3.4.76 \(\int (1+4 x-7 x^2) (2+5 x+x^2) \sqrt {3+2 x+5 x^2} \, dx\) [376]

3.4.76.1 Optimal result

Integrand size = 33, antiderivative size = 124 \[ \int \left (1+4 x-7 x^2\right ) \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx=-\frac {4633 (1+5 x) \sqrt {3+2 x+5 x^2}}{12500}+\frac {7819 \left (3+2 x+5 x^2\right )^{3/2}}{7500}+\frac {2149 x \left (3+2 x+5 x^2\right )^{3/2}}{2500}-\frac {289}{250} x^2 \left (3+2 x+5 x^2\right )^{3/2}-\frac {7}{30} x^3 \left (3+2 x+5 x^2\right )^{3/2}-\frac {32431 \text {arcsinh}\left (\frac {1+5 x}{\sqrt {14}}\right )}{6250 \sqrt {5}} \]

output

7819/7500*(5*x^2+2*x+3)^(3/2)+2149/2500*x*(5*x^2+2*x+3)^(3/2)-289/250*x^2* 
(5*x^2+2*x+3)^(3/2)-7/30*x^3*(5*x^2+2*x+3)^(3/2)-32431/31250*arcsinh(1/14* 
(1+5*x)*14^(1/2))*5^(1/2)-4633/12500*(1+5*x)*(5*x^2+2*x+3)^(1/2)
 

3.4.76.2 Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.64 \[ \int \left (1+4 x-7 x^2\right ) \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx=\frac {\sqrt {3+2 x+5 x^2} \left (103386+105400 x+129895 x^2+48225 x^3-234250 x^4-43750 x^5\right )}{37500}+\frac {32431 \log \left (-1-5 x+\sqrt {5} \sqrt {3+2 x+5 x^2}\right )}{6250 \sqrt {5}} \]

input

Integrate[(1 + 4*x - 7*x^2)*(2 + 5*x + x^2)*Sqrt[3 + 2*x + 5*x^2],x]
 

output

(Sqrt[3 + 2*x + 5*x^2]*(103386 + 105400*x + 129895*x^2 + 48225*x^3 - 23425 
0*x^4 - 43750*x^5))/37500 + (32431*Log[-1 - 5*x + Sqrt[5]*Sqrt[3 + 2*x + 5 
*x^2]])/(6250*Sqrt[5])
 

3.4.76.3 Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.17, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {2192, 27, 2192, 27, 2192, 25, 1160, 1087, 1090, 222}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \left (-7 x^2+4 x+1\right ) \left (x^2+5 x+2\right ) \sqrt {5 x^2+2 x+3} \, dx\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{30} \int 3 \sqrt {5 x^2+2 x+3} \left (-289 x^3+91 x^2+130 x+20\right )dx-\frac {7}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} \int \sqrt {5 x^2+2 x+3} \left (-289 x^3+91 x^2+130 x+20\right )dx-\frac {7}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{10} \left (\frac {1}{25} \int 2 \sqrt {5 x^2+2 x+3} \left (2149 x^2+2492 x+250\right )dx-\frac {289}{25} x^2 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {7}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} \left (\frac {2}{25} \int \sqrt {5 x^2+2 x+3} \left (2149 x^2+2492 x+250\right )dx-\frac {289}{25} x^2 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {7}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{10} \left (\frac {2}{25} \left (\frac {1}{20} \int -\left ((1447-39095 x) \sqrt {5 x^2+2 x+3}\right )dx+\frac {2149}{20} x \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {289}{25} x^2 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {7}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{10} \left (\frac {2}{25} \left (\frac {2149}{20} x \left (5 x^2+2 x+3\right )^{3/2}-\frac {1}{20} \int (1447-39095 x) \sqrt {5 x^2+2 x+3}dx\right )-\frac {289}{25} x^2 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {7}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {1}{10} \left (\frac {2}{25} \left (\frac {1}{20} \left (\frac {7819}{3} \left (5 x^2+2 x+3\right )^{3/2}-9266 \int \sqrt {5 x^2+2 x+3}dx\right )+\frac {2149}{20} x \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {289}{25} x^2 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {7}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{10} \left (\frac {2}{25} \left (\frac {1}{20} \left (\frac {7819}{3} \left (5 x^2+2 x+3\right )^{3/2}-9266 \left (\frac {7}{5} \int \frac {1}{\sqrt {5 x^2+2 x+3}}dx+\frac {1}{10} \sqrt {5 x^2+2 x+3} (5 x+1)\right )\right )+\frac {2149}{20} x \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {289}{25} x^2 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {7}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {1}{10} \left (\frac {2}{25} \left (\frac {1}{20} \left (\frac {7819}{3} \left (5 x^2+2 x+3\right )^{3/2}-9266 \left (\frac {1}{10} \sqrt {\frac {7}{10}} \int \frac {1}{\sqrt {\frac {1}{56} (10 x+2)^2+1}}d(10 x+2)+\frac {1}{10} \sqrt {5 x^2+2 x+3} (5 x+1)\right )\right )+\frac {2149}{20} x \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {289}{25} x^2 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {7}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{10} \left (\frac {2}{25} \left (\frac {1}{20} \left (\frac {7819}{3} \left (5 x^2+2 x+3\right )^{3/2}-9266 \left (\frac {7 \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )}{5 \sqrt {5}}+\frac {1}{10} \sqrt {5 x^2+2 x+3} (5 x+1)\right )\right )+\frac {2149}{20} x \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {289}{25} x^2 \left (5 x^2+2 x+3\right )^{3/2}\right )-\frac {7}{30} x^3 \left (5 x^2+2 x+3\right )^{3/2}\)

