\(\int \frac {(a+b x+c x^2)^2}{(d+e x)^8} \, dx\) [2130]

Optimal result

Integrand size = 20, antiderivative size = 156 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^8} \, dx=-\frac {\left (c d^2-b d e+a e^2\right )^2}{7 e^5 (d+e x)^7}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )}{3 e^5 (d+e x)^6}-\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{5 e^5 (d+e x)^5}+\frac {c (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac {c^2}{3 e^5 (d+e x)^3} \]

[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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {712} \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^8} \, dx=-\frac {-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{5 e^5 (d+e x)^5}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5 (d+e x)^6}-\frac {\left (a e^2-b d e+c d^2\right )^2}{7 e^5 (d+e x)^7}+\frac {c (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac {c^2}{3 e^5 (d+e x)^3} \]

[In]

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

[Out]

-1/7*(c*d^2 - b*d*e + a*e^2)^2/(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 712

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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^8} \, dx=-\frac {2 c^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+2 e^2 \left (15 a^2 e^2+5 a b e (d+7 e x)+b^2 \left (d^2+7 d e x+21 e^2 x^2\right )\right )+c e \left (4 a e \left (d^2+7 d e x+21 e^2 x^2\right )+3 b \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )}{210 e^5 (d+e x)^7} \]

[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)

Maple [A] (verified)

Time = 2.94 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.17

method	result	size
risch	\(\frac {-\frac {c^{2} x^{4}}{3 e}-\frac {c \left (3 b e +2 c d \right ) x^{3}}{6 e^{2}}-\frac {\left (4 a c \,e^{2}+2 b^{2} e^{2}+3 b c d e +2 c^{2} d^{2}\right ) x^{2}}{10 e^{3}}-\frac {\left (10 a b \,e^{3}+4 d \,e^{2} a c +2 b^{2} d \,e^{2}+3 b c e \,d^{2}+2 c^{2} d^{3}\right ) x}{30 e^{4}}-\frac {30 a^{2} e^{4}+10 a b d \,e^{3}+4 a c \,d^{2} e^{2}+2 b^{2} d^{2} e^{2}+3 d^{3} e b c +2 c^{2} d^{4}}{210 e^{5}}}{\left (e x +d \right )^{7}}\)	\(183\)
gosper	\(-\frac {70 c^{2} x^{4} e^{4}+105 x^{3} b c \,e^{4}+70 x^{3} c^{2} d \,e^{3}+84 x^{2} a c \,e^{4}+42 x^{2} b^{2} e^{4}+63 x^{2} b c d \,e^{3}+42 x^{2} c^{2} d^{2} e^{2}+70 x a b \,e^{4}+28 x a c d \,e^{3}+14 x \,b^{2} d \,e^{3}+21 x b c \,d^{2} e^{2}+14 x \,c^{2} d^{3} e +30 a^{2} e^{4}+10 a b d \,e^{3}+4 a c \,d^{2} e^{2}+2 b^{2} d^{2} e^{2}+3 d^{3} e b c +2 c^{2} d^{4}}{210 e^{5} \left (e x +d \right )^{7}}\)	\(194\)
default	\(-\frac {2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}}{5 e^{5} \left (e x +d \right )^{5}}-\frac {c^{2}}{3 e^{5} \left (e x +d \right )^{3}}-\frac {a^{2} e^{4}-2 a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 d^{3} e b c +c^{2} d^{4}}{7 e^{5} \left (e x +d \right )^{7}}-\frac {c \left (b e -2 c d \right )}{2 e^{5} \left (e x +d \right )^{4}}-\frac {2 a b \,e^{3}-4 d \,e^{2} a c -2 b^{2} d \,e^{2}+6 b c e \,d^{2}-4 c^{2} d^{3}}{6 e^{5} \left (e x +d \right )^{6}}\)	\(195\)
parallelrisch	\(\frac {-70 x^{4} c^{2} e^{6}-105 b c \,e^{6} x^{3}-70 c^{2} d \,e^{5} x^{3}-84 a c \,e^{6} x^{2}-42 b^{2} e^{6} x^{2}-63 b c d \,e^{5} x^{2}-42 c^{2} d^{2} e^{4} x^{2}-70 a b \,e^{6} x -28 a c d \,e^{5} x -14 b^{2} d \,e^{5} x -21 b c \,d^{2} e^{4} x -14 c^{2} d^{3} e^{3} x -30 a^{2} e^{6}-10 a b d \,e^{5}-4 a c \,d^{2} e^{4}-2 b^{2} d^{2} e^{4}-3 b c \,d^{3} e^{3}-2 c^{2} d^{4} e^{2}}{210 e^{7} \left (e x +d \right )^{7}}\)	\(201\)
norman	\(\frac {-\frac {c^{2} x^{4}}{3 e}-\frac {\left (3 e^{3} b c +2 d \,e^{2} c^{2}\right ) x^{3}}{6 e^{4}}-\frac {\left (4 a c \,e^{4}+2 b^{2} e^{4}+3 d \,e^{3} b c +2 c^{2} d^{2} e^{2}\right ) x^{2}}{10 e^{5}}-\frac {\left (10 a b \,e^{5}+4 a c d \,e^{4}+2 b^{2} d \,e^{4}+3 b c \,d^{2} e^{3}+2 c^{2} d^{3} e^{2}\right ) x}{30 e^{6}}-\frac {30 a^{2} e^{6}+10 a b d \,e^{5}+4 a c \,d^{2} e^{4}+2 b^{2} d^{2} e^{4}+3 b c \,d^{3} e^{3}+2 c^{2} d^{4} e^{2}}{210 e^{7}}}{\left (e x +d \right )^{7}}\)	\(205\)

