3.3.46 \(\int \frac {(1+2 x)^3 (1+3 x+4 x^2)}{(2-x+3 x^2)^{3/2}} \, dx\) [246]

Optimal. Leaf size=103 \[ \frac {2 (12839-3871 x)}{1863 \sqrt {2-x+3 x^2}}+\frac {746}{81} \sqrt {2-x+3 x^2}+\frac {412}{81} x \sqrt {2-x+3 x^2}+\frac {32}{27} x^2 \sqrt {2-x+3 x^2}+\frac {353 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{81 \sqrt {3}} \]

[Out]

353/243*arcsinh(1/23*(1-6*x)*23^(1/2))*3^(1/2)+2/1863*(12839-3871*x)/(3*x^2-x+2)^(1/2)+746/81*(3*x^2-x+2)^(1/2
)+412/81*x*(3*x^2-x+2)^(1/2)+32/27*x^2*(3*x^2-x+2)^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {1674, 1675, 654, 633, 221} \begin {gather*} \frac {32}{27} \sqrt {3 x^2-x+2} x^2+\frac {412}{81} \sqrt {3 x^2-x+2} x+\frac {746}{81} \sqrt {3 x^2-x+2}+\frac {2 (12839-3871 x)}{1863 \sqrt {3 x^2-x+2}}+\frac {353 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{81 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 + 2*x)^3*(1 + 3*x + 4*x^2))/(2 - x + 3*x^2)^(3/2),x]

[Out]

(2*(12839 - 3871*x))/(1863*Sqrt[2 - x + 3*x^2]) + (746*Sqrt[2 - x + 3*x^2])/81 + (412*x*Sqrt[2 - x + 3*x^2])/8
1 + (32*x^2*Sqrt[2 - x + 3*x^2])/27 + (353*ArcSinh[(1 - 6*x)/Sqrt[23]])/(81*Sqrt[3])

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 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 1674

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*
x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[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[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (
2*p + 3)*(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[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 \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{3/2}} \, dx &=\frac {2 (12839-3871 x)}{1863 \sqrt {2-x+3 x^2}}+\frac {2}{23} \int \frac {\frac {1127}{81}+\frac {7682 x}{27}+\frac {2852 x^2}{9}+\frac {368 x^3}{3}}{\sqrt {2-x+3 x^2}} \, dx\\ &=\frac {2 (12839-3871 x)}{1863 \sqrt {2-x+3 x^2}}+\frac {32}{27} x^2 \sqrt {2-x+3 x^2}+\frac {2}{207} \int \frac {\frac {1127}{9}+2070 x+\frac {9476 x^2}{3}}{\sqrt {2-x+3 x^2}} \, dx\\ &=\frac {2 (12839-3871 x)}{1863 \sqrt {2-x+3 x^2}}+\frac {412}{81} x \sqrt {2-x+3 x^2}+\frac {32}{27} x^2 \sqrt {2-x+3 x^2}+\frac {1}{621} \int \frac {-5566+17158 x}{\sqrt {2-x+3 x^2}} \, dx\\ &=\frac {2 (12839-3871 x)}{1863 \sqrt {2-x+3 x^2}}+\frac {746}{81} \sqrt {2-x+3 x^2}+\frac {412}{81} x \sqrt {2-x+3 x^2}+\frac {32}{27} x^2 \sqrt {2-x+3 x^2}-\frac {353}{81} \int \frac {1}{\sqrt {2-x+3 x^2}} \, dx\\ &=\frac {2 (12839-3871 x)}{1863 \sqrt {2-x+3 x^2}}+\frac {746}{81} \sqrt {2-x+3 x^2}+\frac {412}{81} x \sqrt {2-x+3 x^2}+\frac {32}{27} x^2 \sqrt {2-x+3 x^2}-\frac {353 \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+6 x\right )}{81 \sqrt {69}}\\ &=\frac {2 (12839-3871 x)}{1863 \sqrt {2-x+3 x^2}}+\frac {746}{81} \sqrt {2-x+3 x^2}+\frac {412}{81} x \sqrt {2-x+3 x^2}+\frac {32}{27} x^2 \sqrt {2-x+3 x^2}+\frac {353 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{81 \sqrt {3}}\\ \end {align*}

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Mathematica [A]
time = 0.49, size = 70, normalized size = 0.68 \begin {gather*} \frac {\frac {6 \left (29997-2974 x+23207 x^2+13110 x^3+3312 x^4\right )}{\sqrt {2-x+3 x^2}}+8119 \sqrt {3} \log \left (1-6 x+2 \sqrt {6-3 x+9 x^2}\right )}{5589} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + 2*x)^3*(1 + 3*x + 4*x^2))/(2 - x + 3*x^2)^(3/2),x]

[Out]

((6*(29997 - 2974*x + 23207*x^2 + 13110*x^3 + 3312*x^4))/Sqrt[2 - x + 3*x^2] + 8119*Sqrt[3]*Log[1 - 6*x + 2*Sq
rt[6 - 3*x + 9*x^2]])/5589