input

Int[(1 + 4*x - 7*x^2)*(2 + 5*x + x^2)*Sqrt[3 + 2*x + 5*x^2],x]
 

output

(-7*x^3*(3 + 2*x + 5*x^2)^(3/2))/30 + ((-289*x^2*(3 + 2*x + 5*x^2)^(3/2))/ 
25 + (2*((2149*x*(3 + 2*x + 5*x^2)^(3/2))/20 + ((7819*(3 + 2*x + 5*x^2)^(3 
/2))/3 - 9266*(((1 + 5*x)*Sqrt[3 + 2*x + 5*x^2])/10 + (7*ArcSinh[(2 + 10*x 
)/(2*Sqrt[14])])/(5*Sqrt[5])))/20))/25)/10
 

3.4.76.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 1087

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] - Simp[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] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1090

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

rule 1160

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] + Simp[(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[p, -1]
 

rule 2192

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = 
Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x + 
 c*x^2)^(p + 1)/(c*(q + 2*p + 1))), x] + Simp[1/(c*(q + 2*p + 1))   Int[(a 
+ b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b 
*e*(q + p)*x^(q - 1) - c*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c 
, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]
 

3.4.76.4 Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.44

method	result	size
risch	\(-\frac {\left (43750 x^{5}+234250 x^{4}-48225 x^{3}-129895 x^{2}-105400 x -103386\right ) \sqrt {5 x^{2}+2 x +3}}{37500}-\frac {32431 \sqrt {5}\, \operatorname {arcsinh}\left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{31250}\)	\(55\)
trager	\(\left (-\frac {7}{6} x^{5}-\frac {937}{150} x^{4}+\frac {643}{500} x^{3}+\frac {25979}{7500} x^{2}+\frac {1054}{375} x +\frac {17231}{6250}\right ) \sqrt {5 x^{2}+2 x +3}-\frac {32431 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )+5 \sqrt {5 x^{2}+2 x +3}\right )}{31250}\)	\(79\)
default	\(-\frac {4633 \left (10 x +2\right ) \sqrt {5 x^{2}+2 x +3}}{25000}-\frac {32431 \sqrt {5}\, \operatorname {arcsinh}\left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{31250}+\frac {7819 \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}}}{7500}-\frac {7 x^{3} \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}}}{30}-\frac {289 x^{2} \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}}}{250}+\frac {2149 x \left (5 x^{2}+2 x +3\right )^{\frac {3}{2}}}{2500}\)	\(98\)

input

int((-7*x^2+4*x+1)*(x^2+5*x+2)*(5*x^2+2*x+3)^(1/2),x,method=_RETURNVERBOSE 
)
 

output

-1/37500*(43750*x^5+234250*x^4-48225*x^3-129895*x^2-105400*x-103386)*(5*x^ 
2+2*x+3)^(1/2)-32431/31250*5^(1/2)*arcsinh(5/14*14^(1/2)*(x+1/5))
 

3.4.76.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.62 \[ \int \left (1+4 x-7 x^2\right ) \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx=-\frac {1}{37500} \, {\left (43750 \, x^{5} + 234250 \, x^{4} - 48225 \, x^{3} - 129895 \, x^{2} - 105400 \, x - 103386\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {32431}{62500} \, \sqrt {5} \log \left (\sqrt {5} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) \]

input

integrate((-7*x^2+4*x+1)*(x^2+5*x+2)*(5*x^2+2*x+3)^(1/2),x, algorithm="fri 
cas")
 

output

-1/37500*(43750*x^5 + 234250*x^4 - 48225*x^3 - 129895*x^2 - 105400*x - 103 
386)*sqrt(5*x^2 + 2*x + 3) + 32431/62500*sqrt(5)*log(sqrt(5)*sqrt(5*x^2 + 
2*x + 3)*(5*x + 1) - 25*x^2 - 10*x - 8)
 

