3.1477 \(\int \frac {(a^2+2 a b x+b^2 x^2)^2}{(d+e x)^8} \, dx\)

Optimal. Leaf size=89 \[ \frac {b^2 (a+b x)^5}{105 (d+e x)^5 (b d-a e)^3}+\frac {b (a+b x)^5}{21 (d+e x)^6 (b d-a e)^2}+\frac {(a+b x)^5}{7 (d+e x)^7 (b d-a e)} \]

[Out]

1/7*(b*x+a)^5/(-a*e+b*d)/(e*x+d)^7+1/21*b*(b*x+a)^5/(-a*e+b*d)^2/(e*x+d)^6+1/105*b^2*(b*x+a)^5/(-a*e+b*d)^3/(e
*x+d)^5

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Rubi [A]  time = 0.02, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {27, 45, 37} \[ \frac {b^2 (a+b x)^5}{105 (d+e x)^5 (b d-a e)^3}+\frac {b (a+b x)^5}{21 (d+e x)^6 (b d-a e)^2}+\frac {(a+b x)^5}{7 (d+e x)^7 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*x)^5/(7*(b*d - a*e)*(d + e*x)^7) + (b*(a + b*x)^5)/(21*(b*d - a*e)^2*(d + e*x)^6) + (b^2*(a + b*x)^5)/(
105*(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])

Rubi steps

\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^8} \, dx &=\int \frac {(a+b x)^4}{(d+e x)^8} \, dx\\ &=\frac {(a+b x)^5}{7 (b d-a e) (d+e x)^7}+\frac {(2 b) \int \frac {(a+b x)^4}{(d+e x)^7} \, dx}{7 (b d-a e)}\\ &=\frac {(a+b x)^5}{7 (b d-a e) (d+e x)^7}+\frac {b (a+b x)^5}{21 (b d-a e)^2 (d+e x)^6}+\frac {b^2 \int \frac {(a+b x)^4}{(d+e x)^6} \, dx}{21 (b d-a e)^2}\\ &=\frac {(a+b x)^5}{7 (b d-a e) (d+e x)^7}+\frac {b (a+b x)^5}{21 (b d-a e)^2 (d+e x)^6}+\frac {b^2 (a+b x)^5}{105 (b d-a e)^3 (d+e x)^5}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 144, normalized size = 1.62 \[ -\frac {15 a^4 e^4+10 a^3 b e^3 (d+7 e x)+6 a^2 b^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+3 a b^3 e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+b^4 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )}{105 e^5 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/105*(15*a^4*e^4 + 10*a^3*b*e^3*(d + 7*e*x) + 6*a^2*b^2*e^2*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*a*b^3*e*(d^3 +
7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + b^4*(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))/(
e^5*(d + e*x)^7)

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fricas [B]  time = 0.65, size = 247, normalized size = 2.78 \[ -\frac {35 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 3 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 10 \, a^{3} b d e^{3} + 15 \, a^{4} e^{4} + 35 \, {\left (b^{4} d e^{3} + 3 \, a b^{3} e^{4}\right )} x^{3} + 21 \, {\left (b^{4} d^{2} e^{2} + 3 \, a b^{3} d e^{3} + 6 \, a^{2} b^{2} e^{4}\right )} x^{2} + 7 \, {\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + 10 \, a^{3} b e^{4}\right )} x}{105 \, {\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(35*b^4*e^4*x^4 + b^4*d^4 + 3*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 10*a^3*b*d*e^3 + 15*a^4*e^4 + 35*(b^4*d
*e^3 + 3*a*b^3*e^4)*x^3 + 21*(b^4*d^2*e^2 + 3*a*b^3*d*e^3 + 6*a^2*b^2*e^4)*x^2 + 7*(b^4*d^3*e + 3*a*b^3*d^2*e^
2 + 6*a^2*b^2*d*e^3 + 10*a^3*b*e^4)*x)/(e^12*x^7 + 7*d*e^11*x^6 + 21*d^2*e^10*x^5 + 35*d^3*e^9*x^4 + 35*d^4*e^
8*x^3 + 21*d^5*e^7*x^2 + 7*d^6*e^6*x + d^7*e^5)

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giac [B]  time = 0.16, size = 174, normalized size = 1.96 \[ -\frac {{\left (35 \, b^{4} x^{4} e^{4} + 35 \, b^{4} d x^{3} e^{3} + 21 \, b^{4} d^{2} x^{2} e^{2} + 7 \, b^{4} d^{3} x e + b^{4} d^{4} + 105 \, a b^{3} x^{3} e^{4} + 63 \, a b^{3} d x^{2} e^{3} + 21 \, a b^{3} d^{2} x e^{2} + 3 \, a b^{3} d^{3} e + 126 \, a^{2} b^{2} x^{2} e^{4} + 42 \, a^{2} b^{2} d x e^{3} + 6 \, a^{2} b^{2} d^{2} e^{2} + 70 \, a^{3} b x e^{4} + 10 \, a^{3} b d e^{3} + 15 \, a^{4} e^{4}\right )} e^{\left (-5\right )}}{105 \, {\left (x e + d\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(35*b^4*x^4*e^4 + 35*b^4*d*x^3*e^3 + 21*b^4*d^2*x^2*e^2 + 7*b^4*d^3*x*e + b^4*d^4 + 105*a*b^3*x^3*e^4 +
63*a*b^3*d*x^2*e^3 + 21*a*b^3*d^2*x*e^2 + 3*a*b^3*d^3*e + 126*a^2*b^2*x^2*e^4 + 42*a^2*b^2*d*x*e^3 + 6*a^2*b^
2*d^2*e^2 + 70*a^3*b*x*e^4 + 10*a^3*b*d*e^3 + 15*a^4*e^4)*e^(-5)/(x*e + d)^7

