3.20.2 \(\int \frac {(d+e x)^4}{(a d e+(c d^2+a e^2) x+c d e x^2)^4} \, dx\) [1902]

Optimal. Leaf size=20 \[ -\frac {1}{3 c d (a e+c d x)^3} \]

[Out]

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

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

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^4/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

-1/3*1/(c*d*(a*e + c*d*x)^3)

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

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Mathematica [A]
time = 0.00, size = 20, normalized size = 1.00 \begin {gather*} -\frac {1}{3 c d (a e+c d x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^4/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

-1/3*1/(c*d*(a*e + c*d*x)^3)

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Maple [A]
time = 0.70, size = 19, normalized size = 0.95

method	result	size
gosper	\(-\frac {1}{3 c d \left (c d x +a e \right )^{3}}\)	\(19\)
default	\(-\frac {1}{3 c d \left (c d x +a e \right )^{3}}\)	\(19\)
risch	\(-\frac {1}{3 c d \left (c d x +a e \right )^{3}}\)	\(19\)
norman	\(\frac {-\frac {d e x}{c}-\frac {e^{2} x^{2}}{c}-\frac {d^{2}}{3 c}-\frac {e^{3} x^{3}}{3 c d}}{\left (c d x +a e \right )^{3} \left (e x +d \right )^{3}}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (19) = 38\).
time = 0.28, size = 51, normalized size = 2.55 \begin {gather*} -\frac {1}{3 \, {\left (c^{4} d^{4} x^{3} + 3 \, a c^{3} d^{3} x^{2} e + 3 \, a^{2} c^{2} d^{2} x e^{2} + a^{3} c d e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (19) = 38\).
time = 4.57, size = 51, normalized size = 2.55 \begin {gather*} -\frac {1}{3 \, {\left (c^{4} d^{4} x^{3} + 3 \, a c^{3} d^{3} x^{2} e + 3 \, a^{2} c^{2} d^{2} x e^{2} + a^{3} c d e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (17) = 34\).
time = 0.17, size = 58, normalized size = 2.90 \begin {gather*} - \frac {1}{3 a^{3} c d e^{3} + 9 a^{2} c^{2} d^{2} e^{2} x + 9 a c^{3} d^{3} e x^{2} + 3 c^{4} d^{4} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)

[Out]

-1/(3*a**3*c*d*e**3 + 9*a**2*c**2*d**2*e**2*x + 9*a*c**3*d**3*e*x**2 + 3*c**4*d**4*x**3)

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Giac [A]
time = 0.61, size = 19, normalized size = 0.95 \begin {gather*} -\frac {1}{3 \, {\left (c d x + a e\right )}^{3} c d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="giac")

[Out]

-1/3/((c*d*x + a*e)^3*c*d)

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Mupad [B]
time = 0.57, size = 54, normalized size = 2.70 \begin {gather*} -\frac {1}{3\,a^3\,c\,d\,e^3+9\,a^2\,c^2\,d^2\,e^2\,x+9\,a\,c^3\,d^3\,e\,x^2+3\,c^4\,d^4\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^4/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^4,x)

[Out]

-1/(3*c^4*d^4*x^3 + 3*a^3*c*d*e^3 + 9*a*c^3*d^3*e*x^2 + 9*a^2*c^2*d^2*e^2*x)

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