3.2543 \(\int (5-x) (3+2 x)^{5/2} (2+5 x+3 x^2)^3 \, dx\)

Optimal. Leaf size=105 \[ -\frac{9}{896} (2 x+3)^{21/2}+\frac{567 (2 x+3)^{19/2}}{2432}-\frac{207}{128} (2 x+3)^{17/2}+\frac{2095}{384} (2 x+3)^{15/2}-\frac{17201 (2 x+3)^{13/2}}{1664}+\frac{1455}{128} (2 x+3)^{11/2}-\frac{7925 (2 x+3)^{9/2}}{1152}+\frac{1625}{896} (2 x+3)^{7/2} \]

[Out]

(1625*(3 + 2*x)^(7/2))/896 - (7925*(3 + 2*x)^(9/2))/1152 + (1455*(3 + 2*x)^(11/2))/128 - (17201*(3 + 2*x)^(13/
2))/1664 + (2095*(3 + 2*x)^(15/2))/384 - (207*(3 + 2*x)^(17/2))/128 + (567*(3 + 2*x)^(19/2))/2432 - (9*(3 + 2*
x)^(21/2))/896

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Rubi [A]  time = 0.0376753, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {771} \[ -\frac{9}{896} (2 x+3)^{21/2}+\frac{567 (2 x+3)^{19/2}}{2432}-\frac{207}{128} (2 x+3)^{17/2}+\frac{2095}{384} (2 x+3)^{15/2}-\frac{17201 (2 x+3)^{13/2}}{1664}+\frac{1455}{128} (2 x+3)^{11/2}-\frac{7925 (2 x+3)^{9/2}}{1152}+\frac{1625}{896} (2 x+3)^{7/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1625*(3 + 2*x)^(7/2))/896 - (7925*(3 + 2*x)^(9/2))/1152 + (1455*(3 + 2*x)^(11/2))/128 - (17201*(3 + 2*x)^(13/
2))/1664 + (2095*(3 + 2*x)^(15/2))/384 - (207*(3 + 2*x)^(17/2))/128 + (567*(3 + 2*x)^(19/2))/2432 - (9*(3 + 2*
x)^(21/2))/896

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)^{5/2} \left (2+5 x+3 x^2\right )^3 \, dx &=\int \left (\frac{1625}{128} (3+2 x)^{5/2}-\frac{7925}{128} (3+2 x)^{7/2}+\frac{16005}{128} (3+2 x)^{9/2}-\frac{17201}{128} (3+2 x)^{11/2}+\frac{10475}{128} (3+2 x)^{13/2}-\frac{3519}{128} (3+2 x)^{15/2}+\frac{567}{128} (3+2 x)^{17/2}-\frac{27}{128} (3+2 x)^{19/2}\right ) \, dx\\ &=\frac{1625}{896} (3+2 x)^{7/2}-\frac{7925 (3+2 x)^{9/2}}{1152}+\frac{1455}{128} (3+2 x)^{11/2}-\frac{17201 (3+2 x)^{13/2}}{1664}+\frac{2095}{384} (3+2 x)^{15/2}-\frac{207}{128} (3+2 x)^{17/2}+\frac{567 (3+2 x)^{19/2}}{2432}-\frac{9}{896} (3+2 x)^{21/2}\\ \end{align*}

Mathematica [A]  time = 0.0204353, size = 48, normalized size = 0.46 \[ -\frac{(2 x+3)^{7/2} \left (20007 x^7-22113 x^6-339066 x^5-791700 x^4-871983 x^3-517293 x^2-160006 x-20346\right )}{15561} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((3 + 2*x)^(7/2)*(-20346 - 160006*x - 517293*x^2 - 871983*x^3 - 791700*x^4 - 339066*x^5 - 22113*x^6 + 20007*x
^7))/15561

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Maple [A]  time = 0.006, size = 45, normalized size = 0.4 \begin{align*} -{\frac{20007\,{x}^{7}-22113\,{x}^{6}-339066\,{x}^{5}-791700\,{x}^{4}-871983\,{x}^{3}-517293\,{x}^{2}-160006\,x-20346}{15561} \left ( 3+2\,x \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15561*(20007*x^7-22113*x^6-339066*x^5-791700*x^4-871983*x^3-517293*x^2-160006*x-20346)*(3+2*x)^(7/2)

