3.1103 \(\int (c d^2+2 c d e x+c e^2 x^2)^p \, dx\)

Optimal. Leaf size=38 \[ \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^p}{e (2 p+1)} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {609} \[ \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^p}{e (2 p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 609

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p + 1
)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && NeQ[p, -2^(-1)]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 25, normalized size = 0.66 \[ \frac {(d+e x) \left (c (d+e x)^2\right )^p}{2 e p+e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)*(c*(d + e*x)^2)^p)/(e + 2*e*p)

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fricas [A]  time = 1.08, size = 36, normalized size = 0.95 \[ \frac {{\left (e x + d\right )} {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{2 \, e p + e} \]

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)^p,x, algorithm="fricas")

[Out]

(e*x + d)*(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^p/(2*e*p + e)

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giac [A]  time = 0.19, size = 62, normalized size = 1.63 \[ \frac {{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} x e + {\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} d}{2 \, p e + e} \]

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)^p,x, algorithm="giac")

[Out]

((c*x^2*e^2 + 2*c*d*x*e + c*d^2)^p*x*e + (c*x^2*e^2 + 2*c*d*x*e + c*d^2)^p*d)/(2*p*e + e)

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maple [A]  time = 0.04, size = 39, normalized size = 1.03 \[ \frac {\left (e x +d \right ) \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{p}}{\left (2 p +1\right ) e} \]

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)^p,x)

[Out]

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

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maxima [A]  time = 1.44, size = 32, normalized size = 0.84 \[ \frac {{\left (c^{p} e x + c^{p} d\right )} {\left (e x + d\right )}^{2 \, p}}{e {\left (2 \, p + 1\right )}} \]

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)^p,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.48, size = 45, normalized size = 1.18 \[ \left (\frac {x}{2\,p+1}+\frac {d}{e\,\left (2\,p+1\right )}\right )\,{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^p \]

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)^p,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \frac {x}{\sqrt {c d^{2}}} & \text {for}\: e = 0 \wedge p = - \frac {1}{2} \\x \left (c d^{2}\right )^{p} & \text {for}\: e = 0 \\\int \frac {1}{\sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}\, dx & \text {for}\: p = - \frac {1}{2} \\\frac {d \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p + e} + \frac {e x \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p + e} & \text {otherwise} \end {cases} \]

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)**p,x)

[Out]

Piecewise((x/sqrt(c*d**2), Eq(e, 0) & Eq(p, -1/2)), (x*(c*d**2)**p, Eq(e, 0)), (Integral(1/sqrt(c*d**2 + 2*c*d
*e*x + c*e**2*x**2), x), Eq(p, -1/2)), (d*(c*d**2 + 2*c*d*e*x + c*e**2*x**2)**p/(2*e*p + e) + e*x*(c*d**2 + 2*
c*d*e*x + c*e**2*x**2)**p/(2*e*p + e), True))

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