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

Optimal. Leaf size=172 \[ \frac {5}{96} (2 x+5) \left (2 x^2-x+3\right )^{5/2}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{576 (2 x+5)}-\frac {839}{960} \left (2 x^2-x+3\right )^{5/2}-\frac {(909513-226052 x) \left (2 x^2-x+3\right )^{3/2}}{18432}-\frac {(85448933-14243732 x) \sqrt {2 x^2-x+3}}{32768}+\frac {959625 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{64 \sqrt {2}}-\frac {982669459 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{65536 \sqrt {2}} \]

[Out]

-1/18432*(909513-226052*x)*(2*x^2-x+3)^(3/2)-839/960*(2*x^2-x+3)^(5/2)-3667/576*(2*x^2-x+3)^(5/2)/(5+2*x)+5/96
*(5+2*x)*(2*x^2-x+3)^(5/2)-982669459/131072*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)+959625/128*arctanh(1/24*(17
-22*x)*2^(1/2)/(2*x^2-x+3)^(1/2))*2^(1/2)-1/32768*(85448933-14243732*x)*(2*x^2-x+3)^(1/2)

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Rubi [A]  time = 0.28, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1650, 1653, 814, 843, 619, 215, 724, 206} \[ \frac {5}{96} (2 x+5) \left (2 x^2-x+3\right )^{5/2}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{576 (2 x+5)}-\frac {839}{960} \left (2 x^2-x+3\right )^{5/2}-\frac {(909513-226052 x) \left (2 x^2-x+3\right )^{3/2}}{18432}-\frac {(85448933-14243732 x) \sqrt {2 x^2-x+3}}{32768}+\frac {959625 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{64 \sqrt {2}}-\frac {982669459 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{65536 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((85448933 - 14243732*x)*Sqrt[3 - x + 2*x^2])/32768 - ((909513 - 226052*x)*(3 - x + 2*x^2)^(3/2))/18432 - (83
9*(3 - x + 2*x^2)^(5/2))/960 - (3667*(3 - x + 2*x^2)^(5/2))/(576*(5 + 2*x)) + (5*(5 + 2*x)*(3 - x + 2*x^2)^(5/
2))/96 - (982669459*ArcSinh[(1 - 4*x)/Sqrt[23]])/(65536*Sqrt[2]) + (959625*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqr
t[3 - x + 2*x^2])])/(64*Sqrt[2])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 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]

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 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 814

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*(a + b*x + c*x^
2)^p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 1650

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomia
lQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*
x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^
(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Q + c*d*R*(m + 1) - b*e*R*(m + p + 2)
- c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]

Rule 1653

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq
, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*e^(q - 1)*(
m + q + 2*p + 1)), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^
q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q
- 1) - c*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p +
1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] &&  !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

