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

Optimal. Leaf size=189 \[ \frac{49}{50} \left (5 x^2+2 x+3\right )^{5/2} x^5+\frac{581}{150} \left (5 x^2+2 x+3\right )^{5/2} x^4-\frac{18379 \left (5 x^2+2 x+3\right )^{5/2} x^3}{3000}-\frac{219271 \left (5 x^2+2 x+3\right )^{5/2} x^2}{105000}+\frac{86721 \left (5 x^2+2 x+3\right )^{5/2} x}{21875}+\frac{505667 \left (5 x^2+2 x+3\right )^{5/2}}{2187500}-\frac{690561 (5 x+1) \left (5 x^2+2 x+3\right )^{3/2}}{1250000}-\frac{14501781 (5 x+1) \sqrt{5 x^2+2 x+3}}{6250000}-\frac{101512467 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{3125000 \sqrt{5}} \]

[Out]

(-14501781*(1 + 5*x)*Sqrt[3 + 2*x + 5*x^2])/6250000 - (690561*(1 + 5*x)*(3 + 2*x + 5*x^2)^(3/2))/1250000 + (50
5667*(3 + 2*x + 5*x^2)^(5/2))/2187500 + (86721*x*(3 + 2*x + 5*x^2)^(5/2))/21875 - (219271*x^2*(3 + 2*x + 5*x^2
)^(5/2))/105000 - (18379*x^3*(3 + 2*x + 5*x^2)^(5/2))/3000 + (581*x^4*(3 + 2*x + 5*x^2)^(5/2))/150 + (49*x^5*(
3 + 2*x + 5*x^2)^(5/2))/50 - (101512467*ArcSinh[(1 + 5*x)/Sqrt[14]])/(3125000*Sqrt[5])

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Rubi [A]  time = 0.226532, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {1661, 640, 612, 619, 215} \[ \frac{49}{50} \left (5 x^2+2 x+3\right )^{5/2} x^5+\frac{581}{150} \left (5 x^2+2 x+3\right )^{5/2} x^4-\frac{18379 \left (5 x^2+2 x+3\right )^{5/2} x^3}{3000}-\frac{219271 \left (5 x^2+2 x+3\right )^{5/2} x^2}{105000}+\frac{86721 \left (5 x^2+2 x+3\right )^{5/2} x}{21875}+\frac{505667 \left (5 x^2+2 x+3\right )^{5/2}}{2187500}-\frac{690561 (5 x+1) \left (5 x^2+2 x+3\right )^{3/2}}{1250000}-\frac{14501781 (5 x+1) \sqrt{5 x^2+2 x+3}}{6250000}-\frac{101512467 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{3125000 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-14501781*(1 + 5*x)*Sqrt[3 + 2*x + 5*x^2])/6250000 - (690561*(1 + 5*x)*(3 + 2*x + 5*x^2)^(3/2))/1250000 + (50
5667*(3 + 2*x + 5*x^2)^(5/2))/2187500 + (86721*x*(3 + 2*x + 5*x^2)^(5/2))/21875 - (219271*x^2*(3 + 2*x + 5*x^2
)^(5/2))/105000 - (18379*x^3*(3 + 2*x + 5*x^2)^(5/2))/3000 + (581*x^4*(3 + 2*x + 5*x^2)^(5/2))/150 + (49*x^5*(
3 + 2*x + 5*x^2)^(5/2))/50 - (101512467*ArcSinh[(1 + 5*x)/Sqrt[14]])/(3125000*Sqrt[5])