3.4.76.6 Sympy [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.57 \[ \int \left (1+4 x-7 x^2\right ) \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx=\sqrt {5 x^{2} + 2 x + 3} \left (- \frac {7 x^{5}}{6} - \frac {937 x^{4}}{150} + \frac {643 x^{3}}{500} + \frac {25979 x^{2}}{7500} + \frac {1054 x}{375} + \frac {17231}{6250}\right ) - \frac {32431 \sqrt {5} \operatorname {asinh}{\left (\frac {5 \sqrt {14} \left (x + \frac {1}{5}\right )}{14} \right )}}{31250} \]

input

integrate((-7*x**2+4*x+1)*(x**2+5*x+2)*(5*x**2+2*x+3)**(1/2),x)
 

output

sqrt(5*x**2 + 2*x + 3)*(-7*x**5/6 - 937*x**4/150 + 643*x**3/500 + 25979*x* 
*2/7500 + 1054*x/375 + 17231/6250) - 32431*sqrt(5)*asinh(5*sqrt(14)*(x + 1 
/5)/14)/31250
 

3.4.76.7 Maxima [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.88 \[ \int \left (1+4 x-7 x^2\right ) \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx=-\frac {7}{30} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {289}{250} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {2149}{2500} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} x + \frac {7819}{7500} \, {\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac {3}{2}} - \frac {4633}{2500} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x - \frac {32431}{31250} \, \sqrt {5} \operatorname {arsinh}\left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) - \frac {4633}{12500} \, \sqrt {5 \, x^{2} + 2 \, x + 3} \]

input

integrate((-7*x^2+4*x+1)*(x^2+5*x+2)*(5*x^2+2*x+3)^(1/2),x, algorithm="max 
ima")
 

output

-7/30*(5*x^2 + 2*x + 3)^(3/2)*x^3 - 289/250*(5*x^2 + 2*x + 3)^(3/2)*x^2 + 
2149/2500*(5*x^2 + 2*x + 3)^(3/2)*x + 7819/7500*(5*x^2 + 2*x + 3)^(3/2) - 
4633/2500*sqrt(5*x^2 + 2*x + 3)*x - 32431/31250*sqrt(5)*arcsinh(1/14*sqrt( 
14)*(5*x + 1)) - 4633/12500*sqrt(5*x^2 + 2*x + 3)
 

3.4.76.8 Giac [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.58 \[ \int \left (1+4 x-7 x^2\right ) \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx=-\frac {1}{37500} \, {\left (5 \, {\left ({\left (5 \, {\left (10 \, {\left (175 \, x + 937\right )} x - 1929\right )} x - 25979\right )} x - 21080\right )} x - 103386\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {32431}{31250} \, \sqrt {5} \log \left (-\sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )} - 1\right ) \]

input

integrate((-7*x^2+4*x+1)*(x^2+5*x+2)*(5*x^2+2*x+3)^(1/2),x, algorithm="gia 
c")
 

output

-1/37500*(5*((5*(10*(175*x + 937)*x - 1929)*x - 25979)*x - 21080)*x - 1033 
86)*sqrt(5*x^2 + 2*x + 3) + 32431/31250*sqrt(5)*log(-sqrt(5)*(sqrt(5)*x - 
sqrt(5*x^2 + 2*x + 3)) - 1)
 

3.4.76.9 Mupad [B] (verification not implemented)

Time = 14.39 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.23 \[ \int \left (1+4 x-7 x^2\right ) \left (2+5 x+x^2\right ) \sqrt {3+2 x+5 x^2} \, dx=\frac {7819\,\sqrt {5\,x^2+2\,x+3}\,\left (200\,x^2+20\,x+108\right )}{300000}-\frac {7\,x^3\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{30}-\frac {10129\,\sqrt {5}\,\ln \left (\sqrt {5\,x^2+2\,x+3}+\frac {\sqrt {5}\,\left (5\,x+1\right )}{5}\right )}{62500}-\frac {1447\,\left (\frac {x}{2}+\frac {1}{10}\right )\,\sqrt {5\,x^2+2\,x+3}}{2500}-\frac {289\,x^2\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{250}+\frac {2149\,x\,{\left (5\,x^2+2\,x+3\right )}^{3/2}}{2500}-\frac {54733\,\sqrt {5}\,\ln \left (2\,\sqrt {5\,x^2+2\,x+3}+\frac {\sqrt {5}\,\left (10\,x+2\right )}{5}\right )}{62500} \]

input

int((5*x + x^2 + 2)*(2*x + 5*x^2 + 3)^(1/2)*(4*x - 7*x^2 + 1),x)
 

output

(7819*(2*x + 5*x^2 + 3)^(1/2)*(20*x + 200*x^2 + 108))/300000 - (7*x^3*(2*x 
 + 5*x^2 + 3)^(3/2))/30 - (10129*5^(1/2)*log((2*x + 5*x^2 + 3)^(1/2) + (5^ 
(1/2)*(5*x + 1))/5))/62500 - (1447*(x/2 + 1/10)*(2*x + 5*x^2 + 3)^(1/2))/2 
500 - (289*x^2*(2*x + 5*x^2 + 3)^(3/2))/250 + (2149*x*(2*x + 5*x^2 + 3)^(3 
/2))/2500 - (54733*5^(1/2)*log(2*(2*x + 5*x^2 + 3)^(1/2) + (5^(1/2)*(10*x 
+ 2))/5))/62500