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

Optimal. Leaf size=160 \[ -\frac {27754539-31190998 x}{31986607104 \sqrt {2 x^2-x+3}}+\frac {475357 \sqrt {2 x^2-x+3}}{1934917632 (2 x+5)}-\frac {89137 \sqrt {2 x^2-x+3}}{80621568 (2 x+5)^2}-\frac {3667 \sqrt {2 x^2-x+3}}{559872 (2 x+5)^3}+\frac {369609-175877 x}{463574016 \left (2 x^2-x+3\right )^{3/2}}+\frac {4778789 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{7739670528 \sqrt {2}} \]

[Out]

1/463574016*(369609-175877*x)/(2*x^2-x+3)^(3/2)+4778789/15479341056*arctanh(1/24*(17-22*x)*2^(1/2)/(2*x^2-x+3)
^(1/2))*2^(1/2)+1/31986607104*(-27754539+31190998*x)/(2*x^2-x+3)^(1/2)-3667/559872*(2*x^2-x+3)^(1/2)/(5+2*x)^3
-89137/80621568*(2*x^2-x+3)^(1/2)/(5+2*x)^2+475357/1934917632*(2*x^2-x+3)^(1/2)/(5+2*x)

________________________________________________________________________________________

Rubi [A]  time = 0.28, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1646, 1650, 806, 724, 206} \[ -\frac {27754539-31190998 x}{31986607104 \sqrt {2 x^2-x+3}}+\frac {475357 \sqrt {2 x^2-x+3}}{1934917632 (2 x+5)}-\frac {89137 \sqrt {2 x^2-x+3}}{80621568 (2 x+5)^2}-\frac {3667 \sqrt {2 x^2-x+3}}{559872 (2 x+5)^3}+\frac {369609-175877 x}{463574016 \left (2 x^2-x+3\right )^{3/2}}+\frac {4778789 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{7739670528 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(369609 - 175877*x)/(463574016*(3 - x + 2*x^2)^(3/2)) - (27754539 - 31190998*x)/(31986607104*Sqrt[3 - x + 2*x^
2]) - (3667*Sqrt[3 - x + 2*x^2])/(559872*(5 + 2*x)^3) - (89137*Sqrt[3 - x + 2*x^2])/(80621568*(5 + 2*x)^2) + (
475357*Sqrt[3 - x + 2*x^2])/(1934917632*(5 + 2*x)) + (4778789*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x
^2])])/(7739670528*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 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 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(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] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 1646

