3.2130 \(\int \frac {(a+b x+c x^2)^2}{(d+e x)^8} \, dx\)

Optimal. Leaf size=156 \[ -\frac {-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{5 e^5 (d+e x)^5}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5 (d+e x)^6}-\frac {\left (a e^2-b d e+c d^2\right )^2}{7 e^5 (d+e x)^7}+\frac {c (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac {c^2}{3 e^5 (d+e x)^3} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {698} \[ -\frac {-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{5 e^5 (d+e x)^5}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5 (d+e x)^6}-\frac {\left (a e^2-b d e+c d^2\right )^2}{7 e^5 (d+e x)^7}+\frac {c (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac {c^2}{3 e^5 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 161, normalized size = 1.03 \[ -\frac {2 e^2 \left (15 a^2 e^2+5 a b e (d+7 e x)+b^2 \left (d^2+7 d e x+21 e^2 x^2\right )\right )+c e \left (4 a e \left (d^2+7 d e x+21 e^2 x^2\right )+3 b \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )+2 c^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )}{210 e^5 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/210*(2*c^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) + 2*e^2*(15*a^2*e^2 + 5*a*b*e*(d
+ 7*e*x) + b^2*(d^2 + 7*d*e*x + 21*e^2*x^2)) + c*e*(4*a*e*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*b*(d^3 + 7*d^2*e*x
+ 21*d*e^2*x^2 + 35*e^3*x^3)))/(e^5*(d + e*x)^7)

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fricas [A]  time = 0.85, size = 245, normalized size = 1.57 \[ -\frac {70 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 3 \, b c d^{3} e + 10 \, a b d e^{3} + 30 \, a^{2} e^{4} + 2 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 35 \, {\left (2 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} x^{3} + 21 \, {\left (2 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + 2 \, {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 7 \, {\left (2 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 10 \, a b e^{4} + 2 \, {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{210 \, {\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/210*(70*c^2*e^4*x^4 + 2*c^2*d^4 + 3*b*c*d^3*e + 10*a*b*d*e^3 + 30*a^2*e^4 + 2*(b^2 + 2*a*c)*d^2*e^2 + 35*(2
*c^2*d*e^3 + 3*b*c*e^4)*x^3 + 21*(2*c^2*d^2*e^2 + 3*b*c*d*e^3 + 2*(b^2 + 2*a*c)*e^4)*x^2 + 7*(2*c^2*d^3*e + 3*
b*c*d^2*e^2 + 10*a*b*e^4 + 2*(b^2 + 2*a*c)*d*e^3)*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 [A]  time = 0.18, size = 180, normalized size = 1.15 \[ -\frac {{\left (70 \, c^{2} x^{4} e^{4} + 70 \, c^{2} d x^{3} e^{3} + 42 \, c^{2} d^{2} x^{2} e^{2} + 14 \, c^{2} d^{3} x e + 2 \, c^{2} d^{4} + 105 \, b c x^{3} e^{4} + 63 \, b c d x^{2} e^{3} + 21 \, b c d^{2} x e^{2} + 3 \, b c d^{3} e + 42 \, b^{2} x^{2} e^{4} + 84 \, a c x^{2} e^{4} + 14 \, b^{2} d x e^{3} + 28 \, a c d x e^{3} + 2 \, b^{2} d^{2} e^{2} + 4 \, a c d^{2} e^{2} + 70 \, a b x e^{4} + 10 \, a b d e^{3} + 30 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{210 \, {\left (x e + d\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/210*(70*c^2*x^4*e^4 + 70*c^2*d*x^3*e^3 + 42*c^2*d^2*x^2*e^2 + 14*c^2*d^3*x*e + 2*c^2*d^4 + 105*b*c*x^3*e^4
+ 63*b*c*d*x^2*e^3 + 21*b*c*d^2*x*e^2 + 3*b*c*d^3*e + 42*b^2*x^2*e^4 + 84*a*c*x^2*e^4 + 14*b^2*d*x*e^3 + 28*a*
c*d*x*e^3 + 2*b^2*d^2*e^2 + 4*a*c*d^2*e^2 + 70*a*b*x*e^4 + 10*a*b*d*e^3 + 30*a^2*e^4)*e^(-5)/(x*e + d)^7

