\(\int \frac {(c d^2+2 c d e x+c e^2 x^2)^{3/2}}{(d+e x)^5} \, dx\) [1047]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 32, 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} \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx=-\frac {c^2}{e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

[In]

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

[Out]

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

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}& = c^3 \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx \\ & = -\frac {c^2}{e \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 = 25, normalized size of antiderivative = 0.78 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx=-\frac {\left (c (d+e x)^2\right )^{3/2}}{e (d+e x)^4} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.64 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78

method	result	size
risch	\(-\frac {c \sqrt {c \left (e x +d \right )^{2}}}{\left (e x +d \right )^{2} e}\)	\(25\)
pseudoelliptic	\(-\frac {c \sqrt {c \left (e x +d \right )^{2}}}{\left (e x +d \right )^{2} e}\)	\(25\)
gosper	\(-\frac {\left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {3}{2}}}{\left (e x +d \right )^{4} e}\)	\(35\)
default	\(-\frac {\left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {3}{2}}}{\left (e x +d \right )^{4} e}\)	\(35\)
trager	\(\frac {c x \sqrt {c \,x^{2} e^{2}+2 x c d e +c \,d^{2}}}{d \left (e x +d \right )^{2}}\)	\(36\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.47 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx=-\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c}{e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2)/(e*x+d)^5,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 = 0.66 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx=-\frac {c^{\frac {3}{2}} \mathrm {sgn}\left (e x + d\right )}{{\left (e x + d\right )} e} \]

[In]

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

[Out]

-c^(3/2)*sgn(e*x + d)/((e*x + d)*e)

Mupad [B] (verification not implemented)

Time = 9.42 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^5} \, dx=-\frac {c\,\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{e\,{\left (d+e\,x\right )}^2} \]

[In]

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

[Out]

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