3.74 \(\int (3-x+2 x^2)^{5/2} (2+3 x+5 x^2)^2 \, dx\)

Optimal. Leaf size=170 \[ \frac{5}{4} x^3 \left (2 x^2-x+3\right )^{7/2}+\frac{305}{144} x^2 \left (2 x^2-x+3\right )^{7/2}+\frac{8467 x \left (2 x^2-x+3\right )^{7/2}}{4608}+\frac{23225 \left (2 x^2-x+3\right )^{7/2}}{43008}-\frac{1547 (1-4 x) \left (2 x^2-x+3\right )^{5/2}}{98304}-\frac{177905 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{3145728}-\frac{4091815 (1-4 x) \sqrt{2 x^2-x+3}}{16777216}-\frac{94111745 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{33554432 \sqrt{2}} \]

[Out]

(-4091815*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/16777216 - (177905*(1 - 4*x)*(3 - x + 2*x^2)^(3/2))/3145728 - (1547*(
1 - 4*x)*(3 - x + 2*x^2)^(5/2))/98304 + (23225*(3 - x + 2*x^2)^(7/2))/43008 + (8467*x*(3 - x + 2*x^2)^(7/2))/4
608 + (305*x^2*(3 - x + 2*x^2)^(7/2))/144 + (5*x^3*(3 - x + 2*x^2)^(7/2))/4 - (94111745*ArcSinh[(1 - 4*x)/Sqrt
[23]])/(33554432*Sqrt[2])

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Rubi [A]  time = 0.13038, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1661, 640, 612, 619, 215} \[ \frac{5}{4} x^3 \left (2 x^2-x+3\right )^{7/2}+\frac{305}{144} x^2 \left (2 x^2-x+3\right )^{7/2}+\frac{8467 x \left (2 x^2-x+3\right )^{7/2}}{4608}+\frac{23225 \left (2 x^2-x+3\right )^{7/2}}{43008}-\frac{1547 (1-4 x) \left (2 x^2-x+3\right )^{5/2}}{98304}-\frac{177905 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{3145728}-\frac{4091815 (1-4 x) \sqrt{2 x^2-x+3}}{16777216}-\frac{94111745 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{33554432 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4091815*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/16777216 - (177905*(1 - 4*x)*(3 - x + 2*x^2)^(3/2))/3145728 - (1547*(
1 - 4*x)*(3 - x + 2*x^2)^(5/2))/98304 + (23225*(3 - x + 2*x^2)^(7/2))/43008 + (8467*x*(3 - x + 2*x^2)^(7/2))/4
608 + (305*x^2*(3 - x + 2*x^2)^(7/2))/144 + (5*x^3*(3 - x + 2*x^2)^(7/2))/4 - (94111745*ArcSinh[(1 - 4*x)/Sqrt
[23]])/(33554432*Sqrt[2])

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 (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^2 \, dx &=\frac{5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac{1}{20} \int \left (3-x+2 x^2\right )^{5/2} \left (80+240 x+355 x^2+\frac{1525 x^3}{2}\right ) \, dx\\ &=\frac{305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac{5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac{1}{360} \int \left (3-x+2 x^2\right )^{5/2} \left (1440-255 x+\frac{42335 x^2}{4}\right ) \, dx\\ &=\frac{8467 x \left (3-x+2 x^2\right )^{7/2}}{4608}+\frac{305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac{5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac{\int \left (-\frac{34845}{4}+\frac{348375 x}{8}\right ) \left (3-x+2 x^2\right )^{5/2} \, dx}{5760}\\ &=\frac{23225 \left (3-x+2 x^2\right )^{7/2}}{43008}+\frac{8467 x \left (3-x+2 x^2\right )^{7/2}}{4608}+\frac{305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac{5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac{1547 \int \left (3-x+2 x^2\right )^{5/2} \, dx}{4096}\\ &=-\frac{1547 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{98304}+\frac{23225 \left (3-x+2 x^2\right )^{7/2}}{43008}+\frac{8467 x \left (3-x+2 x^2\right )^{7/2}}{4608}+\frac{305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac{5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac{177905 \int \left (3-x+2 x^2\right )^{3/2} \, dx}{196608}\\ &=-\frac{177905 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{3145728}-\frac{1547 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{98304}+\frac{23225 \left (3-x+2 x^2\right )^{7/2}}{43008}+\frac{8467 x \left (3-x+2 x^2\right )^{7/2}}{4608}+\frac{305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac{5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac{4091815 \int \sqrt{3-x+2 x^2} \, dx}{2097152}\\ &=-\frac{4091815 (1-4 x) \sqrt{3-x+2 x^2}}{16777216}-\frac{177905 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{3145728}-\frac{1547 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{98304}+\frac{23225 \left (3-x+2 x^2\right )^{7/2}}{43008}+\frac{8467 x \left (3-x+2 x^2\right )^{7/2}}{4608}+\frac{305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac{5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac{94111745 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{33554432}\\ &=-\frac{4091815 (1-4 x) \sqrt{3-x+2 x^2}}{16777216}-\frac{177905 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{3145728}-\frac{1547 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{98304}+\frac{23225 \left (3-x+2 x^2\right )^{7/2}}{43008}+\frac{8467 x \left (3-x+2 x^2\right )^{7/2}}{4608}+\frac{305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac{5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac{\left (4091815 \sqrt{\frac{23}{2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{33554432}\\ &=-\frac{4091815 (1-4 x) \sqrt{3-x+2 x^2}}{16777216}-\frac{177905 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{3145728}-\frac{1547 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{98304}+\frac{23225 \left (3-x+2 x^2\right )^{7/2}}{43008}+\frac{8467 x \left (3-x+2 x^2\right )^{7/2}}{4608}+\frac{305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac{5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}-\frac{94111745 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{33554432 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.190448, size = 85, normalized size = 0.5 \[ \frac{4 \sqrt{2 x^2-x+3} \left (10569646080 x^9+2055208960 x^8+44163137536 x^7+26401898496 x^6+75389820928 x^5+57147467776 x^4+77872272000 x^3+42992644128 x^2+39533249652 x+14824182519\right )-5929039935 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4227858432} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(14824182519 + 39533249652*x + 42992644128*x^2 + 77872272000*x^3 + 57147467776*x^4 + 75
389820928*x^5 + 26401898496*x^6 + 44163137536*x^7 + 2055208960*x^8 + 10569646080*x^9) - 5929039935*Sqrt[2]*Arc
Sinh[(1 - 4*x)/Sqrt[23]])/4227858432

