3.9.3 \(\int \frac {(c d^2+2 c d e x+c e^2 x^2)^2}{(d+e x)^2} \, dx\)

Optimal. Leaf size=17 \[ \frac {c^2 (d+e x)^3}{3 e} \]

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {27, 12, 32} \begin {gather*} \frac {c^2 (d+e x)^3}{3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*(d + e*x)^3)/(3*e)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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]

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]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {c^2 (d+e x)^3}{3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*(d + e*x)^3)/(3*e)

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IntegrateAlgebraic [A]  time = 0.04, size = 17, normalized size = 1.00 \begin {gather*} \frac {c^2 (d+e x)^3}{3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2/(d + e*x)^2,x]

[Out]

(c^2*(d + e*x)^3)/(3*e)

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fricas [A]  time = 0.38, size = 29, normalized size = 1.71 \begin {gather*} \frac {1}{3} \, c^{2} e^{2} x^{3} + c^{2} d e x^{2} + c^{2} d^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 15, normalized size = 0.88 \begin {gather*} \frac {1}{3} \, {\left (x e + d\right )}^{3} c^{2} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(x*e + d)^3*c^2*e^(-1)

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maple [A]  time = 0.06, size = 16, normalized size = 0.94 \begin {gather*} \frac {\left (e x +d \right )^{3} c^{2}}{3 e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.36, size = 29, normalized size = 1.71 \begin {gather*} \frac {1}{3} \, c^{2} e^{2} x^{3} + c^{2} d e x^{2} + c^{2} d^{2} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 24, normalized size = 1.41 \begin {gather*} \frac {c^2\,x\,\left (3\,d^2+3\,d\,e\,x+e^2\,x^2\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^2*x*(3*d^2 + e^2*x^2 + 3*d*e*x))/3

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sympy [B]  time = 0.11, size = 29, normalized size = 1.71 \begin {gather*} c^{2} d^{2} x + c^{2} d e x^{2} + \frac {c^{2} e^{2} x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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