3.11.43 \(\int \frac {(c d^2+2 c d e x+c e^2 x^2)^{3/2}}{d+e x} \, dx\) [1043]

Optimal. Leaf size=31 \[ \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 e} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {657, 643} \begin {gather*} \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.00, size = 20, normalized size = 0.65 \begin {gather*} \frac {\left (c (d+e x)^2\right )^{3/2}}{3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(d + e*x)^2)^(3/2)/(3*e)

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Maple [A]
time = 0.66, size = 28, normalized size = 0.90

method	result	size
risch	\(\frac {c \left (e x +d \right )^{2} \sqrt {\left (e x +d \right )^{2} c}}{3 e}\)	\(25\)
default	\(\frac {\left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}}}{3 e}\)	\(28\)
gosper	\(\frac {x \left (e^{2} x^{2}+3 d x e +3 d^{2}\right ) \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}}}{3 \left (e x +d \right )^{3}}\)	\(51\)
trager	\(\frac {c x \left (e^{2} x^{2}+3 d x e +3 d^{2}\right ) \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{3 e x +3 d}\)	\(52\)

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

[Out]

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

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Maxima [A]
time = 0.29, size = 26, normalized size = 0.84 \begin {gather*} \frac {1}{3} \, {\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac {3}{2}} e^{\left (-1\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)/(e*x+d),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (26) = 52\).
time = 2.93, size = 56, normalized size = 1.81 \begin {gather*} \frac {{\left (c x^{3} e^{2} + 3 \, c d x^{2} e + 3 \, c d^{2} x\right )} \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}}{3 \, {\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)/(e*x+d),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 1.41, size = 39, normalized size = 1.26 \begin {gather*} \begin {cases} \frac {\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac {3}{2}}}{3 e} & \text {for}\: e \neq 0 \\\frac {x \left (c d^{2}\right )^{\frac {3}{2}}}{d} & \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)**(3/2)/(e*x+d),x)

[Out]

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

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Giac [A]
time = 0.59, size = 36, normalized size = 1.16 \begin {gather*} \frac {1}{3} \, {\left (c x^{3} e^{2} + 3 \, c d x^{2} e + 3 \, c d^{2} x\right )} \sqrt {c} \mathrm {sgn}\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)/(e*x+d),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.48, size = 16, normalized size = 0.52 \begin {gather*} \frac {{\left (c\,{\left (d+e\,x\right )}^2\right )}^{3/2}}{3\,e} \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)/(d + e*x),x)

[Out]

(c*(d + e*x)^2)^(3/2)/(3*e)

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