3.19.91 \(\int \frac {(d+e x)^3}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\) [1891]

Optimal. Leaf size=20 \[ -\frac {1}{2 c d (a e+c d x)^2} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 32} \begin {gather*} -\frac {1}{2 c d (a e+c d x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*1/(c*d*(a*e + c*d*x)^2)

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]

Rule 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 20, normalized size = 1.00 \begin {gather*} -\frac {1}{2 c d (a e+c d x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*1/(c*d*(a*e + c*d*x)^2)

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Maple [A]
time = 0.71, size = 19, normalized size = 0.95

method	result	size
gosper	\(-\frac {1}{2 c d \left (c d x +a e \right )^{2}}\)	\(19\)
default	\(-\frac {1}{2 c d \left (c d x +a e \right )^{2}}\)	\(19\)
risch	\(-\frac {1}{2 c d \left (c d x +a e \right )^{2}}\)	\(19\)
norman	\(\frac {-\frac {e x}{c}-\frac {d}{2 c}-\frac {e^{2} x^{2}}{2 c d}}{\left (c d x +a e \right )^{2} \left (e x +d \right )^{2}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 35, normalized size = 1.75 \begin {gather*} -\frac {1}{2 \, {\left (c^{3} d^{3} x^{2} + 2 \, a c^{2} d^{2} x e + a^{2} c d e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 3.59, size = 35, normalized size = 1.75 \begin {gather*} -\frac {1}{2 \, {\left (c^{3} d^{3} x^{2} + 2 \, a c^{2} d^{2} x e + a^{2} c d e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (17) = 34\).
time = 0.12, size = 39, normalized size = 1.95 \begin {gather*} - \frac {1}{2 a^{2} c d e^{2} + 4 a c^{2} d^{2} e x + 2 c^{3} d^{3} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.80, size = 19, normalized size = 0.95 \begin {gather*} -\frac {1}{2 \, {\left (c d x + a e\right )}^{2} c d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/((c*d*x + a*e)^2*c*d)

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Mupad [B]
time = 0.03, size = 37, normalized size = 1.85 \begin {gather*} -\frac {1}{2\,a^2\,c\,d\,e^2+4\,a\,c^2\,d^2\,e\,x+2\,c^3\,d^3\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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