3.997 \(\int \frac {(c d^2+2 c d e x+c e^2 x^2)^2}{(d+e x)^8} \, dx\)

Optimal. Leaf size=17 \[ -\frac {c^2}{3 e (d+e x)^3} \]

[Out]

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

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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} \[ -\frac {c^2}{3 e (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^8,x]

[Out]

-c^2/(3*e*(d + e*x)^3)

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 {\left (c d^2+2 c d e x+c e^2 x^2\right )^2}{(d+e x)^8} \, dx &=\int \frac {c^2}{(d+e x)^4} \, dx\\ &=c^2 \int \frac {1}{(d+e x)^4} \, dx\\ &=-\frac {c^2}{3 e (d+e x)^3}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \[ -\frac {c^2}{3 e (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^8,x]

[Out]

-1/3*c^2/(e*(d + e*x)^3)

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fricas [B]  time = 0.67, size = 38, normalized size = 2.24 \[ -\frac {c^{2}}{3 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^2/(e*x+d)^8,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.15, size = 66, normalized size = 3.88 \[ -\frac {{\left (c^{2} x^{4} e^{8} + 4 \, c^{2} d x^{3} e^{7} + 6 \, c^{2} d^{2} x^{2} e^{6} + 4 \, c^{2} d^{3} x e^{5} + c^{2} d^{4} e^{4}\right )} e^{\left (-5\right )}}{3 \, {\left (x e + d\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^2/(e*x+d)^8,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 16, normalized size = 0.94 \[ -\frac {c^{2}}{3 \left (e x +d \right )^{3} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e^2*x^2+2*c*d*e*x+c*d^2)^2/(e*x+d)^8,x)

[Out]

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

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maxima [B]  time = 1.32, size = 38, normalized size = 2.24 \[ -\frac {c^{2}}{3 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^2/(e*x+d)^8,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.03, size = 37, normalized size = 2.18 \[ -\frac {c^2}{3\,e\,\left (d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^2/(d + e*x)^8,x)

[Out]

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

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sympy [B]  time = 0.29, size = 39, normalized size = 2.29 \[ - \frac {c^{2}}{3 d^{3} e + 9 d^{2} e^{2} x + 9 d e^{3} x^{2} + 3 e^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**2/(e*x+d)**8,x)

[Out]

-c**2/(3*d**3*e + 9*d**2*e**2*x + 9*d*e**3*x**2 + 3*e**4*x**3)

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