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

Rubi steps

\begin {align*} \int \frac {2+x+3 x^2-x^3+5 x^4}{(5+2 x)^4 \left (3-x+2 x^2\right )^{5/2}} \, dx &=\frac {369609-175877 x}{463574016 \left (3-x+2 x^2\right )^{3/2}}+\frac {2}{69} \int \frac {\frac {606939313}{26873856}+\frac {727085495 x}{13436928}+\frac {186705485 x^2}{2239488}-\frac {10162483 x^3}{3359232}-\frac {175877 x^4}{419904}}{(5+2 x)^4 \left (3-x+2 x^2\right )^{3/2}} \, dx\\ &=\frac {369609-175877 x}{463574016 \left (3-x+2 x^2\right )^{3/2}}-\frac {27754539-31190998 x}{31986607104 \sqrt {3-x+2 x^2}}+\frac {4 \int \frac {-\frac {4811736919}{40310784}-\frac {3560904781 x}{13436928}-\frac {87176555 x^2}{1679616}-\frac {39913579 x^3}{10077696}}{(5+2 x)^4 \sqrt {3-x+2 x^2}} \, dx}{1587}\\ &=\frac {369609-175877 x}{463574016 \left (3-x+2 x^2\right )^{3/2}}-\frac {27754539-31190998 x}{31986607104 \sqrt {3-x+2 x^2}}-\frac {3667 \sqrt {3-x+2 x^2}}{559872 (5+2 x)^3}-\frac {\int \frac {\frac {86989289}{11664}+\frac {1265556853 x}{186624}+\frac {39913579 x^2}{93312}}{(5+2 x)^3 \sqrt {3-x+2 x^2}} \, dx}{85698}\\ &=\frac {369609-175877 x}{463574016 \left (3-x+2 x^2\right )^{3/2}}-\frac {27754539-31190998 x}{31986607104 \sqrt {3-x+2 x^2}}-\frac {3667 \sqrt {3-x+2 x^2}}{559872 (5+2 x)^3}-\frac {89137 \sqrt {3-x+2 x^2}}{80621568 (5+2 x)^2}+\frac {\int \frac {-\frac {5274322027}{20736}-\frac {301114735 x}{5184}}{(5+2 x)^2 \sqrt {3-x+2 x^2}} \, dx}{12340512}\\ &=\frac {369609-175877 x}{463574016 \left (3-x+2 x^2\right )^{3/2}}-\frac {27754539-31190998 x}{31986607104 \sqrt {3-x+2 x^2}}-\frac {3667 \sqrt {3-x+2 x^2}}{559872 (5+2 x)^3}-\frac {89137 \sqrt {3-x+2 x^2}}{80621568 (5+2 x)^2}+\frac {475357 \sqrt {3-x+2 x^2}}{1934917632 (5+2 x)}-\frac {4778789 \int \frac {1}{(5+2 x) \sqrt {3-x+2 x^2}} \, dx}{1289945088}\\ &=\frac {369609-175877 x}{463574016 \left (3-x+2 x^2\right )^{3/2}}-\frac {27754539-31190998 x}{31986607104 \sqrt {3-x+2 x^2}}-\frac {3667 \sqrt {3-x+2 x^2}}{559872 (5+2 x)^3}-\frac {89137 \sqrt {3-x+2 x^2}}{80621568 (5+2 x)^2}+\frac {475357 \sqrt {3-x+2 x^2}}{1934917632 (5+2 x)}+\frac {4778789 \operatorname {Subst}\left (\int \frac {1}{288-x^2} \, dx,x,\frac {17-22 x}{\sqrt {3-x+2 x^2}}\right )}{644972544}\\ &=\frac {369609-175877 x}{463574016 \left (3-x+2 x^2\right )^{3/2}}-\frac {27754539-31190998 x}{31986607104 \sqrt {3-x+2 x^2}}-\frac {3667 \sqrt {3-x+2 x^2}}{559872 (5+2 x)^3}-\frac {89137 \sqrt {3-x+2 x^2}}{80621568 (5+2 x)^2}+\frac {475357 \sqrt {3-x+2 x^2}}{1934917632 (5+2 x)}+\frac {4778789 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{7739670528 \sqrt {2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.24, size = 89, normalized size = 0.56 \[ \frac {2527979381 \sqrt {2} \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {4 x^2-2 x+6}}\right )+\frac {24 \left (6664404208 x^6+34872810880 x^5+46210466520 x^4+27484986184 x^3-6702882569 x^2+73621973154 x-95241881529\right )}{(2 x+5)^3 \left (2 x^2-x+3\right )^{3/2}}}{8188571418624} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((24*(-95241881529 + 73621973154*x - 6702882569*x^2 + 27484986184*x^3 + 46210466520*x^4 + 34872810880*x^5 + 66
64404208*x^6))/((5 + 2*x)^3*(3 - x + 2*x^2)^(3/2)) + 2527979381*Sqrt[2]*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x +
4*x^2])])/8188571418624

________________________________________________________________________________________

fricas [A]  time = 0.92, size = 170, normalized size = 1.06 \[ \frac {2527979381 \, \sqrt {2} {\left (32 \, x^{7} + 208 \, x^{6} + 464 \, x^{5} + 632 \, x^{4} + 1162 \, x^{3} + 1265 \, x^{2} + 600 \, x + 1125\right )} \log \left (\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} - 1060 \, x^{2} + 1036 \, x - 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 48 \, {\left (6664404208 \, x^{6} + 34872810880 \, x^{5} + 46210466520 \, x^{4} + 27484986184 \, x^{3} - 6702882569 \, x^{2} + 73621973154 \, x - 95241881529\right )} \sqrt {2 \, x^{2} - x + 3}}{16377142837248 \, {\left (32 \, x^{7} + 208 \, x^{6} + 464 \, x^{5} + 632 \, x^{4} + 1162 \, x^{3} + 1265 \, x^{2} + 600 \, x + 1125\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16377142837248*(2527979381*sqrt(2)*(32*x^7 + 208*x^6 + 464*x^5 + 632*x^4 + 1162*x^3 + 1265*x^2 + 600*x + 112
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)) + 48*(6664
404208*x^6 + 34872810880*x^5 + 46210466520*x^4 + 27484986184*x^3 - 6702882569*x^2 + 73621973154*x - 9524188152
9)*sqrt(2*x^2 - x + 3))/(32*x^7 + 208*x^6 + 464*x^5 + 632*x^4 + 1162*x^3 + 1265*x^2 + 600*x + 1125)