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 612

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 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 \left (1+4 x-7 x^2\right )^2 \left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2} \, dx &=\frac{49}{50} x^5 \left (3+2 x+5 x^2\right )^{5/2}+\frac{1}{50} \int \left (3+2 x+5 x^2\right )^{3/2} \left (100+1050 x+2250 x^2-4700 x^3-9735 x^4+8715 x^5\right ) \, dx\\ &=\frac{581}{150} x^4 \left (3+2 x+5 x^2\right )^{5/2}+\frac{49}{50} x^5 \left (3+2 x+5 x^2\right )^{5/2}+\frac{\int \left (3+2 x+5 x^2\right )^{3/2} \left (4500+47250 x+101250 x^2-316080 x^3-551370 x^4\right ) \, dx}{2250}\\ &=-\frac{18379 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{3000}+\frac{581}{150} x^4 \left (3+2 x+5 x^2\right )^{5/2}+\frac{49}{50} x^5 \left (3+2 x+5 x^2\right )^{5/2}+\frac{\int \left (3+2 x+5 x^2\right )^{3/2} \left (180000+1890000 x+9012330 x^2-6578130 x^3\right ) \, dx}{90000}\\ &=-\frac{219271 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{105000}-\frac{18379 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{3000}+\frac{581}{150} x^4 \left (3+2 x+5 x^2\right )^{5/2}+\frac{49}{50} x^5 \left (3+2 x+5 x^2\right )^{5/2}+\frac{\int \left (3+2 x+5 x^2\right )^{3/2} \left (6300000+105618780 x+374634720 x^2\right ) \, dx}{3150000}\\ &=\frac{86721 x \left (3+2 x+5 x^2\right )^{5/2}}{21875}-\frac{219271 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{105000}-\frac{18379 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{3000}+\frac{581}{150} x^4 \left (3+2 x+5 x^2\right )^{5/2}+\frac{49}{50} x^5 \left (3+2 x+5 x^2\right )^{5/2}+\frac{\int (-934904160+546120360 x) \left (3+2 x+5 x^2\right )^{3/2} \, dx}{94500000}\\ &=\frac{505667 \left (3+2 x+5 x^2\right )^{5/2}}{2187500}+\frac{86721 x \left (3+2 x+5 x^2\right )^{5/2}}{21875}-\frac{219271 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{105000}-\frac{18379 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{3000}+\frac{581}{150} x^4 \left (3+2 x+5 x^2\right )^{5/2}+\frac{49}{50} x^5 \left (3+2 x+5 x^2\right )^{5/2}-\frac{690561 \int \left (3+2 x+5 x^2\right )^{3/2} \, dx}{62500}\\ &=-\frac{690561 (1+5 x) \left (3+2 x+5 x^2\right )^{3/2}}{1250000}+\frac{505667 \left (3+2 x+5 x^2\right )^{5/2}}{2187500}+\frac{86721 x \left (3+2 x+5 x^2\right )^{5/2}}{21875}-\frac{219271 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{105000}-\frac{18379 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{3000}+\frac{581}{150} x^4 \left (3+2 x+5 x^2\right )^{5/2}+\frac{49}{50} x^5 \left (3+2 x+5 x^2\right )^{5/2}-\frac{14501781 \int \sqrt{3+2 x+5 x^2} \, dx}{625000}\\ &=-\frac{14501781 (1+5 x) \sqrt{3+2 x+5 x^2}}{6250000}-\frac{690561 (1+5 x) \left (3+2 x+5 x^2\right )^{3/2}}{1250000}+\frac{505667 \left (3+2 x+5 x^2\right )^{5/2}}{2187500}+\frac{86721 x \left (3+2 x+5 x^2\right )^{5/2}}{21875}-\frac{219271 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{105000}-\frac{18379 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{3000}+\frac{581}{150} x^4 \left (3+2 x+5 x^2\right )^{5/2}+\frac{49}{50} x^5 \left (3+2 x+5 x^2\right )^{5/2}-\frac{101512467 \int \frac{1}{\sqrt{3+2 x+5 x^2}} \, dx}{3125000}\\ &=-\frac{14501781 (1+5 x) \sqrt{3+2 x+5 x^2}}{6250000}-\frac{690561 (1+5 x) \left (3+2 x+5 x^2\right )^{3/2}}{1250000}+\frac{505667 \left (3+2 x+5 x^2\right )^{5/2}}{2187500}+\frac{86721 x \left (3+2 x+5 x^2\right )^{5/2}}{21875}-\frac{219271 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{105000}-\frac{18379 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{3000}+\frac{581}{150} x^4 \left (3+2 x+5 x^2\right )^{5/2}+\frac{49}{50} x^5 \left (3+2 x+5 x^2\right )^{5/2}-\frac{\left (14501781 \sqrt{\frac{7}{10}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{56}}} \, dx,x,2+10 x\right )}{6250000}\\ &=-\frac{14501781 (1+5 x) \sqrt{3+2 x+5 x^2}}{6250000}-\frac{690561 (1+5 x) \left (3+2 x+5 x^2\right )^{3/2}}{1250000}+\frac{505667 \left (3+2 x+5 x^2\right )^{5/2}}{2187500}+\frac{86721 x \left (3+2 x+5 x^2\right )^{5/2}}{21875}-\frac{219271 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{105000}-\frac{18379 x^3 \left (3+2 x+5 x^2\right )^{5/2}}{3000}+\frac{581}{150} x^4 \left (3+2 x+5 x^2\right )^{5/2}+\frac{49}{50} x^5 \left (3+2 x+5 x^2\right )^{5/2}-\frac{101512467 \sinh ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{3125000 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.25241, size = 85, normalized size = 0.45 \[ \frac{5 \sqrt{5 x^2+2 x+3} \left (3215625000 x^9+15281875000 x^8-5561281250 x^7-4105593750 x^6-12554262500 x^5-3227597000 x^4+5959365525 x^3+3721040355 x^2+2291675850 x-249003936\right )-4263523614 \sqrt{5} \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{656250000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*Sqrt[3 + 2*x + 5*x^2]*(-249003936 + 2291675850*x + 3721040355*x^2 + 5959365525*x^3 - 3227597000*x^4 - 12554
262500*x^5 - 4105593750*x^6 - 5561281250*x^7 + 15281875000*x^8 + 3215625000*x^9) - 4263523614*Sqrt[5]*ArcSinh[
(1 + 5*x)/Sqrt[14]])/656250000

