\(\int \frac {(1+4 x-7 x^2)^3 (2+5 x+x^2)}{(3+2 x+5 x^2)^{3/2}} \, dx\) [20]

Optimal result

Integrand size = 35, antiderivative size = 166 \[ \int \frac {\left (1+4 x-7 x^2\right )^3 \left (2+5 x+x^2\right )}{\left (3+2 x+5 x^2\right )^{3/2}} \, dx=\frac {16 (6122807-5338217 x)}{546875 \sqrt {3+2 x+5 x^2}}+\frac {15715799 \sqrt {3+2 x+5 x^2}}{156250}-\frac {3192602 x \sqrt {3+2 x+5 x^2}}{46875}-\frac {2583293 x^2 \sqrt {3+2 x+5 x^2}}{187500}+\frac {393659 x^3 \sqrt {3+2 x+5 x^2}}{12500}-\frac {25921 x^4 \sqrt {3+2 x+5 x^2}}{3750}-\frac {343}{150} x^5 \sqrt {3+2 x+5 x^2}+\frac {50047657 \text {arcsinh}\left (\frac {1+5 x}{\sqrt {14}}\right )}{156250 \sqrt {5}} \] Output:

16/546875*(6122807-5338217*x)/(5*x^2+2*x+3)^(1/2)[phone]/156250*(5*x^2+2 
*x+3)^(1/2)-3192602/46875*x*(5*x^2+2*x+3)^(1/2)-2583293/187500*x^2*(5*x^2+ 
2*x+3)^(1/2)+393659/12500*x^3*(5*x^2+2*x+3)^(1/2)-25921/3750*x^4*(5*x^2+2* 
x+3)^(1/2)-343/150*x^5*(5*x^2+2*x+3)^(1/2)[phone]/781250*arcsinh(1/14*(1 
+5*x)*14^(1/2))*5^(1/2)
 

Mathematica [A] (verified)

Time = 0.74 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.54 \[ \int \frac {\left (1+4 x-7 x^2\right )^3 \left (2+5 x+x^2\right )}{\left (3+2 x+5 x^2\right )^{3/2}} \, dx=\frac {3155769618-1045703388 x+2135143465 x^2-1795638985 x^3-174819575 x^4+897612625 x^5-256821250 x^6-75031250 x^7}{6562500 \sqrt {3+2 x+5 x^2}}-\frac {50047657 \log \left (-1-5 x+\sqrt {5} \sqrt {3+2 x+5 x^2}\right )}{156250 \sqrt {5}} \] Input:

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

Output:

(3155769618 - 1045703388*x + 2135143465*x^2 - 1795638985*x^3 - 174819575*x 
^4 + 897612625*x^5 - 256821250*x^6 - 75031250*x^7)/(6562500*Sqrt[3 + 2*x + 
 5*x^2]) - (50047657*Log[-1 - 5*x + Sqrt[5]*Sqrt[3 + 2*x + 5*x^2]])/(15625 
0*Sqrt[5])
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.13, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2191, 27, 2192, 27, 2192, 27, 2192, 27, 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.

