\(\int (d+e x)^2 (c d^2+2 c d e x+c e^2 x^2)^{3/2} \, dx\) [1040]

Optimal result

Integrand size = 32, antiderivative size = 39 \[ \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx=\frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{6 c e} \]

[Out]

1/6*(e*x+d)*(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 = 39, 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} \[ \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx=\frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{6 c e} \]

[In]

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

[Out]

((d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2))/(6*c*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}& = \frac {\int \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx}{c} \\ & = \frac {(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{6 c e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.72 \[ \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx=\frac {(d+e x) \left (c (d+e x)^2\right )^{5/2}}{6 c e} \]

[In]

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

[Out]

((d + e*x)*(c*(d + e*x)^2)^(5/2))/(6*c*e)

Maple [A] (verified)

Time = 2.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.64

method	result	size
risch	\(\frac {c \left (e x +d \right )^{5} \sqrt {c \left (e x +d \right )^{2}}}{6 e}\)	\(25\)
default	\(\frac {\left (e x +d \right )^{3} \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {3}{2}}}{6 e}\)	\(35\)
gosper	\(\frac {x \left (e^{5} x^{5}+6 x^{4} d \,e^{4}+15 d^{2} e^{3} x^{3}+20 d^{3} e^{2} x^{2}+15 d^{4} e x +6 d^{5}\right ) \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {3}{2}}}{6 \left (e x +d \right )^{3}}\)	\(84\)
trager	\(\frac {c x \left (e^{5} x^{5}+6 x^{4} d \,e^{4}+15 d^{2} e^{3} x^{3}+20 d^{3} e^{2} x^{2}+15 d^{4} e x +6 d^{5}\right ) \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{6 e x +6 d}\)	\(85\)

[In]

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

[Out]

1/6*c*(e*x+d)^5*(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. 91 vs. \(2 (35) = 70\).

Time = 0.28 (sec) , antiderivative size = 91, normalized size of antiderivative = 2.33 \[ \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx=\frac {{\left (c e^{5} x^{6} + 6 \, c d e^{4} x^{5} + 15 \, c d^{2} e^{3} x^{4} + 20 \, c d^{3} e^{2} x^{3} + 15 \, c d^{4} e x^{2} + 6 \, c d^{5} x\right )} \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{6 \, {\left (e x + d\right )}} \]

[In]

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

[Out]

1/6*(c*e^5*x^6 + 6*c*d*e^4*x^5 + 15*c*d^2*e^3*x^4 + 20*c*d^3*e^2*x^3 + 15*c*d^4*e*x^2 + 6*c*d^5*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. 197 vs. \(2 (36) = 72\).

Time = 0.90 (sec) , antiderivative size = 197, normalized size of antiderivative = 5.05 \[ \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx=\begin {cases} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}} \left (\frac {c d^{5}}{6 e} + \frac {5 c d^{4} x}{6} + \frac {5 c d^{3} e x^{2}}{3} + \frac {5 c d^{2} e^{2} x^{3}}{3} + \frac {5 c d e^{3} x^{4}}{6} + \frac {c e^{4} x^{5}}{6}\right ) & \text {for}\: c e^{2} \neq 0 \\\frac {\frac {d^{2} \left (c d^{2} + 2 c d e x\right )^{\frac {5}{2}}}{20} + \frac {\left (c d^{2} + 2 c d e x\right )^{\frac {7}{2}}}{14 c} + \frac {\left (c d^{2} + 2 c d e x\right )^{\frac {9}{2}}}{36 c^{2} d^{2}}}{c d e} & \text {for}\: c d e \neq 0 \\\left (c d^{2}\right )^{\frac {3}{2}} \left (\begin {cases} d^{2} x & \text {for}\: e = 0 \\\frac {\left (d + e x\right )^{3}}{3 e} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]

[In]

integrate((e*x+d)**2*(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*d**5/(6*e) + 5*c*d**4*x/6 + 5*c*d**3*e*x**2/3 + 5*c*d**2*
e**2*x**3/3 + 5*c*d*e**3*x**4/6 + c*e**4*x**5/6), Ne(c*e**2, 0)), ((d**2*(c*d**2 + 2*c*d*e*x)**(5/2)/20 + (c*d
**2 + 2*c*d*e*x)**(7/2)/(14*c) + (c*d**2 + 2*c*d*e*x)**(9/2)/(36*c**2*d**2))/(c*d*e), Ne(c*d*e, 0)), ((c*d**2)
**(3/2)*Piecewise((d**2*x, Eq(e, 0)), ((d + e*x)**3/(3*e), True)), True))

Maxima [F(-2)]

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

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (35) = 70\).

Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.31 \[ \int (d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx=\frac {1}{6} \, {\left (3 \, {\left (e x^{2} + 2 \, d x\right )} c d^{4} \mathrm {sgn}\left (e x + d\right ) + \frac {c d^{6} \mathrm {sgn}\left (e x + d\right )}{e} + 3 \, {\left (e x^{2} + 2 \, d x\right )}^{2} c d^{2} e \mathrm {sgn}\left (e x + d\right ) + {\left (e x^{2} + 2 \, d x\right )}^{3} c e^{2} \mathrm {sgn}\left (e x + d\right )\right )} \sqrt {c} \]

[In]

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

[Out]

1/6*(3*(e*x^2 + 2*d*x)*c*d^4*sgn(e*x + d) + c*d^6*sgn(e*x + d)/e + 3*(e*x^2 + 2*d*x)^2*c*d^2*e*sgn(e*x + d) +
(e*x^2 + 2*d*x)^3*c*e^2*sgn(e*x + d))*sqrt(c)

Mupad [F(-1)]

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

[In]

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

[Out]

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