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

Optimal. Leaf size=147 \[ \frac {1235}{448} \left (2 x^2-x+3\right )^{5/2} x^2+\frac {24499 \left (2 x^2-x+3\right )^{5/2} x}{10752}+\frac {73861 \left (2 x^2-x+3\right )^{5/2}}{215040}+\frac {24293 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{196608}+\frac {558739 (1-4 x) \sqrt {2 x^2-x+3}}{1048576}+\frac {25}{16} \left (2 x^2-x+3\right )^{5/2} x^3+\frac {12850997 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2097152 \sqrt {2}} \]

[Out]

24293/196608*(1-4*x)*(2*x^2-x+3)^(3/2)+73861/215040*(2*x^2-x+3)^(5/2)+24499/10752*x*(2*x^2-x+3)^(5/2)+1235/448
*x^2*(2*x^2-x+3)^(5/2)+25/16*x^3*(2*x^2-x+3)^(5/2)[phone]/4194304*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)+558
739/1048576*(1-4*x)*(2*x^2-x+3)^(1/2)

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Rubi [A]  time = 0.12, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {25}{16} \left (2 x^2-x+3\right )^{5/2} x^3+\frac {1235}{448} \left (2 x^2-x+3\right )^{5/2} x^2+\frac {24499 \left (2 x^2-x+3\right )^{5/2} x}{10752}+\frac {73861 \left (2 x^2-x+3\right )^{5/2}}{215040}+\frac {24293 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{196608}+\frac {558739 (1-4 x) \sqrt {2 x^2-x+3}}{1048576}+\frac {12850997 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2097152 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(558739*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/1048576 + (24293*(1 - 4*x)*(3 - x + 2*x^2)^(3/2))/196608 + (73861*(3 -
x + 2*x^2)^(5/2))/215040 + (24499*x*(3 - x + 2*x^2)^(5/2))/10752 + (1235*x^2*(3 - x + 2*x^2)^(5/2))/448 + (25*
x^3*(3 - x + 2*x^2)^(5/2))/16 + (12850997*ArcSinh[(1 - 4*x)/Sqrt[23]])/(2097152*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 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 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 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^2 \, dx &=\frac {25}{16} x^3 \left (3-x+2 x^2\right )^{5/2}+\frac {1}{16} \int \left (3-x+2 x^2\right )^{3/2} \left (64+192 x+239 x^2+\frac {1235 x^3}{2}\right ) \, dx\\ &=\frac {1235}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac {25}{16} x^3 \left (3-x+2 x^2\right )^{5/2}+\frac {1}{224} \int \left (3-x+2 x^2\right )^{3/2} \left (896-1017 x+\frac {24499 x^2}{4}\right ) \, dx\\ &=\frac {24499 x \left (3-x+2 x^2\right )^{5/2}}{10752}+\frac {1235}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac {25}{16} x^3 \left (3-x+2 x^2\right )^{5/2}+\frac {\int \left (-\frac {30489}{4}+\frac {73861 x}{8}\right ) \left (3-x+2 x^2\right )^{3/2} \, dx}{2688}\\ &=\frac {73861 \left (3-x+2 x^2\right )^{5/2}}{215040}+\frac {24499 x \left (3-x+2 x^2\right )^{5/2}}{10752}+\frac {1235}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac {25}{16} x^3 \left (3-x+2 x^2\right )^{5/2}-\frac {24293 \int \left (3-x+2 x^2\right )^{3/2} \, dx}{12288}\\ &=\frac {24293 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{196608}+\frac {73861 \left (3-x+2 x^2\right )^{5/2}}{215040}+\frac {24499 x \left (3-x+2 x^2\right )^{5/2}}{10752}+\frac {1235}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac {25}{16} x^3 \left (3-x+2 x^2\right )^{5/2}-\frac {558739 \int \sqrt {3-x+2 x^2} \, dx}{131072}\\ &=\frac {558739 (1-4 x) \sqrt {3-x+2 x^2}}{1048576}+\frac {24293 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{196608}+\frac {73861 \left (3-x+2 x^2\right )^{5/2}}{215040}+\frac {24499 x \left (3-x+2 x^2\right )^{5/2}}{10752}+\frac {1235}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac {25}{16} x^3 \left (3-x+2 x^2\right )^{5/2}-\frac {12850997 \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx}{2097152}\\ &=\frac {558739 (1-4 x) \sqrt {3-x+2 x^2}}{1048576}+\frac {24293 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{196608}+\frac {73861 \left (3-x+2 x^2\right )^{5/2}}{215040}+\frac {24499 x \left (3-x+2 x^2\right )^{5/2}}{10752}+\frac {1235}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac {25}{16} x^3 \left (3-x+2 x^2\right )^{5/2}-\frac {\left (558739 \sqrt {\frac {23}{2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{2097152}\\ &=\frac {558739 (1-4 x) \sqrt {3-x+2 x^2}}{1048576}+\frac {24293 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{196608}+\frac {73861 \left (3-x+2 x^2\right )^{5/2}}{215040}+\frac {24499 x \left (3-x+2 x^2\right )^{5/2}}{10752}+\frac {1235}{448} x^2 \left (3-x+2 x^2\right )^{5/2}+\frac {25}{16} x^3 \left (3-x+2 x^2\right )^{5/2}+\frac {12850997 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{2097152 \sqrt {2}}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 75, normalized size = 0.51 \[ \frac {4 \sqrt {2 x^2-x+3} \left (688128000 x^7+525926400 x^6+2025840640 x^5+2061273088 x^4+2728413312 x^3+1799647136 x^2+1619403428 x+439831323\right )[phone] \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{440401920} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(439831323 + 1619403428*x + 1799647136*x^2 + 2728413312*x^3 + 2061273088*x^4 + 20258406
40*x^5 + 525926400*x^6 + 688128000*x^7) + 1349354685*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/440401920

