3.2526 \(\int (5-x) (3+2 x)^{7/2} (2+5 x+3 x^2) \, dx\)

Optimal. Leaf size=53 \[ -\frac {1}{40} (2 x+3)^{15/2}+\frac {47}{104} (2 x+3)^{13/2}-\frac {109}{88} (2 x+3)^{11/2}+\frac {65}{72} (2 x+3)^{9/2} \]

[Out]

65/72*(3+2*x)^(9/2)-109/88*(3+2*x)^(11/2)+47/104*(3+2*x)^(13/2)-1/40*(3+2*x)^(15/2)

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Rubi [A]  time = 0.02, 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 = {771} \[ -\frac {1}{40} (2 x+3)^{15/2}+\frac {47}{104} (2 x+3)^{13/2}-\frac {109}{88} (2 x+3)^{11/2}+\frac {65}{72} (2 x+3)^{9/2} \]

Antiderivative was successfully verified.

[In]

Int[(5 - x)*(3 + 2*x)^(7/2)*(2 + 5*x + 3*x^2),x]

[Out]

(65*(3 + 2*x)^(9/2))/72 - (109*(3 + 2*x)^(11/2))/88 + (47*(3 + 2*x)^(13/2))/104 - (3 + 2*x)^(15/2)/40

Rule 771

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)^{7/2} \left (2+5 x+3 x^2\right ) \, dx &=\int \left (\frac {65}{8} (3+2 x)^{7/2}-\frac {109}{8} (3+2 x)^{9/2}+\frac {47}{8} (3+2 x)^{11/2}-\frac {3}{8} (3+2 x)^{13/2}\right ) \, dx\\ &=\frac {65}{72} (3+2 x)^{9/2}-\frac {109}{88} (3+2 x)^{11/2}+\frac {47}{104} (3+2 x)^{13/2}-\frac {1}{40} (3+2 x)^{15/2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 0.53 \[ -\frac {(2 x+3)^{9/2} \left (1287 x^3-5841 x^2-10269 x-3727\right )}{6435} \]

Antiderivative was successfully verified.

[In]

Integrate[(5 - x)*(3 + 2*x)^(7/2)*(2 + 5*x + 3*x^2),x]

[Out]

-1/6435*((3 + 2*x)^(9/2)*(-3727 - 10269*x - 5841*x^2 + 1287*x^3))

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fricas [A]  time = 0.63, size = 44, normalized size = 0.83 \[ -\frac {1}{6435} \, {\left (20592 \, x^{7} + 30096 \, x^{6} - 447048 \, x^{5} - 2029120 \, x^{4} - 3733305 \, x^{3} - 3496257 \, x^{2} - 1636821 \, x - 301887\right )} \sqrt {2 \, x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^(7/2)*(3*x^2+5*x+2),x, algorithm="fricas")

[Out]

-1/6435*(20592*x^7 + 30096*x^6 - 447048*x^5 - 2029120*x^4 - 3733305*x^3 - 3496257*x^2 - 1636821*x - 301887)*sq
rt(2*x + 3)

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giac [A]  time = 0.16, size = 37, normalized size = 0.70 \[ -\frac {1}{40} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} + \frac {47}{104} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} - \frac {109}{88} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {65}{72} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^(7/2)*(3*x^2+5*x+2),x, algorithm="giac")

[Out]

-1/40*(2*x + 3)^(15/2) + 47/104*(2*x + 3)^(13/2) - 109/88*(2*x + 3)^(11/2) + 65/72*(2*x + 3)^(9/2)

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maple [A]  time = 0.00, size = 25, normalized size = 0.47 \[ -\frac {\left (1287 x^{3}-5841 x^{2}-10269 x -3727\right ) \left (2 x +3\right )^{\frac {9}{2}}}{6435} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3+2*x)^(7/2)*(3*x^2+5*x+2),x)

[Out]

-1/6435*(1287*x^3-5841*x^2-10269*x-3727)*(3+2*x)^(9/2)

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maxima [A]  time = 0.54, size = 37, normalized size = 0.70 \[ -\frac {1}{40} \, {\left (2 \, x + 3\right )}^{\frac {15}{2}} + \frac {47}{104} \, {\left (2 \, x + 3\right )}^{\frac {13}{2}} - \frac {109}{88} \, {\left (2 \, x + 3\right )}^{\frac {11}{2}} + \frac {65}{72} \, {\left (2 \, x + 3\right )}^{\frac {9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^(7/2)*(3*x^2+5*x+2),x, algorithm="maxima")

[Out]

-1/40*(2*x + 3)^(15/2) + 47/104*(2*x + 3)^(13/2) - 109/88*(2*x + 3)^(11/2) + 65/72*(2*x + 3)^(9/2)

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mupad [B]  time = 3.03, size = 37, normalized size = 0.70 \[ \frac {65\,{\left (2\,x+3\right )}^{9/2}}{72}-\frac {109\,{\left (2\,x+3\right )}^{11/2}}{88}+\frac {47\,{\left (2\,x+3\right )}^{13/2}}{104}-\frac {{\left (2\,x+3\right )}^{15/2}}{40} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x + 3)^(7/2)*(x - 5)*(5*x + 3*x^2 + 2),x)

[Out]

(65*(2*x + 3)^(9/2))/72 - (109*(2*x + 3)^(11/2))/88 + (47*(2*x + 3)^(13/2))/104 - (2*x + 3)^(15/2)/40

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sympy [B]  time = 3.78, size = 116, normalized size = 2.19 \[ - \frac {16 x^{7} \sqrt {2 x + 3}}{5} - \frac {304 x^{6} \sqrt {2 x + 3}}{65} + \frac {49672 x^{5} \sqrt {2 x + 3}}{715} + \frac {405824 x^{4} \sqrt {2 x + 3}}{1287} + \frac {248887 x^{3} \sqrt {2 x + 3}}{429} + \frac {388473 x^{2} \sqrt {2 x + 3}}{715} + \frac {181869 x \sqrt {2 x + 3}}{715} + \frac {33543 \sqrt {2 x + 3}}{715} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)**(7/2)*(3*x**2+5*x+2),x)

[Out]

-16*x**7*sqrt(2*x + 3)/5 - 304*x**6*sqrt(2*x + 3)/65 + 49672*x**5*sqrt(2*x + 3)/715 + 405824*x**4*sqrt(2*x + 3
)/1287 + 248887*x**3*sqrt(2*x + 3)/429 + 388473*x**2*sqrt(2*x + 3)/715 + 181869*x*sqrt(2*x + 3)/715 + 33543*sq
rt(2*x + 3)/715

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