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

Optimal. Leaf size=139 \[ \frac {26800085 \sqrt {2 x^2-x+3}}{1719926784 (2 x+5)}-\frac {16295969 \sqrt {2 x^2-x+3}}{71663616 (2 x+5)^2}+\frac {513097 \sqrt {2 x^2-x+3}}{497664 (2 x+5)^3}-\frac {3667 \sqrt {2 x^2-x+3}}{2304 (2 x+5)^4}+\frac {2053207 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{20639121408 \sqrt {2}} \]

[Out]

2053207/41278242816*arctanh(1/24*(17-22*x)*2^(1/2)/(2*x^2-x+3)^(1/2))*2^(1/2)-3667/2304*(2*x^2-x+3)^(1/2)/(5+2
*x)^4+513097/497664*(2*x^2-x+3)^(1/2)/(5+2*x)^3-16295969/71663616*(2*x^2-x+3)^(1/2)/(5+2*x)^2+26800085/1719926
784*(2*x^2-x+3)^(1/2)/(5+2*x)

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Rubi [A]  time = 0.19, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1650, 806, 724, 206} \[ \frac {26800085 \sqrt {2 x^2-x+3}}{1719926784 (2 x+5)}-\frac {16295969 \sqrt {2 x^2-x+3}}{71663616 (2 x+5)^2}+\frac {513097 \sqrt {2 x^2-x+3}}{497664 (2 x+5)^3}-\frac {3667 \sqrt {2 x^2-x+3}}{2304 (2 x+5)^4}+\frac {2053207 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{20639121408 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3667*Sqrt[3 - x + 2*x^2])/(2304*(5 + 2*x)^4) + (513097*Sqrt[3 - x + 2*x^2])/(497664*(5 + 2*x)^3) - (16295969
*Sqrt[3 - x + 2*x^2])/(71663616*(5 + 2*x)^2) + (26800085*Sqrt[3 - x + 2*x^2])/(1719926784*(5 + 2*x)) + (205320
7*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(20639121408*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 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)^5 \sqrt {3-x+2 x^2}} \, dx &=-\frac {3667 \sqrt {3-x+2 x^2}}{2304 (5+2 x)^4}-\frac {1}{288} \int \frac {\frac {37027}{16}-\frac {10167 x}{4}+1944 x^2-720 x^3}{(5+2 x)^4 \sqrt {3-x+2 x^2}} \, dx\\ &=-\frac {3667 \sqrt {3-x+2 x^2}}{2304 (5+2 x)^4}+\frac {513097 \sqrt {3-x+2 x^2}}{497664 (5+2 x)^3}+\frac {\int \frac {\frac {2607829}{16}-\frac {295607 x}{2}+77760 x^2}{(5+2 x)^3 \sqrt {3-x+2 x^2}} \, dx}{62208}\\ &=-\frac {3667 \sqrt {3-x+2 x^2}}{2304 (5+2 x)^4}+\frac {513097 \sqrt {3-x+2 x^2}}{497664 (5+2 x)^3}-\frac {16295969 \sqrt {3-x+2 x^2}}{71663616 (5+2 x)^2}-\frac {\int \frac {\frac {19411145}{16}-\frac {6098911 x}{4}}{(5+2 x)^2 \sqrt {3-x+2 x^2}} \, dx}{8957952}\\ &=-\frac {3667 \sqrt {3-x+2 x^2}}{2304 (5+2 x)^4}+\frac {513097 \sqrt {3-x+2 x^2}}{497664 (5+2 x)^3}-\frac {16295969 \sqrt {3-x+2 x^2}}{71663616 (5+2 x)^2}+\frac {26800085 \sqrt {3-x+2 x^2}}{1719926784 (5+2 x)}-\frac {2053207 \int \frac {1}{(5+2 x) \sqrt {3-x+2 x^2}} \, dx}{3439853568}\\ &=-\frac {3667 \sqrt {3-x+2 x^2}}{2304 (5+2 x)^4}+\frac {513097 \sqrt {3-x+2 x^2}}{497664 (5+2 x)^3}-\frac {16295969 \sqrt {3-x+2 x^2}}{71663616 (5+2 x)^2}+\frac {26800085 \sqrt {3-x+2 x^2}}{1719926784 (5+2 x)}+\frac {2053207 \operatorname {Subst}\left (\int \frac {1}{288-x^2} \, dx,x,\frac {17-22 x}{\sqrt {3-x+2 x^2}}\right )}{1719926784}\\ &=-\frac {3667 \sqrt {3-x+2 x^2}}{2304 (5+2 x)^4}+\frac {513097 \sqrt {3-x+2 x^2}}{497664 (5+2 x)^3}-\frac {16295969 \sqrt {3-x+2 x^2}}{71663616 (5+2 x)^2}+\frac {26800085 \sqrt {3-x+2 x^2}}{1719926784 (5+2 x)}+\frac {2053207 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{20639121408 \sqrt {2}}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 81, normalized size = 0.58 \[ \frac {2053207 \sqrt {2} (2 x+5)^4 \tanh ^{-1}\left (\frac {17-22 x}{12 \sqrt {4 x^2-2 x+6}}\right )+24 \sqrt {2 x^2-x+3} \left (214400680 x^3+43592076 x^2-255525906 x-298655447\right )}{41278242816 (2 x+5)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(24*Sqrt[3 - x + 2*x^2]*(-298655447 - 255525906*x + 43592076*x^2 + 214400680*x^3) + 2053207*Sqrt[2]*(5 + 2*x)^
4*ArcTanh[(17 - 22*x)/(12*Sqrt[6 - 2*x + 4*x^2])])/(41278242816*(5 + 2*x)^4)