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Maple [A]
time = 0.11, size = 115, normalized size = 1.12

method	result	size
risch	\(\frac {\frac {32}{9} x^{4}+\frac {380}{27} x^{3}+\frac {2018}{81} x^{2}-\frac {5948}{1863} x +\frac {2222}{69}}{\sqrt {3 x^{2}-x +2}}-\frac {353 \sqrt {3}\, \arcsinh \left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{243}\)	\(50\)
trager	\(\frac {\frac {32}{9} x^{4}+\frac {380}{27} x^{3}+\frac {2018}{81} x^{2}-\frac {5948}{1863} x +\frac {2222}{69}}{\sqrt {3 x^{2}-x +2}}-\frac {353 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}-x +2}-\RootOf \left (\textit {\_Z}^{2}-3\right )\right )}{243}\)	\(77\)
default	\(\frac {557}{18 \sqrt {3 x^{2}-x +2}}-\frac {353 \sqrt {3}\, \arcsinh \left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{243}+\frac {2018 x^{2}}{81 \sqrt {3 x^{2}-x +2}}+\frac {353 x}{81 \sqrt {3 x^{2}-x +2}}+\frac {32 x^{4}}{9 \sqrt {3 x^{2}-x +2}}+\frac {380 x^{3}}{27 \sqrt {3 x^{2}-x +2}}-\frac {521 \left (6 x -1\right )}{414 \sqrt {3 x^{2}-x +2}}\)	\(115\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x+1)^3*(4*x^2+3*x+1)/(3*x^2-x+2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

557/18/(3*x^2-x+2)^(1/2)-353/243*3^(1/2)*arcsinh(6/23*23^(1/2)*(x-1/6))+2018/81*x^2/(3*x^2-x+2)^(1/2)+353/81*x
/(3*x^2-x+2)^(1/2)+32/9*x^4/(3*x^2-x+2)^(1/2)+380/27*x^3/(3*x^2-x+2)^(1/2)-521/414*(6*x-1)/(3*x^2-x+2)^(1/2)

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Maxima [A]
time = 0.51, size = 97, normalized size = 0.94 \begin {gather*} \frac {32 \, x^{4}}{9 \, \sqrt {3 \, x^{2} - x + 2}} + \frac {380 \, x^{3}}{27 \, \sqrt {3 \, x^{2} - x + 2}} + \frac {2018 \, x^{2}}{81 \, \sqrt {3 \, x^{2} - x + 2}} - \frac {353}{243} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (6 \, x - 1\right )}\right ) - \frac {5948 \, x}{1863 \, \sqrt {3 \, x^{2} - x + 2}} + \frac {2222}{69 \, \sqrt {3 \, x^{2} - x + 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)^3*(4*x^2+3*x+1)/(3*x^2-x+2)^(3/2),x, algorithm="maxima")

[Out]

32/9*x^4/sqrt(3*x^2 - x + 2) + 380/27*x^3/sqrt(3*x^2 - x + 2) + 2018/81*x^2/sqrt(3*x^2 - x + 2) - 353/243*sqrt
(3)*arcsinh(1/23*sqrt(23)*(6*x - 1)) - 5948/1863*x/sqrt(3*x^2 - x + 2) + 2222/69/sqrt(3*x^2 - x + 2)

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Fricas [A]
time = 0.37, size = 97, normalized size = 0.94 \begin {gather*} \frac {8119 \, \sqrt {3} {\left (3 \, x^{2} - x + 2\right )} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} - x + 2} {\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) + 12 \, {\left (3312 \, x^{4} + 13110 \, x^{3} + 23207 \, x^{2} - 2974 \, x + 29997\right )} \sqrt {3 \, x^{2} - x + 2}}{11178 \, {\left (3 \, x^{2} - x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)^3*(4*x^2+3*x+1)/(3*x^2-x+2)^(3/2),x, algorithm="fricas")

[Out]

1/11178*(8119*sqrt(3)*(3*x^2 - x + 2)*log(4*sqrt(3)*sqrt(3*x^2 - x + 2)*(6*x - 1) - 72*x^2 + 24*x - 25) + 12*(
3312*x^4 + 13110*x^3 + 23207*x^2 - 2974*x + 29997)*sqrt(3*x^2 - x + 2))/(3*x^2 - x + 2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2 x + 1\right )^{3} \cdot \left (4 x^{2} + 3 x + 1\right )}{\left (3 x^{2} - x + 2\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)**3*(4*x**2+3*x+1)/(3*x**2-x+2)**(3/2),x)

[Out]

Integral((2*x + 1)**3*(4*x**2 + 3*x + 1)/(3*x**2 - x + 2)**(3/2), x)

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Giac [A]
time = 4.12, size = 67, normalized size = 0.65 \begin {gather*} \frac {353}{243} \, \sqrt {3} \log \left (-2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} - x + 2}\right )} + 1\right ) + \frac {2 \, {\left ({\left (23 \, {\left (6 \, {\left (24 \, x + 95\right )} x + 1009\right )} x - 2974\right )} x + 29997\right )}}{1863 \, \sqrt {3 \, x^{2} - x + 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)^3*(4*x^2+3*x+1)/(3*x^2-x+2)^(3/2),x, algorithm="giac")

[Out]

353/243*sqrt(3)*log(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 - x + 2)) + 1) + 2/1863*((23*(6*(24*x + 95)*x + 1009)*x
- 2974)*x + 29997)/sqrt(3*x^2 - x + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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