3.15.69 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^2}{(d+e x)^9} \, dx\)

Optimal. Leaf size=185 \[ \frac {b^2 (a+b x)^5 (-8 a B e+3 A b e+5 b B d)}{840 e (d+e x)^5 (b d-a e)^4}+\frac {b (a+b x)^5 (-8 a B e+3 A b e+5 b B d)}{168 e (d+e x)^6 (b d-a e)^3}+\frac {(a+b x)^5 (-8 a B e+3 A b e+5 b B d)}{56 e (d+e x)^7 (b d-a e)^2}-\frac {(a+b x)^5 (B d-A e)}{8 e (d+e x)^8 (b d-a e)} \]

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Rubi [A]  time = 0.09, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 5, 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^2 (a+b x)^5 (-8 a B e+3 A b e+5 b B d)}{840 e (d+e x)^5 (b d-a e)^4}+\frac {b (a+b x)^5 (-8 a B e+3 A b e+5 b B d)}{168 e (d+e x)^6 (b d-a e)^3}+\frac {(a+b x)^5 (-8 a B e+3 A b e+5 b B d)}{56 e (d+e x)^7 (b d-a e)^2}-\frac {(a+b x)^5 (B d-A e)}{8 e (d+e x)^8 (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)^9,x]

[Out]

-((B*d - A*e)*(a + b*x)^5)/(8*e*(b*d - a*e)*(d + e*x)^8) + ((5*b*B*d + 3*A*b*e - 8*a*B*e)*(a + b*x)^5)/(56*e*(
b*d - a*e)^2*(d + e*x)^7) + (b*(5*b*B*d + 3*A*b*e - 8*a*B*e)*(a + b*x)^5)/(168*e*(b*d - a*e)^3*(d + e*x)^6) +
(b^2*(5*b*B*d + 3*A*b*e - 8*a*B*e)*(a + b*x)^5)/(840*e*(b*d - a*e)^4*(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)^9} \, dx &=\int \frac {(a+b x)^4 (A+B x)}{(d+e x)^9} \, dx\\ &=-\frac {(B d-A e) (a+b x)^5}{8 e (b d-a e) (d+e x)^8}+\frac {(5 b B d+3 A b e-8 a B e) \int \frac {(a+b x)^4}{(d+e x)^8} \, dx}{8 e (b d-a e)}\\ &=-\frac {(B d-A e) (a+b x)^5}{8 e (b d-a e) (d+e x)^8}+\frac {(5 b B d+3 A b e-8 a B e) (a+b x)^5}{56 e (b d-a e)^2 (d+e x)^7}+\frac {(b (5 b B d+3 A b e-8 a B e)) \int \frac {(a+b x)^4}{(d+e x)^7} \, dx}{28 e (b d-a e)^2}\\ &=-\frac {(B d-A e) (a+b x)^5}{8 e (b d-a e) (d+e x)^8}+\frac {(5 b B d+3 A b e-8 a B e) (a+b x)^5}{56 e (b d-a e)^2 (d+e x)^7}+\frac {b (5 b B d+3 A b e-8 a B e) (a+b x)^5}{168 e (b d-a e)^3 (d+e x)^6}+\frac {\left (b^2 (5 b B d+3 A b e-8 a B e)\right ) \int \frac {(a+b x)^4}{(d+e x)^6} \, dx}{168 e (b d-a e)^3}\\ &=-\frac {(B d-A e) (a+b x)^5}{8 e (b d-a e) (d+e x)^8}+\frac {(5 b B d+3 A b e-8 a B e) (a+b x)^5}{56 e (b d-a e)^2 (d+e x)^7}+\frac {b (5 b B d+3 A b e-8 a B e) (a+b x)^5}{168 e (b d-a e)^3 (d+e x)^6}+\frac {b^2 (5 b B d+3 A b e-8 a B e) (a+b x)^5}{840 e (b d-a e)^4 (d+e x)^5}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 320, normalized size = 1.73 \begin {gather*} -\frac {15 a^4 e^4 (7 A e+B (d+8 e x))+20 a^3 b e^3 \left (3 A e (d+8 e x)+B \left (d^2+8 d e x+28 e^2 x^2\right )\right )+6 a^2 b^2 e^2 \left (5 A e \left (d^2+8 d e x+28 e^2 x^2\right )+3 B \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )+12 a b^3 e \left (A e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+B \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )+b^4 \left (3 A e \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 B \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )\right )}{840 e^6 (d+e x)^8} \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)^9,x]

[Out]

