\(\int \frac {(c d^2+2 c d e x+c e^2 x^2)^{3/2}}{(d+e x)^6} \, dx\) [1048]

Optimal result

Integrand size = 32, antiderivative size = 41 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx=-\frac {c^2}{2 e (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 41, 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, 621} \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx=-\frac {c^2}{2 e (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

[In]

Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)/(d + e*x)^6,x]

[Out]

-1/2*c^2/(e*(d + e*x)*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2])

Rule 621

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[2*((a + b*x + c*x^2)^(p + 1)/((2*p + 1)*(b + 2*
c*x))), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && LtQ[p, -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}& = c^3 \int \frac {1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx \\ & = -\frac {c^2}{2 e (d+e x) \sqrt {c d^2+2 c d e x+c e^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.66 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx=-\frac {\left (c (d+e x)^2\right )^{3/2}}{2 e (d+e x)^5} \]

[In]

Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2)/(d + e*x)^6,x]

[Out]

-1/2*(c*(d + e*x)^2)^(3/2)/(e*(d + e*x)^5)

Maple [A] (verified)

Time = 2.88 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.61

method	result	size
risch	\(-\frac {c \sqrt {c \left (e x +d \right )^{2}}}{2 \left (e x +d \right )^{3} e}\)	\(25\)
gosper	\(-\frac {\left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {3}{2}}}{2 \left (e x +d \right )^{5} e}\)	\(35\)
default	\(-\frac {\left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {3}{2}}}{2 \left (e x +d \right )^{5} e}\)	\(35\)
trager	\(\frac {c \left (e x +2 d \right ) x \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{2 d^{2} \left (e x +d \right )^{3}}\)	\(44\)

[In]

int((c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)/(e*x+d)^6,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.41 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx=-\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c}{2 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \]

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)/(e*x+d)^6,x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**(3/2)/(e*x+d)**6,x)

[Out]

Integral((c*(d + e*x)**2)**(3/2)/(d + e*x)**6, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)/(e*x+d)^6,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.51 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx=-\frac {c^{\frac {3}{2}} \mathrm {sgn}\left (e x + d\right )}{2 \, {\left (e x + d\right )}^{2} e} \]

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)/(e*x+d)^6,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.73 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.85 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^6} \, dx=-\frac {c\,\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{2\,e\,{\left (d+e\,x\right )}^3} \]

[In]

int((c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(3/2)/(d + e*x)^6,x)

[Out]

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