3.25.51 \(\int (5-x) (2+5 x+3 x^2)^{7/2} \, dx\) [2451]

Optimal. Leaf size=149 \[ -\frac {1225 (5+6 x) \sqrt {2+5 x+3 x^2}}{7962624}+\frac {1225 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{995328}-\frac {245 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{20736}+\frac {35}{288} (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{27} \left (2+5 x+3 x^2\right )^{9/2}+\frac {1225 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{15925248 \sqrt {3}} \]

[Out]

1225/995328*(5+6*x)*(3*x^2+5*x+2)^(3/2)-245/20736*(5+6*x)*(3*x^2+5*x+2)^(5/2)+35/288*(5+6*x)*(3*x^2+5*x+2)^(7/
2)-1/27*(3*x^2+5*x+2)^(9/2)+1225/47775744*arctanh(1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x+2)^(1/2))*3^(1/2)-1225/796262
4*(5+6*x)*(3*x^2+5*x+2)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {654, 626, 635, 212} \begin {gather*} -\frac {1}{27} \left (3 x^2+5 x+2\right )^{9/2}+\frac {35}{288} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {245 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{20736}+\frac {1225 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{995328}-\frac {1225 (6 x+5) \sqrt {3 x^2+5 x+2}}{7962624}+\frac {1225 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{15925248 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1225*(5 + 6*x)*Sqrt[2 + 5*x + 3*x^2])/7962624 + (1225*(5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/995328 - (245*(5 +
6*x)*(2 + 5*x + 3*x^2)^(5/2))/20736 + (35*(5 + 6*x)*(2 + 5*x + 3*x^2)^(7/2))/288 - (2 + 5*x + 3*x^2)^(9/2)/27
+ (1225*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/(15925248*Sqrt[3])

Rule 212

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

Rule 626

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 635

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 654

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 )^{7/2} \, dx &=-\frac {1}{27} \left (2+5 x+3 x^2\right )^{9/2}+\frac {35}{6} \int \left (2+5 x+3 x^2\right )^{7/2} \, dx\\ &=\frac {35}{288} (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{27} \left (2+5 x+3 x^2\right )^{9/2}-\frac {245}{576} \int \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=-\frac {245 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{20736}+\frac {35}{288} (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{27} \left (2+5 x+3 x^2\right )^{9/2}+\frac {1225 \int \left (2+5 x+3 x^2\right )^{3/2} \, dx}{41472}\\ &=\frac {1225 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{995328}-\frac {245 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{20736}+\frac {35}{288} (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{27} \left (2+5 x+3 x^2\right )^{9/2}-\frac {1225 \int \sqrt {2+5 x+3 x^2} \, dx}{663552}\\ &=-\frac {1225 (5+6 x) \sqrt {2+5 x+3 x^2}}{7962624}+\frac {1225 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{995328}-\frac {245 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{20736}+\frac {35}{288} (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{27} \left (2+5 x+3 x^2\right )^{9/2}+\frac {1225 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{15925248}\\ &=-\frac {1225 (5+6 x) \sqrt {2+5 x+3 x^2}}{7962624}+\frac {1225 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{995328}-\frac {245 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{20736}+\frac {35}{288} (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{27} \left (2+5 x+3 x^2\right )^{9/2}+\frac {1225 \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{7962624}\\ &=-\frac {1225 (5+6 x) \sqrt {2+5 x+3 x^2}}{7962624}+\frac {1225 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{995328}-\frac {245 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{20736}+\frac {35}{288} (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{27} \left (2+5 x+3 x^2\right )^{9/2}+\frac {1225 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{15925248 \sqrt {3}}\\ \end {align*}

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Mathematica [A]
time = 0.56, size = 91, normalized size = 0.61 \begin {gather*} \frac {-3 \sqrt {2+5 x+3 x^2} \left (-32198883-278256050 x-1014795048 x^2-2013572880 x^3-2320737408 x^4-1507127040 x^5-452625408 x^6+2488320 x^7+23887872 x^8\right )+1225 \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{23887872} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Sqrt[2 + 5*x + 3*x^2]*(-32198883 - 278256050*x - 1014795048*x^2 - 2013572880*x^3 - 2320737408*x^4 - 150712
7040*x^5 - 452625408*x^6 + 2488320*x^7 + 23887872*x^8) + 1225*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x
)])/23887872