-1/840*(15*a^4*e^4*(7*A*e + B*(d + 8*e*x)) + 20*a^3*b*e^3*(3*A*e*(d + 8*e*x) + B*(d^2 + 8*d*e*x + 28*e^2*x^2))
+ 6*a^2*b^2*e^2*(5*A*e*(d^2 + 8*d*e*x + 28*e^2*x^2) + 3*B*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3)) + 12
*a*b^3*e*(A*e*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + B*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x
^3 + 70*e^4*x^4)) + b^4*(3*A*e*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 5*B*(d^5 + 8*d
^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5)))/(e^6*(d + e*x)^8)

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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)^9} \, 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)^9,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^9, x]

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fricas [B]  time = 0.40, size = 489, normalized size = 2.64 \begin {gather*} -\frac {280 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 105 \, A a^{4} e^{5} + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 10 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 15 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 70 \, {\left (5 \, B b^{4} d e^{4} + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 56 \, {\left (5 \, B b^{4} d^{2} e^{3} + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 28 \, {\left (5 \, B b^{4} d^{3} e^{2} + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 10 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 8 \, {\left (5 \, B b^{4} d^{4} e + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 10 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 15 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{840 \, {\left (e^{14} x^{8} + 8 \, d e^{13} x^{7} + 28 \, d^{2} e^{12} x^{6} + 56 \, d^{3} e^{11} x^{5} + 70 \, d^{4} e^{10} x^{4} + 56 \, d^{5} e^{9} x^{3} + 28 \, d^{6} e^{8} x^{2} + 8 \, d^{7} e^{7} x + d^{8} 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)^9,x, algorithm="fricas")

[Out]

-1/840*(280*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 105*A*a^4*e^5 + 3*(4*B*a*b^3 + A*b^4)*d^4*e + 6*(3*B*a^2*b^2 + 2*A*a
*b^3)*d^3*e^2 + 10*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 + 15*(B*a^4 + 4*A*a^3*b)*d*e^4 + 70*(5*B*b^4*d*e^4 + 3*(4
*B*a*b^3 + A*b^4)*e^5)*x^4 + 56*(5*B*b^4*d^2*e^3 + 3*(4*B*a*b^3 + A*b^4)*d*e^4 + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*e
^5)*x^3 + 28*(5*B*b^4*d^3*e^2 + 3*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + 10*(2*B*a^
3*b + 3*A*a^2*b^2)*e^5)*x^2 + 8*(5*B*b^4*d^4*e + 3*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d
^2*e^3 + 10*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + 15*(B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^14*x^8 + 8*d*e^13*x^7 + 28*d^2
*e^12*x^6 + 56*d^3*e^11*x^5 + 70*d^4*e^10*x^4 + 56*d^5*e^9*x^3 + 28*d^6*e^8*x^2 + 8*d^7*e^7*x + d^8*e^6)

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giac [B]  time = 0.16, size = 440, normalized size = 2.38 \begin {gather*} -\frac {{\left (280 \, B b^{4} x^{5} e^{5} + 350 \, B b^{4} d x^{4} e^{4} + 280 \, B b^{4} d^{2} x^{3} e^{3} + 140 \, B b^{4} d^{3} x^{2} e^{2} + 40 \, B b^{4} d^{4} x e + 5 \, B b^{4} d^{5} + 840 \, B a b^{3} x^{4} e^{5} + 210 \, A b^{4} x^{4} e^{5} + 672 \, B a b^{3} d x^{3} e^{4} + 168 \, A b^{4} d x^{3} e^{4} + 336 \, B a b^{3} d^{2} x^{2} e^{3} + 84 \, A b^{4} d^{2} x^{2} e^{3} + 96 \, B a b^{3} d^{3} x e^{2} + 24 \, A b^{4} d^{3} x e^{2} + 12 \, B a b^{3} d^{4} e + 3 \, A b^{4} d^{4} e + 1008 \, B a^{2} b^{2} x^{3} e^{5} + 672 \, A a b^{3} x^{3} e^{5} + 504 \, B a^{2} b^{2} d x^{2} e^{4} + 336 \, A a b^{3} d x^{2} e^{4} + 144 \, B a^{2} b^{2} d^{2} x e^{3} + 96 \, A a b^{3} d^{2} x e^{3} + 18 \, B a^{2} b^{2} d^{3} e^{2} + 12 \, A a b^{3} d^{3} e^{2} + 560 \, B a^{3} b x^{2} e^{5} + 840 \, A a^{2} b^{2} x^{2} e^{5} + 160 \, B a^{3} b d x e^{4} + 240 \, A a^{2} b^{2} d x e^{4} + 20 \, B a^{3} b d^{2} e^{3} + 30 \, A a^{2} b^{2} d^{2} e^{3} + 120 \, B a^{4} x e^{5} + 480 \, A a^{3} b x e^{5} + 15 \, B a^{4} d e^{4} + 60 \, A a^{3} b d e^{4} + 105 \, A a^{4} e^{5}\right )} e^{\left (-6\right )}}{840 \, {\left (x e + d\right )}^{8}} \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)^9,x, algorithm="giac")