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

\(\Big \downarrow \) 2191

\(\displaystyle \frac {1}{28} \int \frac {28 \left (-5359375 x^6-15465625 x^5+41667500 x^4-5403250 x^3-36448575 x^2+16868255 x+16918718\right )}{78125 \sqrt {5 x^2+2 x+3}}dx+\frac {16 (6122807-5338217 x)}{546875 \sqrt {5 x^2+2 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {-5359375 x^6-15465625 x^5+41667500 x^4-5403250 x^3-36448575 x^2+16868255 x+16918718}{\sqrt {5 x^2+2 x+3}}dx}{78125}+\frac {16 (6122807-5338217 x)}{546875 \sqrt {5 x^2+2 x+3}}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {1}{30} \int \frac {5 \left (-81003125 x^5+266083125 x^4-32419500 x^3-218691450 x^2+101209530 x+101512308\right )}{\sqrt {5 x^2+2 x+3}}dx-\frac {1071875}{6} x^5 \sqrt {5 x^2+2 x+3}}{78125}+\frac {16 (6122807-5338217 x)}{546875 \sqrt {5 x^2+2 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{6} \int \frac {-81003125 x^5+266083125 x^4-32419500 x^3-218691450 x^2+101209530 x+101512308}{\sqrt {5 x^2+2 x+3}}dx-\frac {1071875}{6} x^5 \sqrt {5 x^2+2 x+3}}{78125}+\frac {16 (6122807-5338217 x)}{546875 \sqrt {5 x^2+2 x+3}}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{25} \int \frac {150 \left (49207375 x^4+1077000 x^3-36448575 x^2+16868255 x+16918718\right )}{\sqrt {5 x^2+2 x+3}}dx-3240125 x^4 \sqrt {5 x^2+2 x+3}\right )-\frac {1071875}{6} x^5 \sqrt {5 x^2+2 x+3}}{78125}+\frac {16 (6122807-5338217 x)}{546875 \sqrt {5 x^2+2 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{6} \left (6 \int \frac {49207375 x^4+1077000 x^3-36448575 x^2+16868255 x+16918718}{\sqrt {5 x^2+2 x+3}}dx-3240125 x^4 \sqrt {5 x^2+2 x+3}\right )-\frac {1071875}{6} x^5 \sqrt {5 x^2+2 x+3}}{78125}+\frac {16 (6122807-5338217 x)}{546875 \sqrt {5 x^2+2 x+3}}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {1}{6} \left (6 \left (\frac {1}{20} \int \frac {5 \left (-64582325 x^3-234367575 x^2+67473020 x+67674872\right )}{\sqrt {5 x^2+2 x+3}}dx+\frac {9841475}{4} \sqrt {5 x^2+2 x+3} x^3\right )-3240125 x^4 \sqrt {5 x^2+2 x+3}\right )-\frac {1071875}{6} x^5 \sqrt {5 x^2+2 x+3}}{78125}+\frac {16 (6122807-5338217 x)}{546875 \sqrt {5 x^2+2 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{6} \left (6 \left (\frac {1}{4} \int \frac {-64582325 x^3-234367575 x^2+67473020 x+67674872}{\sqrt {5 x^2+2 x+3}}dx+\frac {9841475}{4} \sqrt {5 x^2+2 x+3} x^3\right )-3240125 x^4 \sqrt {5 x^2+2 x+3}\right )-\frac {1071875}{6} x^5 \sqrt {5 x^2+2 x+3}}{78125}+\frac {16 (6122807-5338217 x)}{546875 \sqrt {5 x^2+2 x+3}}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {1}{6} \left (6 \left (\frac {1}{4} \left (\frac {1}{15} \int \frac {10 \left (-319260200 x^2+139958925 x+101512308\right )}{\sqrt {5 x^2+2 x+3}}dx-\frac {12916465}{3} x^2 \sqrt {5 x^2+2 x+3}\right )+\frac {9841475}{4} \sqrt {5 x^2+2 x+3} x^3\right )-3240125 x^4 \sqrt {5 x^2+2 x+3}\right )-\frac {1071875}{6} x^5 \sqrt {5 x^2+2 x+3}}{78125}+\frac {16 (6122807-5338217 x)}{546875 \sqrt {5 x^2+2 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{6} \left (6 \left (\frac {1}{4} \left (\frac {2}{3} \int \frac {-319260200 x^2+139958925 x+101512308}{\sqrt {5 x^2+2 x+3}}dx-\frac {12916465}{3} x^2 \sqrt {5 x^2+2 x+3}\right )+\frac {9841475}{4} \sqrt {5 x^2+2 x+3} x^3\right )-3240125 x^4 \sqrt {5 x^2+2 x+3}\right )-\frac {1071875}{6} x^5 \sqrt {5 x^2+2 x+3}}{78125}+\frac {16 (6122807-5338217 x)}{546875 \sqrt {5 x^2+2 x+3}}\)

