3.1503 \(\int \frac{(b+2 c x) (a+b x+c x^2)}{(d+e x)^5} \, dx\)

Optimal. Leaf size=122 \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{3 e^4 (d+e x)^3}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{4 e^4 (d+e x)^4}+\frac{3 c (2 c d-b e)}{2 e^4 (d+e x)^2}-\frac{2 c^2}{e^4 (d+e x)} \]

[Out]

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

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Rubi [A]  time = 0.0902768, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{3 e^4 (d+e x)^3}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{4 e^4 (d+e x)^4}+\frac{3 c (2 c d-b e)}{2 e^4 (d+e x)^2}-\frac{2 c^2}{e^4 (d+e x)} \]

Antiderivative was successfully verified.

[In]

Int[((b + 2*c*x)*(a + b*x + c*x^2))/(d + e*x)^5,x]

[Out]

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

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

Mathematica [A]  time = 0.0419135, size = 100, normalized size = 0.82 \[ -\frac{c e \left (2 a e (d+4 e x)+3 b \left (d^2+4 d e x+6 e^2 x^2\right )\right )+b e^2 (3 a e+b (d+4 e x))+6 c^2 \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )}{12 e^4 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((b + 2*c*x)*(a + b*x + c*x^2))/(d + e*x)^5,x]

[Out]

-(6*c^2*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) + b*e^2*(3*a*e + b*(d + 4*e*x)) + c*e*(2*a*e*(d + 4*e*x) +
3*b*(d^2 + 4*d*e*x + 6*e^2*x^2)))/(12*e^4*(d + e*x)^4)

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Maple [A]  time = 0.006, size = 131, normalized size = 1.1 \begin{align*} -{\frac{3\,c \left ( be-2\,cd \right ) }{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{ab{e}^{3}-2\,acd{e}^{2}-{b}^{2}d{e}^{2}+3\,b{d}^{2}ce-2\,{c}^{2}{d}^{3}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-2\,{\frac{{c}^{2}}{{e}^{4} \left ( ex+d \right ) }}-{\frac{2\,ac{e}^{2}+{b}^{2}{e}^{2}-6\,bcde+6\,{c}^{2}{d}^{2}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(c*x^2+b*x+a)/(e*x+d)^5,x)

[Out]

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

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Maxima [A]  time = 1.00831, size = 203, normalized size = 1.66 \begin{align*} -\frac{24 \, c^{2} e^{3} x^{3} + 6 \, c^{2} d^{3} + 3 \, b c d^{2} e + 3 \, a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2} + 18 \,{\left (2 \, c^{2} d e^{2} + b c e^{3}\right )} x^{2} + 4 \,{\left (6 \, c^{2} d^{2} e + 3 \, b c d e^{2} +{\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x}{12 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)/(e*x+d)^5,x, algorithm="maxima")

[Out]

-1/12*(24*c^2*e^3*x^3 + 6*c^2*d^3 + 3*b*c*d^2*e + 3*a*b*e^3 + (b^2 + 2*a*c)*d*e^2 + 18*(2*c^2*d*e^2 + b*c*e^3)
*x^2 + 4*(6*c^2*d^2*e + 3*b*c*d*e^2 + (b^2 + 2*a*c)*e^3)*x)/(e^8*x^4 + 4*d*e^7*x^3 + 6*d^2*e^6*x^2 + 4*d^3*e^5
*x + d^4*e^4)

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Fricas [A]  time = 1.70352, size = 317, normalized size = 2.6 \begin{align*} -\frac{24 \, c^{2} e^{3} x^{3} + 6 \, c^{2} d^{3} + 3 \, b c d^{2} e + 3 \, a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2} + 18 \,{\left (2 \, c^{2} d e^{2} + b c e^{3}\right )} x^{2} + 4 \,{\left (6 \, c^{2} d^{2} e + 3 \, b c d e^{2} +{\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x}{12 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)/(e*x+d)^5,x, algorithm="fricas")

[Out]

-1/12*(24*c^2*e^3*x^3 + 6*c^2*d^3 + 3*b*c*d^2*e + 3*a*b*e^3 + (b^2 + 2*a*c)*d*e^2 + 18*(2*c^2*d*e^2 + b*c*e^3)
*x^2 + 4*(6*c^2*d^2*e + 3*b*c*d*e^2 + (b^2 + 2*a*c)*e^3)*x)/(e^8*x^4 + 4*d*e^7*x^3 + 6*d^2*e^6*x^2 + 4*d^3*e^5
*x + d^4*e^4)

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Sympy [A]  time = 7.56155, size = 168, normalized size = 1.38 \begin{align*} - \frac{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} + 24 c^{2} e^{3} x^{3} + x^{2} \left (18 b c e^{3} + 36 c^{2} d e^{2}\right ) + x \left (8 a c e^{3} + 4 b^{2} e^{3} + 12 b c d e^{2} + 24 c^{2} d^{2} e\right )}{12 d^{4} e^{4} + 48 d^{3} e^{5} x + 72 d^{2} e^{6} x^{2} + 48 d e^{7} x^{3} + 12 e^{8} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x**2+b*x+a)/(e*x+d)**5,x)

[Out]

-(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 + 24*c**2*e**3*x**3 + x**2*(18*b*c*e**3
+ 36*c**2*d*e**2) + x*(8*a*c*e**3 + 4*b**2*e**3 + 12*b*c*d*e**2 + 24*c**2*d**2*e))/(12*d**4*e**4 + 48*d**3*e*
*5*x + 72*d**2*e**6*x**2 + 48*d*e**7*x**3 + 12*e**8*x**4)

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Giac [A]  time = 1.26921, size = 251, normalized size = 2.06 \begin{align*} -\frac{1}{12} \,{\left (\frac{24 \, c^{2} e^{\left (-1\right )}}{x e + d} - \frac{36 \, c^{2} d e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}} + \frac{24 \, c^{2} d^{2} e^{\left (-1\right )}}{{\left (x e + d\right )}^{3}} - \frac{6 \, c^{2} d^{3} e^{\left (-1\right )}}{{\left (x e + d\right )}^{4}} + \frac{18 \, b c}{{\left (x e + d\right )}^{2}} - \frac{24 \, b c d}{{\left (x e + d\right )}^{3}} + \frac{9 \, b c d^{2}}{{\left (x e + d\right )}^{4}} + \frac{4 \, b^{2} e}{{\left (x e + d\right )}^{3}} + \frac{8 \, a c e}{{\left (x e + d\right )}^{3}} - \frac{3 \, b^{2} d e}{{\left (x e + d\right )}^{4}} - \frac{6 \, a c d e}{{\left (x e + d\right )}^{4}} + \frac{3 \, a b e^{2}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-3\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)/(e*x+d)^5,x, algorithm="giac")

[Out]

-1/12*(24*c^2*e^(-1)/(x*e + d) - 36*c^2*d*e^(-1)/(x*e + d)^2 + 24*c^2*d^2*e^(-1)/(x*e + d)^3 - 6*c^2*d^3*e^(-1
)/(x*e + d)^4 + 18*b*c/(x*e + d)^2 - 24*b*c*d/(x*e + d)^3 + 9*b*c*d^2/(x*e + d)^4 + 4*b^2*e/(x*e + d)^3 + 8*a*
c*e/(x*e + d)^3 - 3*b^2*d*e/(x*e + d)^4 - 6*a*c*d*e/(x*e + d)^4 + 3*a*b*e^2/(x*e + d)^4)*e^(-3)