3.1009 \(\int \frac {(d+e x)^5}{(c d^2+2 c d e x+c e^2 x^2)^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {27, 9} \[ \frac {(d+e x)^2}{2 c^2 e} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^5/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2,x]

[Out]

(d + e*x)^2/(2*c^2*e)

Rule 9

Int[(a_)*((b_) + (c_.)*(x_)), x_Symbol] :> Simp[(a*(b + c*x)^2)/(2*c), x] /; FreeQ[{a, b, c}, 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]

Rubi steps

\begin {align*} \int \frac {(d+e x)^5}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx &=\int \frac {d+e x}{c^2} \, dx\\ &=\frac {(d+e x)^2}{2 c^2 e}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 0.94 \[ \frac {d x+\frac {e x^2}{2}}{c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^5/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2,x]

[Out]

(d*x + (e*x^2)/2)/c^2

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fricas [A]  time = 1.02, size = 15, normalized size = 0.88 \[ \frac {e x^{2} + 2 \, d x}{2 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(c*e^2*x^2+2*c*d*e*x+c*d^2)^2,x, algorithm="fricas")

[Out]

1/2*(e*x^2 + 2*d*x)/c^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(c*e^2*x^2+2*c*d*e*x+c*d^2)^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: (1/2*x^2*c^2*exp(2)^2*exp(1)^5+5*x*c^2*e
xp(2)^2*d*exp(1)^4-4*x*c^2*exp(2)*d*exp(1)^6)/c^4/exp(2)^4+(-4*exp(2)^3*d^4*exp(1)+16*exp(2)^2*d^4*exp(1)^3-20
*exp(2)*d^4*exp(1)^5+8*d^4*exp(1)^7+(exp(2)^4*d^3-14*exp(2)^3*d^3*exp(1)^2+41*exp(2)^2*d^3*exp(1)^4-44*exp(2)*
d^3*exp(1)^6+16*d^3*exp(1)^8)*x)/2/exp(2)^4/c^2/(2*exp(1)*d*x+exp(2)*x^2+d^2)+(5*exp(2)^2*d^2*exp(1)^3-11*exp(
2)*d^2*exp(1)^5+6*d^2*exp(1)^7)/c^2/exp(2)^4*ln(x^2*exp(2)+2*x*d*exp(1)+d^2)+(exp(2)^4*d^3+6*exp(2)^3*d^3*exp(
1)^2-39*exp(2)^2*d^3*exp(1)^4+56*exp(2)*d^3*exp(1)^6-24*d^3*exp(1)^8)/c^2/exp(2)^4*1/2/d/sqrt(-exp(1)^2+exp(2)
)*atan((d*exp(1)+x*exp(2))/d/sqrt(-exp(1)^2+exp(2)))

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maple [A]  time = 0.04, size = 15, normalized size = 0.88 \[ \frac {\frac {1}{2} e \,x^{2}+d x}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5/(c*e^2*x^2+2*c*d*e*x+c*d^2)^2,x)

[Out]

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

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maxima [A]  time = 1.29, size = 15, normalized size = 0.88 \[ \frac {e x^{2} + 2 \, d x}{2 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(c*e^2*x^2+2*c*d*e*x+c*d^2)^2,x, algorithm="maxima")

[Out]

1/2*(e*x^2 + 2*d*x)/c^2

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mupad [B]  time = 0.02, size = 13, normalized size = 0.76 \[ \frac {x\,\left (2\,d+e\,x\right )}{2\,c^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^5/(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^2,x)

[Out]

(x*(2*d + e*x))/(2*c^2)

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sympy [A]  time = 0.11, size = 15, normalized size = 0.88 \[ \frac {d x}{c^{2}} + \frac {e x^{2}}{2 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5/(c*e**2*x**2+2*c*d*e*x+c*d**2)**2,x)

[Out]

d*x/c**2 + e*x**2/(2*c**2)

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