3.15.68 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^2}{(d+e x)^8} \, dx\)

Optimal. Leaf size=135 \[ \frac {b (a+b x)^5 (-7 a B e+2 A b e+5 b B d)}{210 e (d+e x)^5 (b d-a e)^3}+\frac {(a+b x)^5 (-7 a B e+2 A b e+5 b B d)}{42 e (d+e x)^6 (b d-a e)^2}-\frac {(a+b x)^5 (B d-A e)}{7 e (d+e x)^7 (b d-a e)} \]

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Rubi [A]  time = 0.06, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {27, 78, 45, 37} \begin {gather*} \frac {b (a+b x)^5 (-7 a B e+2 A b e+5 b B d)}{210 e (d+e x)^5 (b d-a e)^3}+\frac {(a+b x)^5 (-7 a B e+2 A b e+5 b B d)}{42 e (d+e x)^6 (b d-a e)^2}-\frac {(a+b x)^5 (B d-A e)}{7 e (d+e x)^7 (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^8,x]

[Out]

-((B*d - A*e)*(a + b*x)^5)/(7*e*(b*d - a*e)*(d + e*x)^7) + ((5*b*B*d + 2*A*b*e - 7*a*B*e)*(a + b*x)^5)/(42*e*(
b*d - a*e)^2*(d + e*x)^6) + (b*(5*b*B*d + 2*A*b*e - 7*a*B*e)*(a + b*x)^5)/(210*e*(b*d - a*e)^3*(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 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) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^8} \, dx &=\int \frac {(a+b x)^4 (A+B x)}{(d+e x)^8} \, dx\\ &=-\frac {(B d-A e) (a+b x)^5}{7 e (b d-a e) (d+e x)^7}+\frac {(5 b B d+2 A b e-7 a B e) \int \frac {(a+b x)^4}{(d+e x)^7} \, dx}{7 e (b d-a e)}\\ &=-\frac {(B d-A e) (a+b x)^5}{7 e (b d-a e) (d+e x)^7}+\frac {(5 b B d+2 A b e-7 a B e) (a+b x)^5}{42 e (b d-a e)^2 (d+e x)^6}+\frac {(b (5 b B d+2 A b e-7 a B e)) \int \frac {(a+b x)^4}{(d+e x)^6} \, dx}{42 e (b d-a e)^2}\\ &=-\frac {(B d-A e) (a+b x)^5}{7 e (b d-a e) (d+e x)^7}+\frac {(5 b B d+2 A b e-7 a B e) (a+b x)^5}{42 e (b d-a e)^2 (d+e x)^6}+\frac {b (5 b B d+2 A b e-7 a B e) (a+b x)^5}{210 e (b d-a e)^3 (d+e x)^5}\\ \end {align*}

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Mathematica [B]  time = 0.13, size = 323, normalized size = 2.39 \begin {gather*} -\frac {5 a^4 e^4 (6 A e+B (d+7 e x))+4 a^3 b e^3 \left (5 A e (d+7 e x)+2 B \left (d^2+7 d e x+21 e^2 x^2\right )\right )+3 a^2 b^2 e^2 \left (4 A e \left (d^2+7 d e x+21 e^2 x^2\right )+3 B \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )+2 a b^3 e \left (3 A e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+4 B \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )+b^4 \left (2 A e \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+5 B \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )\right )}{210 e^6 (d+e x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^8,x]

[Out]

-1/210*(5*a^4*e^4*(6*A*e + B*(d + 7*e*x)) + 4*a^3*b*e^3*(5*A*e*(d + 7*e*x) + 2*B*(d^2 + 7*d*e*x + 21*e^2*x^2))
+ 3*a^2*b^2*e^2*(4*A*e*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*B*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3)) + 2*
a*b^3*e*(3*A*e*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 4*B*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^
3*x^3 + 35*e^4*x^4)) + b^4*(2*A*e*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) + 5*B*(d^5 +
7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5)))/(e^6*(d + e*x)^7)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^8,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^8, x]

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fricas [B]  time = 0.40, size = 478, normalized size = 3.54 \begin {gather*} -\frac {105 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 30 \, A a^{4} e^{5} + 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 5 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 35 \, {\left (5 \, B b^{4} d e^{4} + 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 35 \, {\left (5 \, B b^{4} d^{2} e^{3} + 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 21 \, {\left (5 \, B b^{4} d^{3} e^{2} + 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 7 \, {\left (5 \, B b^{4} d^{4} e + 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 5 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{210 \, {\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^8,x, algorithm="fricas")

[Out]

