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

Optimal. Leaf size=103 \[ -\frac {3 c d^2-e (2 b d-a e)}{4 e^4 (d+e x)^4}+\frac {d \left (a e^2-b d e+c d^2\right )}{5 e^4 (d+e x)^5}+\frac {3 c d-b e}{3 e^4 (d+e x)^3}-\frac {c}{2 e^4 (d+e x)^2} \]

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Rubi [A]  time = 0.07, antiderivative size = 103, 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} \begin {gather*} -\frac {3 c d^2-e (2 b d-a e)}{4 e^4 (d+e x)^4}+\frac {d \left (a e^2-b d e+c d^2\right )}{5 e^4 (d+e x)^5}+\frac {3 c d-b e}{3 e^4 (d+e x)^3}-\frac {c}{2 e^4 (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.03, size = 79, normalized size = 0.77 \begin {gather*} -\frac {e \left (3 a e (d+5 e x)+2 b \left (d^2+5 d e x+10 e^2 x^2\right )\right )+3 c \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )}{60 e^4 (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (a+b x+c x^2\right )}{(d+e x)^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 132, normalized size = 1.28 \begin {gather*} -\frac {30 \, c e^{3} x^{3} + 3 \, c d^{3} + 2 \, b d^{2} e + 3 \, a d e^{2} + 10 \, {\left (3 \, c d e^{2} + 2 \, b e^{3}\right )} x^{2} + 5 \, {\left (3 \, c d^{2} e + 2 \, b d e^{2} + 3 \, a e^{3}\right )} x}{60 \, {\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(30*c*e^3*x^3 + 3*c*d^3 + 2*b*d^2*e + 3*a*d*e^2 + 10*(3*c*d*e^2 + 2*b*e^3)*x^2 + 5*(3*c*d^2*e + 2*b*d*e^
2 + 3*a*e^3)*x)/(e^9*x^5 + 5*d*e^8*x^4 + 10*d^2*e^7*x^3 + 10*d^3*e^6*x^2 + 5*d^4*e^5*x + d^5*e^4)

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giac [A]  time = 0.17, size = 80, normalized size = 0.78 \begin {gather*} -\frac {{\left (30 \, c x^{3} e^{3} + 30 \, c d x^{2} e^{2} + 15 \, c d^{2} x e + 3 \, c d^{3} + 20 \, b x^{2} e^{3} + 10 \, b d x e^{2} + 2 \, b d^{2} e + 15 \, a x e^{3} + 3 \, a d e^{2}\right )} e^{\left (-4\right )}}{60 \, {\left (x e + d\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 93, normalized size = 0.90 \begin {gather*} -\frac {c}{2 \left (e x +d \right )^{2} e^{4}}+\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d}{5 \left (e x +d \right )^{5} e^{4}}-\frac {a \,e^{2}-2 b d e +3 c \,d^{2}}{4 \left (e x +d \right )^{4} e^{4}}-\frac {b e -3 c d}{3 \left (e x +d \right )^{3} e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.53, size = 132, normalized size = 1.28 \begin {gather*} -\frac {30 \, c e^{3} x^{3} + 3 \, c d^{3} + 2 \, b d^{2} e + 3 \, a d e^{2} + 10 \, {\left (3 \, c d e^{2} + 2 \, b e^{3}\right )} x^{2} + 5 \, {\left (3 \, c d^{2} e + 2 \, b d e^{2} + 3 \, a e^{3}\right )} x}{60 \, {\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(30*c*e^3*x^3 + 3*c*d^3 + 2*b*d^2*e + 3*a*d*e^2 + 10*(3*c*d*e^2 + 2*b*e^3)*x^2 + 5*(3*c*d^2*e + 2*b*d*e^
2 + 3*a*e^3)*x)/(e^9*x^5 + 5*d*e^8*x^4 + 10*d^2*e^7*x^3 + 10*d^3*e^6*x^2 + 5*d^4*e^5*x + d^5*e^4)

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mupad [B]  time = 0.06, size = 128, normalized size = 1.24 \begin {gather*} -\frac {\frac {c\,x^3}{2\,e}+\frac {d\,\left (3\,c\,d^2+2\,b\,d\,e+3\,a\,e^2\right )}{60\,e^4}+\frac {x\,\left (3\,c\,d^2+2\,b\,d\,e+3\,a\,e^2\right )}{12\,e^3}+\frac {x^2\,\left (2\,b\,e+3\,c\,d\right )}{6\,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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((c*x^3)/(2*e) + (d*(3*a*e^2 + 3*c*d^2 + 2*b*d*e))/(60*e^4) + (x*(3*a*e^2 + 3*c*d^2 + 2*b*d*e))/(12*e^3) + (x
^2*(2*b*e + 3*c*d))/(6*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)

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sympy [A]  time = 3.66, size = 141, normalized size = 1.37 \begin {gather*} \frac {- 3 a d e^{2} - 2 b d^{2} e - 3 c d^{3} - 30 c e^{3} x^{3} + x^{2} \left (- 20 b e^{3} - 30 c d e^{2}\right ) + x \left (- 15 a e^{3} - 10 b d e^{2} - 15 c d^{2} e\right )}{60 d^{5} e^{4} + 300 d^{4} e^{5} x + 600 d^{3} e^{6} x^{2} + 600 d^{2} e^{7} x^{3} + 300 d e^{8} x^{4} + 60 e^{9} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-3*a*d*e**2 - 2*b*d**2*e - 3*c*d**3 - 30*c*e**3*x**3 + x**2*(-20*b*e**3 - 30*c*d*e**2) + x*(-15*a*e**3 - 10*b
*d*e**2 - 15*c*d**2*e))/(60*d**5*e**4 + 300*d**4*e**5*x + 600*d**3*e**6*x**2 + 600*d**2*e**7*x**3 + 300*d*e**8
*x**4 + 60*e**9*x**5)

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