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

Optimal. Leaf size=166 \[ -\frac{343}{150} \sqrt{5 x^2+2 x+3} x^5-\frac{25921 \sqrt{5 x^2+2 x+3} x^4}{3750}+\frac{393659 \sqrt{5 x^2+2 x+3} x^3}{12500}-\frac{2583293 \sqrt{5 x^2+2 x+3} x^2}{187500}-\frac{3192602 \sqrt{5 x^2+2 x+3} x}{46875}+\frac{15715799 \sqrt{5 x^2+2 x+3}}{156250}+\frac{16 (6122807-5338217 x)}{546875 \sqrt{5 x^2+2 x+3}}+\frac{50047657 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{156250 \sqrt{5}} \]

[Out]

(16*(6122807 - 5338217*x))/(546875*Sqrt[3 + 2*x + 5*x^2]) + (15715799*Sqrt[3 + 2*x + 5*x^2])/156250 - (3192602
*x*Sqrt[3 + 2*x + 5*x^2])/46875 - (2583293*x^2*Sqrt[3 + 2*x + 5*x^2])/187500 + (393659*x^3*Sqrt[3 + 2*x + 5*x^
2])/12500 - (25921*x^4*Sqrt[3 + 2*x + 5*x^2])/3750 - (343*x^5*Sqrt[3 + 2*x + 5*x^2])/150 + (50047657*ArcSinh[(
1 + 5*x)/Sqrt[14]])/(156250*Sqrt[5])

________________________________________________________________________________________

Rubi [A]  time = 0.241002, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {1660, 1661, 640, 619, 215} \[ -\frac{343}{150} \sqrt{5 x^2+2 x+3} x^5-\frac{25921 \sqrt{5 x^2+2 x+3} x^4}{3750}+\frac{393659 \sqrt{5 x^2+2 x+3} x^3}{12500}-\frac{2583293 \sqrt{5 x^2+2 x+3} x^2}{187500}-\frac{3192602 \sqrt{5 x^2+2 x+3} x}{46875}+\frac{15715799 \sqrt{5 x^2+2 x+3}}{156250}+\frac{16 (6122807-5338217 x)}{546875 \sqrt{5 x^2+2 x+3}}+\frac{50047657 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{156250 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*(6122807 - 5338217*x))/(546875*Sqrt[3 + 2*x + 5*x^2]) + (15715799*Sqrt[3 + 2*x + 5*x^2])/156250 - (3192602
*x*Sqrt[3 + 2*x + 5*x^2])/46875 - (2583293*x^2*Sqrt[3 + 2*x + 5*x^2])/187500 + (393659*x^3*Sqrt[3 + 2*x + 5*x^
2])/12500 - (25921*x^4*Sqrt[3 + 2*x + 5*x^2])/3750 - (343*x^5*Sqrt[3 + 2*x + 5*x^2])/150 + (50047657*ArcSinh[(
1 + 5*x)/Sqrt[14]])/(156250*Sqrt[5])

Rule 1660

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[Pq, 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] + Dist[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 1661

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expo
n[Pq, x]]}, Simp[(e*x^(q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*(q + 2*p + 1)), x] + Dist[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]

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]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[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 215

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]

