\(\int \frac {(5-x) (2+5 x+3 x^2)}{(3+2 x)^{3/2}} \, dx\) [2531]

Optimal result

Integrand size = 25, antiderivative size = 53 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{(3+2 x)^{3/2}} \, dx=-\frac {65}{8 \sqrt {3+2 x}}-\frac {109}{8} \sqrt {3+2 x}+\frac {47}{24} (3+2 x)^{3/2}-\frac {3}{40} (3+2 x)^{5/2} \]

[Out]

47/24*(3+2*x)^(3/2)-3/40*(3+2*x)^(5/2)-65/8/(3+2*x)^(1/2)-109/8*(3+2*x)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {785} \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{(3+2 x)^{3/2}} \, dx=-\frac {3}{40} (2 x+3)^{5/2}+\frac {47}{24} (2 x+3)^{3/2}-\frac {109}{8} \sqrt {2 x+3}-\frac {65}{8 \sqrt {2 x+3}} \]

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2))/(3 + 2*x)^(3/2),x]

[Out]

-65/(8*Sqrt[3 + 2*x]) - (109*Sqrt[3 + 2*x])/8 + (47*(3 + 2*x)^(3/2))/24 - (3*(3 + 2*x)^(5/2))/40

Rule 785

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {65}{8 (3+2 x)^{3/2}}-\frac {109}{8 \sqrt {3+2 x}}+\frac {47}{8} \sqrt {3+2 x}-\frac {3}{8} (3+2 x)^{3/2}\right ) \, dx \\ & = -\frac {65}{8 \sqrt {3+2 x}}-\frac {109}{8} \sqrt {3+2 x}+\frac {47}{24} (3+2 x)^{3/2}-\frac {3}{40} (3+2 x)^{5/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.53 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{(3+2 x)^{3/2}} \, dx=-\frac {501+117 x-77 x^2+9 x^3}{15 \sqrt {3+2 x}} \]

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2))/(3 + 2*x)^(3/2),x]

[Out]

-1/15*(501 + 117*x - 77*x^2 + 9*x^3)/Sqrt[3 + 2*x]

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.47

method	result	size
gosper	\(-\frac {9 x^{3}-77 x^{2}+117 x +501}{15 \sqrt {3+2 x}}\)	\(25\)
trager	\(-\frac {9 x^{3}-77 x^{2}+117 x +501}{15 \sqrt {3+2 x}}\)	\(25\)
risch	\(-\frac {9 x^{3}-77 x^{2}+117 x +501}{15 \sqrt {3+2 x}}\)	\(25\)
pseudoelliptic	\(\frac {-9 x^{3}+77 x^{2}-117 x -501}{15 \sqrt {3+2 x}}\)	\(25\)
derivativedivides	\(\frac {47 \left (3+2 x \right )^{\frac {3}{2}}}{24}-\frac {3 \left (3+2 x \right )^{\frac {5}{2}}}{40}-\frac {65}{8 \sqrt {3+2 x}}-\frac {109 \sqrt {3+2 x}}{8}\)	\(38\)
default	\(\frac {47 \left (3+2 x \right )^{\frac {3}{2}}}{24}-\frac {3 \left (3+2 x \right )^{\frac {5}{2}}}{40}-\frac {65}{8 \sqrt {3+2 x}}-\frac {109 \sqrt {3+2 x}}{8}\)	\(38\)
meijerg	\(\frac {10 \sqrt {3}\, \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1+\frac {2 x}{3}}}\right )}{3 \sqrt {\pi }}+\frac {15 \sqrt {3}\, \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-\frac {8}{9} x^{2}+\frac {16}{3} x +16\right )}{6 \sqrt {1+\frac {2 x}{3}}}\right )}{2 \sqrt {\pi }}+\frac {23 \sqrt {3}\, \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (\frac {8 x}{3}+8\right )}{4 \sqrt {1+\frac {2 x}{3}}}\right )}{2 \sqrt {\pi }}-\frac {27 \sqrt {3}\, \left (-\frac {16 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (\frac {64}{27} x^{3}-\frac {64}{9} x^{2}+\frac {128}{3} x +128\right )}{40 \sqrt {1+\frac {2 x}{3}}}\right )}{8 \sqrt {\pi }}\)	\(134\)

[In]

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

[Out]

-1/15*(9*x^3-77*x^2+117*x+501)/(3+2*x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.45 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{(3+2 x)^{3/2}} \, dx=-\frac {9 \, x^{3} - 77 \, x^{2} + 117 \, x + 501}{15 \, \sqrt {2 \, x + 3}} \]

[In]

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

[Out]

-1/15*(9*x^3 - 77*x^2 + 117*x + 501)/sqrt(2*x + 3)

Sympy [A] (verification not implemented)

Time = 0.91 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.87 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{(3+2 x)^{3/2}} \, dx=- \frac {3 \left (2 x + 3\right )^{\frac {5}{2}}}{40} + \frac {47 \left (2 x + 3\right )^{\frac {3}{2}}}{24} - \frac {109 \sqrt {2 x + 3}}{8} - \frac {65}{8 \sqrt {2 x + 3}} \]

[In]

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

[Out]

-3*(2*x + 3)**(5/2)/40 + 47*(2*x + 3)**(3/2)/24 - 109*sqrt(2*x + 3)/8 - 65/(8*sqrt(2*x + 3))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{(3+2 x)^{3/2}} \, dx=-\frac {3}{40} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {47}{24} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - \frac {109}{8} \, \sqrt {2 \, x + 3} - \frac {65}{8 \, \sqrt {2 \, x + 3}} \]

[In]

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

[Out]

-3/40*(2*x + 3)^(5/2) + 47/24*(2*x + 3)^(3/2) - 109/8*sqrt(2*x + 3) - 65/8/sqrt(2*x + 3)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{(3+2 x)^{3/2}} \, dx=-\frac {3}{40} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} + \frac {47}{24} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - \frac {109}{8} \, \sqrt {2 \, x + 3} - \frac {65}{8 \, \sqrt {2 \, x + 3}} \]

[In]

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

[Out]

-3/40*(2*x + 3)^(5/2) + 47/24*(2*x + 3)^(3/2) - 109/8*sqrt(2*x + 3) - 65/8/sqrt(2*x + 3)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{(3+2 x)^{3/2}} \, dx=\frac {47\,{\left (2\,x+3\right )}^{3/2}}{24}-\frac {109\,\sqrt {2\,x+3}}{8}-\frac {65}{8\,\sqrt {2\,x+3}}-\frac {3\,{\left (2\,x+3\right )}^{5/2}}{40} \]

[In]

int(-((x - 5)*(5*x + 3*x^2 + 2))/(2*x + 3)^(3/2),x)

[Out]

(47*(2*x + 3)^(3/2))/24 - (109*(2*x + 3)^(1/2))/8 - 65/(8*(2*x + 3)^(1/2)) - (3*(2*x + 3)^(5/2))/40