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

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

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Rubi [A]  time = 0.09, antiderivative size = 122, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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)} \end {gather*}

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*}

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Mathematica [A]  time = 0.04, size = 100, normalized size = 0.82 \begin {gather*} -\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 (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )}{12 e^4 (d+e x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*(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)))/(e^4*(d + e*x)^4)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.44, size = 150, normalized size = 1.23 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.19, size = 186, normalized size = 1.52 \begin {gather*} -\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 {gather*}

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)

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maple [A]  time = 0.05, size = 131, normalized size = 1.07 \begin {gather*} -\frac {2 c^{2}}{\left (e x +d \right ) e^{4}}-\frac {3 \left (b e -2 c d \right ) c}{2 \left (e x +d \right )^{2} e^{4}}-\frac {2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}}{3 \left (e x +d \right )^{3} e^{4}}-\frac {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}}{4 \left (e x +d \right )^{4} e^{4}} \end {gather*}

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]

-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)^3-3/2*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

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maxima [A]  time = 0.53, size = 150, normalized size = 1.23 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 1.81, size = 151, normalized size = 1.24 \begin {gather*} -\frac {\frac {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}{12\,e^4}+\frac {x\,\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 {2\,c^2\,x^3}{e}+\frac {3\,c\,x^2\,\left (b\,e+2\,c\,d\right )}{2\,e^2}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 7.37, size = 170, normalized size = 1.39 \begin {gather*} \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 {gather*}

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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