\(\Big \downarrow \) 2192

\(\displaystyle \frac {\frac {1}{6} \left (6 \left (\frac {1}{4} \left (\frac {2}{3} \left (\frac {1}{10} \int \frac {330 (7143545 x+5978496)}{\sqrt {5 x^2+2 x+3}}dx-31926020 x \sqrt {5 x^2+2 x+3}\right )-\frac {12916465}{3} x^2 \sqrt {5 x^2+2 x+3}\right )+\frac {9841475}{4} \sqrt {5 x^2+2 x+3} x^3\right )-3240125 x^4 \sqrt {5 x^2+2 x+3}\right )-\frac {1071875}{6} x^5 \sqrt {5 x^2+2 x+3}}{78125}+\frac {16 (6122807-5338217 x)}{546875 \sqrt {5 x^2+2 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{6} \left (6 \left (\frac {1}{4} \left (\frac {2}{3} \left (33 \int \frac {7143545 x+5978496}{\sqrt {5 x^2+2 x+3}}dx-31926020 x \sqrt {5 x^2+2 x+3}\right )-\frac {12916465}{3} x^2 \sqrt {5 x^2+2 x+3}\right )+\frac {9841475}{4} \sqrt {5 x^2+2 x+3} x^3\right )-3240125 x^4 \sqrt {5 x^2+2 x+3}\right )-\frac {1071875}{6} x^5 \sqrt {5 x^2+2 x+3}}{78125}+\frac {16 (6122807-5338217 x)}{546875 \sqrt {5 x^2+2 x+3}}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {\frac {1}{6} \left (6 \left (\frac {1}{4} \left (\frac {2}{3} \left (33 \left (4549787 \int \frac {1}{\sqrt {5 x^2+2 x+3}}dx+1428709 \sqrt {5 x^2+2 x+3}\right )-31926020 x \sqrt {5 x^2+2 x+3}\right )-\frac {12916465}{3} x^2 \sqrt {5 x^2+2 x+3}\right )+\frac {9841475}{4} \sqrt {5 x^2+2 x+3} x^3\right )-3240125 x^4 \sqrt {5 x^2+2 x+3}\right )-\frac {1071875}{6} x^5 \sqrt {5 x^2+2 x+3}}{78125}+\frac {16 (6122807-5338217 x)}{546875 \sqrt {5 x^2+2 x+3}}\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {\frac {1}{6} \left (6 \left (\frac {1}{4} \left (\frac {2}{3} \left (33 \left (\frac {4549787 \int \frac {1}{\sqrt {\frac {1}{56} (10 x+2)^2+1}}d(10 x+2)}{2 \sqrt {70}}+1428709 \sqrt {5 x^2+2 x+3}\right )-31926020 x \sqrt {5 x^2+2 x+3}\right )-\frac {12916465}{3} x^2 \sqrt {5 x^2+2 x+3}\right )+\frac {9841475}{4} \sqrt {5 x^2+2 x+3} x^3\right )-3240125 x^4 \sqrt {5 x^2+2 x+3}\right )-\frac {1071875}{6} x^5 \sqrt {5 x^2+2 x+3}}{78125}+\frac {16 (6122807-5338217 x)}{546875 \sqrt {5 x^2+2 x+3}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {\frac {1}{6} \left (6 \left (\frac {1}{4} \left (\frac {2}{3} \left (33 \left (\frac {4549787 \text {arcsinh}\left (\frac {10 x+2}{2 \sqrt {14}}\right )}{\sqrt {5}}+1428709 \sqrt {5 x^2+2 x+3}\right )-31926020 x \sqrt {5 x^2+2 x+3}\right )-\frac {12916465}{3} x^2 \sqrt {5 x^2+2 x+3}\right )+\frac {9841475}{4} \sqrt {5 x^2+2 x+3} x^3\right )-3240125 x^4 \sqrt {5 x^2+2 x+3}\right )-\frac {1071875}{6} x^5 \sqrt {5 x^2+2 x+3}}{78125}+\frac {16 (6122807-5338217 x)}{546875 \sqrt {5 x^2+2 x+3}}\)