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fricas [A]  time = 0.95, size = 125, normalized size = 0.90 \[ \frac {2053207 \, \sqrt {2} {\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )} \log \left (\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} - 1060 \, x^{2} + 1036 \, x - 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 48 \, {\left (214400680 \, x^{3} + 43592076 \, x^{2} - 255525906 \, x - 298655447\right )} \sqrt {2 \, x^{2} - x + 3}}{82556485632 \, {\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} \]

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)^5/(2*x^2-x+3)^(1/2),x, algorithm="fricas")

[Out]

1/82556485632*(2053207*sqrt(2)*(16*x^4 + 160*x^3 + 600*x^2 + 1000*x + 625)*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*(214400680*x^3 + 43592076*x^2 - 255525906*x
- 298655447)*sqrt(2*x^2 - x + 3))/(16*x^4 + 160*x^3 + 600*x^2 + 1000*x + 625)

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giac [A]  time = 0.28, size = 164, normalized size = 1.18 \[ \frac {1}{41278242816} \, \sqrt {2} {\left (12 \, {\left (\frac {24 \, {\left (\frac {144 \, {\left (\frac {513097}{\mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right )} - \frac {792072}{{\left (2 \, x + 5\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right )}\right )}}{2 \, x + 5} - \frac {16295969}{\mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right )}\right )}}{2 \, x + 5} + \frac {26800085}{\mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right )}\right )} \sqrt {-\frac {11}{2 \, x + 5} + \frac {36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac {2053207 \, \log \left (12 \, \sqrt {-\frac {11}{2 \, x + 5} + \frac {36}{{\left (2 \, x + 5\right )}^{2}} + 1} + \frac {72}{2 \, x + 5} - 11\right )}{\mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right )} - 321601020 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 5}\right )\right )} \]

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)^5/(2*x^2-x+3)^(1/2),x, algorithm="giac")

[Out]

1/41278242816*sqrt(2)*(12*(24*(144*(513097/sgn(1/(2*x + 5)) - 792072/((2*x + 5)*sgn(1/(2*x + 5))))/(2*x + 5) -
16295969/sgn(1/(2*x + 5)))/(2*x + 5) + 26800085/sgn(1/(2*x + 5)))*sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) +
2053207*log(12*sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 72/(2*x + 5) - 11)/sgn(1/(2*x + 5)) - 321601020*sgn(
1/(2*x + 5)))

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maple [A]  time = 0.01, size = 116, normalized size = 0.83 \[ \frac {2053207 \sqrt {2}\, \arctanh \left (\frac {\left (-11 x +\frac {17}{2}\right ) \sqrt {2}}{12 \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}\right )}{41278242816}+\frac {26800085 \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}{3439853568 \left (x +\frac {5}{2}\right )}-\frac {16295969 \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}{286654464 \left (x +\frac {5}{2}\right )^{2}}-\frac {3667 \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}{36864 \left (x +\frac {5}{2}\right )^{4}}+\frac {513097 \sqrt {-11 x +2 \left (x +\frac {5}{2}\right )^{2}-\frac {19}{2}}}{3981312 \left (x +\frac {5}{2}\right )^{3}} \]

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)^5/(2*x^2-x+3)^(1/2),x)

[Out]

26800085/3439853568/(x+5/2)*(-11*x+2*(x+5/2)^2-19/2)^(1/2)+2053207/41278242816*2^(1/2)*arctanh(1/12*(-11*x+17/
2)*2^(1/2)/(-11*x+2*(x+5/2)^2-19/2)^(1/2))-16295969/286654464/(x+5/2)^2*(-11*x+2*(x+5/2)^2-19/2)^(1/2)-3667/36
864/(x+5/2)^4*(-11*x+2*(x+5/2)^2-19/2)^(1/2)+513097/3981312/(x+5/2)^3*(-11*x+2*(x+5/2)^2-19/2)^(1/2)

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maxima [A]  time = 1.02, size = 149, normalized size = 1.07 \[ -\frac {2053207}{41278242816} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) - \frac {3667 \, \sqrt {2 \, x^{2} - x + 3}}{2304 \, {\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} + \frac {513097 \, \sqrt {2 \, x^{2} - x + 3}}{497664 \, {\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} - \frac {16295969 \, \sqrt {2 \, x^{2} - x + 3}}{71663616 \, {\left (4 \, x^{2} + 20 \, x + 25\right )}} + \frac {26800085 \, \sqrt {2 \, x^{2} - x + 3}}{1719926784 \, {\left (2 \, x + 5\right )}} \]

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)^5/(2*x^2-x+3)^(1/2),x, algorithm="maxima")

[Out]

-2053207/41278242816*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*sqrt(23)/abs(2*x + 5)) - 3667/2304*
sqrt(2*x^2 - x + 3)/(16*x^4 + 160*x^3 + 600*x^2 + 1000*x + 625) + 513097/497664*sqrt(2*x^2 - x + 3)/(8*x^3 + 6
0*x^2 + 150*x + 125) - 16295969/71663616*sqrt(2*x^2 - x + 3)/(4*x^2 + 20*x + 25) + 26800085/1719926784*sqrt(2*
x^2 - x + 3)/(2*x + 5)

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

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)^5*(2*x^2 - x + 3)^(1/2)),x)

[Out]

int((x + 3*x^2 - x^3 + 5*x^4 + 2)/((2*x + 5)^5*(2*x^2 - x + 3)^(1/2)), x)

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

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)**5/(2*x**2-x+3)**(1/2),x)

[Out]

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

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