[Out]

-1/840*(280*B*b^4*x^5*e^5 + 350*B*b^4*d*x^4*e^4 + 280*B*b^4*d^2*x^3*e^3 + 140*B*b^4*d^3*x^2*e^2 + 40*B*b^4*d^4
*x*e + 5*B*b^4*d^5 + 840*B*a*b^3*x^4*e^5 + 210*A*b^4*x^4*e^5 + 672*B*a*b^3*d*x^3*e^4 + 168*A*b^4*d*x^3*e^4 + 3
36*B*a*b^3*d^2*x^2*e^3 + 84*A*b^4*d^2*x^2*e^3 + 96*B*a*b^3*d^3*x*e^2 + 24*A*b^4*d^3*x*e^2 + 12*B*a*b^3*d^4*e +
3*A*b^4*d^4*e + 1008*B*a^2*b^2*x^3*e^5 + 672*A*a*b^3*x^3*e^5 + 504*B*a^2*b^2*d*x^2*e^4 + 336*A*a*b^3*d*x^2*e^
4 + 144*B*a^2*b^2*d^2*x*e^3 + 96*A*a*b^3*d^2*x*e^3 + 18*B*a^2*b^2*d^3*e^2 + 12*A*a*b^3*d^3*e^2 + 560*B*a^3*b*x
^2*e^5 + 840*A*a^2*b^2*x^2*e^5 + 160*B*a^3*b*d*x*e^4 + 240*A*a^2*b^2*d*x*e^4 + 20*B*a^3*b*d^2*e^3 + 30*A*a^2*b
^2*d^2*e^3 + 120*B*a^4*x*e^5 + 480*A*a^3*b*x*e^5 + 15*B*a^4*d*e^4 + 60*A*a^3*b*d*e^4 + 105*A*a^4*e^5)*e^(-6)/(
x*e + d)^8

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maple [B]  time = 0.05, size = 430, normalized size = 2.32 \begin {gather*} -\frac {B \,b^{4}}{3 \left (e x +d \right )^{3} e^{6}}-\frac {\left (A b e +4 a B e -5 B b d \right ) b^{3}}{4 \left (e x +d \right )^{4} e^{6}}-\frac {2 \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}}{5 \left (e x +d \right )^{5} e^{6}}-\frac {\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}{3 \left (e x +d \right )^{6} 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}}{7 \left (e x +d \right )^{7} 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}}{8 \left (e x +d \right )^{8} 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)^9,x)

[Out]

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

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maxima [B]  time = 0.77, size = 489, normalized size = 2.64 \begin {gather*} -\frac {280 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 105 \, A a^{4} e^{5} + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 10 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 15 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 70 \, {\left (5 \, B b^{4} d e^{4} + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 56 \, {\left (5 \, B b^{4} d^{2} e^{3} + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 28 \, {\left (5 \, B b^{4} d^{3} e^{2} + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 10 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 8 \, {\left (5 \, B b^{4} d^{4} e + 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 10 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 15 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{840 \, {\left (e^{14} x^{8} + 8 \, d e^{13} x^{7} + 28 \, d^{2} e^{12} x^{6} + 56 \, d^{3} e^{11} x^{5} + 70 \, d^{4} e^{10} x^{4} + 56 \, d^{5} e^{9} x^{3} + 28 \, d^{6} e^{8} x^{2} + 8 \, d^{7} e^{7} x + d^{8} 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)^9,x, algorithm="maxima")

[Out]

-1/840*(280*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 105*A*a^4*e^5 + 3*(4*B*a*b^3 + A*b^4)*d^4*e + 6*(3*B*a^2*b^2 + 2*A*a
*b^3)*d^3*e^2 + 10*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 + 15*(B*a^4 + 4*A*a^3*b)*d*e^4 + 70*(5*B*b^4*d*e^4 + 3*(4
*B*a*b^3 + A*b^4)*e^5)*x^4 + 56*(5*B*b^4*d^2*e^3 + 3*(4*B*a*b^3 + A*b^4)*d*e^4 + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*e
^5)*x^3 + 28*(5*B*b^4*d^3*e^2 + 3*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + 10*(2*B*a^
3*b + 3*A*a^2*b^2)*e^5)*x^2 + 8*(5*B*b^4*d^4*e + 3*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d
^2*e^3 + 10*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + 15*(B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^14*x^8 + 8*d*e^13*x^7 + 28*d^2
*e^12*x^6 + 56*d^3*e^11*x^5 + 70*d^4*e^10*x^4 + 56*d^5*e^9*x^3 + 28*d^6*e^8*x^2 + 8*d^7*e^7*x + d^8*e^6)