-1/210*(105*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 30*A*a^4*e^5 + 2*(4*B*a*b^3 + A*b^4)*d^4*e + 3*(3*B*a^2*b^2 + 2*A*a*
b^3)*d^3*e^2 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 + 5*(B*a^4 + 4*A*a^3*b)*d*e^4 + 35*(5*B*b^4*d*e^4 + 2*(4*B*
a*b^3 + A*b^4)*e^5)*x^4 + 35*(5*B*b^4*d^2*e^3 + 2*(4*B*a*b^3 + A*b^4)*d*e^4 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*e^5)
*x^3 + 21*(5*B*b^4*d^3*e^2 + 2*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + 4*(2*B*a^3*b
+ 3*A*a^2*b^2)*e^5)*x^2 + 7*(5*B*b^4*d^4*e + 2*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e
^3 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + 5*(B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^13*x^7 + 7*d*e^12*x^6 + 21*d^2*e^11*
x^5 + 35*d^3*e^10*x^4 + 35*d^4*e^9*x^3 + 21*d^5*e^8*x^2 + 7*d^6*e^7*x + d^7*e^6)

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giac [B]  time = 0.16, size = 440, normalized size = 3.26 \begin {gather*} -\frac {{\left (105 \, B b^{4} x^{5} e^{5} + 175 \, B b^{4} d x^{4} e^{4} + 175 \, B b^{4} d^{2} x^{3} e^{3} + 105 \, B b^{4} d^{3} x^{2} e^{2} + 35 \, B b^{4} d^{4} x e + 5 \, B b^{4} d^{5} + 280 \, B a b^{3} x^{4} e^{5} + 70 \, A b^{4} x^{4} e^{5} + 280 \, B a b^{3} d x^{3} e^{4} + 70 \, A b^{4} d x^{3} e^{4} + 168 \, B a b^{3} d^{2} x^{2} e^{3} + 42 \, A b^{4} d^{2} x^{2} e^{3} + 56 \, B a b^{3} d^{3} x e^{2} + 14 \, A b^{4} d^{3} x e^{2} + 8 \, B a b^{3} d^{4} e + 2 \, A b^{4} d^{4} e + 315 \, B a^{2} b^{2} x^{3} e^{5} + 210 \, A a b^{3} x^{3} e^{5} + 189 \, B a^{2} b^{2} d x^{2} e^{4} + 126 \, A a b^{3} d x^{2} e^{4} + 63 \, B a^{2} b^{2} d^{2} x e^{3} + 42 \, A a b^{3} d^{2} x e^{3} + 9 \, B a^{2} b^{2} d^{3} e^{2} + 6 \, A a b^{3} d^{3} e^{2} + 168 \, B a^{3} b x^{2} e^{5} + 252 \, A a^{2} b^{2} x^{2} e^{5} + 56 \, B a^{3} b d x e^{4} + 84 \, A a^{2} b^{2} d x e^{4} + 8 \, B a^{3} b d^{2} e^{3} + 12 \, A a^{2} b^{2} d^{2} e^{3} + 35 \, B a^{4} x e^{5} + 140 \, A a^{3} b x e^{5} + 5 \, B a^{4} d e^{4} + 20 \, A a^{3} b d e^{4} + 30 \, A a^{4} e^{5}\right )} e^{\left (-6\right )}}{210 \, {\left (x e + d\right )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^8,x, algorithm="giac")

[Out]

-1/210*(105*B*b^4*x^5*e^5 + 175*B*b^4*d*x^4*e^4 + 175*B*b^4*d^2*x^3*e^3 + 105*B*b^4*d^3*x^2*e^2 + 35*B*b^4*d^4
*x*e + 5*B*b^4*d^5 + 280*B*a*b^3*x^4*e^5 + 70*A*b^4*x^4*e^5 + 280*B*a*b^3*d*x^3*e^4 + 70*A*b^4*d*x^3*e^4 + 168
*B*a*b^3*d^2*x^2*e^3 + 42*A*b^4*d^2*x^2*e^3 + 56*B*a*b^3*d^3*x*e^2 + 14*A*b^4*d^3*x*e^2 + 8*B*a*b^3*d^4*e + 2*
A*b^4*d^4*e + 315*B*a^2*b^2*x^3*e^5 + 210*A*a*b^3*x^3*e^5 + 189*B*a^2*b^2*d*x^2*e^4 + 126*A*a*b^3*d*x^2*e^4 +
63*B*a^2*b^2*d^2*x*e^3 + 42*A*a*b^3*d^2*x*e^3 + 9*B*a^2*b^2*d^3*e^2 + 6*A*a*b^3*d^3*e^2 + 168*B*a^3*b*x^2*e^5
+ 252*A*a^2*b^2*x^2*e^5 + 56*B*a^3*b*d*x*e^4 + 84*A*a^2*b^2*d*x*e^4 + 8*B*a^3*b*d^2*e^3 + 12*A*a^2*b^2*d^2*e^3
+ 35*B*a^4*x*e^5 + 140*A*a^3*b*x*e^5 + 5*B*a^4*d*e^4 + 20*A*a^3*b*d*e^4 + 30*A*a^4*e^5)*e^(-6)/(x*e + d)^7

