\(\int \frac {(c d^2+2 c d e x+c e^2 x^2)^{5/2}}{(d+e x)^3} \, dx\) [1056]

Optimal result

Integrand size = 32, antiderivative size = 32 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {c \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 32, 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} \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {c \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 e} \]

[In]

Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(d + e*x)^3,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.66 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {c \left (c (d+e x)^2\right )^{3/2}}{3 e} \]

[In]

Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(d + e*x)^3,x]

[Out]

(c*(c*(d + e*x)^2)^(3/2))/(3*e)

Maple [A] (verified)

Time = 2.96 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84

method	result	size
risch	\(\frac {c^{2} \left (e x +d \right )^{2} \sqrt {c \left (e x +d \right )^{2}}}{3 e}\)	\(27\)
pseudoelliptic	\(\frac {c^{2} \left (e x +d \right )^{2} \sqrt {c \left (e x +d \right )^{2}}}{3 e}\)	\(27\)
default	\(\frac {\left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {5}{2}}}{3 \left (e x +d \right )^{2} e}\)	\(35\)
gosper	\(\frac {x \left (x^{2} e^{2}+3 d e x +3 d^{2}\right ) \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {5}{2}}}{3 \left (e x +d \right )^{5}}\)	\(51\)
trager	\(\frac {c^{2} x \left (x^{2} e^{2}+3 d e x +3 d^{2}\right ) \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{3 e x +3 d}\)	\(54\)

[In]

int((c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (28) = 56\).

Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.91 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {{\left (c^{2} e^{2} x^{3} + 3 \, c^{2} d e x^{2} + 3 \, c^{2} d^{2} x\right )} \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{3 \, {\left (e x + d\right )}} \]

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^3,x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (29) = 58\).

Time = 2.08 (sec) , antiderivative size = 201, normalized size of antiderivative = 6.28 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=c^{2} d \left (\begin {cases} \left (\frac {d}{2 e} + \frac {x}{2}\right ) \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}} & \text {for}\: c e^{2} \neq 0 \\\frac {\left (c d^{2} + 2 c d e x\right )^{\frac {3}{2}}}{3 c d e} & \text {for}\: c d e \neq 0 \\x \sqrt {c d^{2}} & \text {otherwise} \end {cases}\right ) + c^{2} e \left (\begin {cases} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}} \left (- \frac {d^{2}}{6 e^{2}} + \frac {d x}{6 e} + \frac {x^{2}}{3}\right ) & \text {for}\: c e^{2} \neq 0 \\\frac {- \frac {c d^{2} \left (c d^{2} + 2 c d e x\right )^{\frac {3}{2}}}{3} + \frac {\left (c d^{2} + 2 c d e x\right )^{\frac {5}{2}}}{5}}{2 c^{2} d^{2} e^{2}} & \text {for}\: c d e \neq 0 \\\frac {x^{2} \sqrt {c d^{2}}}{2} & \text {otherwise} \end {cases}\right ) \]

[In]

integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**(5/2)/(e*x+d)**3,x)

[Out]

c**2*d*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)) + c**2*e*Piecewise((sqrt(c*d**2 + 2*c*d*e*x + c*e**
2*x**2)*(-d**2/(6*e**2) + d*x/(6*e) + x**2/3), Ne(c*e**2, 0)), ((-c*d**2*(c*d**2 + 2*c*d*e*x)**(3/2)/3 + (c*d*
*2 + 2*c*d*e*x)**(5/2)/5)/(2*c**2*d**2*e**2), Ne(c*d*e, 0)), (x**2*sqrt(c*d**2)/2, True))

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^3,x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (28) = 56\).

Time = 0.29 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.16 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {1}{3} \, {\left (c^{2} e^{2} x^{3} \mathrm {sgn}\left (e x + d\right ) + 3 \, c^{2} d e x^{2} \mathrm {sgn}\left (e x + d\right ) + 3 \, c^{2} d^{2} x \mathrm {sgn}\left (e x + d\right ) + \frac {c^{2} d^{3} \mathrm {sgn}\left (e x + d\right )}{e}\right )} \sqrt {c} \]

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^3,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx=\int \frac {{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^3} \,d x \]

[In]

int((c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(5/2)/(d + e*x)^3,x)

[Out]

int((c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(5/2)/(d + e*x)^3, x)