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Maxima [A]  time = 1.02053, size = 99, normalized size = 0.94 \begin{align*} -\frac{9}{896} \,{\left (2 \, x + 3\right )}^{\frac{21}{2}} + \frac{567}{2432} \,{\left (2 \, x + 3\right )}^{\frac{19}{2}} - \frac{207}{128} \,{\left (2 \, x + 3\right )}^{\frac{17}{2}} + \frac{2095}{384} \,{\left (2 \, x + 3\right )}^{\frac{15}{2}} - \frac{17201}{1664} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} + \frac{1455}{128} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} - \frac{7925}{1152} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{1625}{896} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/896*(2*x + 3)^(21/2) + 567/2432*(2*x + 3)^(19/2) - 207/128*(2*x + 3)^(17/2) + 2095/384*(2*x + 3)^(15/2) - 1
7201/1664*(2*x + 3)^(13/2) + 1455/128*(2*x + 3)^(11/2) - 7925/1152*(2*x + 3)^(9/2) + 1625/896*(2*x + 3)^(7/2)

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Fricas [A]  time = 1.74205, size = 238, normalized size = 2.27 \begin{align*} -\frac{1}{15561} \,{\left (160056 \, x^{10} + 543348 \, x^{9} - 2428218 \, x^{8} - 19193889 \, x^{7} - 54383679 \, x^{6} - 87436314 \, x^{5} - 88365578 \, x^{4} - 57400347 \, x^{3} - 23339691 \, x^{2} - 5418846 \, x - 549342\right )} \sqrt{2 \, x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15561*(160056*x^10 + 543348*x^9 - 2428218*x^8 - 19193889*x^7 - 54383679*x^6 - 87436314*x^5 - 88365578*x^4 -
57400347*x^3 - 23339691*x^2 - 5418846*x - 549342)*sqrt(2*x + 3)

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Sympy [A]  time = 38.6205, size = 94, normalized size = 0.9 \begin{align*} - \frac{9 \left (2 x + 3\right )^{\frac{21}{2}}}{896} + \frac{567 \left (2 x + 3\right )^{\frac{19}{2}}}{2432} - \frac{207 \left (2 x + 3\right )^{\frac{17}{2}}}{128} + \frac{2095 \left (2 x + 3\right )^{\frac{15}{2}}}{384} - \frac{17201 \left (2 x + 3\right )^{\frac{13}{2}}}{1664} + \frac{1455 \left (2 x + 3\right )^{\frac{11}{2}}}{128} - \frac{7925 \left (2 x + 3\right )^{\frac{9}{2}}}{1152} + \frac{1625 \left (2 x + 3\right )^{\frac{7}{2}}}{896} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9*(2*x + 3)**(21/2)/896 + 567*(2*x + 3)**(19/2)/2432 - 207*(2*x + 3)**(17/2)/128 + 2095*(2*x + 3)**(15/2)/384
- 17201*(2*x + 3)**(13/2)/1664 + 1455*(2*x + 3)**(11/2)/128 - 7925*(2*x + 3)**(9/2)/1152 + 1625*(2*x + 3)**(7
/2)/896

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Giac [A]  time = 1.09489, size = 99, normalized size = 0.94 \begin{align*} -\frac{9}{896} \,{\left (2 \, x + 3\right )}^{\frac{21}{2}} + \frac{567}{2432} \,{\left (2 \, x + 3\right )}^{\frac{19}{2}} - \frac{207}{128} \,{\left (2 \, x + 3\right )}^{\frac{17}{2}} + \frac{2095}{384} \,{\left (2 \, x + 3\right )}^{\frac{15}{2}} - \frac{17201}{1664} \,{\left (2 \, x + 3\right )}^{\frac{13}{2}} + \frac{1455}{128} \,{\left (2 \, x + 3\right )}^{\frac{11}{2}} - \frac{7925}{1152} \,{\left (2 \, x + 3\right )}^{\frac{9}{2}} + \frac{1625}{896} \,{\left (2 \, x + 3\right )}^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/896*(2*x + 3)^(21/2) + 567/2432*(2*x + 3)^(19/2) - 207/128*(2*x + 3)^(17/2) + 2095/384*(2*x + 3)^(15/2) - 1
7201/1664*(2*x + 3)^(13/2) + 1455/128*(2*x + 3)^(11/2) - 7925/1152*(2*x + 3)^(9/2) + 1625/896*(2*x + 3)^(7/2)