*B*a^3*b^3*d^4*e^3 +
15*B*a^4*b^2*d^3*e^4 + 42*A*a^5*b*d*e^6 + 12*B*a*b^5*d^6*e)/(504*e^8) + (x*(7*B*a^6*e^6 + 7*B*b^6*d^6 + 42*A*a
^5*b*e^6 + 2*A*b^6*d^5*e + 6*A*a*b^5*d^4*e^2 + 30*A*a^4*b^2*d*e^5 + 12*A*a^2*b^4*d^3*e^3 + 20*A*a^3*b^3*d^2*e^
4 + 15*B*a^2*b^4*d^4*e^2 + 16*B*a^3*b^3*d^3*e^3 + 15*B*a^4*b^2*d^2*e^4 + 12*B*a*b^5*d^5*e + 12*B*a^5*b*d*e^5))
/(56*e^7) + (b^3*x^4*(16*B*a^3*e^3 + 7*B*b^3*d^3 + 12*A*a^2*b*e^3 + 2*A*b^3*d^2*e + 6*A*a*b^2*d*e^2 + 12*B*a*b
^2*d^2*e + 15*B*a^2*b*d*e^2))/(4*e^4) + (b^5*x^6*(2*A*b*e + 12*B*a*e + 7*B*b*d))/(6*e^2) + (b*x^2*(12*B*a^5*e^
5 + 7*B*b^5*d^5 + 30*A*a^4*b*e^5 + 2*A*b^5*d^4*e + 6*A*a*b^4*d^3*e^2 + 20*A*a^3*b^2*d*e^4 + 12*A*a^2*b^3*d^2*e
^3 + 15*B*a^2*b^3*d^3*e^2 + 16*B*a^3*b^2*d^2*e^3 + 12*B*a*b^4*d^4*e + 15*B*a^4*b*d*e^4))/(14*e^6) + (b^2*x^3*(
15*B*a^4*e^4 + 7*B*b^4*d^4 + 20*A*a^3*b*e^4 + 2*A*b^4*d^3*e + 6*A*a*b^3*d^2*e^2 + 12*A*a^2*b^2*d*e^3 + 15*B*a^
2*b^2*d^2*e^2 + 12*B*a*b^3*d^3*e + 16*B*a^3*b*d*e^3))/(6*e^5) + (b^4*x^5*(15*B*a^2*e^2 + 7*B*b^2*d^2 + 6*A*a*b
*e^2 + 2*A*b^2*d*e + 12*B*a*b*d*e))/(4*e^3) + (B*b^6*x^7)/(2*e))/(d^9 + e^9*x^9 + 9*d*e^8*x^8 + 36*d^7*e^2*x^2
+ 84*d^6*e^3*x^3 + 126*d^5*e^4*x^4 + 126*d^4*e^5*x^5 + 84*d^3*e^6*x^6 + 36*d^2*e^7*x^7 + 9*d^8*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)**10,x)

[Out]

Timed out

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