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

Optimal. Leaf size=156 \[ -\frac {1}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}+\frac {(4298 x+10211) \left (3 x^2+5 x+2\right )^{7/2}}{4536}+\frac {4507 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{15552}-\frac {22535 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{746496}+\frac {22535 (6 x+5) \sqrt {3 x^2+5 x+2}}{5971968}-\frac {22535 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{11943936 \sqrt {3}} \]

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Rubi [A]  time = 0.07, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {832, 779, 612, 621, 206} \begin {gather*} -\frac {1}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}+\frac {(4298 x+10211) \left (3 x^2+5 x+2\right )^{7/2}}{4536}+\frac {4507 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{15552}-\frac {22535 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{746496}+\frac {22535 (6 x+5) \sqrt {3 x^2+5 x+2}}{5971968}-\frac {22535 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{11943936 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(22535*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/5971968 - (22535*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/746496 + (4507*(5
+ 6*x)*(2 + 5*x + 3*x^2)^(5/2))/15552 - ((3 + 2*x)^2*(2 + 5*x + 3*x^2)^(7/2))/27 + ((10211 + 4298*x)*(2 + 5*x
+ 3*x^2)^(7/2))/4536 - (22535*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(11943936*Sqrt[3])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

Rubi steps

\begin {align*} \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2} \, dx &=-\frac {1}{27} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}+\frac {1}{27} \int (3+2 x) \left (\frac {931}{2}+307 x\right ) \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=-\frac {1}{27} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(10211+4298 x) \left (2+5 x+3 x^2\right )^{7/2}}{4536}+\frac {4507}{432} \int \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=\frac {4507 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{15552}-\frac {1}{27} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(10211+4298 x) \left (2+5 x+3 x^2\right )^{7/2}}{4536}-\frac {22535 \int \left (2+5 x+3 x^2\right )^{3/2} \, dx}{31104}\\ &=-\frac {22535 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{746496}+\frac {4507 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{15552}-\frac {1}{27} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(10211+4298 x) \left (2+5 x+3 x^2\right )^{7/2}}{4536}+\frac {22535 \int \sqrt {2+5 x+3 x^2} \, dx}{497664}\\ &=\frac {22535 (5+6 x) \sqrt {2+5 x+3 x^2}}{5971968}-\frac {22535 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{746496}+\frac {4507 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{15552}-\frac {1}{27} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(10211+4298 x) \left (2+5 x+3 x^2\right )^{7/2}}{4536}-\frac {22535 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{11943936}\\ &=\frac {22535 (5+6 x) \sqrt {2+5 x+3 x^2}}{5971968}-\frac {22535 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{746496}+\frac {4507 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{15552}-\frac {1}{27} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(10211+4298 x) \left (2+5 x+3 x^2\right )^{7/2}}{4536}-\frac {22535 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{5971968}\\ &=\frac {22535 (5+6 x) \sqrt {2+5 x+3 x^2}}{5971968}-\frac {22535 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{746496}+\frac {4507 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{15552}-\frac {1}{27} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(10211+4298 x) \left (2+5 x+3 x^2\right )^{7/2}}{4536}-\frac {22535 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{11943936 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 92, normalized size = 0.59 \begin {gather*} \frac {-157745 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )-6 \sqrt {3 x^2+5 x+2} \left (167215104 x^8+268240896 x^7-3275873280 x^6-15455860992 x^5-30355761024 x^4-32476001904 x^3-19762157208 x^2-6434937470 x-871825317\right )}{250822656} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[2 + 5*x + 3*x^2]*(-871825317 - 6434937470*x - 19762157208*x^2 - 32476001904*x^3 - 30355761024*x^4 - 1
5455860992*x^5 - 3275873280*x^6 + 268240896*x^7 + 167215104*x^8) - 157745*Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6
+ 15*x + 9*x^2])])/250822656

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IntegrateAlgebraic [A]  time = 0.83, size = 94, normalized size = 0.60 \begin {gather*} \frac {\sqrt {3 x^2+5 x+2} \left (-167215104 x^8-268240896 x^7+3275873280 x^6+15455860992 x^5+30355761024 x^4+32476001904 x^3+19762157208 x^2+6434937470 x+871825317\right )}{41803776}-\frac {22535 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{5971968 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(871825317 + 6434937470*x + 19762157208*x^2 + 32476001904*x^3 + 30355761024*x^4 + 15455
860992*x^5 + 3275873280*x^6 - 268240896*x^7 - 167215104*x^8))/41803776 - (22535*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/
(Sqrt[3]*(1 + x))])/(5971968*Sqrt[3])

