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]

1/7*(b*x+a)^7/(-a*e+b*d)/(e*x+d)^7

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Rubi [A]  time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, 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*}

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Mathematica [B]  time = 0.10, size = 271, normalized size = 9.68 \[ -\frac {a^6 e^6+a^5 b e^5 (d+7 e x)+a^4 b^2 e^4 \left (d^2+7 d e x+21 e^2 x^2\right )+a^3 b^3 e^3 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+a^2 b^4 e^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+a b^5 e \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )+b^6 \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )}{7 e^7 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/7*(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
+ 21*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))/(e^7*(d + e*x)^7)

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fricas [B]  time = 0.56, size = 398, normalized size = 14.21 \[ -\frac {7 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + a b^{5} d^{5} e + a^{2} b^{4} d^{4} e^{2} + a^{3} b^{3} d^{3} e^{3} + a^{4} b^{2} d^{2} e^{4} + a^{5} b d e^{5} + a^{6} e^{6} + 21 \, {\left (b^{6} d e^{5} + a b^{5} e^{6}\right )} x^{5} + 35 \, {\left (b^{6} d^{2} e^{4} + a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 35 \, {\left (b^{6} d^{3} e^{3} + a b^{5} d^{2} e^{4} + a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} + 21 \, {\left (b^{6} d^{4} e^{2} + a b^{5} d^{3} e^{3} + a^{2} b^{4} d^{2} e^{4} + a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 7 \, {\left (b^{6} d^{5} e + a b^{5} d^{4} e^{2} + a^{2} b^{4} d^{3} e^{3} + a^{3} b^{3} d^{2} e^{4} + a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x}{7 \, {\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \]

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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giac [B]  time = 0.18, size = 346, normalized size = 12.36 \[ -\frac {{\left (7 \, b^{6} x^{6} e^{6} + 21 \, b^{6} d x^{5} e^{5} + 35 \, b^{6} d^{2} x^{4} e^{4} + 35 \, b^{6} d^{3} x^{3} e^{3} + 21 \, b^{6} d^{4} x^{2} e^{2} + 7 \, b^{6} d^{5} x e + b^{6} d^{6} + 21 \, a b^{5} x^{5} e^{6} + 35 \, a b^{5} d x^{4} e^{5} + 35 \, a b^{5} d^{2} x^{3} e^{4} + 21 \, a b^{5} d^{3} x^{2} e^{3} + 7 \, a b^{5} d^{4} x e^{2} + a b^{5} d^{5} e + 35 \, a^{2} b^{4} x^{4} e^{6} + 35 \, a^{2} b^{4} d x^{3} e^{5} + 21 \, a^{2} b^{4} d^{2} x^{2} e^{4} + 7 \, a^{2} b^{4} d^{3} x e^{3} + a^{2} b^{4} d^{4} e^{2} + 35 \, a^{3} b^{3} x^{3} e^{6} + 21 \, a^{3} b^{3} d x^{2} e^{5} + 7 \, a^{3} b^{3} d^{2} x e^{4} + a^{3} b^{3} d^{3} e^{3} + 21 \, a^{4} b^{2} x^{2} e^{6} + 7 \, a^{4} b^{2} d x e^{5} + a^{4} b^{2} d^{2} e^{4} + 7 \, a^{5} b x e^{6} + a^{5} b d e^{5} + a^{6} e^{6}\right )} e^{\left (-7\right )}}{7 \, {\left (x e + d\right )}^{7}} \]

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

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maple [B]  time = 0.05, size = 357, normalized size = 12.75 \[ -\frac {b^{6}}{\left (e x +d \right ) e^{7}}-\frac {3 \left (a e -b d \right ) b^{5}}{\left (e x +d \right )^{2} e^{7}}-\frac {5 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}}{\left (e x +d \right )^{3} e^{7}}-\frac {5 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) b^{3}}{\left (e x +d \right )^{4} e^{7}}-\frac {3 \left (e^{4} a^{4}-4 d \,e^{3} a^{3} b +6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) b^{2}}{\left (e x +d \right )^{5} e^{7}}-\frac {\left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right ) b}{\left (e x +d \right )^{6} e^{7}}-\frac {a^{6} e^{6}-6 d \,e^{5} a^{5} b +15 d^{2} e^{4} a^{4} b^{2}-20 d^{3} e^{3} a^{3} b^{3}+15 d^{4} a^{2} b^{4} e^{2}-6 d^{5} e a \,b^{5}+b^{6} d^{6}}{7 \left (e x +d \right )^{7} e^{7}} \]

