3.12.100 \(\int \frac {(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^9} \, dx\)

Optimal. Leaf size=58 \[ \frac {b (a+b x)^7}{56 (d+e x)^7 (b d-a e)^2}+\frac {(a+b x)^7}{8 (d+e x)^8 (b d-a e)} \]

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Rubi [A]  time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {27, 45, 37} \begin {gather*} \frac {b (a+b x)^7}{56 (d+e x)^7 (b d-a e)^2}+\frac {(a+b x)^7}{8 (d+e x)^8 (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^9,x]

[Out]

(a + b*x)^7/(8*(b*d - a*e)*(d + e*x)^8) + (b*(a + b*x)^7)/(56*(b*d - a*e)^2*(d + e*x)^7)

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])

Rubi steps

\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^9} \, dx &=\int \frac {(a+b x)^6}{(d+e x)^9} \, dx\\ &=\frac {(a+b x)^7}{8 (b d-a e) (d+e x)^8}+\frac {b \int \frac {(a+b x)^6}{(d+e x)^8} \, dx}{8 (b d-a e)}\\ &=\frac {(a+b x)^7}{8 (b d-a e) (d+e x)^8}+\frac {b (a+b x)^7}{56 (b d-a e)^2 (d+e x)^7}\\ \end {align*}

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Mathematica [B]  time = 0.09, size = 277, normalized size = 4.78 \begin {gather*} -\frac {7 a^6 e^6+6 a^5 b e^5 (d+8 e x)+5 a^4 b^2 e^4 \left (d^2+8 d e x+28 e^2 x^2\right )+4 a^3 b^3 e^3 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+3 a^2 b^4 e^2 \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 )+2 a b^5 e \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 )+b^6 \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )}{56 e^7 (d+e x)^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^9,x]

[Out]

-1/56*(7*a^6*e^6 + 6*a^5*b*e^5*(d + 8*e*x) + 5*a^4*b^2*e^4*(d^2 + 8*d*e*x + 28*e^2*x^2) + 4*a^3*b^3*e^3*(d^3 +
8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + 3*a^2*b^4*e^2*(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) + 2*a*b^5*e*(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) + b^6*(d^
6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5*x^5 + 28*e^6*x^6))/(e^7*(d + e*x)^
8)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^9} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^9,x]

[Out]

IntegrateAlgebraic[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^9, x]

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fricas [B]  time = 0.38, size = 430, normalized size = 7.41 \begin {gather*} -\frac {28 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 2 \, a b^{5} d^{5} e + 3 \, a^{2} b^{4} d^{4} e^{2} + 4 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} + 6 \, a^{5} b d e^{5} + 7 \, a^{6} e^{6} + 56 \, {\left (b^{6} d e^{5} + 2 \, a b^{5} e^{6}\right )} x^{5} + 70 \, {\left (b^{6} d^{2} e^{4} + 2 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} + 56 \, {\left (b^{6} d^{3} e^{3} + 2 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + 4 \, a^{3} b^{3} e^{6}\right )} x^{3} + 28 \, {\left (b^{6} d^{4} e^{2} + 2 \, a b^{5} d^{3} e^{3} + 3 \, a^{2} b^{4} d^{2} e^{4} + 4 \, a^{3} b^{3} d e^{5} + 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 8 \, {\left (b^{6} d^{5} e + 2 \, a b^{5} d^{4} e^{2} + 3 \, a^{2} b^{4} d^{3} e^{3} + 4 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 6 \, a^{5} b e^{6}\right )} x}{56 \, {\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^9,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.17, size = 352, normalized size = 6.07 \begin {gather*} -\frac {{\left (28 \, b^{6} x^{6} e^{6} + 56 \, b^{6} d x^{5} e^{5} + 70 \, b^{6} d^{2} x^{4} e^{4} + 56 \, b^{6} d^{3} x^{3} e^{3} + 28 \, b^{6} d^{4} x^{2} e^{2} + 8 \, b^{6} d^{5} x e + b^{6} d^{6} + 112 \, a b^{5} x^{5} e^{6} + 140 \, a b^{5} d x^{4} e^{5} + 112 \, a b^{5} d^{2} x^{3} e^{4} + 56 \, a b^{5} d^{3} x^{2} e^{3} + 16 \, a b^{5} d^{4} x e^{2} + 2 \, a b^{5} d^{5} e + 210 \, a^{2} b^{4} x^{4} e^{6} + 168 \, a^{2} b^{4} d x^{3} e^{5} + 84 \, a^{2} b^{4} d^{2} x^{2} e^{4} + 24 \, a^{2} b^{4} d^{3} x e^{3} + 3 \, a^{2} b^{4} d^{4} e^{2} + 224 \, a^{3} b^{3} x^{3} e^{6} + 112 \, a^{3} b^{3} d x^{2} e^{5} + 32 \, a^{3} b^{3} d^{2} x e^{4} + 4 \, a^{3} b^{3} d^{3} e^{3} + 140 \, a^{4} b^{2} x^{2} e^{6} + 40 \, a^{4} b^{2} d x e^{5} + 5 \, a^{4} b^{2} d^{2} e^{4} + 48 \, a^{5} b x e^{6} + 6 \, a^{5} b d e^{5} + 7 \, a^{6} e^{6}\right )} e^{\left (-7\right )}}{56 \, {\left (x e + d\right )}^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^9,x, algorithm="giac")

