\(\int \frac {(a+b x+c x^2)^2}{(d+e x)^6} \, dx\) [2128]

Optimal result

Integrand size = 20, antiderivative size = 151 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx=-\frac {\left (c d^2-b d e+a e^2\right )^2}{5 e^5 (d+e x)^5}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )}{2 e^5 (d+e x)^4}-\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{3 e^5 (d+e x)^3}+\frac {c (2 c d-b e)}{e^5 (d+e x)^2}-\frac {c^2}{e^5 (d+e x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 151, 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} \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx=-\frac {-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{3 e^5 (d+e x)^3}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{2 e^5 (d+e x)^4}-\frac {\left (a e^2-b d e+c d^2\right )^2}{5 e^5 (d+e x)^5}+\frac {c (2 c d-b e)}{e^5 (d+e x)^2}-\frac {c^2}{e^5 (d+e x)} \]

[In]

Int[(a + b*x + c*x^2)^2/(d + e*x)^6,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx=-\frac {6 c^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )+e^2 \left (6 a^2 e^2+3 a b e (d+5 e x)+b^2 \left (d^2+5 d e x+10 e^2 x^2\right )\right )+c e \left (2 a e \left (d^2+5 d e x+10 e^2 x^2\right )+3 b \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )}{30 e^5 (d+e x)^5} \]

[In]

Integrate[(a + b*x + c*x^2)^2/(d + e*x)^6,x]

[Out]

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

Maple [A] (verified)

Time = 5.11 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.19

method	result	size
risch	\(\frac {-\frac {c^{2} x^{4}}{e}-\frac {c \left (b e +2 c d \right ) x^{3}}{e^{2}}-\frac {\left (2 a c \,e^{2}+b^{2} e^{2}+3 b c d e +6 c^{2} d^{2}\right ) x^{2}}{3 e^{3}}-\frac {\left (3 a b \,e^{3}+2 d \,e^{2} a c +b^{2} d \,e^{2}+3 b c e \,d^{2}+6 c^{2} d^{3}\right ) x}{6 e^{4}}-\frac {6 a^{2} e^{4}+3 a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}+3 d^{3} e b c +6 c^{2} d^{4}}{30 e^{5}}}{\left (e x +d \right )^{5}}\)	\(179\)
norman	\(\frac {-\frac {c^{2} x^{4}}{e}-\frac {\left (b c e +2 c^{2} d \right ) x^{3}}{e^{2}}-\frac {\left (2 a c \,e^{2}+b^{2} e^{2}+3 b c d e +6 c^{2} d^{2}\right ) x^{2}}{3 e^{3}}-\frac {\left (3 a b \,e^{3}+2 d \,e^{2} a c +b^{2} d \,e^{2}+3 b c e \,d^{2}+6 c^{2} d^{3}\right ) x}{6 e^{4}}-\frac {6 a^{2} e^{4}+3 a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}+3 d^{3} e b c +6 c^{2} d^{4}}{30 e^{5}}}{\left (e x +d \right )^{5}}\)	\(181\)
gosper	\(-\frac {30 c^{2} x^{4} e^{4}+30 x^{3} b c \,e^{4}+60 x^{3} c^{2} d \,e^{3}+20 x^{2} a c \,e^{4}+10 x^{2} b^{2} e^{4}+30 x^{2} b c d \,e^{3}+60 x^{2} c^{2} d^{2} e^{2}+15 x a b \,e^{4}+10 x a c d \,e^{3}+5 x \,b^{2} d \,e^{3}+15 x b c \,d^{2} e^{2}+30 x \,c^{2} d^{3} e +6 a^{2} e^{4}+3 a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}+3 d^{3} e b c +6 c^{2} d^{4}}{30 \left (e x +d \right )^{5} e^{5}}\)	\(193\)
parallelrisch	\(\frac {-30 c^{2} x^{4} e^{4}-30 x^{3} b c \,e^{4}-60 x^{3} c^{2} d \,e^{3}-20 x^{2} a c \,e^{4}-10 x^{2} b^{2} e^{4}-30 x^{2} b c d \,e^{3}-60 x^{2} c^{2} d^{2} e^{2}-15 x a b \,e^{4}-10 x a c d \,e^{3}-5 x \,b^{2} d \,e^{3}-15 x b c \,d^{2} e^{2}-30 x \,c^{2} d^{3} e -6 a^{2} e^{4}-3 a b d \,e^{3}-2 a c \,d^{2} e^{2}-b^{2} d^{2} e^{2}-3 d^{3} e b c -6 c^{2} d^{4}}{30 e^{5} \left (e x +d \right )^{5}}\)	\(194\)
default	\(-\frac {a^{2} e^{4}-2 a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 d^{3} e b c +c^{2} d^{4}}{5 e^{5} \left (e x +d \right )^{5}}-\frac {c^{2}}{e^{5} \left (e x +d \right )}-\frac {2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}}{3 e^{5} \left (e x +d \right )^{3}}-\frac {2 a b \,e^{3}-4 d \,e^{2} a c -2 b^{2} d \,e^{2}+6 b c e \,d^{2}-4 c^{2} d^{3}}{4 e^{5} \left (e x +d \right )^{4}}-\frac {c \left (b e -2 c d \right )}{e^{5} \left (e x +d \right )^{2}}\)	\(195\)

