3.1.76 \(\int \frac {(2+3 x+5 x^2)^4}{\sqrt {3-x+2 x^2}} \, dx\)

Optimal. Leaf size=185 \[ -\frac {15428243 \sqrt {2 x^2-x+3} x^2}{131072}+\frac {1572007407 \sqrt {2 x^2-x+3} x}{7340032}+\frac {16493087661 \sqrt {2 x^2-x+3}}{29360128}+\frac {625}{16} \sqrt {2 x^2-x+3} x^7+\frac {57375}{448} \sqrt {2 x^2-x+3} x^6+\frac {2116475 \sqrt {2 x^2-x+3} x^5}{10752}+\frac {686531 \sqrt {2 x^2-x+3} x^4}{6144}-\frac {19750457 \sqrt {2 x^2-x+3} x^3}{229376}+\frac {2899366573 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{8388608 \sqrt {2}} \]

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Rubi [A]  time = 0.31, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1661, 640, 619, 215} \begin {gather*} \frac {625}{16} \sqrt {2 x^2-x+3} x^7+\frac {57375}{448} \sqrt {2 x^2-x+3} x^6+\frac {2116475 \sqrt {2 x^2-x+3} x^5}{10752}+\frac {686531 \sqrt {2 x^2-x+3} x^4}{6144}-\frac {19750457 \sqrt {2 x^2-x+3} x^3}{229376}-\frac {15428243 \sqrt {2 x^2-x+3} x^2}{131072}+\frac {1572007407 \sqrt {2 x^2-x+3} x}{7340032}+\frac {16493087661 \sqrt {2 x^2-x+3}}{29360128}+\frac {2899366573 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{8388608 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(16493087661*Sqrt[3 - x + 2*x^2])/29360128 + (1572007407*x*Sqrt[3 - x + 2*x^2])/7340032 - (15428243*x^2*Sqrt[3
- x + 2*x^2])/131072 - (19750457*x^3*Sqrt[3 - x + 2*x^2])/229376 + (686531*x^4*Sqrt[3 - x + 2*x^2])/6144 + (2
116475*x^5*Sqrt[3 - x + 2*x^2])/10752 + (57375*x^6*Sqrt[3 - x + 2*x^2])/448 + (625*x^7*Sqrt[3 - x + 2*x^2])/16
+ (2899366573*ArcSinh[(1 - 4*x)/Sqrt[23]])/(8388608*Sqrt[2])

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]

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 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 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]

