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

Optimal. Leaf size=126 \[ -\frac {1}{21} \left (3 x^2+5 x+2\right )^{7/2}+\frac {35}{216} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {175 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{10368}+\frac {175 (6 x+5) \sqrt {3 x^2+5 x+2}}{82944}-\frac {175 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{165888 \sqrt {3}} \]

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Rubi [A]  time = 0.04, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {640, 612, 621, 206} \begin {gather*} -\frac {1}{21} \left (3 x^2+5 x+2\right )^{7/2}+\frac {35}{216} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {175 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{10368}+\frac {175 (6 x+5) \sqrt {3 x^2+5 x+2}}{82944}-\frac {175 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{165888 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(175*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/82944 - (175*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/10368 + (35*(5 + 6*x)*(2
+ 5*x + 3*x^2)^(5/2))/216 - (2 + 5*x + 3*x^2)^(7/2)/21 - (175*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x
^2])])/(165888*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 640

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

Rubi steps

\begin {align*} \int (5-x) \left (2+5 x+3 x^2\right )^{5/2} \, dx &=-\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}+\frac {35}{6} \int \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=\frac {35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}-\frac {175}{432} \int \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=-\frac {175 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{10368}+\frac {35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}+\frac {175 \int \sqrt {2+5 x+3 x^2} \, dx}{6912}\\ &=\frac {175 (5+6 x) \sqrt {2+5 x+3 x^2}}{82944}-\frac {175 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{10368}+\frac {35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}-\frac {175 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{165888}\\ &=\frac {175 (5+6 x) \sqrt {2+5 x+3 x^2}}{82944}-\frac {175 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{10368}+\frac {35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}-\frac {175 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{82944}\\ &=\frac {175 (5+6 x) \sqrt {2+5 x+3 x^2}}{82944}-\frac {175 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{10368}+\frac {35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}-\frac {175 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{165888 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 108, normalized size = 0.86 \begin {gather*} -\frac {1}{21} \left (3 x^2+5 x+2\right )^{7/2}+\frac {35}{216} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {175 \left (\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 (144 x^3+360 x^2+290 x+75\right )\right )}{497664} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(35*(5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/216 - (2 + 5*x + 3*x^2)^(7/2)/21 - (175*(6*Sqrt[2 + 5*x + 3*x^2]*(75 +
290*x + 360*x^2 + 144*x^3) + Sqrt[3]*ArcTanh[(5 + 6*x)/(2*Sqrt[6 + 15*x + 9*x^2])]))/497664

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IntegrateAlgebraic [A]  time = 0.65, size = 84, normalized size = 0.67 \begin {gather*} \frac {\sqrt {3 x^2+5 x+2} \left (-746496 x^6+1347840 x^5+13454208 x^4+26388720 x^3+23110872 x^2+9651790 x+1568541\right )}{580608}-\frac {175 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{82944 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2 + 5*x + 3*x^2]*(1568541 + 9651790*x + 23110872*x^2 + 26388720*x^3 + 13454208*x^4 + 1347840*x^5 - 74649
6*x^6))/580608 - (175*ArcTanh[Sqrt[2 + 5*x + 3*x^2]/(Sqrt[3]*(1 + x))])/(82944*Sqrt[3])

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fricas [A]  time = 0.40, size = 83, normalized size = 0.66 \begin {gather*} -\frac {1}{580608} \, {\left (746496 \, x^{6} - 1347840 \, x^{5} - 13454208 \, x^{4} - 26388720 \, x^{3} - 23110872 \, x^{2} - 9651790 \, x - 1568541\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {175}{995328} \, \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*x^2+5*x+2)^(5/2),x, algorithm="fricas")

[Out]

-1/580608*(746496*x^6 - 1347840*x^5 - 13454208*x^4 - 26388720*x^3 - 23110872*x^2 - 9651790*x - 1568541)*sqrt(3
*x^2 + 5*x + 2) + 175/995328*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 = 0.23, size = 79, normalized size = 0.63 \begin {gather*} -\frac {1}{580608} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (8 \, {\left (6 \, {\left (36 \, x - 65\right )} x - 3893\right )} x - 61085\right )} x - 962953\right )} x - 4825895\right )} x - 1568541\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {175}{497664} \, \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*x^2+5*x+2)^(5/2),x, algorithm="giac")

[Out]

-1/580608*(2*(12*(18*(8*(6*(36*x - 65)*x - 3893)*x - 61085)*x - 962953)*x - 4825895)*x - 1568541)*sqrt(3*x^2 +
5*x + 2) + 175/497664*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 = 0.04, size = 102, normalized size = 0.81 \begin {gather*} -\frac {175 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{497664}+\frac {35 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{216}-\frac {175 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{10368}+\frac {175 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{82944}-\frac {\left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{21} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

35/216*(6*x+5)*(3*x^2+5*x+2)^(5/2)-175/10368*(6*x+5)*(3*x^2+5*x+2)^(3/2)+175/82944*(6*x+5)*(3*x^2+5*x+2)^(1/2)
-175/497664*3^(1/2)*ln(1/3*(3*x+5/2)*3^(1/2)+(3*x^2+5*x+2)^(1/2))-1/21*(3*x^2+5*x+2)^(7/2)

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maxima [A]  time = 1.21, size = 130, normalized size = 1.03 \begin {gather*} -\frac {1}{21} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} + \frac {35}{36} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {175}{216} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {175}{1728} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {875}{10368} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {175}{13824} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {175}{497664} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {875}{82944} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(3*x^2 + 5*x + 2)^(7/2) + 35/36*(3*x^2 + 5*x + 2)^(5/2)*x + 175/216*(3*x^2 + 5*x + 2)^(5/2) - 175/1728*(
3*x^2 + 5*x + 2)^(3/2)*x - 875/10368*(3*x^2 + 5*x + 2)^(3/2) + 175/13824*sqrt(3*x^2 + 5*x + 2)*x - 175/497664*
sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + 875/82944*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 (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(-(x - 5)*(5*x + 3*x^2 + 2)^(5/2),x)

[Out]

int(-(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 (- 96 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 165 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 113 x^{3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 15 x^{4} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 9 x^{5} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 20 \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*x**2+5*x+2)**(5/2),x)

[Out]

-Integral(-96*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-165*x**2*sqrt(3*x**2 + 5*x + 2), x) - Integral(-113*x**
3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-15*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(9*x**5*sqrt(3*x**2 + 5*
x + 2), x) - Integral(-20*sqrt(3*x**2 + 5*x + 2), x)

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