[In]

int((c*x^2+b*x+a)^2/(e*x+d)^6,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx=-\frac {30 \, c^{2} e^{4} x^{4} + 6 \, c^{2} d^{4} + 3 \, b c d^{3} e + 3 \, a b d e^{3} + 6 \, a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 30 \, {\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 10 \, {\left (6 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 5 \, {\left (6 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 3 \, a b e^{4} + {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{30 \, {\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \]

[In]

integrate((c*x^2+b*x+a)^2/(e*x+d)^6,x, algorithm="fricas")

[Out]

-1/30*(30*c^2*e^4*x^4 + 6*c^2*d^4 + 3*b*c*d^3*e + 3*a*b*d*e^3 + 6*a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2 + 30*(2*c^2*
d*e^3 + b*c*e^4)*x^3 + 10*(6*c^2*d^2*e^2 + 3*b*c*d*e^3 + (b^2 + 2*a*c)*e^4)*x^2 + 5*(6*c^2*d^3*e + 3*b*c*d^2*e
^2 + 3*a*b*e^4 + (b^2 + 2*a*c)*d*e^3)*x)/(e^10*x^5 + 5*d*e^9*x^4 + 10*d^2*e^8*x^3 + 10*d^3*e^7*x^2 + 5*d^4*e^6
*x + d^5*e^5)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx=\text {Timed out} \]

[In]

integrate((c*x**2+b*x+a)**2/(e*x+d)**6,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx=-\frac {30 \, c^{2} e^{4} x^{4} + 6 \, c^{2} d^{4} + 3 \, b c d^{3} e + 3 \, a b d e^{3} + 6 \, a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 30 \, {\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 10 \, {\left (6 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 5 \, {\left (6 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 3 \, a b e^{4} + {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{30 \, {\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \]

[In]

integrate((c*x^2+b*x+a)^2/(e*x+d)^6,x, algorithm="maxima")

[Out]

-1/30*(30*c^2*e^4*x^4 + 6*c^2*d^4 + 3*b*c*d^3*e + 3*a*b*d*e^3 + 6*a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2 + 30*(2*c^2*
d*e^3 + b*c*e^4)*x^3 + 10*(6*c^2*d^2*e^2 + 3*b*c*d*e^3 + (b^2 + 2*a*c)*e^4)*x^2 + 5*(6*c^2*d^3*e + 3*b*c*d^2*e
^2 + 3*a*b*e^4 + (b^2 + 2*a*c)*d*e^3)*x)/(e^10*x^5 + 5*d*e^9*x^4 + 10*d^2*e^8*x^3 + 10*d^3*e^7*x^2 + 5*d^4*e^6
*x + d^5*e^5)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.27 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx=-\frac {30 \, c^{2} e^{4} x^{4} + 60 \, c^{2} d e^{3} x^{3} + 30 \, b c e^{4} x^{3} + 60 \, c^{2} d^{2} e^{2} x^{2} + 30 \, b c d e^{3} x^{2} + 10 \, b^{2} e^{4} x^{2} + 20 \, a c e^{4} x^{2} + 30 \, c^{2} d^{3} e x + 15 \, b c d^{2} e^{2} x + 5 \, b^{2} d e^{3} x + 10 \, a c d e^{3} x + 15 \, a b e^{4} x + 6 \, c^{2} d^{4} + 3 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} + 3 \, a b d e^{3} + 6 \, a^{2} e^{4}}{30 \, {\left (e x + d\right )}^{5} e^{5}} \]

[In]

integrate((c*x^2+b*x+a)^2/(e*x+d)^6,x, algorithm="giac")

[Out]

-1/30*(30*c^2*e^4*x^4 + 60*c^2*d*e^3*x^3 + 30*b*c*e^4*x^3 + 60*c^2*d^2*e^2*x^2 + 30*b*c*d*e^3*x^2 + 10*b^2*e^4
*x^2 + 20*a*c*e^4*x^2 + 30*c^2*d^3*e*x + 15*b*c*d^2*e^2*x + 5*b^2*d*e^3*x + 10*a*c*d*e^3*x + 15*a*b*e^4*x + 6*
c^2*d^4 + 3*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 + 3*a*b*d*e^3 + 6*a^2*e^4)/((e*x + d)^5*e^5)

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.46 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx=-\frac {\frac {6\,a^2\,e^4+3\,a\,b\,d\,e^3+2\,a\,c\,d^2\,e^2+b^2\,d^2\,e^2+3\,b\,c\,d^3\,e+6\,c^2\,d^4}{30\,e^5}+\frac {x\,\left (b^2\,d\,e^2+3\,b\,c\,d^2\,e+3\,a\,b\,e^3+6\,c^2\,d^3+2\,a\,c\,d\,e^2\right )}{6\,e^4}+\frac {c^2\,x^4}{e}+\frac {x^2\,\left (b^2\,e^2+3\,b\,c\,d\,e+6\,c^2\,d^2+2\,a\,c\,e^2\right )}{3\,e^3}+\frac {c\,x^3\,\left (b\,e+2\,c\,d\right )}{e^2}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \]

[In]

int((a + b*x + c*x^2)^2/(d + e*x)^6,x)

[Out]

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