Rubi steps

\begin {align*} \int \frac {\left (2+3 x+5 x^2\right )^4}{\sqrt {3-x+2 x^2}} \, dx &=\frac {625}{16} x^7 \sqrt {3-x+2 x^2}+\frac {1}{16} \int \frac {256+1536 x+6016 x^2+14976 x^3+28176 x^4+37440 x^5+24475 x^6+\frac {57375 x^7}{2}}{\sqrt {3-x+2 x^2}} \, dx\\ &=\frac {57375}{448} x^6 \sqrt {3-x+2 x^2}+\frac {625}{16} x^7 \sqrt {3-x+2 x^2}+\frac {1}{224} \int \frac {3584+21504 x+84224 x^2+209664 x^3+394464 x^4+7785 x^5+\frac {2116475 x^6}{4}}{\sqrt {3-x+2 x^2}} \, dx\\ &=\frac {2116475 x^5 \sqrt {3-x+2 x^2}}{10752}+\frac {57375}{448} x^6 \sqrt {3-x+2 x^2}+\frac {625}{16} x^7 \sqrt {3-x+2 x^2}+\frac {\int \frac {43008+258048 x+1010688 x^2+2515968 x^3-\frac {12812853 x^4}{4}+\frac {24028585 x^5}{8}}{\sqrt {3-x+2 x^2}} \, dx}{2688}\\ &=\frac {686531 x^4 \sqrt {3-x+2 x^2}}{6144}+\frac {2116475 x^5 \sqrt {3-x+2 x^2}}{10752}+\frac {57375}{448} x^6 \sqrt {3-x+2 x^2}+\frac {625}{16} x^7 \sqrt {3-x+2 x^2}+\frac {\int \frac {430080+2580480 x+10106880 x^2-\frac {21766395 x^3}{2}-\frac {296256855 x^4}{16}}{\sqrt {3-x+2 x^2}} \, dx}{26880}\\ &=-\frac {19750457 x^3 \sqrt {3-x+2 x^2}}{229376}+\frac {686531 x^4 \sqrt {3-x+2 x^2}}{6144}+\frac {2116475 x^5 \sqrt {3-x+2 x^2}}{10752}+\frac {57375}{448} x^6 \sqrt {3-x+2 x^2}+\frac {625}{16} x^7 \sqrt {3-x+2 x^2}+\frac {\int \frac {3440640+20643840 x+\frac {3959992335 x^2}{16}-\frac {4859896545 x^3}{32}}{\sqrt {3-x+2 x^2}} \, dx}{215040}\\ &=-\frac {15428243 x^2 \sqrt {3-x+2 x^2}}{131072}-\frac {19750457 x^3 \sqrt {3-x+2 x^2}}{229376}+\frac {686531 x^4 \sqrt {3-x+2 x^2}}{6144}+\frac {2116475 x^5 \sqrt {3-x+2 x^2}}{10752}+\frac {57375}{448} x^6 \sqrt {3-x+2 x^2}+\frac {625}{16} x^7 \sqrt {3-x+2 x^2}+\frac {\int \frac {20643840+\frac {16561498275 x}{16}+\frac {70740333315 x^2}{64}}{\sqrt {3-x+2 x^2}} \, dx}{1290240}\\ &=\frac {1572007407 x \sqrt {3-x+2 x^2}}{7340032}-\frac {15428243 x^2 \sqrt {3-x+2 x^2}}{131072}-\frac {19750457 x^3 \sqrt {3-x+2 x^2}}{229376}+\frac {686531 x^4 \sqrt {3-x+2 x^2}}{6144}+\frac {2116475 x^5 \sqrt {3-x+2 x^2}}{10752}+\frac {57375}{448} x^6 \sqrt {3-x+2 x^2}+\frac {625}{16} x^7 \sqrt {3-x+2 x^2}+\frac {\int \frac {-\frac {206936176905}{64}+\frac {742188944745 x}{128}}{\sqrt {3-x+2 x^2}} \, dx}{5160960}\\ &=\frac {16493087661 \sqrt {3-x+2 x^2}}{29360128}+\frac {1572007407 x \sqrt {3-x+2 x^2}}{7340032}-\frac {15428243 x^2 \sqrt {3-x+2 x^2}}{131072}-\frac {19750457 x^3 \sqrt {3-x+2 x^2}}{229376}+\frac {686531 x^4 \sqrt {3-x+2 x^2}}{6144}+\frac {2116475 x^5 \sqrt {3-x+2 x^2}}{10752}+\frac {57375}{448} x^6 \sqrt {3-x+2 x^2}+\frac {625}{16} x^7 \sqrt {3-x+2 x^2}-\frac {2899366573 \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx}{8388608}\\ &=\frac {16493087661 \sqrt {3-x+2 x^2}}{29360128}+\frac {1572007407 x \sqrt {3-x+2 x^2}}{7340032}-\frac {15428243 x^2 \sqrt {3-x+2 x^2}}{131072}-\frac {19750457 x^3 \sqrt {3-x+2 x^2}}{229376}+\frac {686531 x^4 \sqrt {3-x+2 x^2}}{6144}+\frac {2116475 x^5 \sqrt {3-x+2 x^2}}{10752}+\frac {57375}{448} x^6 \sqrt {3-x+2 x^2}+\frac {625}{16} x^7 \sqrt {3-x+2 x^2}-\frac {2899366573 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{8388608 \sqrt {46}}\\ &=\frac {16493087661 \sqrt {3-x+2 x^2}}{29360128}+\frac {1572007407 x \sqrt {3-x+2 x^2}}{7340032}-\frac {15428243 x^2 \sqrt {3-x+2 x^2}}{131072}-\frac {19750457 x^3 \sqrt {3-x+2 x^2}}{229376}+\frac {686531 x^4 \sqrt {3-x+2 x^2}}{6144}+\frac {2116475 x^5 \sqrt {3-x+2 x^2}}{10752}+\frac {57375}{448} x^6 \sqrt {3-x+2 x^2}+\frac {625}{16} x^7 \sqrt {3-x+2 x^2}+\frac {2899366573 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{8388608 \sqrt {2}}\\ \end {align*}

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Mathematica [A]  time = 0.24, size = 75, normalized size = 0.41 \begin {gather*} \frac {4 \sqrt {2 x^2-x+3} \left (3440640000 x^7+11280384000 x^6+17338163200 x^5+9842108416 x^4-7584175488 x^3-10367779296 x^2+18864088884 x+49479262983\right )[phone] \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{352321536} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(49479262983 + 18864088884*x - 10367779296*x^2 - 7584175488*x^3 + 9842108416*x^4 + 1733
8163200*x^5 + 11280384000*x^6 + 3440640000*x^7) + 60886698033*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/352321536

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IntegrateAlgebraic [A]  time = 0.80, size = 90, normalized size = 0.49 \begin {gather*} \frac {2899366573 \log \left (2 \sqrt {2} \sqrt {2 x^2-x+3}-4 x+1\right )}{8388608 \sqrt {2}}+\frac {\sqrt {2 x^2-x+3} \left (3440640000 x^7+11280384000 x^6+17338163200 x^5+9842108416 x^4-7584175488 x^3-10367779296 x^2+18864088884 x+49479262983\right )}{88080384} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(2 + 3*x + 5*x^2)^4/Sqrt[3 - x + 2*x^2],x]

[Out]

