\(\int (d+e x)^3 (c d^2+2 c d e x+c e^2 x^2)^{5/2} \, dx\) [1050]

Optimal result

Integrand size = 32, antiderivative size = 34 \[ \int (d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx=\frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{9/2}}{9 c^2 e} \]

[Out]

1/9*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(9/2)/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} \[ \int (d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx=\frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{9/2}}{9 c^2 e} \]

[In]

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

[Out]

(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(9/2)/(9*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 )^{7/2} \, dx}{c} \\ & = \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{9/2}}{9 c^2 e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.79 \[ \int (d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx=\frac {(d+e x)^4 \left (c (d+e x)^2\right )^{5/2}}{9 e} \]

[In]

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

[Out]

((d + e*x)^4*(c*(d + e*x)^2)^(5/2))/(9*e)

Maple [A] (verified)

Time = 3.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.79

method	result	size
risch	\(\frac {c^{2} \left (e x +d \right )^{8} \sqrt {c \left (e x +d \right )^{2}}}{9 e}\)	\(27\)
pseudoelliptic	\(\frac {c^{2} \left (e x +d \right )^{8} \sqrt {c \left (e x +d \right )^{2}}}{9 e}\)	\(27\)
default	\(\frac {\left (e x +d \right )^{4} \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {5}{2}}}{9 e}\)	\(35\)
gosper	\(\frac {x \left (e^{8} x^{8}+9 d \,e^{7} x^{7}+36 d^{2} e^{6} x^{6}+84 d^{3} e^{5} x^{5}+126 d^{4} e^{4} x^{4}+126 d^{5} e^{3} x^{3}+84 d^{6} e^{2} x^{2}+36 d^{7} e x +9 d^{8}\right ) \left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {5}{2}}}{9 \left (e x +d \right )^{5}}\)	\(117\)
trager	\(\frac {c^{2} x \left (e^{8} x^{8}+9 d \,e^{7} x^{7}+36 d^{2} e^{6} x^{6}+84 d^{3} e^{5} x^{5}+126 d^{4} e^{4} x^{4}+126 d^{5} e^{3} x^{3}+84 d^{6} e^{2} x^{2}+36 d^{7} e x +9 d^{8}\right ) \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{9 e x +9 d}\)	\(120\)

[In]

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

[Out]

1/9*c^2*(e*x+d)^8*(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. 145 vs. \(2 (30) = 60\).

Time = 0.38 (sec) , antiderivative size = 145, normalized size of antiderivative = 4.26 \[ \int (d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx=\frac {{\left (c^{2} e^{8} x^{9} + 9 \, c^{2} d e^{7} x^{8} + 36 \, c^{2} d^{2} e^{6} x^{7} + 84 \, c^{2} d^{3} e^{5} x^{6} + 126 \, c^{2} d^{4} e^{4} x^{5} + 126 \, c^{2} d^{5} e^{3} x^{4} + 84 \, c^{2} d^{6} e^{2} x^{3} + 36 \, c^{2} d^{7} e x^{2} + 9 \, c^{2} d^{8} x\right )} \sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{9 \, {\left (e x + d\right )}} \]

[In]

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

[Out]

1/9*(c^2*e^8*x^9 + 9*c^2*d*e^7*x^8 + 36*c^2*d^2*e^6*x^7 + 84*c^2*d^3*e^5*x^6 + 126*c^2*d^4*e^4*x^5 + 126*c^2*d
^5*e^3*x^4 + 84*c^2*d^6*e^2*x^3 + 36*c^2*d^7*e*x^2 + 9*c^2*d^8*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. 374 vs. \(2 (31) = 62\).

Time = 0.49 (sec) , antiderivative size = 374, normalized size of antiderivative = 11.00 \[ \int (d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx=\begin {cases} \frac {c^{2} d^{8} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9 e} + \frac {8 c^{2} d^{7} x \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac {28 c^{2} d^{6} e x^{2} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac {56 c^{2} d^{5} e^{2} x^{3} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac {70 c^{2} d^{4} e^{3} x^{4} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac {56 c^{2} d^{3} e^{4} x^{5} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac {28 c^{2} d^{2} e^{5} x^{6} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac {8 c^{2} d e^{6} x^{7} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac {c^{2} e^{7} x^{8} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} & \text {for}\: e \neq 0 \\d^{3} x \left (c d^{2}\right )^{\frac {5}{2}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((c**2*d**8*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(9*e) + 8*c**2*d**7*x*sqrt(c*d**2 + 2*c*d*e*x + c*
e**2*x**2)/9 + 28*c**2*d**6*e*x**2*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/9 + 56*c**2*d**5*e**2*x**3*sqrt(c*d*
*2 + 2*c*d*e*x + c*e**2*x**2)/9 + 70*c**2*d**4*e**3*x**4*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/9 + 56*c**2*d*
*3*e**4*x**5*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/9 + 28*c**2*d**2*e**5*x**6*sqrt(c*d**2 + 2*c*d*e*x + c*e**
2*x**2)/9 + 8*c**2*d*e**6*x**7*sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/9 + c**2*e**7*x**8*sqrt(c*d**2 + 2*c*d*e
*x + c*e**2*x**2)/9, Ne(e, 0)), (d**3*x*(c*d**2)**(5/2), True))

Maxima [F(-2)]

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

[In]

integrate((e*x+d)^3*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),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. 189 vs. \(2 (30) = 60\).

Time = 0.27 (sec) , antiderivative size = 189, normalized size of antiderivative = 5.56 \[ \int (d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx=\frac {1}{9} \, {\left (c^{2} e^{8} x^{9} \mathrm {sgn}\left (e x + d\right ) + 9 \, c^{2} d e^{7} x^{8} \mathrm {sgn}\left (e x + d\right ) + 36 \, c^{2} d^{2} e^{6} x^{7} \mathrm {sgn}\left (e x + d\right ) + 84 \, c^{2} d^{3} e^{5} x^{6} \mathrm {sgn}\left (e x + d\right ) + 126 \, c^{2} d^{4} e^{4} x^{5} \mathrm {sgn}\left (e x + d\right ) + 126 \, c^{2} d^{5} e^{3} x^{4} \mathrm {sgn}\left (e x + d\right ) + 84 \, c^{2} d^{6} e^{2} x^{3} \mathrm {sgn}\left (e x + d\right ) + 36 \, c^{2} d^{7} e x^{2} \mathrm {sgn}\left (e x + d\right ) + 9 \, c^{2} d^{8} x \mathrm {sgn}\left (e x + d\right ) + \frac {c^{2} d^{9} \mathrm {sgn}\left (e x + d\right )}{e}\right )} \sqrt {c} \]

[In]

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

[Out]

1/9*(c^2*e^8*x^9*sgn(e*x + d) + 9*c^2*d*e^7*x^8*sgn(e*x + d) + 36*c^2*d^2*e^6*x^7*sgn(e*x + d) + 84*c^2*d^3*e^
5*x^6*sgn(e*x + d) + 126*c^2*d^4*e^4*x^5*sgn(e*x + d) + 126*c^2*d^5*e^3*x^4*sgn(e*x + d) + 84*c^2*d^6*e^2*x^3*
sgn(e*x + d) + 36*c^2*d^7*e*x^2*sgn(e*x + d) + 9*c^2*d^8*x*sgn(e*x + d) + c^2*d^9*sgn(e*x + d)/e)*sqrt(c)

Mupad [F(-1)]

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

[In]

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

[Out]

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