3.19.43 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^2}{(d+e x)^2} \, dx\) [1843]

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

[Out]

1/3*(c*d*x+a*e)^3/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)^3}{3 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)^2/(d + e*x)^2,x]

[Out]

(a*e + c*d*x)^3/(3*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 )^2}{(d+e x)^2} \, dx &=\int (a e+c d x)^2 \, dx\\ &=\frac {(a e+c d x)^3}{3 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)^3}{3 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)^2/(d + e*x)^2,x]

[Out]

(a*e + c*d*x)^3/(3*c*d)

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Maple [A]
time = 0.67, size = 19, normalized size = 0.95

method	result	size
default	\(\frac {\left (c d x +a e \right )^{3}}{3 c d}\)	\(19\)
gosper	\(\frac {x \left (c^{2} d^{2} x^{2}+3 a c d e x +3 a^{2} e^{2}\right )}{3}\)	\(30\)
risch	\(\frac {c^{2} d^{2} x^{3}}{3}+a c d e \,x^{2}+a^{2} e^{2} x +\frac {e^{3} a^{3}}{3 d c}\)	\(43\)
norman	\(\frac {\left (a d \,e^{2} c +\frac {1}{3} c^{2} d^{3}\right ) x^{3}+\left (a^{2} e^{3}+d^{2} e a c \right ) x^{2}+a^{2} d \,e^{2} x +\frac {e \,c^{2} d^{2} x^{4}}{3}}{e x +d}\)	\(70\)

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)^2/(e*x+d)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 28, normalized size = 1.40 \begin {gather*} \frac {1}{3} \, c^{2} d^{2} x^{3} + a c d x^{2} e + a^{2} x e^{2} \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)^2/(e*x+d)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.96, size = 28, normalized size = 1.40 \begin {gather*} \frac {1}{3} \, c^{2} d^{2} x^{3} + a c d x^{2} e + a^{2} x e^{2} \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)^2/(e*x+d)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (14) = 28\).
time = 0.03, size = 29, normalized size = 1.45 \begin {gather*} a^{2} e^{2} x + a c d e x^{2} + \frac {c^{2} d^{2} x^{3}}{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)**2/(e*x+d)**2,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (19) = 38\).
time = 1.73, size = 99, normalized size = 4.95 \begin {gather*} \frac {1}{3} \, {\left (c^{2} d^{2} - \frac {3 \, c^{2} d^{3}}{x e + d} + \frac {3 \, c^{2} d^{4}}{{\left (x e + d\right )}^{2}} + \frac {3 \, a c d e^{2}}{x e + d} - \frac {6 \, a c d^{2} e^{2}}{{\left (x e + d\right )}^{2}} + \frac {3 \, a^{2} e^{4}}{{\left (x e + d\right )}^{2}}\right )} {\left (x e + d\right )}^{3} e^{\left (-3\right )} \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)^2/(e*x+d)^2,x, algorithm="giac")

[Out]

1/3*(c^2*d^2 - 3*c^2*d^3/(x*e + d) + 3*c^2*d^4/(x*e + d)^2 + 3*a*c*d*e^2/(x*e + d) - 6*a*c*d^2*e^2/(x*e + d)^2
+ 3*a^2*e^4/(x*e + d)^2)*(x*e + d)^3*e^(-3)

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Mupad [B]
time = 0.04, size = 28, normalized size = 1.40 \begin {gather*} a^2\,e^2\,x+a\,c\,d\,e\,x^2+\frac {c^2\,d^2\,x^3}{3} \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)^2/(d + e*x)^2,x)

[Out]

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

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