Rubi steps

\begin {align*} \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^2} \, dx &=-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{576 (5+2 x)}-\frac {1}{72} \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (\frac {26675}{16}-4990 x+486 x^2-180 x^3\right )}{5+2 x} \, dx\\ &=-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{576 (5+2 x)}+\frac {5}{96} (5+2 x) \left (3-x+2 x^2\right )^{5/2}-\frac {\int \frac {\left (3-x+2 x^2\right )^{3/2} \left (148350-406320 x+120816 x^2\right )}{5+2 x} \, dx}{6912}\\ &=-\frac {839}{960} \left (3-x+2 x^2\right )^{5/2}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{576 (5+2 x)}+\frac {5}{96} (5+2 x) \left (3-x+2 x^2\right )^{5/2}-\frac {\int \frac {(8954400-27126240 x) \left (3-x+2 x^2\right )^{3/2}}{5+2 x} \, dx}{276480}\\ &=-\frac {(909513-226052 x) \left (3-x+2 x^2\right )^{3/2}}{18432}-\frac {839}{960} \left (3-x+2 x^2\right )^{5/2}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{576 (5+2 x)}+\frac {5}{96} (5+2 x) \left (3-x+2 x^2\right )^{5/2}+\frac {\int \frac {(-11522887200+30766461120 x) \sqrt {3-x+2 x^2}}{5+2 x} \, dx}{17694720}\\ &=-\frac {(85448933-14243732 x) \sqrt {3-x+2 x^2}}{32768}-\frac {(909513-226052 x) \left (3-x+2 x^2\right )^{3/2}}{18432}-\frac {839}{960} \left (3-x+2 x^2\right )^{5/2}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{576 (5+2 x)}+\frac {5}{96} (5+2 x) \left (3-x+2 x^2\right )^{5/2}-\frac {\int \frac {8489566411200-16980528251520 x}{(5+2 x) \sqrt {3-x+2 x^2}} \, dx}{566231040}\\ &=-\frac {(85448933-14243732 x) \sqrt {3-x+2 x^2}}{32768}-\frac {(909513-226052 x) \left (3-x+2 x^2\right )^{3/2}}{18432}-\frac {839}{960} \left (3-x+2 x^2\right )^{5/2}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{576 (5+2 x)}+\frac {5}{96} (5+2 x) \left (3-x+2 x^2\right )^{5/2}+\frac {982669459 \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx}{65536}-\frac {2878875}{32} \int \frac {1}{(5+2 x) \sqrt {3-x+2 x^2}} \, dx\\ &=-\frac {(85448933-14243732 x) \sqrt {3-x+2 x^2}}{32768}-\frac {(909513-226052 x) \left (3-x+2 x^2\right )^{3/2}}{18432}-\frac {839}{960} \left (3-x+2 x^2\right )^{5/2}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{576 (5+2 x)}+\frac {5}{96} (5+2 x) \left (3-x+2 x^2\right )^{5/2}+\frac {2878875}{16} \operatorname {Subst}\left (\int \frac {1}{288-x^2} \, dx,x,\frac {17-22 x}{\sqrt {3-x+2 x^2}}\right )+\frac {982669459 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{65536 \sqrt {46}}\\ &=-\frac {(85448933-14243732 x) \sqrt {3-x+2 x^2}}{32768}-\frac {(909513-226052 x) \left (3-x+2 x^2\right )^{3/2}}{18432}-\frac {839}{960} \left (3-x+2 x^2\right )^{5/2}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{576 (5+2 x)}+\frac {5}{96} (5+2 x) \left (3-x+2 x^2\right )^{5/2}-\frac {982669459 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{65536 \sqrt {2}}+\frac {959625 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{64 \sqrt {2}}\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 108, normalized size = 0.63 \[ \frac {14739840000 \sqrt {2} \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {4 x^2-2 x+6}}\right )+\frac {4 \sqrt {2 x^2-x+3} \left (409600 x^6-1798144 x^5+8283904 x^4-35369408 x^3+182033816 x^2-1404323114 x-6814208295\right )}{2 x+5}-14740041885 \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{1966080} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*Sqrt[3 - x + 2*x^2]*(-6814208295 - 1404323114*x + 182033816*x^2 - 35369408*x^3 + 8283904*x^4 - 1798144*x^5
+ 409600*x^6))/(5 + 2*x) - 14740041885*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]] + 14739840000*Sqrt[2]*ArcTanh[(17
- 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/1966080

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fricas [A]  time = 0.88, size = 153, normalized size = 0.89 \[ \frac {14740041885 \, \sqrt {2} {\left (2 \, x + 5\right )} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 14739840000 \, \sqrt {2} {\left (2 \, x + 5\right )} \log \left (\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} - 1060 \, x^{2} + 1036 \, x - 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 8 \, {\left (409600 \, x^{6} - 1798144 \, x^{5} + 8283904 \, x^{4} - 35369408 \, x^{3} + 182033816 \, x^{2} - 1404323114 \, x - 6814208295\right )} \sqrt {2 \, x^{2} - x + 3}}{3932160 \, {\left (2 \, x + 5\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3932160*(14740041885*sqrt(2)*(2*x + 5)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25) +
14739840000*sqrt(2)*(2*x + 5)*log((24*sqrt(2)*sqrt(2*x^2 - x + 3)*(22*x - 17) - 1060*x^2 + 1036*x - 1153)/(4*x
^2 + 20*x + 25)) + 8*(409600*x^6 - 1798144*x^5 + 8283904*x^4 - 35369408*x^3 + 182033816*x^2 - 1404323114*x - 6
814208295)*sqrt(2*x^2 - x + 3))/(2*x + 5)

