\(\int (d+e x) (c d^2+2 c d e x+c e^2 x^2)^{3/2} \, dx\) [1041]

Optimal result

Integrand size = 30, antiderivative size = 34 \[ \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx=\frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{5 c e} \]

[Out]

1/5*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/c/e

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {643} \[ \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx=\frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{5 c e} \]

[In]

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

[Out]

(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(5*c*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]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{5 c e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.68 \[ \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx=\frac {\left (c (d+e x)^2\right )^{5/2}}{5 c e} \]

[In]

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

[Out]

(c*(d + e*x)^2)^(5/2)/(5*c*e)

Maple [A] (verified)

Time = 2.55 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.74

method	result	size
risch	\(\frac {c \left (e x +d \right )^{4} \sqrt {c \left (e x +d \right )^{2}}}{5 e}\)	\(25\)
pseudoelliptic	\(\frac {c \left (e x +d \right )^{4} \sqrt {c \left (e x +d \right )^{2}}}{5 e}\)	\(25\)
default	\(\frac {\left (e x +d \right )^{2} \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {3}{2}}}{5 e}\)	\(35\)
gosper	\(\frac {x \left (e^{4} x^{4}+5 d \,e^{3} x^{3}+10 d^{2} e^{2} x^{2}+10 d^{3} e x +5 d^{4}\right ) \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {3}{2}}}{5 \left (e x +d \right )^{3}}\)	\(73\)
trager	\(\frac {c x \left (e^{4} x^{4}+5 d \,e^{3} x^{3}+10 d^{2} e^{2} x^{2}+10 d^{3} e x +5 d^{4}\right ) \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{5 e x +5 d}\)	\(74\)

[In]

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

[Out]

1/5*c*(e*x+d)^4*(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. 79 vs. \(2 (30) = 60\).

Time = 0.27 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.32 \[ \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx=\frac {{\left (c e^{4} x^{5} + 5 \, c d e^{3} x^{4} + 10 \, c d^{2} e^{2} x^{3} + 10 \, c d^{3} e x^{2} + 5 \, c d^{4} x\right )} \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{5 \, {\left (e x + d\right )}} \]

[In]

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

[Out]

1/5*(c*e^4*x^5 + 5*c*d*e^3*x^4 + 10*c*d^2*e^2*x^3 + 10*c*d^3*e*x^2 + 5*c*d^4*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. 194 vs. \(2 (29) = 58\).

Time = 0.17 (sec) , antiderivative size = 194, normalized size of antiderivative = 5.71 \[ \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx=\begin {cases} \frac {c d^{4} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5 e} + \frac {4 c d^{3} x \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} + \frac {6 c d^{2} e x^{2} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} + \frac {4 c d e^{2} x^{3} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} + \frac {c e^{3} x^{4} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} & \text {for}\: e \neq 0 \\d x \left (c d^{2}\right )^{\frac {3}{2}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((c*d**4*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(5*e) + 4*c*d**3*x*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x
**2)/5 + 6*c*d**2*e*x**2*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/5 + 4*c*d*e**2*x**3*sqrt(c*d**2 + 2*c*d*e*x +
c*e**2*x**2)/5 + c*e**3*x**4*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/5, Ne(e, 0)), (d*x*(c*d**2)**(3/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx=\frac {{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {5}{2}}}{5 \, c e} \]

[In]

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

[Out]

1/5*(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(5/2)/(c*e)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (30) = 60\).

Time = 0.29 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.85 \[ \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx=\frac {1}{5} \, {\left (c e^{4} x^{5} \mathrm {sgn}\left (e x + d\right ) + 5 \, c d e^{3} x^{4} \mathrm {sgn}\left (e x + d\right ) + 10 \, c d^{2} e^{2} x^{3} \mathrm {sgn}\left (e x + d\right ) + 10 \, c d^{3} e x^{2} \mathrm {sgn}\left (e x + d\right ) + 5 \, c d^{4} x \mathrm {sgn}\left (e x + d\right ) + \frac {c d^{5} \mathrm {sgn}\left (e x + d\right )}{e}\right )} \sqrt {c} \]

[In]

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

[Out]

1/5*(c*e^4*x^5*sgn(e*x + d) + 5*c*d*e^3*x^4*sgn(e*x + d) + 10*c*d^2*e^2*x^3*sgn(e*x + d) + 10*c*d^3*e*x^2*sgn(
e*x + d) + 5*c*d^4*x*sgn(e*x + d) + c*d^5*sgn(e*x + d)/e)*sqrt(c)

Mupad [B] (verification not implemented)

Time = 9.90 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx=\frac {{\left (d+e\,x\right )}^2\,{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{3/2}}{5\,e} \]

[In]

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

[Out]

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