\(\int \frac {(1+4 x-7 x^2) (2+5 x+x^2)}{\sqrt {3+2 x+5 x^2}} \, dx\) [16]

Optimal result

Integrand size = 33, antiderivative size = 101 \[ \int \frac {\left (1+4 x-7 x^2\right ) \left (2+5 x+x^2\right )}{\sqrt {3+2 x+5 x^2}} \, dx=\frac {463}{125} \sqrt {3+2 x+5 x^2}+\frac {59}{30} x \sqrt {3+2 x+5 x^2}-\frac {571}{300} x^2 \sqrt {3+2 x+5 x^2}-\frac {7}{20} x^3 \sqrt {3+2 x+5 x^2}-\frac {1901 \text {arcsinh}\left (\frac {1+5 x}{\sqrt {14}}\right )}{250 \sqrt {5}} \] Output:

463/125*(5*x^2+2*x+3)^(1/2)+59/30*x*(5*x^2+2*x+3)^(1/2)-571/300*x^2*(5*x^2 
+2*x+3)^(1/2)-7/20*x^3*(5*x^2+2*x+3)^(1/2)-1901/1250*arcsinh(1/14*(1+5*x)* 
14^(1/2))*5^(1/2)
 

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.68 \[ \int \frac {\left (1+4 x-7 x^2\right ) \left (2+5 x+x^2\right )}{\sqrt {3+2 x+5 x^2}} \, dx=\frac {\sqrt {3+2 x+5 x^2} \left (5556+2950 x-2855 x^2-525 x^3\right )}{1500}+\frac {1901 \log \left (-1-5 x+\sqrt {5} \sqrt {3+2 x+5 x^2}\right )}{250 \sqrt {5}} \] 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]*(5556 + 2950*x - 2855*x^2 - 525*x^3))/1500 + (1901* 
Log[-1 - 5*x + Sqrt[5]*Sqrt[3 + 2*x + 5*x^2]])/(250*Sqrt[5])
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {2192, 2192, 27, 2192, 27, 1160, 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.

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

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{20} \int \frac {-571 x^3+203 x^2+260 x+40}{\sqrt {5 x^2+2 x+3}}dx-\frac {7}{20} x^3 \sqrt {5 x^2+2 x+3}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{20} \left (\frac {1}{15} \int \frac {2 \left (2950 x^2+3663 x+300\right )}{\sqrt {5 x^2+2 x+3}}dx-\frac {571}{15} x^2 \sqrt {5 x^2+2 x+3}\right )-\frac {7}{20} x^3 \sqrt {5 x^2+2 x+3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{20} \left (\frac {2}{15} \int \frac {2950 x^2+3663 x+300}{\sqrt {5 x^2+2 x+3}}dx-\frac {571}{15} x^2 \sqrt {5 x^2+2 x+3}\right )-\frac {7}{20} x^3 \sqrt {5 x^2+2 x+3}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {1}{20} \left (\frac {2}{15} \left (\frac {1}{10} \int -\frac {30 (195-926 x)}{\sqrt {5 x^2+2 x+3}}dx+295 \sqrt {5 x^2+2 x+3} x\right )-\frac {571}{15} x^2 \sqrt {5 x^2+2 x+3}\right )-\frac {7}{20} x^3 \sqrt {5 x^2+2 x+3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{20} \left (\frac {2}{15} \left (295 x \sqrt {5 x^2+2 x+3}-3 \int \frac {195-926 x}{\sqrt {5 x^2+2 x+3}}dx\right )-\frac {571}{15} x^2 \sqrt {5 x^2+2 x+3}\right )-\frac {7}{20} x^3 \sqrt {5 x^2+2 x+3}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {1}{20} \left (\frac {2}{15} \left (295 x \sqrt {5 x^2+2 x+3}-3 \left (\frac {1901}{5} \int \frac {1}{\sqrt {5 x^2+2 x+3}}dx-\frac {926}{5} \sqrt {5 x^2+2 x+3}\right )\right )-\frac {571}{15} x^2 \sqrt {5 x^2+2 x+3}\right )-\frac {7}{20} x^3 \sqrt {5 x^2+2 x+3}\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {1}{20} \left (\frac {2}{15} \left (295 x \sqrt {5 x^2+2 x+3}-3 \left (\frac {1901 \int \frac {1}{\sqrt {\frac {1}{56} (10 x+2)^2+1}}d(10 x+2)}{10 \sqrt {70}}-\frac {926}{5} \sqrt {5 x^2+2 x+3}\right )\right )-\frac {571}{15} x^2 \sqrt {5 x^2+2 x+3}\right )-\frac {7}{20} x^3 \sqrt {5 x^2+2 x+3}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{20} \left (\frac {2}{15} \left (295 x \sqrt {5 x^2+2 x+3}-3 \left (\frac {1901 \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )}{5 \sqrt {5}}-\frac {926}{5} \sqrt {5 x^2+2 x+3}\right )\right )-\frac {571}{15} x^2 \sqrt {5 x^2+2 x+3}\right )-\frac {7}{20} x^3 \sqrt {5 x^2+2 x+3}\)

