3.1.67 \(\int (3-x+2 x^2)^{3/2} (2+3 x+5 x^2)^2 \, dx\) [67]

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

[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)

________________________________________________________________________________________

Rubi [A]
time = 0.07, 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 = {1675, 654, 626, 633, 221} \begin {gather*} \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}} \end {gather*}

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 221

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 626

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 633

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 654

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 1675

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 ) \text {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*}

________________________________________________________________________________________

Mathematica [A]
time = 0.57, size = 85, normalized size = 0.58 \begin {gather*} \frac {4 \sqrt {3-x+2 x^2} \left (439831323+1619403428 x+1799647136 x^2+2728413312 x^3+2061273088 x^4+2025840640 x^5+525926400 x^6+688128000 x^7\right )[phone] \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{440401920} \end {gather*}

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]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/440401920

________________________________________________________________________________________

Maple [A]
time = 0.12, size = 117, normalized size = 0.80

method	result	size
risch	\(\frac {\left (688128000 x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {2 x^{2}-x +3}}{110100480}-\frac {12850997 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{4194304}\)	\(65\)
trager	\(\left (\frac {25}{4} x^{7}+\frac {535}{112} x^{6}+\frac {49459}{2688} x^{5}+\frac {143783}{7680} x^{4}+\frac {7105243}{286720} x^{3}+\frac {8034139}{491520} x^{2}+\frac {404850857}{27525120} x +\frac {146610441}{36700160}\right ) \sqrt {2 x^{2}-x +3}-\frac {12850997 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \RootOf \left (\textit {\_Z}^{2}-2\right ) x -\RootOf \left (\textit {\_Z}^{2}-2\right )+4 \sqrt {2 x^{2}-x +3}\right )}{4194304}\)	\(91\)
default	\(-\frac {558739 \left (4 x -1\right ) \sqrt {2 x^{2}-x +3}}{1048576}-\frac {12850997 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{4194304}-\frac {24293 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{196608}+\frac {24499 x \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{10752}+\frac {1235 x^{2} \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{448}+\frac {25 x^{3} \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{16}+\frac {73861 \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{215040}\)	\(117\)

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,method=_RETURNVERBOSE)

[Out]

-558739/1048576*(4*x-1)*(2*x^2-x+3)^(1/2)-12850997/4194304*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))-24293/196608
*(4*x-1)*(2*x^2-x+3)^(3/2)+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)+73861/215040*(2*x^2-x+3)^(5/2)

________________________________________________________________________________________

Maxima [A]
time = 0.51, size = 138, normalized size = 0.94 \begin {gather*} \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} \end {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 4.10, size = 88, normalized size = 0.60 \begin {gather*} \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 ) \end {gather*}

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Giac [A]
time = 2.98, size = 83, normalized size = 0.56 \begin {gather*} \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 ) \end {gather*}

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________