\(\int \frac {(a+b x+c x^2)^2}{(d+e x)^4} \, dx\) [2126]

Optimal result

Integrand size = 20, antiderivative size = 139 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^4} \, dx=\frac {c^2 x}{e^4}-\frac {\left (c d^2-b d e+a e^2\right )^2}{3 e^5 (d+e x)^3}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )}{e^5 (d+e x)^2}-\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^5 (d+e x)}-\frac {2 c (2 c d-b e) \log (d+e x)}{e^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 139, 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} \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^4} \, dx=-\frac {-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{e^5 (d+e x)}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^5 (d+e x)^2}-\frac {\left (a e^2-b d e+c d^2\right )^2}{3 e^5 (d+e x)^3}-\frac {2 c (2 c d-b e) \log (d+e x)}{e^5}+\frac {c^2 x}{e^4} \]

[In]

Int[(a + b*x + c*x^2)^2/(d + e*x)^4,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.27 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^4} \, dx=\frac {c^2 \left (-13 d^4-27 d^3 e x-9 d^2 e^2 x^2+9 d e^3 x^3+3 e^4 x^4\right )-e^2 \left (a^2 e^2+a b e (d+3 e x)+b^2 \left (d^2+3 d e x+3 e^2 x^2\right )\right )+c e \left (-2 a e \left (d^2+3 d e x+3 e^2 x^2\right )+b d \left (11 d^2+27 d e x+18 e^2 x^2\right )\right )-6 c (2 c d-b e) (d+e x)^3 \log (d+e x)}{3 e^5 (d+e x)^3} \]

[In]

Integrate[(a + b*x + c*x^2)^2/(d + e*x)^4,x]

[Out]

(c^2*(-13*d^4 - 27*d^3*e*x - 9*d^2*e^2*x^2 + 9*d*e^3*x^3 + 3*e^4*x^4) - e^2*(a^2*e^2 + a*b*e*(d + 3*e*x) + b^2
*(d^2 + 3*d*e*x + 3*e^2*x^2)) + c*e*(-2*a*e*(d^2 + 3*d*e*x + 3*e^2*x^2) + b*d*(11*d^2 + 27*d*e*x + 18*e^2*x^2)
) - 6*c*(2*c*d - b*e)*(d + e*x)^3*Log[d + e*x])/(3*e^5*(d + e*x)^3)

Maple [A] (verified)

Time = 3.00 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.29

method	result	size
norman	\(\frac {\frac {c^{2} x^{4}}{e}-\frac {a^{2} e^{4}+a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-11 d^{3} e b c +22 c^{2} d^{4}}{3 e^{5}}-\frac {\left (2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +12 c^{2} d^{2}\right ) x^{2}}{e^{3}}-\frac {\left (a b \,e^{3}+2 d \,e^{2} a c +b^{2} d \,e^{2}-9 b c e \,d^{2}+18 c^{2} d^{3}\right ) x}{e^{4}}}{\left (e x +d \right )^{3}}+\frac {2 c \left (b e -2 c d \right ) \ln \left (e x +d \right )}{e^{5}}\)	\(179\)
risch	\(\frac {c^{2} x}{e^{4}}+\frac {\left (-2 a c \,e^{3}-b^{2} e^{3}+6 b c d \,e^{2}-6 d^{2} e \,c^{2}\right ) x^{2}+\left (-a b \,e^{3}-2 d \,e^{2} a c -b^{2} d \,e^{2}+9 b c e \,d^{2}-10 c^{2} d^{3}\right ) x -\frac {a^{2} e^{4}+a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-11 d^{3} e b c +13 c^{2} d^{4}}{3 e}}{e^{4} \left (e x +d \right )^{3}}+\frac {2 c \ln \left (e x +d \right ) b}{e^{4}}-\frac {4 c^{2} d \ln \left (e x +d \right )}{e^{5}}\)	\(186\)
default	\(\frac {c^{2} x}{e^{4}}-\frac {2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}}{e^{5} \left (e x +d \right )}-\frac {a^{2} e^{4}-2 a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 d^{3} e b c +c^{2} d^{4}}{3 e^{5} \left (e x +d \right )^{3}}-\frac {2 a b \,e^{3}-4 d \,e^{2} a c -2 b^{2} d \,e^{2}+6 b c e \,d^{2}-4 c^{2} d^{3}}{2 e^{5} \left (e x +d \right )^{2}}+\frac {2 c \left (b e -2 c d \right ) \ln \left (e x +d \right )}{e^{5}}\)	\(187\)
parallelrisch	\(\frac {-6 x^{2} a c \,e^{4}-a b d \,e^{3}-b^{2} d^{2} e^{2}-a^{2} e^{4}-12 \ln \left (e x +d \right ) c^{2} d^{4}-36 x^{2} c^{2} d^{2} e^{2}-54 x \,c^{2} d^{3} e +18 \ln \left (e x +d \right ) x b c \,d^{2} e^{2}-22 c^{2} d^{4}+3 c^{2} x^{4} e^{4}+18 \ln \left (e x +d \right ) x^{2} b c d \,e^{3}-36 \ln \left (e x +d \right ) x \,c^{2} d^{3} e +11 d^{3} e b c -6 x a c d \,e^{3}-3 x^{2} b^{2} e^{4}-3 x a b \,e^{4}-3 x \,b^{2} d \,e^{3}+18 x^{2} b c d \,e^{3}+6 \ln \left (e x +d \right ) b c \,d^{3} e +27 x b c \,d^{2} e^{2}-36 \ln \left (e x +d \right ) x^{2} c^{2} d^{2} e^{2}+6 \ln \left (e x +d \right ) x^{3} b c \,e^{4}-12 \ln \left (e x +d \right ) x^{3} c^{2} d \,e^{3}-2 a c \,d^{2} e^{2}}{3 e^{5} \left (e x +d \right )^{3}}\)	\(304\)

