3.26.28 \(\int (5-x) (3+2 x)^{3/2} (2+5 x+3 x^2) \, dx\) [2528]

Optimal. Leaf size=53 \[ \frac {13}{8} (3+2 x)^{5/2}-\frac {109}{56} (3+2 x)^{7/2}+\frac {47}{72} (3+2 x)^{9/2}-\frac {3}{88} (3+2 x)^{11/2} \]

[Out]

13/8*(3+2*x)^(5/2)-109/56*(3+2*x)^(7/2)+47/72*(3+2*x)^(9/2)-3/88*(3+2*x)^(11/2)

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {785} \begin {gather*} -\frac {3}{88} (2 x+3)^{11/2}+\frac {47}{72} (2 x+3)^{9/2}-\frac {109}{56} (2 x+3)^{7/2}+\frac {13}{8} (2 x+3)^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(13*(3 + 2*x)^(5/2))/8 - (109*(3 + 2*x)^(7/2))/56 + (47*(3 + 2*x)^(9/2))/72 - (3*(3 + 2*x)^(11/2))/88

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*} \int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right ) \, dx &=\int \left (\frac {65}{8} (3+2 x)^{3/2}-\frac {109}{8} (3+2 x)^{5/2}+\frac {47}{8} (3+2 x)^{7/2}-\frac {3}{8} (3+2 x)^{9/2}\right ) \, dx\\ &=\frac {13}{8} (3+2 x)^{5/2}-\frac {109}{56} (3+2 x)^{7/2}+\frac {47}{72} (3+2 x)^{9/2}-\frac {3}{88} (3+2 x)^{11/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 28, normalized size = 0.53 \begin {gather*} -\frac {1}{693} (3+2 x)^{5/2} \left (-513-1455 x-959 x^2+189 x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/693*((3 + 2*x)^(5/2)*(-513 - 1455*x - 959*x^2 + 189*x^3))

________________________________________________________________________________________

Maple [A]
time = 0.07, size = 38, normalized size = 0.72

method	result	size
gosper	\(-\frac {\left (189 x^{3}-959 x^{2}-1455 x -513\right ) \left (3+2 x \right )^{\frac {5}{2}}}{693}\)	\(25\)
trager	\(\left (-\frac {12}{11} x^{5}+\frac {224}{99} x^{4}+\frac {5209}{231} x^{3}+\frac {3127}{77} x^{2}+\frac {2139}{77} x +\frac {513}{77}\right ) \sqrt {3+2 x}\)	\(34\)
risch	\(-\frac {\left (756 x^{5}-1568 x^{4}-15627 x^{3}-28143 x^{2}-19251 x -4617\right ) \sqrt {3+2 x}}{693}\)	\(35\)
derivativedivides	\(\frac {13 \left (3+2 x \right )^{\frac {5}{2}}}{8}-\frac {109 \left (3+2 x \right )^{\frac {7}{2}}}{56}+\frac {47 \left (3+2 x \right )^{\frac {9}{2}}}{72}-\frac {3 \left (3+2 x \right )^{\frac {11}{2}}}{88}\)	\(38\)
default	\(\frac {13 \left (3+2 x \right )^{\frac {5}{2}}}{8}-\frac {109 \left (3+2 x \right )^{\frac {7}{2}}}{56}+\frac {47 \left (3+2 x \right )^{\frac {9}{2}}}{72}-\frac {3 \left (3+2 x \right )^{\frac {11}{2}}}{88}\)	\(38\)
meijerg	\(\frac {1215 \sqrt {3}\, \left (-\frac {64 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (\frac {1120}{81} x^{4}+\frac {800}{27} x^{3}+\frac {8}{3} x^{2}-\frac {16}{3} x +16\right ) \sqrt {1+\frac {2 x}{3}}}{945}\right )}{16 \sqrt {\pi }}+\frac {1863 \sqrt {3}\, \left (\frac {16 \sqrt {\pi }}{105}-\frac {2 \sqrt {\pi }\, \left (-\frac {160}{27} x^{3}-\frac {128}{9} x^{2}-\frac {8}{3} x +8\right ) \sqrt {1+\frac {2 x}{3}}}{105}\right )}{16 \sqrt {\pi }}+\frac {135 \sqrt {3}\, \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (\frac {8}{9} x^{2}+\frac {8}{3} x +2\right ) \sqrt {1+\frac {2 x}{3}}}{15}\right )}{4 \sqrt {\pi }}-\frac {2187 \sqrt {3}\, \left (\frac {128 \sqrt {\pi }}{3465}-\frac {\sqrt {\pi }\, \left (-\frac {8960}{81} x^{5}-\frac {17920}{81} x^{4}-\frac {320}{27} x^{3}+\frac {64}{3} x^{2}-\frac {128}{3} x +128\right ) \sqrt {1+\frac {2 x}{3}}}{3465}\right )}{64 \sqrt {\pi }}\)	\(176\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

13/8*(3+2*x)^(5/2)-109/56*(3+2*x)^(7/2)+47/72*(3+2*x)^(9/2)-3/88*(3+2*x)^(11/2)

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 37, normalized size = 0.70 \begin {gather*} -\frac {3}{88} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {47}{72} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} - \frac {109}{56} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {13}{8} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/88*(2*x + 3)^(11/2) + 47/72*(2*x + 3)^(9/2) - 109/56*(2*x + 3)^(7/2) + 13/8*(2*x + 3)^(5/2)

________________________________________________________________________________________

Fricas [A]
time = 2.66, size = 34, normalized size = 0.64 \begin {gather*} -\frac {1}{693} \, {\left (756 \, x^{5} - 1568 \, x^{4} - 15627 \, x^{3} - 28143 \, x^{2} - 19251 \, x - 4617\right )} \sqrt {2 \, x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/693*(756*x^5 - 1568*x^4 - 15627*x^3 - 28143*x^2 - 19251*x - 4617)*sqrt(2*x + 3)

________________________________________________________________________________________

Sympy [A]
time = 7.48, size = 46, normalized size = 0.87 \begin {gather*} - \frac {3 \left (2 x + 3\right )^{\frac {11}{2}}}{88} + \frac {47 \left (2 x + 3\right )^{\frac {9}{2}}}{72} - \frac {109 \left (2 x + 3\right )^{\frac {7}{2}}}{56} + \frac {13 \left (2 x + 3\right )^{\frac {5}{2}}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*(2*x + 3)**(11/2)/88 + 47*(2*x + 3)**(9/2)/72 - 109*(2*x + 3)**(7/2)/56 + 13*(2*x + 3)**(5/2)/8

________________________________________________________________________________________

Giac [A]
time = 1.38, size = 37, normalized size = 0.70 \begin {gather*} -\frac {3}{88} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {47}{72} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} - \frac {109}{56} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {13}{8} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/88*(2*x + 3)^(11/2) + 47/72*(2*x + 3)^(9/2) - 109/56*(2*x + 3)^(7/2) + 13/8*(2*x + 3)^(5/2)

________________________________________________________________________________________

Mupad [B]
time = 0.04, size = 37, normalized size = 0.70 \begin {gather*} \frac {13\,{\left (2\,x+3\right )}^{5/2}}{8}-\frac {109\,{\left (2\,x+3\right )}^{7/2}}{56}+\frac {47\,{\left (2\,x+3\right )}^{9/2}}{72}-\frac {3\,{\left (2\,x+3\right )}^{11/2}}{88} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(13*(2*x + 3)^(5/2))/8 - (109*(2*x + 3)^(7/2))/56 + (47*(2*x + 3)^(9/2))/72 - (3*(2*x + 3)^(11/2))/88

________________________________________________________________________________________