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maple [B]  time = 0.06, size = 430, normalized size = 3.19 \begin {gather*} -\frac {B \,b^{4}}{2 \left (e x +d \right )^{2} e^{6}}-\frac {\left (A b e +4 a B e -5 B b d \right ) b^{3}}{3 \left (e x +d \right )^{3} e^{6}}-\frac {\left (2 A a b \,e^{2}-2 A \,b^{2} d e +3 B \,a^{2} e^{2}-8 B d a b e +5 B \,b^{2} d^{2}\right ) b^{2}}{2 \left (e x +d \right )^{4} e^{6}}-\frac {2 \left (3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +2 B \,a^{3} e^{3}-9 B \,a^{2} b d \,e^{2}+12 B a \,b^{2} d^{2} e -5 B \,b^{3} d^{3}\right ) b}{5 \left (e x +d \right )^{5} e^{6}}-\frac {A \,a^{4} e^{5}-4 A \,a^{3} b d \,e^{4}+6 A \,a^{2} b^{2} d^{2} e^{3}-4 A a \,b^{3} d^{3} e^{2}+A \,b^{4} d^{4} e -B d \,a^{4} e^{4}+4 B \,d^{2} a^{3} b \,e^{3}-6 B \,d^{3} a^{2} b^{2} e^{2}+4 B \,d^{4} a \,b^{3} e -b^{4} B \,d^{5}}{7 \left (e x +d \right )^{7} e^{6}}-\frac {4 A \,a^{3} b \,e^{4}-12 A \,a^{2} b^{2} d \,e^{3}+12 A \,d^{2} a \,b^{3} e^{2}-4 A \,b^{4} d^{3} e +B \,a^{4} e^{4}-8 B d \,a^{3} b \,e^{3}+18 B \,d^{2} a^{2} b^{2} e^{2}-16 B \,d^{3} a \,b^{3} e +5 b^{4} B \,d^{4}}{6 \left (e x +d \right )^{6} e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^8,x)

[Out]

-2/5*b*(3*A*a^2*b*e^3-6*A*a*b^2*d*e^2+3*A*b^3*d^2*e+2*B*a^3*e^3-9*B*a^2*b*d*e^2+12*B*a*b^2*d^2*e-5*B*b^3*d^3)/
e^6/(e*x+d)^5-1/2*b^2*(2*A*a*b*e^2-2*A*b^2*d*e+3*B*a^2*e^2-8*B*a*b*d*e+5*B*b^2*d^2)/e^6/(e*x+d)^4-1/3*b^3*(A*b
*e+4*B*a*e-5*B*b*d)/e^6/(e*x+d)^3-1/2*b^4*B/e^6/(e*x+d)^2-1/7*(A*a^4*e^5-4*A*a^3*b*d*e^4+6*A*a^2*b^2*d^2*e^3-4
*A*a*b^3*d^3*e^2+A*b^4*d^4*e-B*a^4*d*e^4+4*B*a^3*b*d^2*e^3-6*B*a^2*b^2*d^3*e^2+4*B*a*b^3*d^4*e-B*b^4*d^5)/e^6/
(e*x+d)^7-1/6*(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+B*a^4*e^4-8*B*a^3*b*d*e^3+18*
B*a^2*b^2*d^2*e^2-16*B*a*b^3*d^3*e+5*B*b^4*d^4)/e^6/(e*x+d)^6

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maxima [B]  time = 0.74, size = 478, normalized size = 3.54 \begin {gather*} -\frac {105 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 30 \, A a^{4} e^{5} + 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 5 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 35 \, {\left (5 \, B b^{4} d e^{4} + 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 35 \, {\left (5 \, B b^{4} d^{2} e^{3} + 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 21 \, {\left (5 \, B b^{4} d^{3} e^{2} + 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 7 \, {\left (5 \, B b^{4} d^{4} e + 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 5 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{210 \, {\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^8,x, algorithm="maxima")

[Out]

