\(\int \frac {(d+e x)^3}{(c d^2+2 c d e x+c e^2 x^2)^{3/2}} \, dx\) [1072]

Optimal result

Integrand size = 32, antiderivative size = 31 \[ \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{c^2 e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {657, 643} \[ \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{c^2 e} \]

[In]

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

[Out]

Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]/(c^2*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*} \text {integral}& = \frac {\int \frac {d+e x}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx}{c} \\ & = \frac {\sqrt {c d^2+2 c d e x+c e^2 x^2}}{c^2 e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {c (d+e x)^2}}{c^2 e} \]

[In]

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

[Out]

Sqrt[c*(d + e*x)^2]/(c^2*e)

Maple [A] (verified)

Time = 2.50 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.61

method	result	size
pseudoelliptic	\(\frac {\sqrt {c \left (e x +d \right )^{2}}}{c^{2} e}\)	\(19\)
risch	\(\frac {\left (e x +d \right ) x}{c \sqrt {c \left (e x +d \right )^{2}}}\)	\(22\)
default	\(\frac {x \left (e x +d \right )^{3}}{\left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {3}{2}}}\)	\(32\)
trager	\(\frac {x \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{c^{2} \left (e x +d \right )}\)	\(35\)

[In]

int((e*x+d)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.23 \[ \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} x}{c^{2} e x + c^{2} d} \]

[In]

integrate((e*x+d)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\begin {cases} \frac {\sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{c^{2} e} & \text {for}\: e \neq 0 \\\frac {d^{3} x}{\left (c d^{2}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (29) = 58\).

Time = 0.19 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.06 \[ \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {e x^{2}}{\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c} - \frac {d^{2}}{\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c e} \]

[In]

integrate((e*x+d)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.42 \[ \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {x}{c^{\frac {3}{2}} \mathrm {sgn}\left (e x + d\right )} \]

[In]

integrate((e*x+d)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x, algorithm="giac")

[Out]

x/(c^(3/2)*sgn(e*x + d))

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^3}{{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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