3.1497 \(\int \frac{(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^8} \, dx\)

Optimal. Leaf size=28 \[ \frac{(a+b x)^7}{7 (d+e x)^7 (b d-a e)} \]

[Out]

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

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Rubi [A]  time = 0.0046189, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {27, 37} \[ \frac{(a+b x)^7}{7 (d+e x)^7 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^8,x]

[Out]

(a + b*x)^7/(7*(b*d - a*e)*(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]

Rubi steps

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

Mathematica [B]  time = 0.0989133, size = 271, normalized size = 9.68 \[ -\frac{a^2 b^4 e^2 \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )+a^3 b^3 e^3 \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )+a^4 b^2 e^4 \left (d^2+7 d e x+21 e^2 x^2\right )+a^5 b e^5 (d+7 e x)+a^6 e^6+a b^5 e \left (21 d^3 e^2 x^2+35 d^2 e^3 x^3+7 d^4 e x+d^5+35 d e^4 x^4+21 e^5 x^5\right )+b^6 \left (21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+7 d^5 e x+d^6+21 d e^5 x^5+7 e^6 x^6\right )}{7 e^7 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^8,x]

[Out]

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

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Maple [B]  time = 0.048, size = 357, normalized size = 12.8 \begin{align*} -5\,{\frac{{b}^{4} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{{e}^{7} \left ( ex+d \right ) ^{3}}}-{\frac{b \left ({a}^{5}{e}^{5}-5\,{a}^{4}bd{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 ) }{{e}^{7} \left ( ex+d \right ) ^{6}}}-5\,{\frac{{b}^{3} \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) }{{e}^{7} \left ( ex+d \right ) ^{4}}}-3\,{\frac{{b}^{5} \left ( ae-bd \right ) }{{e}^{7} \left ( ex+d \right ) ^{2}}}-{\frac{{e}^{6}{a}^{6}-6\,{a}^{5}bd{e}^{5}+15\,{d}^{2}{e}^{4}{a}^{4}{b}^{2}-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+{d}^{6}{b}^{6}}{7\,{e}^{7} \left ( ex+d \right ) ^{7}}}-{\frac{{b}^{6}}{{e}^{7} \left ( ex+d \right ) }}-3\,{\frac{{b}^{2} \left ({a}^{4}{e}^{4}-4\,{a}^{3}bd{e}^{3}+6\,{d}^{2}{e}^{2}{b}^{2}{a}^{2}-4\,a{b}^{3}{d}^{3}e+{b}^{4}{d}^{4} \right ) }{{e}^{7} \left ( ex+d \right ) ^{5}}} \end{align*}

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)^8,x)

[Out]

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

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Maxima [B]  time = 1.2449, size = 537, normalized size = 19.18 \begin{align*} -\frac{7 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + a b^{5} d^{5} e + a^{2} b^{4} d^{4} e^{2} + a^{3} b^{3} d^{3} e^{3} + a^{4} b^{2} d^{2} e^{4} + a^{5} b d e^{5} + a^{6} e^{6} + 21 \,{\left (b^{6} d e^{5} + a b^{5} e^{6}\right )} x^{5} + 35 \,{\left (b^{6} d^{2} e^{4} + a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 35 \,{\left (b^{6} d^{3} e^{3} + a b^{5} d^{2} e^{4} + a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} + 21 \,{\left (b^{6} d^{4} e^{2} + a b^{5} d^{3} e^{3} + a^{2} b^{4} d^{2} e^{4} + a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 7 \,{\left (b^{6} d^{5} e + a b^{5} d^{4} e^{2} + a^{2} b^{4} d^{3} e^{3} + a^{3} b^{3} d^{2} e^{4} + a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x}{7 \,{\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \end{align*}

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)^8,x, algorithm="maxima")

[Out]

-1/7*(7*b^6*e^6*x^6 + b^6*d^6 + a*b^5*d^5*e + a^2*b^4*d^4*e^2 + a^3*b^3*d^3*e^3 + a^4*b^2*d^2*e^4 + a^5*b*d*e^
5 + a^6*e^6 + 21*(b^6*d*e^5 + a*b^5*e^6)*x^5 + 35*(b^6*d^2*e^4 + a*b^5*d*e^5 + a^2*b^4*e^6)*x^4 + 35*(b^6*d^3*
e^3 + a*b^5*d^2*e^4 + a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 + 21*(b^6*d^4*e^2 + a*b^5*d^3*e^3 + a^2*b^4*d^2*e^4 + a
^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 7*(b^6*d^5*e + a*b^5*d^4*e^2 + a^2*b^4*d^3*e^3 + a^3*b^3*d^2*e^4 + a^4*b^2*d
*e^5 + a^5*b*e^6)*x)/(e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5 + 35*d^3*e^11*x^4 + 35*d^4*e^10*x^3 + 21*d^5*e
^9*x^2 + 7*d^6*e^8*x + d^7*e^7)