[In]

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

[Out]

(-1/3*c^2*x^4/e-1/6/e^2*c*(3*b*e+2*c*d)*x^3-1/10/e^3*(4*a*c*e^2+2*b^2*e^2+3*b*c*d*e+2*c^2*d^2)*x^2-1/30/e^4*(1
0*a*b*e^3+4*a*c*d*e^2+2*b^2*d*e^2+3*b*c*d^2*e+2*c^2*d^3)*x-1/210/e^5*(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))/(e*x+d)^7

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.57 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^8} \, dx=-\frac {70 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 3 \, b c d^{3} e + 10 \, a b d e^{3} + 30 \, a^{2} e^{4} + 2 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 35 \, {\left (2 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} x^{3} + 21 \, {\left (2 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + 2 \, {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 7 \, {\left (2 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 10 \, a b e^{4} + 2 \, {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{210 \, {\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^8} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.57 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^8} \, dx=-\frac {70 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 3 \, b c d^{3} e + 10 \, a b d e^{3} + 30 \, a^{2} e^{4} + 2 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 35 \, {\left (2 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} x^{3} + 21 \, {\left (2 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + 2 \, {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 7 \, {\left (2 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 10 \, a b e^{4} + 2 \, {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{210 \, {\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^8} \, dx=-\frac {70 \, c^{2} e^{4} x^{4} + 70 \, c^{2} d e^{3} x^{3} + 105 \, b c e^{4} x^{3} + 42 \, c^{2} d^{2} e^{2} x^{2} + 63 \, b c d e^{3} x^{2} + 42 \, b^{2} e^{4} x^{2} + 84 \, a c e^{4} x^{2} + 14 \, c^{2} d^{3} e x + 21 \, b c d^{2} e^{2} x + 14 \, b^{2} d e^{3} x + 28 \, a c d e^{3} x + 70 \, a b e^{4} x + 2 \, c^{2} d^{4} + 3 \, b c d^{3} e + 2 \, b^{2} d^{2} e^{2} + 4 \, a c d^{2} e^{2} + 10 \, a b d e^{3} + 30 \, a^{2} e^{4}}{210 \, {\left (e x + d\right )}^{7} e^{5}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^8} \, dx=-\frac {\frac {30\,a^2\,e^4+10\,a\,b\,d\,e^3+4\,a\,c\,d^2\,e^2+2\,b^2\,d^2\,e^2+3\,b\,c\,d^3\,e+2\,c^2\,d^4}{210\,e^5}+\frac {x\,\left (2\,b^2\,d\,e^2+3\,b\,c\,d^2\,e+10\,a\,b\,e^3+2\,c^2\,d^3+4\,a\,c\,d\,e^2\right )}{30\,e^4}+\frac {c^2\,x^4}{3\,e}+\frac {x^2\,\left (2\,b^2\,e^2+3\,b\,c\,d\,e+2\,c^2\,d^2+4\,a\,c\,e^2\right )}{10\,e^3}+\frac {c\,x^3\,\left (3\,b\,e+2\,c\,d\right )}{6\,e^2}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \]

[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)