Rubi steps

\begin{align*} \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{1}{28} \int \frac{\frac{473724104}{78125}+\frac{94462228 x}{15625}-\frac{40822404 x^2}{3125}-\frac{1210328 x^3}{625}+\frac{1866704 x^4}{125}-\frac{138572 x^5}{25}-\frac{9604 x^6}{5}}{\sqrt{3+2 x+5 x^2}} \, dx\\ &=\frac{16 (6122807-5338217 x)}{546875 \sqrt{3+2 x+5 x^2}}-\frac{343}{150} x^5 \sqrt{3+2 x+5 x^2}+\frac{1}{840} \int \frac{\frac{2842344624}{15625}+\frac{566773368 x}{3125}-\frac{244934424 x^2}{625}-\frac{7261968 x^3}{125}+\frac{11920524 x^4}{25}-\frac{725788 x^5}{5}}{\sqrt{3+2 x+5 x^2}} \, dx\\ &=\frac{16 (6122807-5338217 x)}{546875 \sqrt{3+2 x+5 x^2}}-\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{\int \frac{\frac{2842344624}{625}+\frac{566773368 x}{125}-\frac{244934424 x^2}{25}+\frac{1447488 x^3}{5}+\frac{66134712 x^4}{5}}{\sqrt{3+2 x+5 x^2}} \, dx}{21000}\\ &=\frac{16 (6122807-5338217 x)}{546875 \sqrt{3+2 x+5 x^2}}+\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{\int \frac{\frac{11369378496}{125}+\frac{2267093472 x}{25}-\frac{1574950104 x^2}{5}-\frac{433993224 x^3}{5}}{\sqrt{3+2 x+5 x^2}} \, dx}{420000}\\ &=\frac{16 (6122807-5338217 x)}{546875 \sqrt{3+2 x+5 x^2}}-\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{\int \frac{\frac{34108135488}{25}[phone] x-4290857088 x^2}{\sqrt{3+2 x+5 x^2}} \, dx}{6300000}\\ &=\frac{16 (6122807-5338217 x)}{546875 \sqrt{3+2 x+5 x^2}}-\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{\int \frac{\frac{132579127296}{5}[phone] x}{\sqrt{3+2 x+5 x^2}} \, dx}{63000000}\\ &=\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 \int \frac{1}{\sqrt{3+2 x+5 x^2}} \, dx}{156250}\\ &=\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 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{56}}} \, dx,x,2+10 x\right )}{312500 \sqrt{70}}\\ &=\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 \sinh ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{156250 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.460733, size = 75, normalized size = 0.45 \[ \frac{2102001594 \sqrt{5} \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )-\frac{5 \left (75031250 x^7+256821250 x^6-897612625 x^5+174819575 x^4+1795638985 x^3-2135143465 x^2+1045703388 x-3155769618\right )}{\sqrt{5 x^2+2 x+3}}}{32812500} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-5*(-3155769618 + 1045703388*x - 2135143465*x^2 + 1795638985*x^3 + 174819575*x^4 - 897612625*x^5 + 256821250
*x^6 + 75031250*x^7))/Sqrt[3 + 2*x + 5*x^2] + 2102001594*Sqrt[5]*ArcSinh[(1 + 5*x)/Sqrt[14]])/32812500

________________________________________________________________________________________

Maple [A]  time = 0.069, size = 166, normalized size = 1. \begin{align*}{\frac{175268451}{390625}{\frac{1}{\sqrt{5\,{x}^{2}+2\,x+3}}}}+{\frac{1025843\,{x}^{5}}{7500}{\frac{1}{\sqrt{5\,{x}^{2}+2\,x+3}}}}-{\frac{998969\,{x}^{4}}{37500}{\frac{1}{\sqrt{5\,{x}^{2}+2\,x+3}}}}+{\frac{61004099\,{x}^{2}}{187500}{\frac{1}{\sqrt{5\,{x}^{2}+2\,x+3}}}}+{\frac{50047657\,\sqrt{5}}{781250}{\it Arcsinh} \left ({\frac{5\,\sqrt{14}}{14} \left ( x+{\frac{1}{5}} \right ) } \right ) }+{\frac{1760497010\,x+352099402}{10937500}{\frac{1}{\sqrt{5\,{x}^{2}+2\,x+3}}}}-{\frac{50047657\,x}{156250}{\frac{1}{\sqrt{5\,{x}^{2}+2\,x+3}}}}-{\frac{51303971\,{x}^{3}}{187500}{\frac{1}{\sqrt{5\,{x}^{2}+2\,x+3}}}}-{\frac{343\,{x}^{7}}{30}{\frac{1}{\sqrt{5\,{x}^{2}+2\,x+3}}}}-{\frac{29351\,{x}^{6}}{750}{\frac{1}{\sqrt{5\,{x}^{2}+2\,x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

175268451/390625/(5*x^2+2*x+3)^(1/2)+1025843/7500*x^5/(5*x^2+2*x+3)^(1/2)-998969/37500*x^4/(5*x^2+2*x+3)^(1/2)
[phone]/187500*x^2/(5*x^2+2*x+3)^(1/2)[phone]/781250*5^(1/2)*arcsinh(5/14*14^(1/2)*(x+1/5))[phone]/1093
7500*(10*x+2)/(5*x^2+2*x+3)^(1/2)-50047657/156250*x/(5*x^2+2*x+3)^(1/2)-51303971/187500*x^3/(5*x^2+2*x+3)^(1/2
)-343/30*x^7/(5*x^2+2*x+3)^(1/2)-29351/750*x^6/(5*x^2+2*x+3)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.48461, size = 200, normalized size = 1.2 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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/
sqrt(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)

________________________________________________________________________________________

Fricas [A]  time = 1.44196, size = 386, normalized size = 2.33 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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 + 104
5703388*x - 3155769618)*sqrt(5*x^2 + 2*x + 3))/(5*x^2 + 2*x + 3)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \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}}\, dx - \int - \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}}\, 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 - \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}}\, dx - \int - \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}}\, 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 - \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}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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*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(-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)

________________________________________________________________________________________

Giac [A]  time = 1.18346, size = 109, normalized size = 0.66 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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