-1/210*(105*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 30*A*a^4*e^5 + 2*(4*B*a*b^3 + A*b^4)*d^4*e + 3*(3*B*a^2*b^2 + 2*A*a*
b^3)*d^3*e^2 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 + 5*(B*a^4 + 4*A*a^3*b)*d*e^4 + 35*(5*B*b^4*d*e^4 + 2*(4*B*
a*b^3 + A*b^4)*e^5)*x^4 + 35*(5*B*b^4*d^2*e^3 + 2*(4*B*a*b^3 + A*b^4)*d*e^4 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*e^5)
*x^3 + 21*(5*B*b^4*d^3*e^2 + 2*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + 4*(2*B*a^3*b
+ 3*A*a^2*b^2)*e^5)*x^2 + 7*(5*B*b^4*d^4*e + 2*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e
^3 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + 5*(B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^13*x^7 + 7*d*e^12*x^6 + 21*d^2*e^11*
x^5 + 35*d^3*e^10*x^4 + 35*d^4*e^9*x^3 + 21*d^5*e^8*x^2 + 7*d^6*e^7*x + d^7*e^6)

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mupad [B]  time = 2.36, size = 479, normalized size = 3.55 \begin {gather*} -\frac {\frac {5\,B\,a^4\,d\,e^4+30\,A\,a^4\,e^5+8\,B\,a^3\,b\,d^2\,e^3+20\,A\,a^3\,b\,d\,e^4+9\,B\,a^2\,b^2\,d^3\,e^2+12\,A\,a^2\,b^2\,d^2\,e^3+8\,B\,a\,b^3\,d^4\,e+6\,A\,a\,b^3\,d^3\,e^2+5\,B\,b^4\,d^5+2\,A\,b^4\,d^4\,e}{210\,e^6}+\frac {x\,\left (5\,B\,a^4\,e^4+8\,B\,a^3\,b\,d\,e^3+20\,A\,a^3\,b\,e^4+9\,B\,a^2\,b^2\,d^2\,e^2+12\,A\,a^2\,b^2\,d\,e^3+8\,B\,a\,b^3\,d^3\,e+6\,A\,a\,b^3\,d^2\,e^2+5\,B\,b^4\,d^4+2\,A\,b^4\,d^3\,e\right )}{30\,e^5}+\frac {b^3\,x^4\,\left (2\,A\,b\,e+8\,B\,a\,e+5\,B\,b\,d\right )}{6\,e^2}+\frac {b\,x^2\,\left (8\,B\,a^3\,e^3+9\,B\,a^2\,b\,d\,e^2+12\,A\,a^2\,b\,e^3+8\,B\,a\,b^2\,d^2\,e+6\,A\,a\,b^2\,d\,e^2+5\,B\,b^3\,d^3+2\,A\,b^3\,d^2\,e\right )}{10\,e^4}+\frac {b^2\,x^3\,\left (9\,B\,a^2\,e^2+8\,B\,a\,b\,d\,e+6\,A\,a\,b\,e^2+5\,B\,b^2\,d^2+2\,A\,b^2\,d\,e\right )}{6\,e^3}+\frac {B\,b^4\,x^5}{2\,e}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^2)/(d + e*x)^8,x)

[Out]

-((30*A*a^4*e^5 + 5*B*b^4*d^5 + 2*A*b^4*d^4*e + 5*B*a^4*d*e^4 + 6*A*a*b^3*d^3*e^2 + 8*B*a^3*b*d^2*e^3 + 12*A*a
^2*b^2*d^2*e^3 + 9*B*a^2*b^2*d^3*e^2 + 20*A*a^3*b*d*e^4 + 8*B*a*b^3*d^4*e)/(210*e^6) + (x*(5*B*a^4*e^4 + 5*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 + 9*B*a^2*b^2*d^2*e^2 + 8*B*a
*b^3*d^3*e + 8*B*a^3*b*d*e^3))/(30*e^5) + (b^3*x^4*(2*A*b*e + 8*B*a*e + 5*B*b*d))/(6*e^2) + (b*x^2*(8*B*a^3*e^
3 + 5*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 + 8*B*a*b^2*d^2*e + 9*B*a^2*b*d*e^2))/(10*e
^4) + (b^2*x^3*(9*B*a^2*e^2 + 5*B*b^2*d^2 + 6*A*a*b*e^2 + 2*A*b^2*d*e + 8*B*a*b*d*e))/(6*e^3) + (B*b^4*x^5)/(2
*e))/(d^7 + e^7*x^7 + 7*d*e^6*x^6 + 21*d^5*e^2*x^2 + 35*d^4*e^3*x^3 + 35*d^3*e^4*x^4 + 21*d^2*e^5*x^5 + 7*d^6*
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)*(b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**8,x)

[Out]

Timed out

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