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giac [B]  time = 0.45, size = 707, normalized size = 4.11 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1966080*sqrt(2)*(14739840000*log(12*sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 72/(2*x + 5) - 11)*sgn(1/(2*x
+ 5)) + 14740041885*log(abs(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5) + 1))*sgn(1/(2*x + 5)) - 1
4740041885*log(abs(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5) - 1))*sgn(1/(2*x + 5)) - 2027704320*
sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1)*sgn(1/(2*x + 5)) + 2*(45496763235*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^
2 + 1) + 6/(2*x + 5))^11*sgn(1/(2*x + 5)) - 126553743360*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x +
5))^10*sgn(1/(2*x + 5)) + 44062768335*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^9*sgn(1/(2*x +
5)) + 33178982400*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^8*sgn(1/(2*x + 5)) + 294206421582*(
sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^7*sgn(1/(2*x + 5)) - 463672074240*(sqrt(-11/(2*x + 5)
+ 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^6*sgn(1/(2*x + 5)) + 35099942478*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1
) + 6/(2*x + 5))^5*sgn(1/(2*x + 5)) + 171324610560*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^4*
sgn(1/(2*x + 5)) + 60059281615*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^3*sgn(1/(2*x + 5)) - 1
05051009024*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^2*sgn(1/(2*x + 5)) - 5210329245*(sqrt(-11
/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))*sgn(1/(2*x + 5)) + 17058392064*sgn(1/(2*x + 5)))/((sqrt(-11/(2
*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^2 - 1)^6)

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maple [A]  time = 0.01, size = 208, normalized size = 1.21 \[ \frac {5 \left (2 x^{2}-x +3\right )^{\frac {5}{2}} x}{48}+\frac {982669459 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{131072}+\frac {959625 \sqrt {2}\, \arctanh \left (\frac {\left (-11 x +\frac {17}{2}\right ) \sqrt {2}}{12 \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}\right )}{128}-\frac {589 \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{960}+\frac {9059 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{6144}+\frac {208357 \left (4 x -1\right ) \sqrt {2 x^{2}-x +3}}{32768}-\frac {3667 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {5}{2}}}{1152 \left (x +\frac {5}{2}\right )}-\frac {106625 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}{2304}+\frac {1637 \left (4 x -1\right ) \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}{16}-\frac {319875 \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}{128}+\frac {3667 \left (4 x -1\right ) \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}{2304} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/48*(2*x^2-x+3)^(5/2)*x-589/960*(2*x^2-x+3)^(5/2)+9059/6144*(4*x-1)*(2*x^2-x+3)^(3/2)+208357/32768*(4*x-1)*(2
*x^2-x+3)^(1/2)[phone]/131072*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))-3667/1152/(x+5/2)*(-11*x+2*(x+5/2)^2-1
9/2)^(5/2)-106625/2304*(-11*x+2*(x+5/2)^2-19/2)^(3/2)+1637/16*(4*x-1)*(-11*x+2*(x+5/2)^2-19/2)^(1/2)-319875/12
8*(-11*x+2*(x+5/2)^2-19/2)^(1/2)+959625/128*2^(1/2)*arctanh(1/12*(-11*x+17/2)*2^(1/2)/(-11*x+2*(x+5/2)^2-19/2)
^(1/2))+3667/2304*(4*x-1)*(-11*x+2*(x+5/2)^2-19/2)^(3/2)

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maxima [A]  time = 1.01, size = 161, normalized size = 0.94 \[ \frac {5}{48} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x - \frac {589}{960} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {9059}{1536} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {185827}{6144} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {3560933}{8192} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {982669459}{131072} \, \sqrt {2} \operatorname {arsinh}\left (\frac {4}{23} \, \sqrt {23} x - \frac {1}{23} \, \sqrt {23}\right ) - \frac {959625}{128} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) - \frac {85448933}{32768} \, \sqrt {2 \, x^{2} - x + 3} - \frac {3667 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{32 \, {\left (2 \, x + 5\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/48*(2*x^2 - x + 3)^(5/2)*x - 589/960*(2*x^2 - x + 3)^(5/2) + 9059/1536*(2*x^2 - x + 3)^(3/2)*x - 185827/6144
*(2*x^2 - x + 3)^(3/2) + 3560933/8192*sqrt(2*x^2 - x + 3)*x + 982669459/131072*sqrt(2)*arcsinh(4/23*sqrt(23)*x
- 1/23*sqrt(23)) - 959625/128*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*sqrt(23)/abs(2*x + 5)) -
85448933/32768*sqrt(2*x^2 - x + 3) - 3667/32*(2*x^2 - x + 3)^(3/2)/(2*x + 5)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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