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mupad [B]  time = 2.26, size = 490, normalized size = 2.65 \begin {gather*} -\frac {\frac {15\,B\,a^4\,d\,e^4+105\,A\,a^4\,e^5+20\,B\,a^3\,b\,d^2\,e^3+60\,A\,a^3\,b\,d\,e^4+18\,B\,a^2\,b^2\,d^3\,e^2+30\,A\,a^2\,b^2\,d^2\,e^3+12\,B\,a\,b^3\,d^4\,e+12\,A\,a\,b^3\,d^3\,e^2+5\,B\,b^4\,d^5+3\,A\,b^4\,d^4\,e}{840\,e^6}+\frac {x\,\left (15\,B\,a^4\,e^4+20\,B\,a^3\,b\,d\,e^3+60\,A\,a^3\,b\,e^4+18\,B\,a^2\,b^2\,d^2\,e^2+30\,A\,a^2\,b^2\,d\,e^3+12\,B\,a\,b^3\,d^3\,e+12\,A\,a\,b^3\,d^2\,e^2+5\,B\,b^4\,d^4+3\,A\,b^4\,d^3\,e\right )}{105\,e^5}+\frac {b^3\,x^4\,\left (3\,A\,b\,e+12\,B\,a\,e+5\,B\,b\,d\right )}{12\,e^2}+\frac {b\,x^2\,\left (20\,B\,a^3\,e^3+18\,B\,a^2\,b\,d\,e^2+30\,A\,a^2\,b\,e^3+12\,B\,a\,b^2\,d^2\,e+12\,A\,a\,b^2\,d\,e^2+5\,B\,b^3\,d^3+3\,A\,b^3\,d^2\,e\right )}{30\,e^4}+\frac {b^2\,x^3\,\left (18\,B\,a^2\,e^2+12\,B\,a\,b\,d\,e+12\,A\,a\,b\,e^2+5\,B\,b^2\,d^2+3\,A\,b^2\,d\,e\right )}{15\,e^3}+\frac {B\,b^4\,x^5}{3\,e}}{d^8+8\,d^7\,e\,x+28\,d^6\,e^2\,x^2+56\,d^5\,e^3\,x^3+70\,d^4\,e^4\,x^4+56\,d^3\,e^5\,x^5+28\,d^2\,e^6\,x^6+8\,d\,e^7\,x^7+e^8\,x^8} \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)^9,x)

[Out]

-((105*A*a^4*e^5 + 5*B*b^4*d^5 + 3*A*b^4*d^4*e + 15*B*a^4*d*e^4 + 12*A*a*b^3*d^3*e^2 + 20*B*a^3*b*d^2*e^3 + 30
*A*a^2*b^2*d^2*e^3 + 18*B*a^2*b^2*d^3*e^2 + 60*A*a^3*b*d*e^4 + 12*B*a*b^3*d^4*e)/(840*e^6) + (x*(15*B*a^4*e^4
+ 5*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 + 18*B*a^2*b^2*d^2*e^
2 + 12*B*a*b^3*d^3*e + 20*B*a^3*b*d*e^3))/(105*e^5) + (b^3*x^4*(3*A*b*e + 12*B*a*e + 5*B*b*d))/(12*e^2) + (b*x
^2*(20*B*a^3*e^3 + 5*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 + 12*B*a*b^2*d^2*e + 18*B*a
^2*b*d*e^2))/(30*e^4) + (b^2*x^3*(18*B*a^2*e^2 + 5*B*b^2*d^2 + 12*A*a*b*e^2 + 3*A*b^2*d*e + 12*B*a*b*d*e))/(15
*e^3) + (B*b^4*x^5)/(3*e))/(d^8 + e^8*x^8 + 8*d*e^7*x^7 + 28*d^6*e^2*x^2 + 56*d^5*e^3*x^3 + 70*d^4*e^4*x^4 + 5
6*d^3*e^5*x^5 + 28*d^2*e^6*x^6 + 8*d^7*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)**9,x)

[Out]

Timed out

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