[In]

int((c*x^2+b*x+a)^2/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

(c^2*x^4/e-1/3*(a^2*e^4+a*b*d*e^3+2*a*c*d^2*e^2+b^2*d^2*e^2-11*b*c*d^3*e+22*c^2*d^4)/e^5-(2*a*c*e^2+b^2*e^2-6*
b*c*d*e+12*c^2*d^2)/e^3*x^2-(a*b*e^3+2*a*c*d*e^2+b^2*d*e^2-9*b*c*d^2*e+18*c^2*d^3)/e^4*x)/(e*x+d)^3+2/e^5*c*(b
*e-2*c*d)*ln(e*x+d)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (137) = 274\).

Time = 0.36 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.03 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^4} \, dx=\frac {3 \, c^{2} e^{4} x^{4} + 9 \, c^{2} d e^{3} x^{3} - 13 \, c^{2} d^{4} + 11 \, b c d^{3} e - a b d e^{3} - a^{2} e^{4} - {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} - 3 \, {\left (3 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} + {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} - 3 \, {\left (9 \, c^{2} d^{3} e - 9 \, b c d^{2} e^{2} + a b e^{4} + {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x - 6 \, {\left (2 \, c^{2} d^{4} - b c d^{3} e + {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x^{3} + 3 \, {\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} x^{2} + 3 \, {\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2}\right )} x\right )} \log \left (e x + d\right )}{3 \, {\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \]

[In]

integrate((c*x^2+b*x+a)^2/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/3*(3*c^2*e^4*x^4 + 9*c^2*d*e^3*x^3 - 13*c^2*d^4 + 11*b*c*d^3*e - a*b*d*e^3 - a^2*e^4 - (b^2 + 2*a*c)*d^2*e^2
- 3*(3*c^2*d^2*e^2 - 6*b*c*d*e^3 + (b^2 + 2*a*c)*e^4)*x^2 - 3*(9*c^2*d^3*e - 9*b*c*d^2*e^2 + a*b*e^4 + (b^2 +
2*a*c)*d*e^3)*x - 6*(2*c^2*d^4 - b*c*d^3*e + (2*c^2*d*e^3 - b*c*e^4)*x^3 + 3*(2*c^2*d^2*e^2 - b*c*d*e^3)*x^2
+ 3*(2*c^2*d^3*e - b*c*d^2*e^2)*x)*log(e*x + d))/(e^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5)

Sympy [A] (verification not implemented)

Time = 3.18 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.57 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^4} \, dx=\frac {c^{2} x}{e^{4}} + \frac {2 c \left (b e - 2 c d\right ) \log {\left (d + e x \right )}}{e^{5}} + \frac {- a^{2} e^{4} - a b d e^{3} - 2 a c d^{2} e^{2} - b^{2} d^{2} e^{2} + 11 b c d^{3} e - 13 c^{2} d^{4} + x^{2} \left (- 6 a c e^{4} - 3 b^{2} e^{4} + 18 b c d e^{3} - 18 c^{2} d^{2} e^{2}\right ) + x \left (- 3 a b e^{4} - 6 a c d e^{3} - 3 b^{2} d e^{3} + 27 b c d^{2} e^{2} - 30 c^{2} d^{3} e\right )}{3 d^{3} e^{5} + 9 d^{2} e^{6} x + 9 d e^{7} x^{2} + 3 e^{8} x^{3}} \]

[In]

integrate((c*x**2+b*x+a)**2/(e*x+d)**4,x)