[Out]

-1/56*(28*b^6*x^6*e^6 + 56*b^6*d*x^5*e^5 + 70*b^6*d^2*x^4*e^4 + 56*b^6*d^3*x^3*e^3 + 28*b^6*d^4*x^2*e^2 + 8*b^
6*d^5*x*e + b^6*d^6 + 112*a*b^5*x^5*e^6 + 140*a*b^5*d*x^4*e^5 + 112*a*b^5*d^2*x^3*e^4 + 56*a*b^5*d^3*x^2*e^3 +
16*a*b^5*d^4*x*e^2 + 2*a*b^5*d^5*e + 210*a^2*b^4*x^4*e^6 + 168*a^2*b^4*d*x^3*e^5 + 84*a^2*b^4*d^2*x^2*e^4 + 2
4*a^2*b^4*d^3*x*e^3 + 3*a^2*b^4*d^4*e^2 + 224*a^3*b^3*x^3*e^6 + 112*a^3*b^3*d*x^2*e^5 + 32*a^3*b^3*d^2*x*e^4 +
4*a^3*b^3*d^3*e^3 + 140*a^4*b^2*x^2*e^6 + 40*a^4*b^2*d*x*e^5 + 5*a^4*b^2*d^2*e^4 + 48*a^5*b*x*e^6 + 6*a^5*b*d
*e^5 + 7*a^6*e^6)*e^(-7)/(x*e + d)^8

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maple [B]  time = 0.05, size = 357, normalized size = 6.16 \begin {gather*} -\frac {b^{6}}{2 \left (e x +d \right )^{2} e^{7}}-\frac {2 \left (a e -b d \right ) b^{5}}{\left (e x +d \right )^{3} e^{7}}-\frac {15 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}}{4 \left (e x +d \right )^{4} e^{7}}-\frac {4 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) b^{3}}{\left (e x +d \right )^{5} e^{7}}-\frac {5 \left (e^{4} a^{4}-4 d \,e^{3} a^{3} b +6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) b^{2}}{2 \left (e x +d \right )^{6} e^{7}}-\frac {6 \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right ) b}{7 \left (e x +d \right )^{7} e^{7}}-\frac {a^{6} e^{6}-6 d \,e^{5} a^{5} b +15 d^{2} e^{4} a^{4} b^{2}-20 d^{3} e^{3} a^{3} b^{3}+15 d^{4} a^{2} b^{4} e^{2}-6 d^{5} e a \,b^{5}+b^{6} d^{6}}{8 \left (e x +d \right )^{8} e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^9,x)

[Out]

-6/7*b*(a^5*e^5-5*a^4*b*d*e^4+10*a^3*b^2*d^2*e^3-10*a^2*b^3*d^3*e^2+5*a*b^4*d^4*e-b^5*d^5)/e^7/(e*x+d)^7-1/2*b
^6/e^7/(e*x+d)^2-15/4*b^4*(a^2*e^2-2*a*b*d*e+b^2*d^2)/e^7/(e*x+d)^4-2*b^5*(a*e-b*d)/e^7/(e*x+d)^3-5/2*b^2*(a^4
*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)/e^7/(e*x+d)^6-4*b^3*(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2
*d^2*e-b^3*d^3)/e^7/(e*x+d)^5-1/8*(a^6*e^6-6*a^5*b*d*e^5+15*a^4*b^2*d^2*e^4-20*a^3*b^3*d^3*e^3+15*a^2*b^4*d^4*
e^2-6*a*b^5*d^5*e+b^6*d^6)/e^7/(e*x+d)^8