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fricas [A]  time = 0.40, size = 93, normalized size = 0.60 \begin {gather*} -\frac {1}{41803776} \, {\left (167215104 \, x^{8} + 268240896 \, x^{7} - 3275873280 \, x^{6} - 15455860992 \, x^{5} - 30355761024 \, x^{4} - 32476001904 \, x^{3} - 19762157208 \, x^{2} - 6434937470 \, x - 871825317\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {22535}{71663616} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/41803776*(167215104*x^8 + 268240896*x^7 - 3275873280*x^6 - 15455860992*x^5 - 30355761024*x^4 - 32476001904*
x^3 - 19762157208*x^2 - 6434937470*x - 871825317)*sqrt(3*x^2 + 5*x + 2) + 22535/71663616*sqrt(3)*log(-4*sqrt(3
)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49)

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giac [A]  time = 0.44, size = 89, normalized size = 0.57 \begin {gather*} -\frac {1}{41803776} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (14 \, {\left (48 \, x + 77\right )} x - 13165\right )} x - 2236091\right )} x - 26350487\right )} x - 225527791\right )} x - 823423217\right )} x - 3217468735\right )} x - 871825317\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {22535}{35831808} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/41803776*(2*(12*(6*(8*(6*(36*(14*(48*x + 77)*x - 13165)*x - 2236091)*x - 26350487)*x - 225527791)*x - 82342
3217)*x - 3217468735)*x - 871825317)*sqrt(3*x^2 + 5*x + 2) + 22535/35831808*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3
)*x - sqrt(3*x^2 + 5*x + 2)) - 5))

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maple [A]  time = 0.05, size = 134, normalized size = 0.86 \begin {gather*} -\frac {4 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}} x^{2}}{27}+\frac {163 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}} x}{324}-\frac {22535 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{35831808}+\frac {8699 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{4536}+\frac {4507 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{15552}-\frac {22535 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{746496}+\frac {22535 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{5971968} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/27*(3*x^2+5*x+2)^(7/2)*x^2+163/324*(3*x^2+5*x+2)^(7/2)*x+8699/4536*(3*x^2+5*x+2)^(7/2)+4507/15552*(6*x+5)*(
3*x^2+5*x+2)^(5/2)-22535/746496*(6*x+5)*(3*x^2+5*x+2)^(3/2)+22535/5971968*(6*x+5)*(3*x^2+5*x+2)^(1/2)-22535/35
831808*3^(1/2)*ln(1/3*(3*x+5/2)*3^(1/2)+(3*x^2+5*x+2)^(1/2))

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maxima [A]  time = 1.30, size = 162, normalized size = 1.04 \begin {gather*} -\frac {4}{27} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x^{2} + \frac {163}{324} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x + \frac {8699}{4536} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} + \frac {4507}{2592} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {22535}{15552} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {22535}{124416} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {112675}{746496} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {22535}{995328} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {22535}{35831808} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {112675}{5971968} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/27*(3*x^2 + 5*x + 2)^(7/2)*x^2 + 163/324*(3*x^2 + 5*x + 2)^(7/2)*x + 8699/4536*(3*x^2 + 5*x + 2)^(7/2) + 45
07/2592*(3*x^2 + 5*x + 2)^(5/2)*x + 22535/15552*(3*x^2 + 5*x + 2)^(5/2) - 22535/124416*(3*x^2 + 5*x + 2)^(3/2)
*x - 112675/746496*(3*x^2 + 5*x + 2)^(3/2) + 22535/995328*sqrt(3*x^2 + 5*x + 2)*x - 22535/35831808*sqrt(3)*log
(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + 112675/5971968*sqrt(3*x^2 + 5*x + 2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int {\left (2\,x+3\right )}^2\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- 1104 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 2717 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 3381 x^{3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 2151 x^{4} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 551 x^{5} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 48 x^{6} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int 36 x^{7} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 180 \sqrt {3 x^{2} + 5 x + 2}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-1104*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-2717*x**2*sqrt(3*x**2 + 5*x + 2), x) - Integral(-3381
*x**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-2151*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(-551*x**5*sqrt(3*
x**2 + 5*x + 2), x) - Integral(48*x**6*sqrt(3*x**2 + 5*x + 2), x) - Integral(36*x**7*sqrt(3*x**2 + 5*x + 2), x
) - Integral(-180*sqrt(3*x**2 + 5*x + 2), x)

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