3.11.9 \(\int \frac {(d+e x)^5}{(c d^2+2 c d e x+c e^2 x^2)^2} \, dx\) [1009]

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} \begin {gather*} \frac {(d+e x)^2}{2 c^2 e} \end {gather*}

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 \begin {gather*} \frac {d x+\frac {e x^2}{2}}{c^2} \end {gather*}

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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Maple [A]
time = 0.58, size = 15, normalized size = 0.88

method	result	size
gosper	\(\frac {x \left (e x +2 d \right )}{2 c^{2}}\)	\(14\)
default	\(\frac {\frac {1}{2} e \,x^{2}+d x}{c^{2}}\)	\(15\)
risch	\(\frac {e \,x^{2}}{2 c^{2}}+\frac {d x}{c^{2}}\)	\(17\)
norman	\(\frac {-\frac {10 d^{3} e \,x^{2}}{c}+\frac {e^{4} x^{5}}{2 c}+\frac {5 d \,e^{3} x^{4}}{2 c}-\frac {9 d^{5}}{2 c e}-\frac {25 d^{4} x}{2 c}}{c \left (e x +d \right )^{3}}\)	\(68\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 16, normalized size = 0.94 \begin {gather*} \frac {x^{2} e + 2 \, d x}{2 \, c^{2}} \end {gather*}

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*(x^2*e + 2*d*x)/c^2

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Fricas [A]
time = 3.22, size = 16, normalized size = 0.94 \begin {gather*} \frac {x^{2} e + 2 \, d x}{2 \, c^{2}} \end {gather*}

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*(x^2*e + 2*d*x)/c^2

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Sympy [A]
time = 0.03, size = 15, normalized size = 0.88 \begin {gather*} \frac {d x}{c^{2}} + \frac {e x^{2}}{2 c^{2}} \end {gather*}

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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Giac [A]
time = 0.59, size = 16, normalized size = 0.94 \begin {gather*} \frac {x^{2} e + 2 \, d x}{2 \, c^{2}} \end {gather*}

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]

1/2*(x^2*e + 2*d*x)/c^2

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

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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