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*Sqrt[3 + 2*x + 5*x^2])/20 + ((-571*x^2*Sqrt[3 + 2*x + 5*x^2])/15 + 
 (2*(295*x*Sqrt[3 + 2*x + 5*x^2] - 3*((-926*Sqrt[3 + 2*x + 5*x^2])/5 + (19 
01*ArcSinh[(2 + 10*x)/(2*Sqrt[14])])/(5*Sqrt[5]))))/15)/20
 

Defintions of rubi rules used

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 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]
 

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.45

method	result	size
risch	\(-\frac {\left (525 x^{3}+2855 x^{2}-2950 x -5556\right ) \sqrt {5 x^{2}+2 x +3}}{1500}-\frac {1901 \sqrt {5}\, \operatorname {arcsinh}\left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{1250}\)	\(45\)
trager	\(\left (-\frac {7}{20} x^{3}-\frac {571}{300} x^{2}+\frac {59}{30} x +\frac {463}{125}\right ) \sqrt {5 x^{2}+2 x +3}+\frac {1901 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )+5 \sqrt {5 x^{2}+2 x +3}\right )}{1250}\)	\(71\)
default	\(-\frac {1901 \sqrt {5}\, \operatorname {arcsinh}\left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{1250}+\frac {463 \sqrt {5 x^{2}+2 x +3}}{125}+\frac {59 x \sqrt {5 x^{2}+2 x +3}}{30}-\frac {571 x^{2} \sqrt {5 x^{2}+2 x +3}}{300}-\frac {7 x^{3} \sqrt {5 x^{2}+2 x +3}}{20}\)	\(79\)

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:

-1/1500*(525*x^3+2855*x^2-2950*x-5556)*(5*x^2+2*x+3)^(1/2)-1901/1250*5^(1/ 
2)*arcsinh(5/14*14^(1/2)*(x+1/5))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.66 \[ \int \frac {\left (1+4 x-7 x^2\right ) \left (2+5 x+x^2\right )}{\sqrt {3+2 x+5 x^2}} \, dx=-\frac {1}{1500} \, {\left (525 \, x^{3} + 2855 \, x^{2} - 2950 \, x - 5556\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {1901}{2500} \, \sqrt {5} \log \left (\sqrt {5} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) \] 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:

