3.19.56 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^3} \, dx\) [1856]

Optimal. Leaf size=20 \[ \frac {(a e+c d x)^4}{4 c d} \]

[Out]

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

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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 {(a e+c d x)^4}{4 c d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*e + c*d*x)^4/(4*c*d)

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

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Mathematica [A]
time = 0.00, size = 20, normalized size = 1.00 \begin {gather*} \frac {(a e+c d x)^4}{4 c d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*e + c*d*x)^4/(4*c*d)

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

method	result	size
default	\(\frac {\left (c d x +a e \right )^{4}}{4 c d}\)	\(19\)
gosper	\(\frac {x \left (c^{3} d^{3} x^{3}+4 a \,c^{2} d^{2} e \,x^{2}+6 a^{2} c d \,e^{2} x +4 e^{3} a^{3}\right )}{4}\)	\(47\)
risch	\(\frac {c^{3} d^{3} x^{4}}{4}+a \,c^{2} d^{2} e \,x^{3}+\frac {3 a^{2} c d \,e^{2} x^{2}}{2}+e^{3} a^{3} x +\frac {e^{4} a^{4}}{4 d c}\)	\(60\)
norman	\(\frac {\left (e^{3} c^{2} d^{2} a +\frac {1}{2} d^{4} e \,c^{3}\right ) x^{5}+\left (\frac {3}{2} d \,e^{4} a^{2} c +2 d^{3} e^{2} c^{2} a +\frac {1}{4} d^{5} c^{3}\right ) x^{4}+\left (a^{3} e^{5}+3 d^{2} e^{3} a^{2} c +d^{4} c^{2} a e \right ) x^{3}-\frac {d^{2} \left (4 d \,e^{4} a^{3}+3 d^{3} e^{2} a^{2} c \right )}{2 e^{2}}-\frac {d \left (3 d \,e^{4} a^{3}+3 d^{3} e^{2} a^{2} c \right ) x}{e}+\frac {e^{2} c^{3} d^{3} x^{6}}{4}}{\left (e x +d \right )^{2}}\)	\(176\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19) = 38\).
time = 0.27, size = 44, normalized size = 2.20 \begin {gather*} \frac {1}{4} \, c^{3} d^{3} x^{4} + a c^{2} d^{2} x^{3} e + \frac {3}{2} \, a^{2} c d x^{2} e^{2} + a^{3} x e^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19) = 38\).
time = 2.24, size = 44, normalized size = 2.20 \begin {gather*} \frac {1}{4} \, c^{3} d^{3} x^{4} + a c^{2} d^{2} x^{3} e + \frac {3}{2} \, a^{2} c d x^{2} e^{2} + a^{3} x e^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (14) = 28\).
time = 0.05, size = 49, normalized size = 2.45 \begin {gather*} a^{3} e^{3} x + \frac {3 a^{2} c d e^{2} x^{2}}{2} + a c^{2} d^{2} e x^{3} + \frac {c^{3} d^{3} x^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*e**3*x + 3*a**2*c*d*e**2*x**2/2 + a*c**2*d**2*e*x**3 + c**3*d**3*x**4/4

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19) = 38\).
time = 0.94, size = 44, normalized size = 2.20 \begin {gather*} \frac {1}{4} \, c^{3} d^{3} x^{4} + a c^{2} d^{2} x^{3} e + \frac {3}{2} \, a^{2} c d x^{2} e^{2} + a^{3} x e^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.05, size = 45, normalized size = 2.25 \begin {gather*} a^3\,e^3\,x+\frac {3\,a^2\,c\,d\,e^2\,x^2}{2}+a\,c^2\,d^2\,e\,x^3+\frac {c^3\,d^3\,x^4}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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