\(\int \frac {(5-x) (2+5 x+3 x^2)}{\sqrt {3+2 x}} \, dx\) [2530]

Optimal result

Integrand size = 25, antiderivative size = 53 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{\sqrt {3+2 x}} \, dx=\frac {65}{8} \sqrt {3+2 x}-\frac {109}{24} (3+2 x)^{3/2}+\frac {47}{40} (3+2 x)^{5/2}-\frac {3}{56} (3+2 x)^{7/2} \]

[Out]

-109/24*(3+2*x)^(3/2)+47/40*(3+2*x)^(5/2)-3/56*(3+2*x)^(7/2)+65/8*(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} \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{\sqrt {3+2 x}} \, dx=-\frac {3}{56} (2 x+3)^{7/2}+\frac {47}{40} (2 x+3)^{5/2}-\frac {109}{24} (2 x+3)^{3/2}+\frac {65}{8} \sqrt {2 x+3} \]

[In]

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

[Out]

(65*Sqrt[3 + 2*x])/8 - (109*(3 + 2*x)^(3/2))/24 + (47*(3 + 2*x)^(5/2))/40 - (3*(3 + 2*x)^(7/2))/56

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 \sqrt {3+2 x}}-\frac {109}{8} \sqrt {3+2 x}+\frac {47}{8} (3+2 x)^{3/2}-\frac {3}{8} (3+2 x)^{5/2}\right ) \, dx \\ & = \frac {65}{8} \sqrt {3+2 x}-\frac {109}{24} (3+2 x)^{3/2}+\frac {47}{40} (3+2 x)^{5/2}-\frac {3}{56} (3+2 x)^{7/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.53 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{\sqrt {3+2 x}} \, dx=-\frac {1}{105} \sqrt {3+2 x} \left (-381-223 x-291 x^2+45 x^3\right ) \]

[In]

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

[Out]

-1/105*(Sqrt[3 + 2*x]*(-381 - 223*x - 291*x^2 + 45*x^3))

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.45

method	result	size
trager	\(\left (-\frac {3}{7} x^{3}+\frac {97}{35} x^{2}+\frac {223}{105} x +\frac {127}{35}\right ) \sqrt {3+2 x}\)	\(24\)
gosper	\(-\frac {\left (45 x^{3}-291 x^{2}-223 x -381\right ) \sqrt {3+2 x}}{105}\)	\(25\)
risch	\(-\frac {\left (45 x^{3}-291 x^{2}-223 x -381\right ) \sqrt {3+2 x}}{105}\)	\(25\)
pseudoelliptic	\(-\frac {\left (45 x^{3}-291 x^{2}-223 x -381\right ) \sqrt {3+2 x}}{105}\)	\(25\)
derivativedivides	\(-\frac {109 \left (3+2 x \right )^{\frac {3}{2}}}{24}+\frac {47 \left (3+2 x \right )^{\frac {5}{2}}}{40}-\frac {3 \left (3+2 x \right )^{\frac {7}{2}}}{56}+\frac {65 \sqrt {3+2 x}}{8}\)	\(38\)
default	\(-\frac {109 \left (3+2 x \right )^{\frac {3}{2}}}{24}+\frac {47 \left (3+2 x \right )^{\frac {5}{2}}}{40}-\frac {3 \left (3+2 x \right )^{\frac {7}{2}}}{56}+\frac {65 \sqrt {3+2 x}}{8}\)	\(38\)
meijerg	\(\frac {5 \sqrt {3}\, \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1+\frac {2 x}{3}}\right )}{\sqrt {\pi }}+\frac {45 \sqrt {3}\, \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (\frac {8}{3} x^{2}-\frac {16}{3} x +16\right ) \sqrt {1+\frac {2 x}{3}}}{15}\right )}{4 \sqrt {\pi }}+\frac {69 \sqrt {3}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-\frac {8 x}{3}+8\right ) \sqrt {1+\frac {2 x}{3}}}{6}\right )}{4 \sqrt {\pi }}-\frac {81 \sqrt {3}\, \left (\frac {32 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-\frac {320}{27} x^{3}+\frac {64}{3} x^{2}-\frac {128}{3} x +128\right ) \sqrt {1+\frac {2 x}{3}}}{140}\right )}{16 \sqrt {\pi }}\)	\(136\)

[In]

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

[Out]

(-3/7*x^3+97/35*x^2+223/105*x+127/35)*(3+2*x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.45 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{\sqrt {3+2 x}} \, dx=-\frac {1}{105} \, {\left (45 \, x^{3} - 291 \, x^{2} - 223 \, x - 381\right )} \sqrt {2 \, x + 3} \]

[In]

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

[Out]

-1/105*(45*x^3 - 291*x^2 - 223*x - 381)*sqrt(2*x + 3)

Sympy [A] (verification not implemented)

Time = 0.75 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.87 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{\sqrt {3+2 x}} \, dx=- \frac {3 \left (2 x + 3\right )^{\frac {7}{2}}}{56} + \frac {47 \left (2 x + 3\right )^{\frac {5}{2}}}{40} - \frac {109 \left (2 x + 3\right )^{\frac {3}{2}}}{24} + \frac {65 \sqrt {2 x + 3}}{8} \]

[In]

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

[Out]

-3*(2*x + 3)**(7/2)/56 + 47*(2*x + 3)**(5/2)/40 - 109*(2*x + 3)**(3/2)/24 + 65*sqrt(2*x + 3)/8

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{\sqrt {3+2 x}} \, dx=-\frac {3}{56} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {47}{40} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - \frac {109}{24} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {65}{8} \, \sqrt {2 \, x + 3} \]

[In]

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

[Out]

-3/56*(2*x + 3)^(7/2) + 47/40*(2*x + 3)^(5/2) - 109/24*(2*x + 3)^(3/2) + 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 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{\sqrt {3+2 x}} \, dx=-\frac {3}{56} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {47}{40} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - \frac {109}{24} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {65}{8} \, \sqrt {2 \, x + 3} \]

[In]

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

[Out]

-3/56*(2*x + 3)^(7/2) + 47/40*(2*x + 3)^(5/2) - 109/24*(2*x + 3)^(3/2) + 65/8*sqrt(2*x + 3)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.70 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{\sqrt {3+2 x}} \, dx=\frac {65\,\sqrt {2\,x+3}}{8}-\frac {109\,{\left (2\,x+3\right )}^{3/2}}{24}+\frac {47\,{\left (2\,x+3\right )}^{5/2}}{40}-\frac {3\,{\left (2\,x+3\right )}^{7/2}}{56} \]

[In]

int(-((x - 5)*(5*x + 3*x^2 + 2))/(2*x + 3)^(1/2),x)

[Out]

(65*(2*x + 3)^(1/2))/8 - (109*(2*x + 3)^(3/2))/24 + (47*(2*x + 3)^(5/2))/40 - (3*(2*x + 3)^(7/2))/56