3.1.94 \(\int \frac {(2+3 x+5 x^2)^3}{(3-x+2 x^2)^{5/2}} \, dx\) [94]

Optimal. Leaf size=105 \[ -\frac {1331 (17-45 x)}{1104 \left (3-x+2 x^2\right )^{3/2}}+\frac {121 (10679-6744 x)}{8464 \sqrt {3-x+2 x^2}}+\frac {3175}{64} \sqrt {3-x+2 x^2}+\frac {125}{16} x \sqrt {3-x+2 x^2}-\frac {7495 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{128 \sqrt {2}} \]

[Out]

-1331/1104*(17-45*x)/(2*x^2-x+3)^(3/2)-7495/256*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)+121/8464*(10679-6744*x)
/(2*x^2-x+3)^(1/2)+3175/64*(2*x^2-x+3)^(1/2)+125/16*x*(2*x^2-x+3)^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1674, 1675, 654, 633, 221} \begin {gather*} \frac {121 (10679-6744 x)}{8464 \sqrt {2 x^2-x+3}}+\frac {125}{16} x \sqrt {2 x^2-x+3}+\frac {3175}{64} \sqrt {2 x^2-x+3}-\frac {1331 (17-45 x)}{1104 \left (2 x^2-x+3\right )^{3/2}}-\frac {7495 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{128 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1331*(17 - 45*x))/(1104*(3 - x + 2*x^2)^(3/2)) + (121*(10679 - 6744*x))/(8464*Sqrt[3 - x + 2*x^2]) + (3175*S
qrt[3 - x + 2*x^2])/64 + (125*x*Sqrt[3 - x + 2*x^2])/16 - (7495*ArcSinh[(1 - 4*x)/Sqrt[23]])/(128*Sqrt[2])

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 {\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^{5/2}} \, dx &=-\frac {1331 (17-45 x)}{1104 \left (3-x+2 x^2\right )^{3/2}}+\frac {2}{69} \int \frac {-\frac {91275}{64}-\frac {57201 x}{32}+\frac {66585 x^2}{16}+\frac {39675 x^3}{8}+\frac {8625 x^4}{4}}{\left (3-x+2 x^2\right )^{3/2}} \, dx\\ &=-\frac {1331 (17-45 x)}{1104 \left (3-x+2 x^2\right )^{3/2}}+\frac {121 (10679-6744 x)}{8464 \sqrt {3-x+2 x^2}}+\frac {4 \int \frac {\frac {1452105}{64}+\frac {277725 x}{8}+\frac {198375 x^2}{16}}{\sqrt {3-x+2 x^2}} \, dx}{1587}\\ &=-\frac {1331 (17-45 x)}{1104 \left (3-x+2 x^2\right )^{3/2}}+\frac {121 (10679-6744 x)}{8464 \sqrt {3-x+2 x^2}}+\frac {125}{16} x \sqrt {3-x+2 x^2}+\frac {\int \frac {\frac {214245}{4}+\frac {5038725 x}{32}}{\sqrt {3-x+2 x^2}} \, dx}{1587}\\ &=-\frac {1331 (17-45 x)}{1104 \left (3-x+2 x^2\right )^{3/2}}+\frac {121 (10679-6744 x)}{8464 \sqrt {3-x+2 x^2}}+\frac {3175}{64} \sqrt {3-x+2 x^2}+\frac {125}{16} x \sqrt {3-x+2 x^2}+\frac {7495}{128} \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx\\ &=-\frac {1331 (17-45 x)}{1104 \left (3-x+2 x^2\right )^{3/2}}+\frac {121 (10679-6744 x)}{8464 \sqrt {3-x+2 x^2}}+\frac {3175}{64} \sqrt {3-x+2 x^2}+\frac {125}{16} x \sqrt {3-x+2 x^2}+\frac {7495 \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{128 \sqrt {46}}\\ &=-\frac {1331 (17-45 x)}{1104 \left (3-x+2 x^2\right )^{3/2}}+\frac {121 (10679-6744 x)}{8464 \sqrt {3-x+2 x^2}}+\frac {3175}{64} \sqrt {3-x+2 x^2}+\frac {125}{16} x \sqrt {3-x+2 x^2}-\frac {7495 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{128 \sqrt {2}}\\ \end {align*}

