\(\int \frac {(c d^2+2 c d e x+c e^2 x^2)^{3/2}}{(d+e x)^2} \, dx\) [1044]

Optimal result

Integrand size = 32, antiderivative size = 37 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx=\frac {c (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 37, 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 = {656, 623} \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx=\frac {c (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 e} \]

[In]

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

[Out]

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

Rule 656

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[e^m/c^(m/2), Int[(a +
b*x + c*x^2)^(p + m/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/2]

Rubi steps \begin{align*} \text {integral}& = c \int \sqrt {c d^2+2 c d e x+c e^2 x^2} \, dx \\ & = \frac {c (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{2 e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx=\frac {c^2 x (d+e x) (2 d+e x)}{2 \sqrt {c (d+e x)^2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.69 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.08

method	result	size
gosper	\(\frac {x \left (e x +2 d \right ) \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {3}{2}}}{2 \left (e x +d \right )^{3}}\)	\(40\)
default	\(\frac {x \left (e x +2 d \right ) \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {3}{2}}}{2 \left (e x +d \right )^{3}}\)	\(40\)
trager	\(\frac {c x \left (e x +2 d \right ) \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{2 e x +2 d}\)	\(41\)
risch	\(\frac {c \sqrt {c \left (e x +d \right )^{2}}\, e \,x^{2}}{2 e x +2 d}+\frac {c \sqrt {c \left (e x +d \right )^{2}}\, d x}{e x +d}\)	\(49\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.16 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx=\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} {\left (c e x^{2} + 2 \, c d x\right )}}{2 \, {\left (e x + d\right )}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 1.07 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.97 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx=c \left (\begin {cases} \left (\frac {d}{2 e} + \frac {x}{2}\right ) \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}} & \text {for}\: c e^{2} \neq 0 \\\frac {\left (c d^{2} + 2 c d e x\right )^{\frac {3}{2}}}{3 c d e} & \text {for}\: c d e \neq 0 \\x \sqrt {c d^{2}} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.16 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx=\frac {1}{2} \, {\left (c e x^{2} \mathrm {sgn}\left (e x + d\right ) + 2 \, c d x \mathrm {sgn}\left (e x + d\right ) + \frac {c d^{2} \mathrm {sgn}\left (e x + d\right )}{e}\right )} \sqrt {c} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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