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

Optimal. Leaf size=39 \[ \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{p+1}}{2 c e (p+1)} \]

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Rubi [A]  time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {629} \begin {gather*} \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{p+1}}{2 c e (p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 629

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]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 28, normalized size = 0.72 \begin {gather*} \frac {\left (c (d+e x)^2\right )^{p+1}}{2 c e (p+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.09, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 47, normalized size = 1.21 \begin {gather*} \frac {{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{2 \, {\left (e p + e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.21, size = 94, normalized size = 2.41 \begin {gather*} \frac {{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} x^{2} e^{2} + 2 \, {\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} d x e + {\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{p} d^{2}}{2 \, {\left (p e + e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((c*x^2*e^2 + 2*c*d*x*e + c*d^2)^p*x^2*e^2 + 2*(c*x^2*e^2 + 2*c*d*x*e + c*d^2)^p*d*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 = 0.05, size = 40, normalized size = 1.03 \begin {gather*} \frac {\left (e x +d \right )^{2} \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{p}}{2 \left (p +1\right ) e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.38, size = 37, normalized size = 0.95 \begin {gather*} \frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p + 1}}{2 \, c e {\left (p + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.44, size = 57, normalized size = 1.46 \begin {gather*} {\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^p\,\left (\frac {d^2}{2\,e\,\left (p+1\right )}+\frac {d\,x}{p+1}+\frac {e\,x^2}{2\,\left (p+1\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.55, size = 139, normalized size = 3.56 \begin {gather*} \begin {cases} \frac {x}{c d} & \text {for}\: e = 0 \wedge p = -1 \\d x \left (c d^{2}\right )^{p} & \text {for}\: e = 0 \\\frac {\log {\left (\frac {d}{e} + x \right )}}{c e} & \text {for}\: p = -1 \\\frac {d^{2} \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p + 2 e} + \frac {2 d e x \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p + 2 e} + \frac {e^{2} x^{2} \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p + 2 e} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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