3.9.29 \(\int \frac {(d+e x)^9}{(c d^2+2 c d e x+c e^2 x^2)^3} \, dx\)

Optimal. Leaf size=17 \[ \frac {(d+e x)^4}{4 c^3 e} \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {27, 12, 32} \begin {gather*} \frac {(d+e x)^4}{4 c^3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^9/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]

[Out]

(d + e*x)^4/(4*c^3*e)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin {align*} \int \frac {(d+e x)^9}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx &=\int \frac {(d+e x)^3}{c^3} \, dx\\ &=\frac {\int (d+e x)^3 \, dx}{c^3}\\ &=\frac {(d+e x)^4}{4 c^3 e}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {(d+e x)^4}{4 c^3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^9/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]

[Out]

(d + e*x)^4/(4*c^3*e)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^9}{\left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^9/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3,x]

[Out]

IntegrateAlgebraic[(d + e*x)^9/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3, x]

________________________________________________________________________________________

fricas [B]  time = 0.38, size = 37, normalized size = 2.18 \begin {gather*} \frac {e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x}{4 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^9/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^9/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: (1/4*x^4*c^9*exp(2)^9*exp(1)^9+3*x^3*c^9
*exp(2)^9*d*exp(1)^8-2*x^3*c^9*exp(2)^8*d*exp(1)^10+18*x^2*c^9*exp(2)^9*d^2*exp(1)^7-57/2*x^2*c^9*exp(2)^8*d^2
*exp(1)^9+12*x^2*c^9*exp(2)^7*d^2*exp(1)^11+84*x*c^9*exp(2)^9*d^3*exp(1)^6-243*x*c^9*exp(2)^8*d^3*exp(1)^8+240
*x*c^9*exp(2)^7*d^3*exp(1)^10-80*x*c^9*exp(2)^6*d^3*exp(1)^12)/c^12/exp(2)^12+((3*exp(2)^8*d^5+15*exp(2)^7*d^5
*exp(1)^2-783*exp(2)^6*d^5*exp(1)^4+4341*exp(2)^5*d^5*exp(1)^6-10056*exp(2)^4*d^5*exp(1)^8+11664*exp(2)^3*d^5*
exp(1)^10-6720*exp(2)^2*d^5*exp(1)^12+1536*exp(2)*d^5*exp(1)^14)*x^3+(9*exp(2)^7*d^6*exp(1)-291*exp(2)^6*d^6*e
xp(1)^3+675*exp(2)^5*d^6*exp(1)^5+2511*exp(2)^4*d^6*exp(1)^7-11624*exp(2)^3*d^6*exp(1)^9+17232*exp(2)^2*d^6*ex
p(1)^11-11328*exp(2)*d^6*exp(1)^13+2816*d^6*exp(1)^15)*x^2+(5*exp(2)^7*d^7-67*exp(2)^6*d^7*exp(1)^2-613*exp(2)
^5*d^7*exp(1)^4+5519*exp(2)^4*d^7*exp(1)^6-15100*exp(2)^3*d^7*exp(1)^8+19216*exp(2)^2*d^7*exp(1)^10-11776*exp(
2)*d^7*exp(1)^12+2816*d^7*exp(1)^14)*x-13*exp(2)^6*d^8*exp(1)-97*exp(2)^5*d^8*exp(1)^3+1121*exp(2)^4*d^8*exp(1
)^5-3355*exp(2)^3*d^8*exp(1)^7+4504*exp(2)^2*d^8*exp(1)^9-2864*exp(2)*d^8*exp(1)^11+704*d^8*exp(1)^13)/8/exp(2
)^7/c^3/(2*exp(1)*d*x+exp(2)*x^2+d^2)^2-(-63*exp(2)^4*d^4*exp(1)^5+306*exp(2)^3*d^4*exp(1)^7-543*exp(2)^2*d^4*
exp(1)^9+420*exp(2)*d^4*exp(1)^11-120*d^4*exp(1)^13)/c^3/exp(2)^7*ln(x^2*exp(2)+2*x*d*exp(1)+d^2)-(-3*exp(2)^7
*d^5-15*exp(2)^6*d^5*exp(1)^2-225*exp(2)^5*d^5*exp(1)^4+2715*exp(2)^4*d^5*exp(1)^6-8520*exp(2)^3*d^5*exp(1)^8+
11808*exp(2)^2*d^5*exp(1)^10-7680*exp(2)*d^5*exp(1)^12+1920*d^5*exp(1)^14)*1/4/c^3/exp(2)^7*1/2/d/sqrt(-exp(1)
^2+exp(2))*atan((d*exp(1)+x*exp(2))/d/sqrt(-exp(1)^2+exp(2)))

________________________________________________________________________________________

maple [A]  time = 0.05, size = 16, normalized size = 0.94 \begin {gather*} \frac {\left (e x +d \right )^{4}}{4 c^{3} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^9/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x)

[Out]

1/4*(e*x+d)^4/c^3/e

________________________________________________________________________________________

maxima [B]  time = 1.36, size = 37, normalized size = 2.18 \begin {gather*} \frac {e^{3} x^{4} + 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} + 4 \, d^{3} x}{4 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^9/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 43, normalized size = 2.53 \begin {gather*} \frac {d^3\,x}{c^3}+\frac {e^3\,x^4}{4\,c^3}+\frac {3\,d^2\,e\,x^2}{2\,c^3}+\frac {d\,e^2\,x^3}{c^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^9/(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^3,x)

[Out]

(d^3*x)/c^3 + (e^3*x^4)/(4*c^3) + (3*d^2*e*x^2)/(2*c^3) + (d*e^2*x^3)/c^3

________________________________________________________________________________________

sympy [B]  time = 0.14, size = 46, normalized size = 2.71 \begin {gather*} \frac {d^{3} x}{c^{3}} + \frac {3 d^{2} e x^{2}}{2 c^{3}} + \frac {d e^{2} x^{3}}{c^{3}} + \frac {e^{3} x^{4}}{4 c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**9/(c*e**2*x**2+2*c*d*e*x+c*d**2)**3,x)

[Out]

d**3*x/c**3 + 3*d**2*e*x**2/(2*c**3) + d*e**2*x**3/c**3 + e**3*x**4/(4*c**3)

________________________________________________________________________________________