[Out]

c**2*x/e**4 + 2*c*(b*e - 2*c*d)*log(d + e*x)/e**5 + (-a**2*e**4 - a*b*d*e**3 - 2*a*c*d**2*e**2 - b**2*d**2*e**
2 + 11*b*c*d**3*e - 13*c**2*d**4 + x**2*(-6*a*c*e**4 - 3*b**2*e**4 + 18*b*c*d*e**3 - 18*c**2*d**2*e**2) + x*(-
3*a*b*e**4 - 6*a*c*d*e**3 - 3*b**2*d*e**3 + 27*b*c*d**2*e**2 - 30*c**2*d**3*e))/(3*d**3*e**5 + 9*d**2*e**6*x +
9*d*e**7*x**2 + 3*e**8*x**3)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^4} \, dx=-\frac {13 \, c^{2} d^{4} - 11 \, b c d^{3} e + a b d e^{3} + a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 3 \, {\left (6 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} + {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 3 \, {\left (10 \, c^{2} d^{3} e - 9 \, b c d^{2} e^{2} + a b e^{4} + {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{3 \, {\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} + \frac {c^{2} x}{e^{4}} - \frac {2 \, {\left (2 \, c^{2} d - b c e\right )} \log \left (e x + d\right )}{e^{5}} \]

[In]

integrate((c*x^2+b*x+a)^2/(e*x+d)^4,x, algorithm="maxima")

[Out]

-1/3*(13*c^2*d^4 - 11*b*c*d^3*e + a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2 + 3*(6*c^2*d^2*e^2 - 6*b*c*d*e^3
+ (b^2 + 2*a*c)*e^4)*x^2 + 3*(10*c^2*d^3*e - 9*b*c*d^2*e^2 + a*b*e^4 + (b^2 + 2*a*c)*d*e^3)*x)/(e^8*x^3 + 3*d
*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5) + c^2*x/e^4 - 2*(2*c^2*d - b*c*e)*log(e*x + d)/e^5

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^4} \, dx=\frac {c^{2} x}{e^{4}} - \frac {2 \, {\left (2 \, c^{2} d - b c e\right )} \log \left ({\left | e x + d \right |}\right )}{e^{5}} - \frac {13 \, c^{2} d^{4} - 11 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} + a b d e^{3} + a^{2} e^{4} + 3 \, {\left (6 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} + b^{2} e^{4} + 2 \, a c e^{4}\right )} x^{2} + 3 \, {\left (10 \, c^{2} d^{3} e - 9 \, b c d^{2} e^{2} + b^{2} d e^{3} + 2 \, a c d e^{3} + a b e^{4}\right )} x}{3 \, {\left (e x + d\right )}^{3} e^{5}} \]

[In]

integrate((c*x^2+b*x+a)^2/(e*x+d)^4,x, algorithm="giac")

[Out]

c^2*x/e^4 - 2*(2*c^2*d - b*c*e)*log(abs(e*x + d))/e^5 - 1/3*(13*c^2*d^4 - 11*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d
^2*e^2 + a*b*d*e^3 + a^2*e^4 + 3*(6*c^2*d^2*e^2 - 6*b*c*d*e^3 + b^2*e^4 + 2*a*c*e^4)*x^2 + 3*(10*c^2*d^3*e - 9
*b*c*d^2*e^2 + b^2*d*e^3 + 2*a*c*d*e^3 + a*b*e^4)*x)/((e*x + d)^3*e^5)

Mupad [B] (verification not implemented)

Time = 9.87 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.46 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^4} \, dx=\frac {c^2\,x}{e^4}-\frac {\frac {a^2\,e^4+a\,b\,d\,e^3+2\,a\,c\,d^2\,e^2+b^2\,d^2\,e^2-11\,b\,c\,d^3\,e+13\,c^2\,d^4}{3\,e}+x\,\left (b^2\,d\,e^2-9\,b\,c\,d^2\,e+a\,b\,e^3+10\,c^2\,d^3+2\,a\,c\,d\,e^2\right )+x^2\,\left (b^2\,e^3-6\,b\,c\,d\,e^2+6\,c^2\,d^2\,e+2\,a\,c\,e^3\right )}{d^3\,e^4+3\,d^2\,e^5\,x+3\,d\,e^6\,x^2+e^7\,x^3}-\frac {\ln \left (d+e\,x\right )\,\left (4\,c^2\,d-2\,b\,c\,e\right )}{e^5} \]

[In]

int((a + b*x + c*x^2)^2/(d + e*x)^4,x)

[Out]

(c^2*x)/e^4 - ((a^2*e^4 + 13*c^2*d^4 + b^2*d^2*e^2 + a*b*d*e^3 - 11*b*c*d^3*e + 2*a*c*d^2*e^2)/(3*e) + x*(10*c
^2*d^3 + b^2*d*e^2 + a*b*e^3 + 2*a*c*d*e^2 - 9*b*c*d^2*e) + x^2*(b^2*e^3 + 6*c^2*d^2*e + 2*a*c*e^3 - 6*b*c*d*e
^2))/(d^3*e^4 + e^7*x^3 + 3*d^2*e^5*x + 3*d*e^6*x^2) - (log(d + e*x)*(4*c^2*d - 2*b*c*e))/e^5