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maple [B]  time = 0.05, size = 186, normalized size = 2.09 \[ -\frac {b^{4}}{3 \left (e x +d \right )^{3} e^{5}}-\frac {\left (a e -b d \right ) b^{3}}{\left (e x +d \right )^{4} e^{5}}-\frac {6 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}}{5 \left (e x +d \right )^{5} e^{5}}-\frac {2 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) b}{3 \left (e x +d \right )^{6} e^{5}}-\frac {e^{4} a^{4}-4 d \,e^{3} a^{3} b +6 d^{2} e^{2} b^{2} a^{2}-4 d^{3} a \,b^{3} e +b^{4} d^{4}}{7 \left (e x +d \right )^{7} e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.54, size = 247, normalized size = 2.78 \[ -\frac {35 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 3 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 10 \, a^{3} b d e^{3} + 15 \, a^{4} e^{4} + 35 \, {\left (b^{4} d e^{3} + 3 \, a b^{3} e^{4}\right )} x^{3} + 21 \, {\left (b^{4} d^{2} e^{2} + 3 \, a b^{3} d e^{3} + 6 \, a^{2} b^{2} e^{4}\right )} x^{2} + 7 \, {\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + 10 \, a^{3} b e^{4}\right )} x}{105 \, {\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(35*b^4*e^4*x^4 + b^4*d^4 + 3*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 10*a^3*b*d*e^3 + 15*a^4*e^4 + 35*(b^4*d
*e^3 + 3*a*b^3*e^4)*x^3 + 21*(b^4*d^2*e^2 + 3*a*b^3*d*e^3 + 6*a^2*b^2*e^4)*x^2 + 7*(b^4*d^3*e + 3*a*b^3*d^2*e^
2 + 6*a^2*b^2*d*e^3 + 10*a^3*b*e^4)*x)/(e^12*x^7 + 7*d*e^11*x^6 + 21*d^2*e^10*x^5 + 35*d^3*e^9*x^4 + 35*d^4*e^
8*x^3 + 21*d^5*e^7*x^2 + 7*d^6*e^6*x + d^7*e^5)

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mupad [B]  time = 0.60, size = 237, normalized size = 2.66 \[ -\frac {\frac {15\,a^4\,e^4+10\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2+3\,a\,b^3\,d^3\,e+b^4\,d^4}{105\,e^5}+\frac {b^4\,x^4}{3\,e}+\frac {b^3\,x^3\,\left (3\,a\,e+b\,d\right )}{3\,e^2}+\frac {b\,x\,\left (10\,a^3\,e^3+6\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{15\,e^4}+\frac {b^2\,x^2\,\left (6\,a^2\,e^2+3\,a\,b\,d\,e+b^2\,d^2\right )}{5\,e^3}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((15*a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 + 3*a*b^3*d^3*e + 10*a^3*b*d*e^3)/(105*e^5) + (b^4*x^4)/(3*e) + (b
^3*x^3*(3*a*e + b*d))/(3*e^2) + (b*x*(10*a^3*e^3 + b^3*d^3 + 3*a*b^2*d^2*e + 6*a^2*b*d*e^2))/(15*e^4) + (b^2*x
^2*(6*a^2*e^2 + b^2*d^2 + 3*a*b*d*e))/(5*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 [B]  time = 9.88, size = 267, normalized size = 3.00 \[ \frac {- 15 a^{4} e^{4} - 10 a^{3} b d e^{3} - 6 a^{2} b^{2} d^{2} e^{2} - 3 a b^{3} d^{3} e - b^{4} d^{4} - 35 b^{4} e^{4} x^{4} + x^{3} \left (- 105 a b^{3} e^{4} - 35 b^{4} d e^{3}\right ) + x^{2} \left (- 126 a^{2} b^{2} e^{4} - 63 a b^{3} d e^{3} - 21 b^{4} d^{2} e^{2}\right ) + x \left (- 70 a^{3} b e^{4} - 42 a^{2} b^{2} d e^{3} - 21 a b^{3} d^{2} e^{2} - 7 b^{4} d^{3} e\right )}{105 d^{7} e^{5} + 735 d^{6} e^{6} x + 2205 d^{5} e^{7} x^{2} + 3675 d^{4} e^{8} x^{3} + 3675 d^{3} e^{9} x^{4} + 2205 d^{2} e^{10} x^{5} + 735 d e^{11} x^{6} + 105 e^{12} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**8,x)

[Out]

(-15*a**4*e**4 - 10*a**3*b*d*e**3 - 6*a**2*b**2*d**2*e**2 - 3*a*b**3*d**3*e - b**4*d**4 - 35*b**4*e**4*x**4 +
x**3*(-105*a*b**3*e**4 - 35*b**4*d*e**3) + x**2*(-126*a**2*b**2*e**4 - 63*a*b**3*d*e**3 - 21*b**4*d**2*e**2) +
x*(-70*a**3*b*e**4 - 42*a**2*b**2*d*e**3 - 21*a*b**3*d**2*e**2 - 7*b**4*d**3*e))/(105*d**7*e**5 + 735*d**6*e*
*6*x + 2205*d**5*e**7*x**2 + 3675*d**4*e**8*x**3 + 3675*d**3*e**9*x**4 + 2205*d**2*e**10*x**5 + 735*d*e**11*x*
*6 + 105*e**12*x**7)

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