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Maple [A]  time = 0.053, size = 136, normalized size = 0.8 \begin{align*}{\frac{5\,{x}^{3}}{4} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{305\,{x}^{2}}{144} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{8467\,x}{4608} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{-4091815+16367260\,x}{16777216}\sqrt{2\,{x}^{2}-x+3}}+{\frac{94111745\,\sqrt{2}}{67108864}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{-1547+6188\,x}{98304} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{-177905+711620\,x}{3145728} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{23225}{43008} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/4*x^3*(2*x^2-x+3)^(7/2)+305/144*x^2*(2*x^2-x+3)^(7/2)+8467/4608*x*(2*x^2-x+3)^(7/2)+4091815/16777216*(-1+4*x
)*(2*x^2-x+3)^(1/2)[phone]/67108864*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))+1547/98304*(-1+4*x)*(2*x^2-x+3)^(
5/2)+177905/3145728*(-1+4*x)*(2*x^2-x+3)^(3/2)+23225/43008*(2*x^2-x+3)^(7/2)

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Maxima [A]  time = 1.47394, size = 225, normalized size = 1.32 \begin{align*} \frac{5}{4} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{3} + \frac{305}{144} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{2} + \frac{8467}{4608} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x + \frac{23225}{43008} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} + \frac{1547}{24576} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x - \frac{1547}{98304} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{177905}{786432} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{177905}{3145728} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{4091815}{4194304} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{94111745}{67108864} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{4091815}{16777216} \, \sqrt{2 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/4*(2*x^2 - x + 3)^(7/2)*x^3 + 305/144*(2*x^2 - x + 3)^(7/2)*x^2 + 8467/4608*(2*x^2 - x + 3)^(7/2)*x + 23225/
43008*(2*x^2 - x + 3)^(7/2) + 1547/24576*(2*x^2 - x + 3)^(5/2)*x - 1547/98304*(2*x^2 - x + 3)^(5/2) + 177905/7
86432*(2*x^2 - x + 3)^(3/2)*x - 177905/3145728*(2*x^2 - x + 3)^(3/2) + 4091815/4194304*sqrt(2*x^2 - x + 3)*x +
94111745/67108864*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 4091815/16777216*sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 1.37715, size = 409, normalized size = 2.41 \begin{align*} \frac{1}{1056964608} \,{\left (10569646080 \, x^{9} + 2055208960 \, x^{8} + 44163137536 \, x^{7} + 26401898496 \, x^{6} + 75389820928 \, x^{5} + 57147467776 \, x^{4} + 77872272000 \, x^{3} + 42992644128 \, x^{2} + 39533249652 \, x + 14824182519\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{94111745}{134217728} \, \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1056964608*(10569646080*x^9 + 2055208960*x^8 + 44163137536*x^7 + 26401898496*x^6 + 75389820928*x^5 + 5714746
7776*x^4 + 77872272000*x^3 + 42992644128*x^2 + 39533249652*x + 14824182519)*sqrt(2*x^2 - x + 3) + 94111745/134
217728*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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (2 x^{2} - x + 3\right )^{\frac{5}{2}} \left (5 x^{2} + 3 x + 2\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1628, size = 126, normalized size = 0.74 \begin{align*} \frac{1}{1056964608} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (4 \,{\left (8 \,{\left (28 \,{\left (160 \,{\left (36 \, x + 7\right )} x + 24067\right )} x + 402861\right )} x + 9202859\right )} x + 27904037\right )} x + 608377125\right )} x + 1343520129\right )} x + 9883312413\right )} x + 14824182519\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{94111745}{67108864} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1056964608*(4*(8*(4*(16*(4*(8*(28*(160*(36*x + 7)*x + 24067)*x + 402861)*x + 9202859)*x + 27904037)*x + 6083
77125)*x + 1343520129)*x + 9883312413)*x + 14824182519)*sqrt(2*x^2 - x + 3) - 94111745/67108864*sqrt(2)*log(-2
*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)