(Sqrt[3 - x + 2*x^2]*(49479262983 + 18864088884*x - 10367779296*x^2 - 7584175488*x^3 + 9842108416*x^4 + 173381
63200*x^5 + 11280384000*x^6 + 3440640000*x^7))/88080384 + (2899366573*Log[1 - 4*x + 2*Sqrt[2]*Sqrt[3 - x + 2*x
^2]])/(8388608*Sqrt[2])

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fricas [A]  time = 0.43, size = 88, normalized size = 0.48 \begin {gather*} \frac {1}{88080384} \, {\left (3440640000 \, x^{7} + 11280384000 \, x^{6} + 17338163200 \, x^{5} + 9842108416 \, x^{4} - 7584175488 \, x^{3} - 10367779296 \, x^{2} + 18864088884 \, x + 49479262983\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {2899366573}{33554432} \, \sqrt {2} \log \left (4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/88080384*(3440640000*x^7 + 11280384000*x^6 + 17338163200*x^5 + 9842108416*x^4 - 7584175488*x^3 - 10367779296
*x^2 + 18864088884*x + 49479262983)*sqrt(2*x^2 - x + 3) + 2899366573/33554432*sqrt(2)*log(4*sqrt(2)*sqrt(2*x^2
- x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25)

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giac [A]  time = 0.50, size = 83, normalized size = 0.45 \begin {gather*} \frac {1}{88080384} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, {\left (120 \, {\left (140 \, x + 459\right )} x + 84659\right )} x + 4805717\right )} x - 59251371\right )} x - 323993103\right )} x + 4716022221\right )} x + 49479262983\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {2899366573}{16777216} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/88080384*(4*(8*(4*(16*(100*(120*(140*x + 459)*x + 84659)*x + 4805717)*x - 59251371)*x - 323993103)*x + 47160
22221)*x + 49479262983)*sqrt(2*x^2 - x + 3) + 2899366573/16777216*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x
^2 - x + 3)) + 1)

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maple [A]  time = 0.02, size = 147, normalized size = 0.79 \begin {gather*} \frac {625 \sqrt {2 x^{2}-x +3}\, x^{7}}{16}+\frac {57375 \sqrt {2 x^{2}-x +3}\, x^{6}}{448}+\frac {2116475 \sqrt {2 x^{2}-x +3}\, x^{5}}{10752}+\frac {686531 \sqrt {2 x^{2}-x +3}\, x^{4}}{6144}-\frac {19750457 \sqrt {2 x^{2}-x +3}\, x^{3}}{229376}-\frac {15428243 \sqrt {2 x^{2}-x +3}\, x^{2}}{131072}+\frac {1572007407 \sqrt {2 x^{2}-x +3}\, x}{7340032}-\frac {2899366573 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{16777216}+\frac {16493087661 \sqrt {2 x^{2}-x +3}}{29360128} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

16493087661/29360128*(2*x^2-x+3)^(1/2)-2899366573/16777216*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))+625/16*x^7*(
2*x^2-x+3)^(1/2)+57375/448*x^6*(2*x^2-x+3)^(1/2)+2116475/10752*x^5*(2*x^2-x+3)^(1/2)+686531/6144*x^4*(2*x^2-x+
3)^(1/2)-19750457/229376*x^3*(2*x^2-x+3)^(1/2)-15428243/131072*x^2*(2*x^2-x+3)^(1/2)[phone]/7340032*x*(2*x
^2-x+3)^(1/2)

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maxima [A]  time = 1.00, size = 148, normalized size = 0.80 \begin {gather*} \frac {625}{16} \, \sqrt {2 \, x^{2} - x + 3} x^{7} + \frac {57375}{448} \, \sqrt {2 \, x^{2} - x + 3} x^{6} + \frac {2116475}{10752} \, \sqrt {2 \, x^{2} - x + 3} x^{5} + \frac {686531}{6144} \, \sqrt {2 \, x^{2} - x + 3} x^{4} - \frac {19750457}{229376} \, \sqrt {2 \, x^{2} - x + 3} x^{3} - \frac {15428243}{131072} \, \sqrt {2 \, x^{2} - x + 3} x^{2} + \frac {1572007407}{7340032} \, \sqrt {2 \, x^{2} - x + 3} x - \frac {2899366573}{16777216} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {16493087661}{29360128} \, \sqrt {2 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

625/16*sqrt(2*x^2 - x + 3)*x^7 + 57375/448*sqrt(2*x^2 - x + 3)*x^6 + 2116475/10752*sqrt(2*x^2 - x + 3)*x^5 + 6
86531/6144*sqrt(2*x^2 - x + 3)*x^4 - 19750457/229376*sqrt(2*x^2 - x + 3)*x^3 - 15428243/131072*sqrt(2*x^2 - x
+ 3)*x^2 + 1572007407/7340032*sqrt(2*x^2 - x + 3)*x - 2899366573/16777216*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x -
1)) + 16493087661/29360128*sqrt(2*x^2 - x + 3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (5\,x^2+3\,x+2\right )}^4}{\sqrt {2\,x^2-x+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (5 x^{2} + 3 x + 2\right )^{4}}{\sqrt {2 x^{2} - x + 3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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