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fricas [A]  time = 0.80, size = 88, normalized size = 0.60 \[ \frac {1}{110100480} \, {\left (688128000 \, x^{7} + 525926400 \, x^{6} + 2025840640 \, x^{5} + 2061273088 \, x^{4} + 2728413312 \, x^{3} + 1799647136 \, x^{2} + 1619403428 \, x + 439831323\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {12850997}{8388608} \, \sqrt {2} \log \left (4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/110100480*(688128000*x^7 + 525926400*x^6 + 2025840640*x^5 + 2061273088*x^4 + 2728413312*x^3 + 1799647136*x^2
+ 1619403428*x + 439831323)*sqrt(2*x^2 - x + 3) + 12850997/8388608*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 = 0.26, size = 83, normalized size = 0.56 \[ \frac {1}{110100480} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (20 \, {\left (120 \, {\left (140 \, x + 107\right )} x + 49459\right )} x + 1006481\right )} x + 21315729\right )} x + 56238973\right )} x + 404850857\right )} x + 439831323\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {12850997}{4194304} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/110100480*(4*(8*(4*(16*(20*(120*(140*x + 107)*x + 49459)*x + 1006481)*x + 21315729)*x + 56238973)*x + 404850
857)*x + 439831323)*sqrt(2*x^2 - x + 3) + 12850997/4194304*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x
+ 3)) + 1)

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maple [A]  time = 0.01, size = 117, normalized size = 0.80 \[ \frac {25 \left (2 x^{2}-x +3\right )^{\frac {5}{2}} x^{3}}{16}+\frac {1235 \left (2 x^{2}-x +3\right )^{\frac {5}{2}} x^{2}}{448}+\frac {24499 \left (2 x^{2}-x +3\right )^{\frac {5}{2}} x}{10752}-\frac {12850997 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{4194304}+\frac {73861 \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{215040}-\frac {558739 \left (4 x -1\right ) \sqrt {2 x^{2}-x +3}}{1048576}-\frac {24293 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{196608} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

73861/215040*(2*x^2-x+3)^(5/2)+25/16*(2*x^2-x+3)^(5/2)*x^3+1235/448*(2*x^2-x+3)^(5/2)*x^2+24499/10752*(2*x^2-x
+3)^(5/2)*x-12850997/4194304*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))-558739/1048576*(4*x-1)*(2*x^2-x+3)^(1/2)-2
4293/196608*(4*x-1)*(2*x^2-x+3)^(3/2)

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maxima [A]  time = 0.98, size = 138, normalized size = 0.94 \[ \frac {25}{16} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{3} + \frac {1235}{448} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{2} + \frac {24499}{10752} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x + \frac {73861}{215040} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} - \frac {24293}{49152} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {24293}{196608} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {558739}{262144} \, \sqrt {2 \, x^{2} - x + 3} x - \frac {12850997}{4194304} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {558739}{1048576} \, \sqrt {2 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/16*(2*x^2 - x + 3)^(5/2)*x^3 + 1235/448*(2*x^2 - x + 3)^(5/2)*x^2 + 24499/10752*(2*x^2 - x + 3)^(5/2)*x + 7
3861/215040*(2*x^2 - x + 3)^(5/2) - 24293/49152*(2*x^2 - x + 3)^(3/2)*x + 24293/196608*(2*x^2 - x + 3)^(3/2) -
558739/262144*sqrt(2*x^2 - x + 3)*x - 12850997/4194304*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) + 558739/1048
576*sqrt(2*x^2 - x + 3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (2\,x^2-x+3\right )}^{3/2}\,{\left (5\,x^2+3\,x+2\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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