3.12.4 \(\int \frac {(c d^2+2 c d e x+c e^2 x^2)^p}{d+e x} \, dx\) [1104]

Optimal. Leaf size=32 \[ \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^p}{2 e p} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {657, 643} \begin {gather*} \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^p}{2 e p} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 643

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

Rule 657

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[e^(m - 1)/c^((m - 1)/2
), Int[(d + e*x)*(a + b*x + c*x^2)^(p + (m - 1)/2), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[b^2 - 4*a*c,
0] &&  !IntegerQ[p] && EqQ[2*c*d - b*e, 0] && IntegerQ[(m - 1)/2]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 21, normalized size = 0.66 \begin {gather*} \frac {\left (c (d+e x)^2\right )^p}{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/(d + e*x),x]

[Out]

(c*(d + e*x)^2)^p/(2*e*p)

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Maple [A]
time = 0.64, size = 20, normalized size = 0.62

method	result	size
risch	\(\frac {\left (\left (e x +d \right )^{2} c \right )^{p}}{2 p e}\)	\(20\)
gosper	\(\frac {\left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{p}}{2 e p}\)	\(31\)
norman	\(\frac {{\mathrm e}^{p \ln \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )}}{2 p e}\)	\(33\)

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

[Out]

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

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Maxima [A]
time = 0.30, size = 20, normalized size = 0.62 \begin {gather*} \frac {{\left (x e + d\right )}^{2 \, p} c^{p} e^{\left (-1\right )}}{2 \, p} \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/(e*x+d),x, algorithm="maxima")

[Out]

1/2*(x*e + d)^(2*p)*c^p*e^(-1)/p

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Fricas [A]
time = 2.66, size = 29, normalized size = 0.91 \begin {gather*} \frac {{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} e^{\left (-1\right )}}{2 \, p} \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/(e*x+d),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.14, size = 48, normalized size = 1.50 \begin {gather*} \begin {cases} \frac {x}{d} & \text {for}\: e = 0 \wedge p = 0 \\\frac {\log {\left (\frac {d}{e} + x \right )}}{e} & \text {for}\: p = 0 \\\frac {x \left (c d^{2}\right )^{p}}{d} & \text {for}\: e = 0 \\\frac {\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p} & \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/(e*x+d),x)

[Out]

Piecewise((x/d, Eq(e, 0) & Eq(p, 0)), (log(d/e + x)/e, Eq(p, 0)), (x*(c*d**2)**p/d, Eq(e, 0)), ((c*d**2 + 2*c*
d*e*x + c*e**2*x**2)**p/(2*e*p), True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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/(e*x+d),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.41, size = 30, normalized size = 0.94 \begin {gather*} \frac {{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^p}{2\,e\,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/(d + e*x),x)

[Out]

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

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