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]

-b^6/e^7/(e*x+d)-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)^7-3*b^5*(a*e-b*d)/e^7/(e*x+d)^2-5*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)^4-5*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)^6-3*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 = 2.10, size = 398, normalized size = 14.21 \[ -\frac {7 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + a b^{5} d^{5} e + a^{2} b^{4} d^{4} e^{2} + a^{3} b^{3} d^{3} e^{3} + a^{4} b^{2} d^{2} e^{4} + a^{5} b d e^{5} + a^{6} e^{6} + 21 \, {\left (b^{6} d e^{5} + a b^{5} e^{6}\right )} x^{5} + 35 \, {\left (b^{6} d^{2} e^{4} + a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 35 \, {\left (b^{6} d^{3} e^{3} + a b^{5} d^{2} e^{4} + a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} + 21 \, {\left (b^{6} d^{4} e^{2} + a b^{5} d^{3} e^{3} + a^{2} b^{4} d^{2} e^{4} + a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 7 \, {\left (b^{6} d^{5} e + a b^{5} d^{4} e^{2} + a^{2} b^{4} d^{3} e^{3} + a^{3} b^{3} d^{2} e^{4} + a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x}{7 \, {\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \]

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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mupad [B]  time = 0.59, size = 378, normalized size = 13.50 \[ -\frac {\frac {a^6\,e^6+a^5\,b\,d\,e^5+a^4\,b^2\,d^2\,e^4+a^3\,b^3\,d^3\,e^3+a^2\,b^4\,d^4\,e^2+a\,b^5\,d^5\,e+b^6\,d^6}{7\,e^7}+\frac {b^6\,x^6}{e}+\frac {5\,b^3\,x^3\,\left (a^3\,e^3+a^2\,b\,d\,e^2+a\,b^2\,d^2\,e+b^3\,d^3\right )}{e^4}+\frac {b\,x\,\left (a^5\,e^5+a^4\,b\,d\,e^4+a^3\,b^2\,d^2\,e^3+a^2\,b^3\,d^3\,e^2+a\,b^4\,d^4\,e+b^5\,d^5\right )}{e^6}+\frac {3\,b^5\,x^5\,\left (a\,e+b\,d\right )}{e^2}+\frac {3\,b^2\,x^2\,\left (a^4\,e^4+a^3\,b\,d\,e^3+a^2\,b^2\,d^2\,e^2+a\,b^3\,d^3\,e+b^4\,d^4\right )}{e^5}+\frac {5\,b^4\,x^4\,\left (a^2\,e^2+a\,b\,d\,e+b^2\,d^2\right )}{e^3}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \]

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)^8,x)

[Out]

-((a^6*e^6 + b^6*d^6 + 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*b^5*d^5*e + a^5*b*d*e^5)/(7*e^7
) + (b^6*x^6)/e + (5*b^3*x^3*(a^3*e^3 + b^3*d^3 + a*b^2*d^2*e + a^2*b*d*e^2))/e^4 + (b*x*(a^5*e^5 + b^5*d^5 +
a^2*b^3*d^3*e^2 + a^3*b^2*d^2*e^3 + a*b^4*d^4*e + a^4*b*d*e^4))/e^6 + (3*b^5*x^5*(a*e + b*d))/e^2 + (3*b^2*x^2
*(a^4*e^4 + b^4*d^4 + a^2*b^2*d^2*e^2 + a*b^3*d^3*e + a^3*b*d*e^3))/e^5 + (5*b^4*x^4*(a^2*e^2 + b^2*d^2 + a*b*
d*e))/e^3)/(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 \[ \text {Timed out} \]

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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