\(\int (d+e x)^3 (c d^2+2 c d e x+c e^2 x^2)^{3/2} \, dx\) [1039]

Optimal result

Integrand size = 32, antiderivative size = 34 \[ \int (d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx=\frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{7 c^2 e} \]

[Out]

1/7*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(7/2)/c^2/e

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 34, 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} \[ \int (d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx=\frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{7 c^2 e} \]

[In]

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

[Out]

(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(7/2)/(7*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 (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx}{c} \\ & = \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}{7 c^2 e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.79 \[ \int (d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx=\frac {(d+e x)^4 \left (c (d+e x)^2\right )^{3/2}}{7 e} \]

[In]

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

[Out]

((d + e*x)^4*(c*(d + e*x)^2)^(3/2))/(7*e)

Maple [A] (verified)

Time = 2.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.74

method	result	size
risch	\(\frac {c \left (e x +d \right )^{6} \sqrt {c \left (e x +d \right )^{2}}}{7 e}\)	\(25\)
pseudoelliptic	\(\frac {c \left (e x +d \right )^{6} \sqrt {c \left (e x +d \right )^{2}}}{7 e}\)	\(25\)
default	\(\frac {\left (e x +d \right )^{4} \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {3}{2}}}{7 e}\)	\(35\)
gosper	\(\frac {x \left (e^{6} x^{6}+7 d \,e^{5} x^{5}+21 d^{2} e^{4} x^{4}+35 x^{3} d^{3} e^{3}+35 d^{4} e^{2} x^{2}+21 d^{5} e x +7 d^{6}\right ) \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {3}{2}}}{7 \left (e x +d \right )^{3}}\)	\(95\)
trager	\(\frac {c x \left (e^{6} x^{6}+7 d \,e^{5} x^{5}+21 d^{2} e^{4} x^{4}+35 x^{3} d^{3} e^{3}+35 d^{4} e^{2} x^{2}+21 d^{5} e x +7 d^{6}\right ) \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{7 e x +7 d}\)	\(96\)

[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]

1/7*c*(e*x+d)^6*(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. 103 vs. \(2 (30) = 60\).

Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 3.03 \[ \int (d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx=\frac {{\left (c e^{6} x^{7} + 7 \, c d e^{5} x^{6} + 21 \, c d^{2} e^{4} x^{5} + 35 \, c d^{3} e^{3} x^{4} + 35 \, c d^{4} e^{2} x^{3} + 21 \, c d^{5} e x^{2} + 7 \, c d^{6} x\right )} \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{7 \, {\left (e x + d\right )}} \]

[In]

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

[Out]

1/7*(c*e^6*x^7 + 7*c*d*e^5*x^6 + 21*c*d^2*e^4*x^5 + 35*c*d^3*e^3*x^4 + 35*c*d^4*e^2*x^3 + 21*c*d^5*e*x^2 + 7*c
*d^6*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. 277 vs. \(2 (31) = 62\).

Time = 0.29 (sec) , antiderivative size = 277, normalized size of antiderivative = 8.15 \[ \int (d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx=\begin {cases} \frac {c d^{6} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7 e} + \frac {6 c d^{5} x \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac {15 c d^{4} e x^{2} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac {20 c d^{3} e^{2} x^{3} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac {15 c d^{2} e^{3} x^{4} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac {6 c d e^{4} x^{5} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} + \frac {c e^{5} x^{6} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{7} & \text {for}\: e \neq 0 \\d^{3} x \left (c d^{2}\right )^{\frac {3}{2}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((c*d**6*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(7*e) + 6*c*d**5*x*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x
**2)/7 + 15*c*d**4*e*x**2*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/7 + 20*c*d**3*e**2*x**3*sqrt(c*d**2 + 2*c*d*e
*x + c*e**2*x**2)/7 + 15*c*d**2*e**3*x**4*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/7 + 6*c*d*e**4*x**5*sqrt(c*d*
*2 + 2*c*d*e*x + c*e**2*x**2)/7 + c*e**5*x**6*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/7, Ne(e, 0)), (d**3*x*(c*
d**2)**(3/2), True))

Maxima [F(-2)]

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

[In]

integrate((e*x+d)^3*(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. 133 vs. \(2 (30) = 60\).

Time = 0.28 (sec) , antiderivative size = 133, normalized size of antiderivative = 3.91 \[ \int (d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx=\frac {1}{7} \, {\left (c e^{6} x^{7} \mathrm {sgn}\left (e x + d\right ) + 7 \, c d e^{5} x^{6} \mathrm {sgn}\left (e x + d\right ) + 21 \, c d^{2} e^{4} x^{5} \mathrm {sgn}\left (e x + d\right ) + 35 \, c d^{3} e^{3} x^{4} \mathrm {sgn}\left (e x + d\right ) + 35 \, c d^{4} e^{2} x^{3} \mathrm {sgn}\left (e x + d\right ) + 21 \, c d^{5} e x^{2} \mathrm {sgn}\left (e x + d\right ) + 7 \, c d^{6} x \mathrm {sgn}\left (e x + d\right ) + \frac {c d^{7} \mathrm {sgn}\left (e x + d\right )}{e}\right )} \sqrt {c} \]

[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]

1/7*(c*e^6*x^7*sgn(e*x + d) + 7*c*d*e^5*x^6*sgn(e*x + d) + 21*c*d^2*e^4*x^5*sgn(e*x + d) + 35*c*d^3*e^3*x^4*sg
n(e*x + d) + 35*c*d^4*e^2*x^3*sgn(e*x + d) + 21*c*d^5*e*x^2*sgn(e*x + d) + 7*c*d^6*x*sgn(e*x + d) + c*d^7*sgn(
e*x + d)/e)*sqrt(c)

Mupad [F(-1)]

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

[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)