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Maple [A]
time = 0.07, size = 121, normalized size = 0.81

method	result	size
risch	\(-\frac {\left (23887872 x^{8}+2488320 x^{7}-452625408 x^{6}-1507127040 x^{5}-2320737408 x^{4}-2013572880 x^{3}-1014795048 x^{2}-278256050 x -32198883\right ) \sqrt {3 x^{2}+5 x +2}}{7962624}+\frac {1225 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{47775744}\)	\(85\)
trager	\(\left (-3 x^{8}-\frac {5}{16} x^{7}+\frac {1819}{32} x^{6}+\frac {218045}{1152} x^{5}+\frac {2014529}{6912} x^{4}+\frac {13983145}{55296} x^{3}+\frac {42283127}{331776} x^{2}+\frac {139128025}{3981312} x +\frac {10732961}{2654208}\right ) \sqrt {3 x^{2}+5 x +2}+\frac {1225 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +5 \RootOf \left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{47775744}\)	\(96\)
default	\(\frac {35 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{288}-\frac {245 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{20736}+\frac {1225 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{995328}-\frac {1225 \left (5+6 x \right ) \sqrt {3 x^{2}+5 x +2}}{7962624}+\frac {1225 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{47775744}-\frac {\left (3 x^{2}+5 x +2\right )^{\frac {9}{2}}}{27}\)	\(121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

35/288*(5+6*x)*(3*x^2+5*x+2)^(7/2)-245/20736*(5+6*x)*(3*x^2+5*x+2)^(5/2)+1225/995328*(5+6*x)*(3*x^2+5*x+2)^(3/
2)-1225/7962624*(5+6*x)*(3*x^2+5*x+2)^(1/2)+1225/47775744*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2
)-1/27*(3*x^2+5*x+2)^(9/2)

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Maxima [A]
time = 0.54, size = 159, normalized size = 1.07 \begin {gather*} -\frac {1}{27} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} + \frac {35}{48} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x + \frac {175}{288} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {245}{3456} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {1225}{20736} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {1225}{165888} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {6125}{995328} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {1225}{1327104} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {1225}{47775744} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {6125}{7962624} \, \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)^(7/2),x, algorithm="maxima")

[Out]

-1/27*(3*x^2 + 5*x + 2)^(9/2) + 35/48*(3*x^2 + 5*x + 2)^(7/2)*x + 175/288*(3*x^2 + 5*x + 2)^(7/2) - 245/3456*(
3*x^2 + 5*x + 2)^(5/2)*x - 1225/20736*(3*x^2 + 5*x + 2)^(5/2) + 1225/165888*(3*x^2 + 5*x + 2)^(3/2)*x + 6125/9
95328*(3*x^2 + 5*x + 2)^(3/2) - 1225/1327104*sqrt(3*x^2 + 5*x + 2)*x + 1225/47775744*sqrt(3)*log(2*sqrt(3)*sqr
t(3*x^2 + 5*x + 2) + 6*x + 5) - 6125/7962624*sqrt(3*x^2 + 5*x + 2)

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Fricas [A]
time = 3.49, size = 93, normalized size = 0.62 \begin {gather*} -\frac {1}{7962624} \, {\left (23887872 \, x^{8} + 2488320 \, x^{7} - 452625408 \, x^{6} - 1507127040 \, x^{5} - 2320737408 \, x^{4} - 2013572880 \, x^{3} - 1014795048 \, x^{2} - 278256050 \, x - 32198883\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {1225}{95551488} \, \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)^(7/2),x, algorithm="fricas")

[Out]

-1/7962624*(23887872*x^8 + 2488320*x^7 - 452625408*x^6 - 1507127040*x^5 - 2320737408*x^4 - 2013572880*x^3 - 10
14795048*x^2 - 278256050*x - 32198883)*sqrt(3*x^2 + 5*x + 2) + 1225/95551488*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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- 292 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 870 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 396 x^{5} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 27 x^{7} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 40 \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)**(7/2),x)

[Out]

-Integral(-292*x*sqrt(3*x**2 + 5*x + 2), x) - Integral(-870*x**2*sqrt(3*x**2 + 5*x + 2), x) - Integral(-1339*x
**3*sqrt(3*x**2 + 5*x + 2), x) - Integral(-1090*x**4*sqrt(3*x**2 + 5*x + 2), x) - Integral(-396*x**5*sqrt(3*x*
*2 + 5*x + 2), x) - Integral(27*x**7*sqrt(3*x**2 + 5*x + 2), x) - Integral(-40*sqrt(3*x**2 + 5*x + 2), x)

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Giac [A]
time = 0.72, size = 89, normalized size = 0.60 \begin {gather*} -\frac {1}{7962624} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (2 \, {\left (48 \, x + 5\right )} x - 1819\right )} x - 218045\right )} x - 2014529\right )} x - 13983145\right )} x - 42283127\right )} x - 139128025\right )} x - 32198883\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {1225}{47775744} \, \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)^(7/2),x, algorithm="giac")

[Out]

-1/7962624*(2*(12*(6*(8*(6*(36*(2*(48*x + 5)*x - 1819)*x - 218045)*x - 2014529)*x - 13983145)*x - 42283127)*x
- 139128025)*x - 32198883)*sqrt(3*x^2 + 5*x + 2) - 1225/47775744*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(
3*x^2 + 5*x + 2)) - 5))

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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 )}^{7/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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