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

Optimal. Leaf size=17 \[ -\frac {1}{7 c^3 e (d+e x)^7} \]

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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}{7 c^3 e (d+e x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(7*c^3*e*(d + e*x)^7)

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)^2 \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)^8} \, dx\\ &=\frac {\int \frac {1}{(d+e x)^8} \, dx}{c^3}\\ &=-\frac {1}{7 c^3 e (d+e x)^7}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -\frac {1}{7 c^3 e (d+e x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/7*1/(c^3*e*(d + e*x)^7)

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

[Out]

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

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fricas [B]  time = 0.38, size = 103, normalized size = 6.06 \begin {gather*} -\frac {1}{7 \, {\left (c^{3} e^{8} x^{7} + 7 \, c^{3} d e^{7} x^{6} + 21 \, c^{3} d^{2} e^{6} x^{5} + 35 \, c^{3} d^{3} e^{5} x^{4} + 35 \, c^{3} d^{4} e^{4} x^{3} + 21 \, c^{3} d^{5} e^{3} x^{2} + 7 \, c^{3} d^{6} e^{2} x + c^{3} d^{7} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7/(c^3*e^8*x^7 + 7*c^3*d*e^7*x^6 + 21*c^3*d^2*e^6*x^5 + 35*c^3*d^3*e^5*x^4 + 35*c^3*d^4*e^4*x^3 + 21*c^3*d^
5*e^3*x^2 + 7*c^3*d^6*e^2*x + c^3*d^7*e)

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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)^2/(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)*x+d)^-1/exp(1)*exp(1)^12/(c^3*d^
6*exp(1)^12-3*c^3*d^6*exp(1)^10*exp(2)+3*c^3*d^6*exp(1)^8*exp(2)^2-c^3*d^6*exp(1)^6*exp(2)^3)-((96*d^3*exp(1)^
12-192*d^3*exp(1)^10*exp(2)+102*d^3*exp(1)^8*exp(2)^2-3*d^3*exp(1)^6*exp(2)^3-3*d^3*exp(1)^4*exp(2)^4)*(-(exp(
1)*x+d)^-1/exp(1))^3+(160*d^2*exp(1)^11-368*d^2*exp(1)^9*exp(2)+226*d^2*exp(1)^7*exp(2)^2-9*d^2*exp(1)^5*exp(2
)^3-9*d^2*exp(1)^3*exp(2)^4)*(-(exp(1)*x+d)^-1/exp(1))^2-(-160*d*exp(1)^8*exp(2)+176*d*exp(1)^6*exp(2)^2-14*d*
exp(1)^4*exp(2)^3-9*d*exp(1)^2*exp(2)^4)*(exp(1)*x+d)^-1/exp(1)+40*exp(1)^5*exp(2)^2-8*exp(1)^3*exp(2)^3-3*exp
(1)*exp(2)^4)/8/d^7/(exp(2)-exp(1)^2)^3/((-(exp(1)*x+d)^-1/exp(1))^2*exp(1)^4*d^2-(-(exp(1)*x+d)^-1/exp(1))^2*
exp(1)^2*d^2*exp(2)-2*(exp(1)*x+d)^-1/exp(1)*exp(1)^3*d+2*(exp(1)*x+d)^-1/exp(1)*exp(1)*d*exp(2)-exp(2))^2/c^3
-3*exp(1)^5/(-c^3*d^7*exp(1)^6+3*c^3*d^7*exp(1)^4*exp(2)-3*c^3*d^7*exp(1)^2*exp(2)^2+c^3*d^7*exp(2)^3)*ln((-(e
xp(1)*x+d)^-1/exp(1))^2*d^2*exp(1)^4-(-(exp(1)*x+d)^-1/exp(1))^2*d^2*exp(1)^2*exp(2)-2*(exp(1)*x+d)^-1/exp(1)*
d*exp(1)^3+2*(exp(1)*x+d)^-1/exp(1)*d*exp(1)*exp(2)-exp(2))+(48*exp(1)^8-24*exp(1)^6*exp(2)-6*exp(1)^4*exp(2)^
2-3*exp(1)^2*exp(2)^3)/2/(-4*c^3*d^6*exp(1)^6+12*c^3*d^6*exp(1)^4*exp(2)-12*c^3*d^6*exp(1)^2*exp(2)^2+4*c^3*d^
6*exp(2)^3)/d/sqrt(-exp(1)^2+exp(2))/exp(1)^2*atan((-d*(exp(1)*x+d)^-1/exp(1)*exp(1)^3+d*(exp(1)*x+d)^-1/exp(1
)*exp(1)*exp(2)+exp(1)^2-exp(2))/sqrt(-exp(1)^2+exp(2))/exp(1))

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maple [A]  time = 0.05, size = 16, normalized size = 0.94 \begin {gather*} -\frac {1}{7 \left (e x +d \right )^{7} c^{3} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7/c^3/e/(e*x+d)^7

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maxima [B]  time = 1.41, size = 103, normalized size = 6.06 \begin {gather*} -\frac {1}{7 \, {\left (c^{3} e^{8} x^{7} + 7 \, c^{3} d e^{7} x^{6} + 21 \, c^{3} d^{2} e^{6} x^{5} + 35 \, c^{3} d^{3} e^{5} x^{4} + 35 \, c^{3} d^{4} e^{4} x^{3} + 21 \, c^{3} d^{5} e^{3} x^{2} + 7 \, c^{3} d^{6} e^{2} x + c^{3} d^{7} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7/(c^3*e^8*x^7 + 7*c^3*d*e^7*x^6 + 21*c^3*d^2*e^6*x^5 + 35*c^3*d^3*e^5*x^4 + 35*c^3*d^4*e^4*x^3 + 21*c^3*d^
5*e^3*x^2 + 7*c^3*d^6*e^2*x + c^3*d^7*e)

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mupad [B]  time = 0.07, size = 105, normalized size = 6.18 \begin {gather*} -\frac {1}{7\,c^3\,d^7\,e+49\,c^3\,d^6\,e^2\,x+147\,c^3\,d^5\,e^3\,x^2+245\,c^3\,d^4\,e^4\,x^3+245\,c^3\,d^3\,e^5\,x^4+147\,c^3\,d^2\,e^6\,x^5+49\,c^3\,d\,e^7\,x^6+7\,c^3\,e^8\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(7*c^3*d^7*e + 7*c^3*e^8*x^7 + 49*c^3*d^6*e^2*x + 49*c^3*d*e^7*x^6 + 147*c^3*d^5*e^3*x^2 + 245*c^3*d^4*e^4*
x^3 + 245*c^3*d^3*e^5*x^4 + 147*c^3*d^2*e^6*x^5)

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sympy [B]  time = 0.55, size = 112, normalized size = 6.59 \begin {gather*} - \frac {1}{7 c^{3} d^{7} e + 49 c^{3} d^{6} e^{2} x + 147 c^{3} d^{5} e^{3} x^{2} + 245 c^{3} d^{4} e^{4} x^{3} + 245 c^{3} d^{3} e^{5} x^{4} + 147 c^{3} d^{2} e^{6} x^{5} + 49 c^{3} d e^{7} x^{6} + 7 c^{3} e^{8} x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(7*c**3*d**7*e + 49*c**3*d**6*e**2*x + 147*c**3*d**5*e**3*x**2 + 245*c**3*d**4*e**4*x**3 + 245*c**3*d**3*e*
*5*x**4 + 147*c**3*d**2*e**6*x**5 + 49*c**3*d*e**7*x**6 + 7*c**3*e**8*x**7)

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