-1/1500*(525*x^3 + 2855*x^2 - 2950*x - 5556)*sqrt(5*x^2 + 2*x + 3) + 1901/ 
2500*sqrt(5)*log(sqrt(5)*sqrt(5*x^2 + 2*x + 3)*(5*x + 1) - 25*x^2 - 10*x - 
 8)
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.57 \[ \int \frac {\left (1+4 x-7 x^2\right ) \left (2+5 x+x^2\right )}{\sqrt {3+2 x+5 x^2}} \, dx=\sqrt {5 x^{2} + 2 x + 3} \left (- \frac {7 x^{3}}{20} - \frac {571 x^{2}}{300} + \frac {59 x}{30} + \frac {463}{125}\right ) - \frac {1901 \sqrt {5} \operatorname {asinh}{\left (\frac {5 \sqrt {14} \left (x + \frac {1}{5}\right )}{14} \right )}}{1250} \] 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**3/20 - 571*x**2/300 + 59*x/30 + 463/125) - 1 
901*sqrt(5)*asinh(5*sqrt(14)*(x + 1/5)/14)/1250
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.79 \[ \int \frac {\left (1+4 x-7 x^2\right ) \left (2+5 x+x^2\right )}{\sqrt {3+2 x+5 x^2}} \, dx=-\frac {7}{20} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x^{3} - \frac {571}{300} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x^{2} + \frac {59}{30} \, \sqrt {5 \, x^{2} + 2 \, x + 3} x - \frac {1901}{1250} \, \sqrt {5} \operatorname {arsinh}\left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) + \frac {463}{125} \, \sqrt {5 \, x^{2} + 2 \, x + 3} \] 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:

-7/20*sqrt(5*x^2 + 2*x + 3)*x^3 - 571/300*sqrt(5*x^2 + 2*x + 3)*x^2 + 59/3 
0*sqrt(5*x^2 + 2*x + 3)*x - 1901/1250*sqrt(5)*arcsinh(1/14*sqrt(14)*(5*x + 
 1)) + 463/125*sqrt(5*x^2 + 2*x + 3)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.61 \[ \int \frac {\left (1+4 x-7 x^2\right ) \left (2+5 x+x^2\right )}{\sqrt {3+2 x+5 x^2}} \, dx=-\frac {1}{1500} \, {\left (5 \, {\left ({\left (105 \, x + 571\right )} x - 590\right )} x - 5556\right )} \sqrt {5 \, x^{2} + 2 \, x + 3} + \frac {1901}{1250} \, \sqrt {5} \log \left (-\sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )} - 1\right ) \] 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:

-1/1500*(5*((105*x + 571)*x - 590)*x - 5556)*sqrt(5*x^2 + 2*x + 3) + 1901/ 
1250*sqrt(5)*log(-sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3)) - 1)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1+4 x-7 x^2\right ) \left (2+5 x+x^2\right )}{\sqrt {3+2 x+5 x^2}} \, dx=\int \frac {\left (x^2+5\,x+2\right )\,\left (-7\,x^2+4\,x+1\right )}{\sqrt {5\,x^2+2\,x+3}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.88 \[ \int \frac {\left (1+4 x-7 x^2\right ) \left (2+5 x+x^2\right )}{\sqrt {3+2 x+5 x^2}} \, dx=-\frac {7 \sqrt {5 x^{2}+2 x +3}\, x^{3}}{20}-\frac {571 \sqrt {5 x^{2}+2 x +3}\, x^{2}}{300}+\frac {59 \sqrt {5 x^{2}+2 x +3}\, x}{30}+\frac {463 \sqrt {5 x^{2}+2 x +3}}{125}-\frac {1901 \sqrt {5}\, \mathrm {log}\left (\frac {\sqrt {5 x^{2}+2 x +3}\, \sqrt {5}+5 x +1}{\sqrt {14}}\right )}{1250} \] Input:

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

Output:

( - 2625*sqrt(5*x**2 + 2*x + 3)*x**3 - 14275*sqrt(5*x**2 + 2*x + 3)*x**2 + 
 14750*sqrt(5*x**2 + 2*x + 3)*x + 27780*sqrt(5*x**2 + 2*x + 3) - 11406*sqr 
t(5)*log((sqrt(5*x**2 + 2*x + 3)*sqrt(5) + 5*x + 1)/sqrt(14)))/7500