\(\int \frac {1}{(5+4 x-3 x^2)^{5/2}} \, dx\) [286]

Optimal result

Integrand size = 14, antiderivative size = 47 \[ \int \frac {1}{\left (5+4 x-3 x^2\right )^{5/2}} \, dx=-\frac {2-3 x}{57 \left (5+4 x-3 x^2\right )^{3/2}}-\frac {2 (2-3 x)}{361 \sqrt {5+4 x-3 x^2}} \]

[Out]

1/57*(-2+3*x)/(-3*x^2+4*x+5)^(3/2)-2/361*(2-3*x)/(-3*x^2+4*x+5)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {628, 627} \[ \int \frac {1}{\left (5+4 x-3 x^2\right )^{5/2}} \, dx=-\frac {2 (2-3 x)}{361 \sqrt {-3 x^2+4 x+5}}-\frac {2-3 x}{57 \left (-3 x^2+4 x+5\right )^{3/2}} \]

[In]

Int[(5 + 4*x - 3*x^2)^(-5/2),x]

[Out]

-1/57*(2 - 3*x)/(5 + 4*x - 3*x^2)^(3/2) - (2*(2 - 3*x))/(361*Sqrt[5 + 4*x - 3*x^2])

Rule 627

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1
)*(b^2 - 4*a*c))), x] - Dist[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rubi steps \begin{align*} \text {integral}& = -\frac {2-3 x}{57 \left (5+4 x-3 x^2\right )^{3/2}}+\frac {2}{19} \int \frac {1}{\left (5+4 x-3 x^2\right )^{3/2}} \, dx \\ & = -\frac {2-3 x}{57 \left (5+4 x-3 x^2\right )^{3/2}}-\frac {2 (2-3 x)}{361 \sqrt {5+4 x-3 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\left (5+4 x-3 x^2\right )^{5/2}} \, dx=\frac {-98+99 x+108 x^2-54 x^3}{1083 \left (5+4 x-3 x^2\right )^{3/2}} \]

[In]

Integrate[(5 + 4*x - 3*x^2)^(-5/2),x]

[Out]

(-98 + 99*x + 108*x^2 - 54*x^3)/(1083*(5 + 4*x - 3*x^2)^(3/2))

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.64

method	result	size
gosper	\(-\frac {54 x^{3}-108 x^{2}-99 x +98}{1083 \left (-3 x^{2}+4 x +5\right )^{\frac {3}{2}}}\)	\(30\)
default	\(-\frac {-6 x +4}{114 \left (-3 x^{2}+4 x +5\right )^{\frac {3}{2}}}-\frac {-6 x +4}{361 \sqrt {-3 x^{2}+4 x +5}}\)	\(40\)
trager	\(-\frac {\left (54 x^{3}-108 x^{2}-99 x +98\right ) \sqrt {-3 x^{2}+4 x +5}}{1083 \left (3 x^{2}-4 x -5\right )^{2}}\)	\(42\)
risch	\(\frac {54 x^{3}-108 x^{2}-99 x +98}{1083 \left (3 x^{2}-4 x -5\right ) \sqrt {-3 x^{2}+4 x +5}}\)	\(42\)

[In]

int(1/(-3*x^2+4*x+5)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/1083/(-3*x^2+4*x+5)^(3/2)*(54*x^3-108*x^2-99*x+98)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (5+4 x-3 x^2\right )^{5/2}} \, dx=-\frac {{\left (54 \, x^{3} - 108 \, x^{2} - 99 \, x + 98\right )} \sqrt {-3 \, x^{2} + 4 \, x + 5}}{1083 \, {\left (9 \, x^{4} - 24 \, x^{3} - 14 \, x^{2} + 40 \, x + 25\right )}} \]

[In]

integrate(1/(-3*x^2+4*x+5)^(5/2),x, algorithm="fricas")

[Out]

-1/1083*(54*x^3 - 108*x^2 - 99*x + 98)*sqrt(-3*x^2 + 4*x + 5)/(9*x^4 - 24*x^3 - 14*x^2 + 40*x + 25)

Sympy [F]

\[ \int \frac {1}{\left (5+4 x-3 x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (- 3 x^{2} + 4 x + 5\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(1/(-3*x**2+4*x+5)**(5/2),x)

[Out]

Integral((-3*x**2 + 4*x + 5)**(-5/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\left (5+4 x-3 x^2\right )^{5/2}} \, dx=\frac {6 \, x}{361 \, \sqrt {-3 \, x^{2} + 4 \, x + 5}} - \frac {4}{361 \, \sqrt {-3 \, x^{2} + 4 \, x + 5}} + \frac {x}{19 \, {\left (-3 \, x^{2} + 4 \, x + 5\right )}^{\frac {3}{2}}} - \frac {2}{57 \, {\left (-3 \, x^{2} + 4 \, x + 5\right )}^{\frac {3}{2}}} \]

[In]

integrate(1/(-3*x^2+4*x+5)^(5/2),x, algorithm="maxima")

[Out]

6/361*x/sqrt(-3*x^2 + 4*x + 5) - 4/361/sqrt(-3*x^2 + 4*x + 5) + 1/19*x/(-3*x^2 + 4*x + 5)^(3/2) - 2/57/(-3*x^2
+ 4*x + 5)^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (5+4 x-3 x^2\right )^{5/2}} \, dx=-\frac {{\left (9 \, {\left (6 \, {\left (x - 2\right )} x - 11\right )} x + 98\right )} \sqrt {-3 \, x^{2} + 4 \, x + 5}}{1083 \, {\left (3 \, x^{2} - 4 \, x - 5\right )}^{2}} \]

[In]

integrate(1/(-3*x^2+4*x+5)^(5/2),x, algorithm="giac")

[Out]

-1/1083*(9*(6*(x - 2)*x - 11)*x + 98)*sqrt(-3*x^2 + 4*x + 5)/(3*x^2 - 4*x - 5)^2

Mupad [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.62 \[ \int \frac {1}{\left (5+4 x-3 x^2\right )^{5/2}} \, dx=\frac {\left (12\,x-8\right )\,\left (-72\,x^2+96\,x+196\right )}{17328\,{\left (-3\,x^2+4\,x+5\right )}^{3/2}} \]

[In]

int(1/(4*x - 3*x^2 + 5)^(5/2),x)

[Out]

((12*x - 8)*(96*x - 72*x^2 + 196))/(17328*(4*x - 3*x^2 + 5)^(3/2))