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Mathematica [A]
time = 0.79, size = 75, normalized size = 0.71 \begin {gather*} \frac {89784565-62463282 x+101546529 x^2-29423976 x^3+16980900 x^4+3174000 x^5}{101568 \left (3-x+2 x^2\right )^{3/2}}-\frac {7495 \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{128 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(89784565 - 62463282*x + 101546529*x^2 - 29423976*x^3 + 16980900*x^4 + 3174000*x^5)/(101568*(3 - x + 2*x^2)^(3
/2)) - (7495*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/(128*Sqrt[2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(179\) vs. \(2(84)=168\).
time = 0.13, size = 180, normalized size = 1.71

method	result	size
risch	\(\frac {3174000 x^{5}[phone] x^{4}-29423976 x^{3}[phone] x^{2}-62463282 x [phone]}{101568 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {7495 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{256}\)	\(55\)
trager	\(\frac {3174000 x^{5}[phone] x^{4}-29423976 x^{3}[phone] x^{2}-62463282 x [phone]}{101568 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {7495 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \RootOf \left (\textit {\_Z}^{2}-2\right ) x -\RootOf \left (\textit {\_Z}^{2}-2\right )+4 \sqrt {2 x^{2}-x +3}\right )}{256}\)	\(82\)
default	\(-\frac {3391139 \left (4 x -1\right )}{203136 \sqrt {2 x^{2}-x +3}}+\frac {125 x^{5}}{4 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {2675 x^{4}}{16 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {7495 x^{3}}{192 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {222809 x^{2}}{256 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {281177 x}{2048 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {7495 x}{128 \sqrt {2 x^{2}-x +3}}-\frac {7495}{512 \sqrt {2 x^{2}-x +3}}+\frac {20961031}{24576 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {14081711 \left (4 x -1\right )}{565248 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {7495 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{256}\)	\(180\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^2+3*x+2)^3/(2*x^2-x+3)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-3391139/203136*(4*x-1)/(2*x^2-x+3)^(1/2)+125/4*x^5/(2*x^2-x+3)^(3/2)+2675/16*x^4/(2*x^2-x+3)^(3/2)-7495/192*x
^3/(2*x^2-x+3)^(3/2)+222809/256*x^2/(2*x^2-x+3)^(3/2)-281177/2048*x/(2*x^2-x+3)^(3/2)-7495/128*x/(2*x^2-x+3)^(
1/2)-7495/512/(2*x^2-x+3)^(1/2)[phone]/24576/(2*x^2-x+3)^(3/2)-14081711/565248*(4*x-1)/(2*x^2-x+3)^(3/2)+749
5/256*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (84) = 168\).
time = 0.52, size = 219, normalized size = 2.09 \begin {gather*} \frac {125 \, x^{5}}{4 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {2675 \, x^{4}}{16 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {7495}{203136} \, x {\left (\frac {284 \, x}{\sqrt {2 \, x^{2} - x + 3}} - \frac {3174 \, x^{2}}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {71}{\sqrt {2 \, x^{2} - x + 3}} + \frac {805 \, x}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {3243}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}\right )} + \frac {7495}{256} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {532145}{101568} \, \sqrt {2 \, x^{2} - x + 3} - \frac {4515389 \, x}{50784 \, \sqrt {2 \, x^{2} - x + 3}} + \frac {7197 \, x^{2}}{8 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {396211}{50784 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {269783 \, x}{1104 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {1002137}{1104 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/4*x^5/(2*x^2 - x + 3)^(3/2) + 2675/16*x^4/(2*x^2 - x + 3)^(3/2) + 7495/203136*x*(284*x/sqrt(2*x^2 - x + 3)
- 3174*x^2/(2*x^2 - x + 3)^(3/2) - 71/sqrt(2*x^2 - x + 3) + 805*x/(2*x^2 - x + 3)^(3/2) - 3243/(2*x^2 - x + 3
)^(3/2)) + 7495/256*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 532145/101568*sqrt(2*x^2 - x + 3) - 4515389/507
84*x/sqrt(2*x^2 - x + 3) + 7197/8*x^2/(2*x^2 - x + 3)^(3/2) + 396211/50784/sqrt(2*x^2 - x + 3) - 269783/1104*x
/(2*x^2 - x + 3)^(3/2) + 1002137/1104/(2*x^2 - x + 3)^(3/2)

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Fricas [A]
time = 1.76, size = 122, normalized size = 1.16 \begin {gather*} \frac {11894565 \, \sqrt {2} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 8 \, {\left (3174000 \, x^{5} + 16980900 \, x^{4} - 29423976 \, x^{3} + 101546529 \, x^{2} - 62463282 \, x + 89784565\right )} \sqrt {2 \, x^{2} - x + 3}}{812544 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/812544*(11894565*sqrt(2)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 3
2*x^2 + 16*x - 25) + 8*(3174000*x^5 + 16980900*x^4 - 29423976*x^3 + 101546529*x^2 - 62463282*x + 89784565)*sqr
t(2*x^2 - x + 3))/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (5 x^{2} + 3 x + 2\right )^{3}}{\left (2 x^{2} - x + 3\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 4.29, size = 72, normalized size = 0.69 \begin {gather*} -\frac {7495}{256} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac {3 \, {\left ({\left (4 \, {\left (13225 \, {\left (20 \, x + 107\right )} x - 2451998\right )} x + 33848843\right )} x - 20821094\right )} x + 89784565}{101568 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7495/256*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) + 1/101568*(3*((4*(13225*(20*x + 107)*
x - 2451998)*x + 33848843)*x - 20821094)*x + 89784565)/(2*x^2 - x + 3)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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