### 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]

(-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])

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Rubi [A]  time = 0.191478, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 40, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.1, 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 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 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 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 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])

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*}

Mathematica [A]  time = 0.136726, size = 81, normalized size = 0.58 $\frac{24 \sqrt{2 x^2-x+3} \left (214400680 x^3+43592076 x^2-255525906 x-298655447\right )+2053207 \sqrt{2} (2 x+5)^4 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )}{41278242816 (2 x+5)^4}$

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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Maple [A]  time = 0.07, size = 116, normalized size = 0.8 \begin{align*} -{\frac{16295969}{286654464}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-2}}+{\frac{26800085}{3439853568}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-1}}+{\frac{2053207\,\sqrt{2}}{41278242816}{\it Artanh} \left ({\frac{\sqrt{2}}{12} \left ({\frac{17}{2}}-11\,x \right ){\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}} \right ) }+{\frac{513097}{3981312}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-3}}-{\frac{3667}{36864}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-4}} \end{align*}

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]

-16295969/286654464/(x+5/2)^2*(2*(x+5/2)^2-11*x-19/2)^(1/2)+26800085/3439853568/(x+5/2)*(2*(x+5/2)^2-11*x-19/2
)^(1/2)+2053207/41278242816*2^(1/2)*arctanh(1/12*(17/2-11*x)*2^(1/2)/(2*(x+5/2)^2-11*x-19/2)^(1/2))+513097/398
1312/(x+5/2)^3*(2*(x+5/2)^2-11*x-19/2)^(1/2)-3667/36864/(x+5/2)^4*(2*(x+5/2)^2-11*x-19/2)^(1/2)

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Maxima [A]  time = 1.56057, size = 201, normalized size = 1.45 \begin{align*} -\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 )}} \end{align*}

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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Fricas [A]  time = 1.33975, size = 405, normalized size = 2.91 \begin{align*} \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 )}} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{5 \, x^{4} - x^{3} + 3 \, x^{2} + x + 2}{\sqrt{2 \, x^{2} - x + 3}{\left (2 \, x + 5\right )}^{5}}\,{d x} \end{align*}

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]

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