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Fricas [B]  time = 1.74631, size = 787, normalized size = 28.11 \begin{align*} -\frac{7 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + a b^{5} d^{5} e + a^{2} b^{4} d^{4} e^{2} + a^{3} b^{3} d^{3} e^{3} + a^{4} b^{2} d^{2} e^{4} + a^{5} b d e^{5} + a^{6} e^{6} + 21 \,{\left (b^{6} d e^{5} + a b^{5} e^{6}\right )} x^{5} + 35 \,{\left (b^{6} d^{2} e^{4} + a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 35 \,{\left (b^{6} d^{3} e^{3} + a b^{5} d^{2} e^{4} + a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} + 21 \,{\left (b^{6} d^{4} e^{2} + a b^{5} d^{3} e^{3} + a^{2} b^{4} d^{2} e^{4} + a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 7 \,{\left (b^{6} d^{5} e + a b^{5} d^{4} e^{2} + a^{2} b^{4} d^{3} e^{3} + a^{3} b^{3} d^{2} e^{4} + a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x}{7 \,{\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \end{align*}

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)^8,x, algorithm="fricas")

[Out]

-1/7*(7*b^6*e^6*x^6 + b^6*d^6 + a*b^5*d^5*e + a^2*b^4*d^4*e^2 + a^3*b^3*d^3*e^3 + a^4*b^2*d^2*e^4 + a^5*b*d*e^
5 + a^6*e^6 + 21*(b^6*d*e^5 + a*b^5*e^6)*x^5 + 35*(b^6*d^2*e^4 + a*b^5*d*e^5 + a^2*b^4*e^6)*x^4 + 35*(b^6*d^3*
e^3 + a*b^5*d^2*e^4 + a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 + 21*(b^6*d^4*e^2 + a*b^5*d^3*e^3 + a^2*b^4*d^2*e^4 + a
^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 7*(b^6*d^5*e + a*b^5*d^4*e^2 + a^2*b^4*d^3*e^3 + a^3*b^3*d^2*e^4 + a^4*b^2*d
*e^5 + a^5*b*e^6)*x)/(e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5 + 35*d^3*e^11*x^4 + 35*d^4*e^10*x^3 + 21*d^5*e
^9*x^2 + 7*d^6*e^8*x + d^7*e^7)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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)**8,x)

[Out]

Timed out

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Giac [B]  time = 1.13582, size = 467, normalized size = 16.68 \begin{align*} -\frac{{\left (7 \, b^{6} x^{6} e^{6} + 21 \, b^{6} d x^{5} e^{5} + 35 \, b^{6} d^{2} x^{4} e^{4} + 35 \, b^{6} d^{3} x^{3} e^{3} + 21 \, b^{6} d^{4} x^{2} e^{2} + 7 \, b^{6} d^{5} x e + b^{6} d^{6} + 21 \, a b^{5} x^{5} e^{6} + 35 \, a b^{5} d x^{4} e^{5} + 35 \, a b^{5} d^{2} x^{3} e^{4} + 21 \, a b^{5} d^{3} x^{2} e^{3} + 7 \, a b^{5} d^{4} x e^{2} + a b^{5} d^{5} e + 35 \, a^{2} b^{4} x^{4} e^{6} + 35 \, a^{2} b^{4} d x^{3} e^{5} + 21 \, a^{2} b^{4} d^{2} x^{2} e^{4} + 7 \, a^{2} b^{4} d^{3} x e^{3} + a^{2} b^{4} d^{4} e^{2} + 35 \, a^{3} b^{3} x^{3} e^{6} + 21 \, a^{3} b^{3} d x^{2} e^{5} + 7 \, a^{3} b^{3} d^{2} x e^{4} + a^{3} b^{3} d^{3} e^{3} + 21 \, a^{4} b^{2} x^{2} e^{6} + 7 \, a^{4} b^{2} d x e^{5} + a^{4} b^{2} d^{2} e^{4} + 7 \, a^{5} b x e^{6} + a^{5} b d e^{5} + a^{6} e^{6}\right )} e^{\left (-7\right )}}{7 \,{\left (x e + d\right )}^{7}} \end{align*}

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)^8,x, algorithm="giac")

[Out]

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