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Maple [A]  time = 0.061, size = 151, normalized size = 0.8 \begin{align*}{\frac{49\,{x}^{5}}{50} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{581\,{x}^{4}}{150} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{5}{2}}}}-{\frac{18379\,{x}^{3}}{3000} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{5}{2}}}}-{\frac{219271\,{x}^{2}}{105000} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{5}{2}}}}-{\frac{145017810\,x+29003562}{12500000}\sqrt{5\,{x}^{2}+2\,x+3}}-{\frac{101512467\,\sqrt{5}}{15625000}{\it Arcsinh} \left ({\frac{5\,\sqrt{14}}{14} \left ( x+{\frac{1}{5}} \right ) } \right ) }+{\frac{86721\,x}{21875} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{5}{2}}}}-{\frac{6905610\,x+1381122}{2500000} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{505667}{2187500} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

49/50*x^5*(5*x^2+2*x+3)^(5/2)+581/150*x^4*(5*x^2+2*x+3)^(5/2)-18379/3000*x^3*(5*x^2+2*x+3)^(5/2)-219271/105000
*x^2*(5*x^2+2*x+3)^(5/2)-14501781/12500000*(10*x+2)*(5*x^2+2*x+3)^(1/2)-101512467/15625000*5^(1/2)*arcsinh(5/1
4*14^(1/2)*(x+1/5))+86721/21875*x*(5*x^2+2*x+3)^(5/2)-690561/2500000*(10*x+2)*(5*x^2+2*x+3)^(3/2)+505667/21875
00*(5*x^2+2*x+3)^(5/2)

