3.9.39 \(\int \frac {1}{(d+e x) (c d^2+2 c d e x+c e^2 x^2)^3} \, dx\)

Optimal. Leaf size=17 \[ -\frac {1}{6 c^3 e (d+e x)^6} \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, 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 {1}{6 c^3 e (d+e x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3),x]

[Out]

-1/(6*c^3*e*(d + e*x)^6)

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 {1}{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx &=\int \frac {1}{c^3 (d+e x)^7} \, dx\\ &=\frac {\int \frac {1}{(d+e x)^7} \, dx}{c^3}\\ &=-\frac {1}{6 c^3 e (d+e x)^6}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -\frac {1}{6 c^3 e (d+e x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3),x]

[Out]

-1/6*1/(c^3*e*(d + e*x)^6)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(d+e x) \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[1/((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3),x]

[Out]

IntegrateAlgebraic[1/((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^3), x]

________________________________________________________________________________________

fricas [B]  time = 0.38, size = 89, normalized size = 5.24 \begin {gather*} -\frac {1}{6 \, {\left (c^{3} e^{7} x^{6} + 6 \, c^{3} d e^{6} x^{5} + 15 \, c^{3} d^{2} e^{5} x^{4} + 20 \, c^{3} d^{3} e^{4} x^{3} + 15 \, c^{3} d^{4} e^{3} x^{2} + 6 \, c^{3} d^{5} e^{2} x + c^{3} d^{6} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x, algorithm="fricas")

[Out]

-1/6/(c^3*e^7*x^6 + 6*c^3*d*e^6*x^5 + 15*c^3*d^2*e^5*x^4 + 20*c^3*d^3*e^4*x^3 + 15*c^3*d^4*e^3*x^2 + 6*c^3*d^5
*e^2*x + c^3*d^6*e)

________________________________________________________________________________________

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(1/(e*x+d)/(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: exp(1)^6/(c^3*exp(2)^3*d^6*exp(1)-3*c^3*
exp(2)^2*d^6*exp(1)^3+3*c^3*exp(2)*d^6*exp(1)^5-c^3*d^6*exp(1)^7)*ln(abs(x*exp(1)+d))-exp(1)^5/(2*c^3*exp(2)^3
*d^6-6*c^3*exp(2)^2*d^6*exp(1)^2+6*c^3*exp(2)*d^6*exp(1)^4-2*c^3*d^6*exp(1)^6)*ln(x^2*exp(2)+2*x*d*exp(1)+d^2)
+(3*exp(2)^2+4*exp(2)*exp(1)^2+8*exp(1)^4)*1/2/(4*c^3*exp(2)^2*d^5-8*c^3*exp(2)*d^5*exp(1)^2+4*c^3*d^5*exp(1)^
4)/d/sqrt(-exp(1)^2+exp(2))*atan((d*exp(1)+x*exp(2))/d/sqrt(-exp(1)^2+exp(2)))+((3*exp(2)^4*d+exp(2)^3*exp(1)^
2*d-4*exp(2)^2*exp(1)^4*d)*x^3+(9*exp(2)^3*exp(1)*d^2+7*exp(2)^2*exp(1)^3*d^2-16*exp(2)*exp(1)^5*d^2)*x^2+(5*e
xp(2)^3*d^3+3*exp(2)^2*exp(1)^2*d^3+8*exp(2)*exp(1)^4*d^3-16*exp(1)^6*d^3)*x+7*exp(2)^2*exp(1)*d^4-3*exp(2)*ex
p(1)^3*d^4-4*exp(1)^5*d^4)/8/d^6/(exp(2)-exp(1)^2)^3/(x^2*exp(2)+2*x*d*exp(1)+d^2)^2/c^3

________________________________________________________________________________________

maple [A]  time = 0.06, size = 16, normalized size = 0.94 \begin {gather*} -\frac {1}{6 \left (e x +d \right )^{6} c^{3} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x)

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 1.38, size = 89, normalized size = 5.24 \begin {gather*} -\frac {1}{6 \, {\left (c^{3} e^{7} x^{6} + 6 \, c^{3} d e^{6} x^{5} + 15 \, c^{3} d^{2} e^{5} x^{4} + 20 \, c^{3} d^{3} e^{4} x^{3} + 15 \, c^{3} d^{4} e^{3} x^{2} + 6 \, c^{3} d^{5} e^{2} x + c^{3} d^{6} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*e^2*x^2+2*c*d*e*x+c*d^2)^3,x, algorithm="maxima")

[Out]

-1/6/(c^3*e^7*x^6 + 6*c^3*d*e^6*x^5 + 15*c^3*d^2*e^5*x^4 + 20*c^3*d^3*e^4*x^3 + 15*c^3*d^4*e^3*x^2 + 6*c^3*d^5
*e^2*x + c^3*d^6*e)

________________________________________________________________________________________

mupad [B]  time = 0.45, size = 91, normalized size = 5.35 \begin {gather*} -\frac {1}{6\,c^3\,d^6\,e+36\,c^3\,d^5\,e^2\,x+90\,c^3\,d^4\,e^3\,x^2+120\,c^3\,d^3\,e^4\,x^3+90\,c^3\,d^2\,e^5\,x^4+36\,c^3\,d\,e^6\,x^5+6\,c^3\,e^7\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x)*(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^3),x)

[Out]

-1/(6*c^3*d^6*e + 6*c^3*e^7*x^6 + 36*c^3*d^5*e^2*x + 36*c^3*d*e^6*x^5 + 90*c^3*d^4*e^3*x^2 + 120*c^3*d^3*e^4*x
^3 + 90*c^3*d^2*e^5*x^4)

________________________________________________________________________________________

sympy [B]  time = 0.49, size = 97, normalized size = 5.71 \begin {gather*} - \frac {1}{6 c^{3} d^{6} e + 36 c^{3} d^{5} e^{2} x + 90 c^{3} d^{4} e^{3} x^{2} + 120 c^{3} d^{3} e^{4} x^{3} + 90 c^{3} d^{2} e^{5} x^{4} + 36 c^{3} d e^{6} x^{5} + 6 c^{3} e^{7} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(c*e**2*x**2+2*c*d*e*x+c*d**2)**3,x)

[Out]

-1/(6*c**3*d**6*e + 36*c**3*d**5*e**2*x + 90*c**3*d**4*e**3*x**2 + 120*c**3*d**3*e**4*x**3 + 90*c**3*d**2*e**5
*x**4 + 36*c**3*d*e**6*x**5 + 6*c**3*e**7*x**6)

________________________________________________________________________________________