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maxima [B]  time = 1.69, size = 430, normalized size = 7.41 \begin {gather*} -\frac {28 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 2 \, a b^{5} d^{5} e + 3 \, a^{2} b^{4} d^{4} e^{2} + 4 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} + 6 \, a^{5} b d e^{5} + 7 \, a^{6} e^{6} + 56 \, {\left (b^{6} d e^{5} + 2 \, a b^{5} e^{6}\right )} x^{5} + 70 \, {\left (b^{6} d^{2} e^{4} + 2 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} + 56 \, {\left (b^{6} d^{3} e^{3} + 2 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + 4 \, a^{3} b^{3} e^{6}\right )} x^{3} + 28 \, {\left (b^{6} d^{4} e^{2} + 2 \, a b^{5} d^{3} e^{3} + 3 \, a^{2} b^{4} d^{2} e^{4} + 4 \, a^{3} b^{3} d e^{5} + 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 8 \, {\left (b^{6} d^{5} e + 2 \, a b^{5} d^{4} e^{2} + 3 \, a^{2} b^{4} d^{3} e^{3} + 4 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 6 \, a^{5} b e^{6}\right )} x}{56 \, {\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^9,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.14, size = 410, normalized size = 7.07 \begin {gather*} -\frac {\frac {7\,a^6\,e^6+6\,a^5\,b\,d\,e^5+5\,a^4\,b^2\,d^2\,e^4+4\,a^3\,b^3\,d^3\,e^3+3\,a^2\,b^4\,d^4\,e^2+2\,a\,b^5\,d^5\,e+b^6\,d^6}{56\,e^7}+\frac {b^6\,x^6}{2\,e}+\frac {b^3\,x^3\,\left (4\,a^3\,e^3+3\,a^2\,b\,d\,e^2+2\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{e^4}+\frac {b\,x\,\left (6\,a^5\,e^5+5\,a^4\,b\,d\,e^4+4\,a^3\,b^2\,d^2\,e^3+3\,a^2\,b^3\,d^3\,e^2+2\,a\,b^4\,d^4\,e+b^5\,d^5\right )}{7\,e^6}+\frac {b^5\,x^5\,\left (2\,a\,e+b\,d\right )}{e^2}+\frac {b^2\,x^2\,\left (5\,a^4\,e^4+4\,a^3\,b\,d\,e^3+3\,a^2\,b^2\,d^2\,e^2+2\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{2\,e^5}+\frac {5\,b^4\,x^4\,\left (3\,a^2\,e^2+2\,a\,b\,d\,e+b^2\,d^2\right )}{4\,e^3}}{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^2 + b^2*x^2 + 2*a*b*x)^3/(d + e*x)^9,x)

[Out]

-((7*a^6*e^6 + b^6*d^6 + 3*a^2*b^4*d^4*e^2 + 4*a^3*b^3*d^3*e^3 + 5*a^4*b^2*d^2*e^4 + 2*a*b^5*d^5*e + 6*a^5*b*d
*e^5)/(56*e^7) + (b^6*x^6)/(2*e) + (b^3*x^3*(4*a^3*e^3 + b^3*d^3 + 2*a*b^2*d^2*e + 3*a^2*b*d*e^2))/e^4 + (b*x*
(6*a^5*e^5 + b^5*d^5 + 3*a^2*b^3*d^3*e^2 + 4*a^3*b^2*d^2*e^3 + 2*a*b^4*d^4*e + 5*a^4*b*d*e^4))/(7*e^6) + (b^5*
x^5*(2*a*e + b*d))/e^2 + (b^2*x^2*(5*a^4*e^4 + b^4*d^4 + 3*a^2*b^2*d^2*e^2 + 2*a*b^3*d^3*e + 4*a^3*b*d*e^3))/(
2*e^5) + (5*b^4*x^4*(3*a^2*e^2 + b^2*d^2 + 2*a*b*d*e))/(4*e^3))/(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 + 56*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**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**9,x)

[Out]

Timed out

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