3.12.3 \(\int (c d^2+2 c d e x+c e^2 x^2)^p \, dx\) [1103]

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

[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 = {623} \begin {gather*} \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^p}{e (2 p+1)} \end {gather*}

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 623

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 = 0.02, size = 25, normalized size = 0.66 \begin {gather*} \frac {(d+e x) \left (c (d+e x)^2\right )^p}{e+2 e p} \end {gather*}

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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Maple [A]
time = 0.60, size = 28, normalized size = 0.74

method	result	size
risch	\(\frac {\left (e x +d \right ) \left (\left (e x +d \right )^{2} c \right )^{p}}{e \left (1+2 p \right )}\)	\(28\)
gosper	\(\frac {\left (e x +d \right ) \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{p}}{e \left (1+2 p \right )}\)	\(39\)
norman	\(\frac {x \,{\mathrm e}^{p \ln \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )}}{1+2 p}+\frac {d \,{\mathrm e}^{p \ln \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )}}{e \left (1+2 p \right )}\)	\(71\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 34, normalized size = 0.89 \begin {gather*} \frac {{\left (c^{p} x e + c^{p} d\right )} e^{\left (2 \, p \log \left (x e + d\right ) - 1\right )}}{2 \, p + 1} \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)^p,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 3.79, size = 38, normalized size = 1.00 \begin {gather*} \frac {{\left (x e + d\right )} {\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} e^{\left (-1\right )}}{2 \, p + 1} \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)^p,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \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)**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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Giac [A]
time = 1.61, size = 62, normalized size = 1.63 \begin {gather*} \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} \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)^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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Mupad [B]
time = 0.48, size = 45, normalized size = 1.18 \begin {gather*} \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 \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)^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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