3.2360 \(\int \frac {x (a+b x+c x^2)}{(d+e x)^4} \, dx\)

Optimal. Leaf size=96 \[ -\frac {3 c d^2-e (2 b d-a e)}{2 e^4 (d+e x)^2}+\frac {d \left (a e^2-b d e+c d^2\right )}{3 e^4 (d+e x)^3}+\frac {3 c d-b e}{e^4 (d+e x)}+\frac {c \log (d+e x)}{e^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {771} \[ -\frac {3 c d^2-e (2 b d-a e)}{2 e^4 (d+e x)^2}+\frac {d \left (a e^2-b d e+c d^2\right )}{3 e^4 (d+e x)^3}+\frac {3 c d-b e}{e^4 (d+e x)}+\frac {c \log (d+e x)}{e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 771

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 86, normalized size = 0.90 \[ \frac {-e \left (a e (d+3 e x)+2 b \left (d^2+3 d e x+3 e^2 x^2\right )\right )+c d \left (11 d^2+27 d e x+18 e^2 x^2\right )+6 c (d+e x)^3 \log (d+e x)}{6 e^4 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.01, size = 141, normalized size = 1.47 \[ \frac {11 \, c d^{3} - 2 \, b d^{2} e - a d e^{2} + 6 \, {\left (3 \, c d e^{2} - b e^{3}\right )} x^{2} + 3 \, {\left (9 \, c d^{2} e - 2 \, b d e^{2} - a e^{3}\right )} x + 6 \, {\left (c e^{3} x^{3} + 3 \, c d e^{2} x^{2} + 3 \, c d^{2} e x + c d^{3}\right )} \log \left (e x + d\right )}{6 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.16, size = 88, normalized size = 0.92 \[ c e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (6 \, {\left (3 \, c d e - b e^{2}\right )} x^{2} + 3 \, {\left (9 \, c d^{2} - 2 \, b d e - a e^{2}\right )} x + {\left (11 \, c d^{3} - 2 \, b d^{2} e - a d e^{2}\right )} e^{\left (-1\right )}\right )} e^{\left (-3\right )}}{6 \, {\left (x e + d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.05, size = 128, normalized size = 1.33 \[ \frac {a d}{3 \left (e x +d \right )^{3} e^{2}}-\frac {b \,d^{2}}{3 \left (e x +d \right )^{3} e^{3}}+\frac {c \,d^{3}}{3 \left (e x +d \right )^{3} e^{4}}-\frac {a}{2 \left (e x +d \right )^{2} e^{2}}+\frac {b d}{\left (e x +d \right )^{2} e^{3}}-\frac {3 c \,d^{2}}{2 \left (e x +d \right )^{2} e^{4}}-\frac {b}{\left (e x +d \right ) e^{3}}+\frac {3 c d}{\left (e x +d \right ) e^{4}}+\frac {c \ln \left (e x +d \right )}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.61, size = 113, normalized size = 1.18 \[ \frac {11 \, c d^{3} - 2 \, b d^{2} e - a d e^{2} + 6 \, {\left (3 \, c d e^{2} - b e^{3}\right )} x^{2} + 3 \, {\left (9 \, c d^{2} e - 2 \, b d e^{2} - a e^{3}\right )} x}{6 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} + \frac {c \log \left (e x + d\right )}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.08, size = 107, normalized size = 1.11 \[ \frac {c\,\ln \left (d+e\,x\right )}{e^4}-\frac {\frac {-11\,c\,d^3+2\,b\,d^2\,e+a\,d\,e^2}{6\,e^4}+\frac {x\,\left (-9\,c\,d^2+2\,b\,d\,e+a\,e^2\right )}{2\,e^3}+\frac {x^2\,\left (b\,e-3\,c\,d\right )}{e^2}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 1.33, size = 114, normalized size = 1.19 \[ \frac {c \log {\left (d + e x \right )}}{e^{4}} + \frac {- a d e^{2} - 2 b d^{2} e + 11 c d^{3} + x^{2} \left (- 6 b e^{3} + 18 c d e^{2}\right ) + x \left (- 3 a e^{3} - 6 b d e^{2} + 27 c d^{2} e\right )}{6 d^{3} e^{4} + 18 d^{2} e^{5} x + 18 d e^{6} x^{2} + 6 e^{7} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________