3.11.31 \(\int (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx\) [1031]

Optimal. Leaf size=34 \[ \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 c e} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {643} \begin {gather*} \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 c e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 23, normalized size = 0.68 \begin {gather*} \frac {\left (c (d+e x)^2\right )^{3/2}}{3 c e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(d + e*x)^2)^(3/2)/(3*c*e)

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Maple [A]
time = 0.48, size = 35, normalized size = 1.03

method	result	size
risch	\(\frac {\left (e x +d \right )^{2} \sqrt {\left (e x +d \right )^{2} c}}{3 e}\)	\(24\)
default	\(\frac {\left (e x +d \right )^{2} \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{3 e}\)	\(35\)
gosper	\(\frac {x \left (e^{2} x^{2}+3 d x e +3 d^{2}\right ) \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{3 e x +3 d}\)	\(51\)
trager	\(\frac {x \left (e^{2} x^{2}+3 d x e +3 d^{2}\right ) \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{3 e x +3 d}\)	\(51\)

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)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 29, normalized size = 0.85 \begin {gather*} \frac {{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac {3}{2}} e^{\left (-1\right )}}{3 \, c} \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)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 2.96, size = 53, normalized size = 1.56 \begin {gather*} \frac {\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}} {\left (x^{3} e^{2} + 3 \, d x^{2} e + 3 \, d^{2} x\right )}}{3 \, {\left (x e + d\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)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (29) = 58\).
time = 0.08, size = 107, normalized size = 3.15 \begin {gather*} \begin {cases} \frac {d^{2} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{3 e} + \frac {2 d x \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{3} + \frac {e x^{2} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{3} & \text {for}\: e \neq 0 \\d x \sqrt {c d^{2}} & \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)**(1/2),x)

[Out]

Piecewise((d**2*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(3*e) + 2*d*x*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/3
+ e*x**2*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/3, Ne(e, 0)), (d*x*sqrt(c*d**2), True))

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Giac [A]
time = 1.47, size = 22, normalized size = 0.65 \begin {gather*} \frac {1}{3} \, {\left (x e + d\right )}^{3} \sqrt {c} e^{\left (-1\right )} \mathrm {sgn}\left (x e + d\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)^(1/2),x, algorithm="giac")

[Out]

1/3*(x*e + d)^3*sqrt(c)*e^(-1)*sgn(x*e + d)

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Mupad [B]
time = 0.53, size = 34, normalized size = 1.00 \begin {gather*} \frac {{\left (d+e\,x\right )}^2\,\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{3\,e} \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)^(1/2),x)

[Out]

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

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