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Maxima [A]  time = 1.58564, size = 232, normalized size = 1.23 \begin{align*} \frac{49}{50} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{5}{2}} x^{5} + \frac{581}{150} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{5}{2}} x^{4} - \frac{18379}{3000} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{5}{2}} x^{3} - \frac{219271}{105000} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{5}{2}} x^{2} + \frac{86721}{21875} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{5}{2}} x + \frac{505667}{2187500} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{5}{2}} - \frac{690561}{250000} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{3}{2}} x - \frac{690561}{1250000} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{3}{2}} - \frac{14501781}{1250000} \, \sqrt{5 \, x^{2} + 2 \, x + 3} x - \frac{101512467}{15625000} \, \sqrt{5} \operatorname{arsinh}\left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) - \frac{14501781}{6250000} \, \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)^2*(x^2+5*x+2)*(5*x^2+2*x+3)^(3/2),x, algorithm="maxima")

[Out]

49/50*(5*x^2 + 2*x + 3)^(5/2)*x^5 + 581/150*(5*x^2 + 2*x + 3)^(5/2)*x^4 - 18379/3000*(5*x^2 + 2*x + 3)^(5/2)*x
^3 - 219271/105000*(5*x^2 + 2*x + 3)^(5/2)*x^2 + 86721/21875*(5*x^2 + 2*x + 3)^(5/2)*x + 505667/2187500*(5*x^2
+ 2*x + 3)^(5/2) - 690561/250000*(5*x^2 + 2*x + 3)^(3/2)*x - 690561/1250000*(5*x^2 + 2*x + 3)^(3/2) - 1450178
1/1250000*sqrt(5*x^2 + 2*x + 3)*x - 101512467/15625000*sqrt(5)*arcsinh(1/14*sqrt(14)*(5*x + 1)) - 14501781/625
0000*sqrt(5*x^2 + 2*x + 3)

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Fricas [A]  time = 1.28599, size = 397, normalized size = 2.1 \begin{align*} \frac{1}{131250000} \,{\left (3215625000 \, x^{9} + 15281875000 \, x^{8} - 5561281250 \, x^{7} - 4105593750 \, x^{6} - 12554262500 \, x^{5} - 3227597000 \, x^{4} + 5959365525 \, x^{3} + 3721040355 \, x^{2} + 2291675850 \, x - 249003936\right )} \sqrt{5 \, x^{2} + 2 \, x + 3} + \frac{101512467}{31250000} \, \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/131250000*(3215625000*x^9 + 15281875000*x^8 - 5561281250*x^7 - 4105593750*x^6 - 12554262500*x^5 - 3227597000
*x^4 + 5959365525*x^3 + 3721040355*x^2 + 2291675850*x - 249003936)*sqrt(5*x^2 + 2*x + 3) + 101512467/31250000*
sqrt(5)*log(sqrt(5)*sqrt(5*x^2 + 2*x + 3)*(5*x + 1) - 25*x^2 - 10*x - 8)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (x^{2} + 5 x + 2\right ) \left (5 x^{2} + 2 x + 3\right )^{\frac{3}{2}} \left (7 x^{2} - 4 x - 1\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.21031, size = 124, normalized size = 0.66 \begin{align*} \frac{1}{131250000} \,{\left (5 \,{\left ({\left (5 \,{\left (10 \,{\left (25 \,{\left (5 \,{\left (7 \,{\left (140 \,{\left (105 \, x + 499\right )} x - 25423\right )} x - 131379\right )} x - 2008682\right )} x - 12910388\right )} x + 238374621\right )} x + 744208071\right )} x + 458335170\right )} x - 249003936\right )} \sqrt{5 \, x^{2} + 2 \, x + 3} + \frac{101512467}{15625000} \, \sqrt{5} \log \left (-\sqrt{5}{\left (\sqrt{5} x - \sqrt{5 \, x^{2} + 2 \, x + 3}\right )} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/131250000*(5*((5*(10*(25*(5*(7*(140*(105*x + 499)*x - 25423)*x - 131379)*x - 2008682)*x - 12910388)*x + 2383
74621)*x + 744208071)*x + 458335170)*x - 249003936)*sqrt(5*x^2 + 2*x + 3) + 101512467/15625000*sqrt(5)*log(-sq
rt(5)*(sqrt(5)*x - sqrt(5*x^2 + 2*x + 3)) - 1)