3.11.42 \(\int (c d^2+2 c d e x+c e^2 x^2)^{3/2} \, dx\) [1042]

Optimal. Leaf size=36 \[ \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 e} \]

[Out]

1/4*(e*x+d)*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)/e

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Rubi [A]
time = 0.00, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {623} \begin {gather*} \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]

[Out]

((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2))/(4*e)

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 )^{3/2} \, dx &=\frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 e}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 25, normalized size = 0.69 \begin {gather*} \frac {(d+e x) \left (c (d+e x)^2\right )^{3/2}}{4 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]

[Out]

((d + e*x)*(c*(d + e*x)^2)^(3/2))/(4*e)

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Maple [A]
time = 0.53, size = 33, normalized size = 0.92

method	result	size
risch	\(\frac {c \left (e x +d \right )^{3} \sqrt {\left (e x +d \right )^{2} c}}{4 e}\)	\(25\)
default	\(\frac {\left (e x +d \right ) \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}}}{4 e}\)	\(33\)
gosper	\(\frac {x \left (e^{3} x^{3}+4 d \,e^{2} x^{2}+6 d^{2} e x +4 d^{3}\right ) \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}}}{4 \left (e x +d \right )^{3}}\)	\(62\)
trager	\(\frac {c x \left (e^{3} x^{3}+4 d \,e^{2} x^{2}+6 d^{2} e x +4 d^{3}\right ) \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{4 e x +4 d}\)	\(63\)

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)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/4*(e*x+d)*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)/e

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Maxima [A]
time = 0.28, size = 53, normalized size = 1.47 \begin {gather*} \frac {1}{4} \, {\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac {3}{2}} d e^{\left (-1\right )} + \frac {1}{4} \, {\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac {3}{2}} x \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)^(3/2),x, algorithm="maxima")

[Out]

1/4*(c*x^2*e^2 + 2*c*d*x*e + c*d^2)^(3/2)*d*e^(-1) + 1/4*(c*x^2*e^2 + 2*c*d*x*e + c*d^2)^(3/2)*x

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (32) = 64\).
time = 2.33, size = 67, normalized size = 1.86 \begin {gather*} \frac {{\left (c x^{4} e^{3} + 4 \, c d x^{3} e^{2} + 6 \, c d^{2} x^{2} e + 4 \, c d^{3} x\right )} \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}}{4 \, {\left (x e + d\right )}} \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)^(3/2),x, algorithm="fricas")

[Out]

1/4*(c*x^4*e^3 + 4*c*d*x^3*e^2 + 6*c*d^2*x^2*e + 4*c*d^3*x)*sqrt(c*x^2*e^2 + 2*c*d*x*e + c*d^2)/(x*e + d)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac {3}{2}}\, dx \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)**(3/2),x)

[Out]

Integral((c*d**2 + 2*c*d*e*x + c*e**2*x**2)**(3/2), x)

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Giac [A]
time = 1.07, size = 54, normalized size = 1.50 \begin {gather*} \frac {1}{4} \, {\left (2 \, {\left (x^{2} e + 2 \, d x\right )} c d^{2} \mathrm {sgn}\left (x e + d\right ) + {\left (x^{2} e + 2 \, d x\right )}^{2} c e \mathrm {sgn}\left (x e + d\right )\right )} \sqrt {c} \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)^(3/2),x, algorithm="giac")

[Out]

1/4*(2*(x^2*e + 2*d*x)*c*d^2*sgn(x*e + d) + (x^2*e + 2*d*x)^2*c*e*sgn(x*e + d))*sqrt(c)

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Mupad [B]
time = 0.41, size = 36, normalized size = 1.00 \begin {gather*} \frac {\left (x\,e^2+d\,e\right )\,{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{3/2}}{4\,e^2} \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)^(3/2),x)

[Out]

((d*e + e^2*x)*(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(3/2))/(4*e^2)

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