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maple [A]  time = 0.06, size = 195, normalized size = 1.25 \[ -\frac {c^{2}}{3 \left (e x +d \right )^{3} e^{5}}-\frac {\left (b e -2 c d \right ) c}{2 \left (e x +d \right )^{4} e^{5}}-\frac {a^{2} e^{4}-2 a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 b c \,d^{3} e +c^{2} d^{4}}{7 \left (e x +d \right )^{7} e^{5}}-\frac {2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}}{5 \left (e x +d \right )^{5} e^{5}}-\frac {2 a b \,e^{3}-4 a c d \,e^{2}-2 b^{2} d \,e^{2}+6 b c \,d^{2} e -4 c^{2} d^{3}}{6 \left (e x +d \right )^{6} e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.13, size = 245, normalized size = 1.57 \[ -\frac {70 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 3 \, b c d^{3} e + 10 \, a b d e^{3} + 30 \, a^{2} e^{4} + 2 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 35 \, {\left (2 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} x^{3} + 21 \, {\left (2 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + 2 \, {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 7 \, {\left (2 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 10 \, a b e^{4} + 2 \, {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{210 \, {\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/210*(70*c^2*e^4*x^4 + 2*c^2*d^4 + 3*b*c*d^3*e + 10*a*b*d*e^3 + 30*a^2*e^4 + 2*(b^2 + 2*a*c)*d^2*e^2 + 35*(2
*c^2*d*e^3 + 3*b*c*e^4)*x^3 + 21*(2*c^2*d^2*e^2 + 3*b*c*d*e^3 + 2*(b^2 + 2*a*c)*e^4)*x^2 + 7*(2*c^2*d^3*e + 3*
b*c*d^2*e^2 + 10*a*b*e^4 + 2*(b^2 + 2*a*c)*d*e^3)*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 = 0.12, size = 249, normalized size = 1.60 \[ -\frac {\frac {30\,a^2\,e^4+10\,a\,b\,d\,e^3+4\,a\,c\,d^2\,e^2+2\,b^2\,d^2\,e^2+3\,b\,c\,d^3\,e+2\,c^2\,d^4}{210\,e^5}+\frac {x\,\left (2\,b^2\,d\,e^2+3\,b\,c\,d^2\,e+10\,a\,b\,e^3+2\,c^2\,d^3+4\,a\,c\,d\,e^2\right )}{30\,e^4}+\frac {c^2\,x^4}{3\,e}+\frac {x^2\,\left (2\,b^2\,e^2+3\,b\,c\,d\,e+2\,c^2\,d^2+4\,a\,c\,e^2\right )}{10\,e^3}+\frac {c\,x^3\,\left (3\,b\,e+2\,c\,d\right )}{6\,e^2}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((30*a^2*e^4 + 2*c^2*d^4 + 2*b^2*d^2*e^2 + 10*a*b*d*e^3 + 3*b*c*d^3*e + 4*a*c*d^2*e^2)/(210*e^5) + (x*(2*c^2*
d^3 + 2*b^2*d*e^2 + 10*a*b*e^3 + 4*a*c*d*e^2 + 3*b*c*d^2*e))/(30*e^4) + (c^2*x^4)/(3*e) + (x^2*(2*b^2*e^2 + 2*
c^2*d^2 + 4*a*c*e^2 + 3*b*c*d*e))/(10*e^3) + (c*x^3*(3*b*e + 2*c*d))/(6*e^2))/(d^7 + e^7*x^7 + 7*d*e^6*x^6 + 2
1*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 [A]  time = 168.16, size = 279, normalized size = 1.79 \[ \frac {- 30 a^{2} e^{4} - 10 a b d e^{3} - 4 a c d^{2} e^{2} - 2 b^{2} d^{2} e^{2} - 3 b c d^{3} e - 2 c^{2} d^{4} - 70 c^{2} e^{4} x^{4} + x^{3} \left (- 105 b c e^{4} - 70 c^{2} d e^{3}\right ) + x^{2} \left (- 84 a c e^{4} - 42 b^{2} e^{4} - 63 b c d e^{3} - 42 c^{2} d^{2} e^{2}\right ) + x \left (- 70 a b e^{4} - 28 a c d e^{3} - 14 b^{2} d e^{3} - 21 b c d^{2} e^{2} - 14 c^{2} d^{3} e\right )}{210 d^{7} e^{5} + 1470 d^{6} e^{6} x + 4410 d^{5} e^{7} x^{2} + 7350 d^{4} e^{8} x^{3} + 7350 d^{3} e^{9} x^{4} + 4410 d^{2} e^{10} x^{5} + 1470 d e^{11} x^{6} + 210 e^{12} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-30*a**2*e**4 - 10*a*b*d*e**3 - 4*a*c*d**2*e**2 - 2*b**2*d**2*e**2 - 3*b*c*d**3*e - 2*c**2*d**4 - 70*c**2*e**
4*x**4 + x**3*(-105*b*c*e**4 - 70*c**2*d*e**3) + x**2*(-84*a*c*e**4 - 42*b**2*e**4 - 63*b*c*d*e**3 - 42*c**2*d
**2*e**2) + x*(-70*a*b*e**4 - 28*a*c*d*e**3 - 14*b**2*d*e**3 - 21*b*c*d**2*e**2 - 14*c**2*d**3*e))/(210*d**7*e
**5 + 1470*d**6*e**6*x + 4410*d**5*e**7*x**2 + 7350*d**4*e**8*x**3 + 7350*d**3*e**9*x**4 + 4410*d**2*e**10*x**
5 + 1470*d*e**11*x**6 + 210*e**12*x**7)

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