Input:

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

Output:

(16*(6122807 - 5338217*x))/(546875*Sqrt[3 + 2*x + 5*x^2]) + ((-1071875*x^5 
*Sqrt[3 + 2*x + 5*x^2])/6 + (-3240125*x^4*Sqrt[3 + 2*x + 5*x^2] + 6*((9841 
475*x^3*Sqrt[3 + 2*x + 5*x^2])/4 + ((-12916465*x^2*Sqrt[3 + 2*x + 5*x^2])/ 
3 + (2*(-31926020*x*Sqrt[3 + 2*x + 5*x^2] + 33*(1428709*Sqrt[3 + 2*x + 5*x 
^2] + (4549787*ArcSinh[(2 + 10*x)/(2*Sqrt[14])])/Sqrt[5])))/3)/4))/6)/7812 
5
 

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 2191

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = 
PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[P 
q, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + 
c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^ 
(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c))   Int 
[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)* 
(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^ 
2 - 4*a*c, 0] && LtQ[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 = 0.41 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.39

method	result	size
risch	\(-\frac {75031250 x^{7}[phone] x^{6}-897612625 x^{5}[phone] x^{4}[phone] x^{3}-2135143465 x^{2}[phone] x -3155769618}{6562500 \sqrt {5 x^{2}+2 x +3}}+\frac {50047657 \sqrt {5}\, \operatorname {arcsinh}\left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{781250}\)	\(65\)
trager	\(-\frac {75031250 x^{7}[phone] x^{6}-897612625 x^{5}[phone] x^{4}[phone] x^{3}-2135143465 x^{2}[phone] x -3155769618}{6562500 \sqrt {5 x^{2}+2 x +3}}-\frac {50047657 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )+5 \sqrt {5 x^{2}+2 x +3}\right )}{781250}\)	\(92\)
default	\(\frac {\frac {176049701 x}{1093750}+\frac {176049701}{5468750}}{\sqrt {5 x^{2}+2 x +3}}+\frac {175268451}{390625 \sqrt {5 x^{2}+2 x +3}}-\frac {50047657 x}{156250 \sqrt {5 x^{2}+2 x +3}}+\frac {50047657 \sqrt {5}\, \operatorname {arcsinh}\left (\frac {5 \sqrt {14}\, \left (x +\frac {1}{5}\right )}{14}\right )}{781250}+\frac {61004099 x^{2}}{187500 \sqrt {5 x^{2}+2 x +3}}-\frac {51303971 x^{3}}{187500 \sqrt {5 x^{2}+2 x +3}}-\frac {998969 x^{4}}{37500 \sqrt {5 x^{2}+2 x +3}}+\frac {1025843 x^{5}}{7500 \sqrt {5 x^{2}+2 x +3}}-\frac {29351 x^{6}}{750 \sqrt {5 x^{2}+2 x +3}}-\frac {343 x^{7}}{30 \sqrt {5 x^{2}+2 x +3}}\)	\(166\)

Input:

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

Output:

-1/6562500*(75031250*x^7+256821250*x^6-897612625*x^5+174819575*x^4+1795638 
985*x^3-2135143465*x^2+1045703388*x-3155769618)/(5*x^2+2*x+3)^(1/2)+500476 
57/781250*5^(1/2)*arcsinh(5/14*14^(1/2)*(x+1/5))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.67 \[ \int \frac {\left (1+4 x-7 x^2\right )^3 \left (2+5 x+x^2\right )}{\left (3+2 x+5 x^2\right )^{3/2}} \, dx=\frac {1051000797 \, \sqrt {5} {\left (5 \, x^{2} + 2 \, x + 3\right )} \log \left (-\sqrt {5} \sqrt {5 \, x^{2} + 2 \, x + 3} {\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) - 5 \, {\left (75031250 \, x^{7} + 256821250 \, x^{6} - 897612625 \, x^{5} + 174819575 \, x^{4} + 1795638985 \, x^{3} - 2135143465 \, x^{2} + 1045703388 \, x - 3155769618\right )} \sqrt {5 \, x^{2} + 2 \, x + 3}}{32812500 \, {\left (5 \, x^{2} + 2 \, x + 3\right )}} \] Input:

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

Output:

1/32812500*(1051000797*sqrt(5)*(5*x^2 + 2*x + 3)*log(-sqrt(5)*sqrt(5*x^2 + 
 2*x + 3)*(5*x + 1) - 25*x^2 - 10*x - 8) - 5*(75031250*x^7 + 256821250*x^6 
 - 897612625*x^5 + 174819575*x^4 + 1795638985*x^3 - 2135143465*x^2 + 10457 
03388*x - 3155769618)*sqrt(5*x^2 + 2*x + 3))/(5*x^2 + 2*x + 3)
 

Sympy [F]

\[ \int \frac {\left (1+4 x-7 x^2\right )^3 \left (2+5 x+x^2\right )}{\left (3+2 x+5 x^2\right )^{3/2}} \, dx=- \int \left (- \frac {29 x}{5 x^{2} \sqrt {5 x^{2} + 2 x + 3} + 2 x \sqrt {5 x^{2} + 2 x + 3} + 3 \sqrt {5 x^{2} + 2 x + 3}}\right )\, dx - \int \left (- \frac {115 x^{2}}{5 x^{2} \sqrt {5 x^{2} + 2 x + 3} + 2 x \sqrt {5 x^{2} + 2 x + 3} + 3 \sqrt {5 x^{2} + 2 x + 3}}\right )\, dx - \int \frac {61 x^{3}}{5 x^{2} \sqrt {5 x^{2} + 2 x + 3} + 2 x \sqrt {5 x^{2} + 2 x + 3} + 3 \sqrt {5 x^{2} + 2 x + 3}}\, dx - \int \frac {871 x^{4}}{5 x^{2} \sqrt {5 x^{2} + 2 x + 3} + 2 x \sqrt {5 x^{2} + 2 x + 3} + 3 \sqrt {5 x^{2} + 2 x + 3}}\, dx - \int \left (- \frac {127 x^{5}}{5 x^{2} \sqrt {5 x^{2} + 2 x + 3} + 2 x \sqrt {5 x^{2} + 2 x + 3} + 3 \sqrt {5 x^{2} + 2 x + 3}}\right )\, dx - \int \left (- \frac {2065 x^{6}}{5 x^{2} \sqrt {5 x^{2} + 2 x + 3} + 2 x \sqrt {5 x^{2} + 2 x + 3} + 3 \sqrt {5 x^{2} + 2 x + 3}}\right )\, dx - \int \frac {1127 x^{7}}{5 x^{2} \sqrt {5 x^{2} + 2 x + 3} + 2 x \sqrt {5 x^{2} + 2 x + 3} + 3 \sqrt {5 x^{2} + 2 x + 3}}\, dx - \int \frac {343 x^{8}}{5 x^{2} \sqrt {5 x^{2} + 2 x + 3} + 2 x \sqrt {5 x^{2} + 2 x + 3} + 3 \sqrt {5 x^{2} + 2 x + 3}}\, dx - \int \left (- \frac {2}{5 x^{2} \sqrt {5 x^{2} + 2 x + 3} + 2 x \sqrt {5 x^{2} + 2 x + 3} + 3 \sqrt {5 x^{2} + 2 x + 3}}\right )\, dx \] Input:

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

Output:

-Integral(-29*x/(5*x**2*sqrt(5*x**2 + 2*x + 3) + 2*x*sqrt(5*x**2 + 2*x + 3 
) + 3*sqrt(5*x**2 + 2*x + 3)), x) - Integral(-115*x**2/(5*x**2*sqrt(5*x**2 
 + 2*x + 3) + 2*x*sqrt(5*x**2 + 2*x + 3) + 3*sqrt(5*x**2 + 2*x + 3)), x) - 
 Integral(61*x**3/(5*x**2*sqrt(5*x**2 + 2*x + 3) + 2*x*sqrt(5*x**2 + 2*x + 
 3) + 3*sqrt(5*x**2 + 2*x + 3)), x) - Integral(871*x**4/(5*x**2*sqrt(5*x** 
2 + 2*x + 3) + 2*x*sqrt(5*x**2 + 2*x + 3) + 3*sqrt(5*x**2 + 2*x + 3)), x) 
- Integral(-127*x**5/(5*x**2*sqrt(5*x**2 + 2*x + 3) + 2*x*sqrt(5*x**2 + 2* 
x + 3) + 3*sqrt(5*x**2 + 2*x + 3)), x) - Integral(-2065*x**6/(5*x**2*sqrt( 
5*x**2 + 2*x + 3) + 2*x*sqrt(5*x**2 + 2*x + 3) + 3*sqrt(5*x**2 + 2*x + 3)) 
, x) - Integral(1127*x**7/(5*x**2*sqrt(5*x**2 + 2*x + 3) + 2*x*sqrt(5*x**2 
 + 2*x + 3) + 3*sqrt(5*x**2 + 2*x + 3)), x) - Integral(343*x**8/(5*x**2*sq 
rt(5*x**2 + 2*x + 3) + 2*x*sqrt(5*x**2 + 2*x + 3) + 3*sqrt(5*x**2 + 2*x + 
3)), x) - Integral(-2/(5*x**2*sqrt(5*x**2 + 2*x + 3) + 2*x*sqrt(5*x**2 + 2 
*x + 3) + 3*sqrt(5*x**2 + 2*x + 3)), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.89 \[ \int \frac {\left (1+4 x-7 x^2\right )^3 \left (2+5 x+x^2\right )}{\left (3+2 x+5 x^2\right )^{3/2}} \, dx=-\frac {343 \, x^{7}}{30 \, \sqrt {5 \, x^{2} + 2 \, x + 3}} - \frac {29351 \, x^{6}}{750 \, \sqrt {5 \, x^{2} + 2 \, x + 3}} + \frac {1025843 \, x^{5}}{7500 \, \sqrt {5 \, x^{2} + 2 \, x + 3}} - \frac {998969 \, x^{4}}{37500 \, \sqrt {5 \, x^{2} + 2 \, x + 3}} - \frac {51303971 \, x^{3}}{187500 \, \sqrt {5 \, x^{2} + 2 \, x + 3}} + \frac {61004099 \, x^{2}}{187500 \, \sqrt {5 \, x^{2} + 2 \, x + 3}} + \frac {50047657}{781250} \, \sqrt {5} \operatorname {arsinh}\left (\frac {1}{14} \, \sqrt {14} {\left (5 \, x + 1\right )}\right ) - \frac {87141949 \, x}{546875 \, \sqrt {5 \, x^{2} + 2 \, x + 3}} + \frac {525961603}{1093750 \, \sqrt {5 \, x^{2} + 2 \, x + 3}} \] Input:

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

Output:

-343/30*x^7/sqrt(5*x^2 + 2*x + 3) - 29351/750*x^6/sqrt(5*x^2 + 2*x + 3) + 
1025843/7500*x^5/sqrt(5*x^2 + 2*x + 3) - 998969/37500*x^4/sqrt(5*x^2 + 2*x 
 + 3) - 51303971/187500*x^3/sqrt(5*x^2 + 2*x + 3) + 61004099/187500*x^2/sq 
rt(5*x^2 + 2*x + 3) + 50047657/781250*sqrt(5)*arcsinh(1/14*sqrt(14)*(5*x + 
 1)) - 87141949/546875*x/sqrt(5*x^2 + 2*x + 3) + 525961603/1093750/sqrt(5* 
x^2 + 2*x + 3)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.49 \[ \int \frac {\left (1+4 x-7 x^2\right )^3 \left (2+5 x+x^2\right )}{\left (3+2 x+5 x^2\right )^{3/2}} \, dx=-\frac {50047657}{781250} \, \sqrt {5} \log \left (-\sqrt {5} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 2 \, x + 3}\right )} - 1\right ) - \frac {{\left (35 \, {\left ({\left (5 \, {\left (35 \, {\left (70 \, {\left (175 \, x + 599\right )} x - 146549\right )} x + 998969\right )} x + 51303971\right )} x - 61004099\right )} x + 1045703388\right )} x - 3155769618}{6562500 \, \sqrt {5 \, x^{2} + 2 \, x + 3}} \] Input:

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

Output:

-50047657/781250*sqrt(5)*log(-sqrt(5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3)) 
- 1) - 1/6562500*((35*((5*(35*(70*(175*x + 599)*x - 146549)*x + 998969)*x 
+ 51303971)*x - 61004099)*x + 1045703388)*x - 3155769618)/sqrt(5*x^2 + 2*x 
 + 3)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.47 \[ \int \frac {\left (1+4 x-7 x^2\right )^3 \left (2+5 x+x^2\right )}{\left (3+2 x+5 x^2\right )^{3/2}} \, dx=\frac {-1500625000 \sqrt {5 x^{2}+2 x +3}\, x^{7}-5136425000 \sqrt {5 x^{2}+2 x +3}\, x^{6}[phone] \sqrt {5 x^{2}+2 x +3}\, x^{5}-3496391500 \sqrt {5 x^{2}+2 x +3}\, x^{4}-35912779700 \sqrt {5 x^{2}+2 x +3}\, x^{3}[phone] \sqrt {5 x^{2}+2 x +3}\, x^{2}-20914067760 \sqrt {5 x^{2}+2 x +3}\, x [phone] \sqrt {5 x^{2}+2 x +3}[phone] \sqrt {5}\, \mathrm {log}\left (\frac {\sqrt {5 x^{2}+2 x +3}\, \sqrt {5}+5 x +1}{\sqrt {14}}\right ) x^{2}[phone] \sqrt {5}\, \mathrm {log}\left (\frac {\sqrt {5 x^{2}+2 x +3}\, \sqrt {5}+5 x +1}{\sqrt {14}}\right ) x [phone] \sqrt {5}\, \mathrm {log}\left (\frac {\sqrt {5 x^{2}+2 x +3}\, \sqrt {5}+5 x +1}{\sqrt {14}}\right )-26833542615 \sqrt {5}\, x^{2}-10733417046 \sqrt {5}\, x -16100125569 \sqrt {5}}{656250000 x^{2}[phone] x [phone]} \] Input:

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

Output:

( - 1500625000*sqrt(5*x**2 + 2*x + 3)*x**7 - 5136425000*sqrt(5*x**2 + 2*x 
+ 3)*x**6 + 17952252500*sqrt(5*x**2 + 2*x + 3)*x**5 - 3496391500*sqrt(5*x* 
*2 + 2*x + 3)*x**4 - 35912779700*sqrt(5*x**2 + 2*x + 3)*x**3 + 42702869300 
*sqrt(5*x**2 + 2*x + 3)*x**2 - 20914067760*sqrt(5*x**2 + 2*x + 3)*x + 6311 
5392360*sqrt(5*x**2 + 2*x + 3) + 42040031880*sqrt(5)*log((sqrt(5*x**2 + 2* 
x + 3)*sqrt(5) + 5*x + 1)/sqrt(14))*x**2 + 16816012752*sqrt(5)*log((sqrt(5 
*x**2 + 2*x + 3)*sqrt(5) + 5*x + 1)/sqrt(14))*x + 25224019128*sqrt(5)*log( 
(sqrt(5*x**2 + 2*x + 3)*sqrt(5) + 5*x + 1)/sqrt(14)) - 26833542615*sqrt(5) 
*x**2 - 10733417046*sqrt(5)*x - 16100125569*sqrt(5))/(131250000*(5*x**2 + 
2*x + 3))