________________________________________________________________________________________

giac [B]  time = 0.28, size = 279, normalized size = 1.74 \[ \frac {4778789}{15479341056} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x + \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) - \frac {4778789}{15479341056} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x - 11 \, \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) + \frac {{\left ({\left (15595499 \, x - 21675019\right )} x + 27298005\right )} x - 14440149}{7996651776 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {\sqrt {2} {\left (38030012 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{5} + 734231900 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{4} + 122834956 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{3} - 2154595396 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 1659431083 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 760577429\right )}}{3869835264 \, {\left (2 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 11\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4778789/15479341056*sqrt(2)*log(abs(-2*sqrt(2)*x + sqrt(2) + 2*sqrt(2*x^2 - x + 3))) - 4778789/15479341056*sqr
t(2)*log(abs(-2*sqrt(2)*x - 11*sqrt(2) + 2*sqrt(2*x^2 - x + 3))) + 1/7996651776*(((15595499*x - 21675019)*x +
27298005)*x - 14440149)/(2*x^2 - x + 3)^(3/2) + 1/3869835264*sqrt(2)*(38030012*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2
- x + 3))^5 + 734231900*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^4 + 122834956*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x +
3))^3 - 2154595396*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 + 1659431083*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))
- 760577429)/(2*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 + 10*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) - 11)^3

________________________________________________________________________________________

maple [A]  time = 0.01, size = 207, normalized size = 1.29 \[ \frac {4778789 \sqrt {2}\, \arctanh \left (\frac {\left (-11 x +\frac {17}{2}\right ) \sqrt {2}}{12 \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}\right )}{15479341056}-\frac {4778789}{429981696 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}-\frac {4778789}{2579890176 \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}-\frac {3667}{13824 \left (x +\frac {5}{2}\right )^{3} \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}+\frac {25951}{110592 \left (x +\frac {5}{2}\right )^{2} \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}-\frac {34861}{3981312 \left (x +\frac {5}{2}\right ) \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}-\frac {72646615 \left (4 x -1\right )}{9889579008 \left (-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}\right )^{\frac {3}{2}}}+\frac {\frac {40 x}{1587}-\frac {10}{1587}}{\sqrt {2 x^{2}-x +3}}+\frac {\frac {5 x}{138}-\frac {5}{552}}{\left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {8183108657 \left (4 x -1\right )}{1364761903104 \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4778789/429981696/(-11*x+2*(x+5/2)^2-19/2)^(3/2)-4778789/2579890176/(-11*x+2*(x+5/2)^2-19/2)^(1/2)-3667/13824
/(x+5/2)^3/(-11*x+2*(x+5/2)^2-19/2)^(3/2)+25951/110592/(x+5/2)^2/(-11*x+2*(x+5/2)^2-19/2)^(3/2)-34861/3981312/
(x+5/2)/(-11*x+2*(x+5/2)^2-19/2)^(3/2)-72646615/9889579008*(4*x-1)/(-11*x+2*(x+5/2)^2-19/2)^(3/2)+4778789/1547
9341056*2^(1/2)*arctanh(1/12*(-11*x+17/2)*2^(1/2)/(-11*x+2*(x+5/2)^2-19/2)^(1/2))+10/1587*(4*x-1)/(2*x^2-x+3)^
(1/2)+5/552*(4*x-1)/(2*x^2-x+3)^(3/2)-8183108657/1364761903104*(4*x-1)/(-11*x+2*(x+5/2)^2-19/2)^(1/2)

________________________________________________________________________________________

maxima [A]  time = 1.01, size = 246, normalized size = 1.54 \[ -\frac {4778789}{15479341056} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) + \frac {416525263 \, x}{341190475776 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {245375387}{113730158592 \, \sqrt {2 \, x^{2} - x + 3}} + \frac {16932905 \, x}{2472394752 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {3667}{1728 \, {\left (8 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 60 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 150 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 125 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {25951}{27648 \, {\left (4 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 20 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 25 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {34861}{1990656 \, {\left (2 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 5 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {10570421}{824131584 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4778789/15479341056*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*sqrt(23)/abs(2*x + 5)) + 416525263/
341190475776*x/sqrt(2*x^2 - x + 3) - 245375387/113730158592/sqrt(2*x^2 - x + 3) + 16932905/2472394752*x/(2*x^2
- x + 3)^(3/2) - 3667/1728/(8*(2*x^2 - x + 3)^(3/2)*x^3 + 60*(2*x^2 - x + 3)^(3/2)*x^2 + 150*(2*x^2 - x + 3)^
(3/2)*x + 125*(2*x^2 - x + 3)^(3/2)) + 25951/27648/(4*(2*x^2 - x + 3)^(3/2)*x^2 + 20*(2*x^2 - x + 3)^(3/2)*x +
25*(2*x^2 - x + 3)^(3/2)) - 34861/1990656/(2*(2*x^2 - x + 3)^(3/2)*x + 5*(2*x^2 - x + 3)^(3/2)) - 10570421/82
4131584/(2*x^2 - x + 3)^(3/2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________