3.1.75 \(\int (3-x+2 x^2)^{5/2} (2+3 x+5 x^2) \, dx\) [75]

Optimal. Leaf size=128 \[ -\frac {732665 (1-4 x) \sqrt {3-x+2 x^2}}{524288}-\frac {31855 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{98304}-\frac {277 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{3072}+\frac {141}{448} \left (3-x+2 x^2\right )^{7/2}+\frac {5}{16} x \left (3-x+2 x^2\right )^{7/2}-\frac {16851295 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{1048576 \sqrt {2}} \]

[Out]

-31855/98304*(1-4*x)*(2*x^2-x+3)^(3/2)-277/3072*(1-4*x)*(2*x^2-x+3)^(5/2)+141/448*(2*x^2-x+3)^(7/2)+5/16*x*(2*
x^2-x+3)^(7/2)-16851295/2097152*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)-732665/524288*(1-4*x)*(2*x^2-x+3)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1675, 654, 626, 633, 221} \begin {gather*} \frac {5}{16} x \left (2 x^2-x+3\right )^{7/2}+\frac {141}{448} \left (2 x^2-x+3\right )^{7/2}-\frac {277 (1-4 x) \left (2 x^2-x+3\right )^{5/2}}{3072}-\frac {31855 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{98304}-\frac {732665 (1-4 x) \sqrt {2 x^2-x+3}}{524288}-\frac {16851295 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{1048576 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-732665*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/524288 - (31855*(1 - 4*x)*(3 - x + 2*x^2)^(3/2))/98304 - (277*(1 - 4*x
)*(3 - x + 2*x^2)^(5/2))/3072 + (141*(3 - x + 2*x^2)^(7/2))/448 + (5*x*(3 - x + 2*x^2)^(7/2))/16 - (16851295*A
rcSinh[(1 - 4*x)/Sqrt[23]])/(1048576*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 626

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

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 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 \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right ) \, dx &=\frac {5}{16} x \left (3-x+2 x^2\right )^{7/2}+\frac {1}{16} \int \left (17+\frac {141 x}{2}\right ) \left (3-x+2 x^2\right )^{5/2} \, dx\\ &=\frac {141}{448} \left (3-x+2 x^2\right )^{7/2}+\frac {5}{16} x \left (3-x+2 x^2\right )^{7/2}+\frac {277}{128} \int \left (3-x+2 x^2\right )^{5/2} \, dx\\ &=-\frac {277 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{3072}+\frac {141}{448} \left (3-x+2 x^2\right )^{7/2}+\frac {5}{16} x \left (3-x+2 x^2\right )^{7/2}+\frac {31855 \int \left (3-x+2 x^2\right )^{3/2} \, dx}{6144}\\ &=-\frac {31855 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{98304}-\frac {277 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{3072}+\frac {141}{448} \left (3-x+2 x^2\right )^{7/2}+\frac {5}{16} x \left (3-x+2 x^2\right )^{7/2}+\frac {732665 \int \sqrt {3-x+2 x^2} \, dx}{65536}\\ &=-\frac {732665 (1-4 x) \sqrt {3-x+2 x^2}}{524288}-\frac {31855 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{98304}-\frac {277 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{3072}+\frac {141}{448} \left (3-x+2 x^2\right )^{7/2}+\frac {5}{16} x \left (3-x+2 x^2\right )^{7/2}+\frac {16851295 \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx}{1048576}\\ &=-\frac {732665 (1-4 x) \sqrt {3-x+2 x^2}}{524288}-\frac {31855 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{98304}-\frac {277 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{3072}+\frac {141}{448} \left (3-x+2 x^2\right )^{7/2}+\frac {5}{16} x \left (3-x+2 x^2\right )^{7/2}+\frac {\left (732665 \sqrt {\frac {23}{2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{1048576}\\ &=-\frac {732665 (1-4 x) \sqrt {3-x+2 x^2}}{524288}-\frac {31855 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{98304}-\frac {277 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{3072}+\frac {141}{448} \left (3-x+2 x^2\right )^{7/2}+\frac {5}{16} x \left (3-x+2 x^2\right )^{7/2}-\frac {16851295 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{1048576 \sqrt {2}}\\ \end {align*}

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Mathematica [A]
time = 0.56, size = 85, normalized size = 0.66 \begin {gather*} \frac {4 \sqrt {3-x+2 x^2} \left (58536675+148957444 x+67272352 x^2+172684416 x^3-1619968 x^4+118808576 x^5-13565952 x^6+27525120 x^7\right )-353877195 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{44040192} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(58536675 + 148957444*x + 67272352*x^2 + 172684416*x^3 - 1619968*x^4 + 118808576*x^5 -
13565952*x^6 + 27525120*x^7) - 353877195*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/44040192

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Maple [A]
time = 0.10, size = 102, normalized size = 0.80

method	result	size
risch	\(\frac {\left (27525120 x^{7}-13565952 x^{6}[phone] x^{5}-1619968 x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {2 x^{2}-x +3}}{11010048}+\frac {16851295 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{2097152}\)	\(65\)
trager	\(\left (\frac {5}{2} x^{7}-\frac {69}{56} x^{6}+\frac {14503}{1344} x^{5}-\frac {113}{768} x^{4}+\frac {449699}{28672} x^{3}+\frac {300323}{49152} x^{2}+\frac {37239361}{2752512} x +\frac {19512225}{3670016}\right ) \sqrt {2 x^{2}-x +3}+\frac {16851295 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \RootOf \left (\textit {\_Z}^{2}-2\right ) x -\RootOf \left (\textit {\_Z}^{2}-2\right )+4 \sqrt {2 x^{2}-x +3}\right )}{2097152}\)	\(91\)
default	\(\frac {5 x \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{16}+\frac {141 \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{448}+\frac {277 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{3072}+\frac {31855 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{98304}+\frac {732665 \left (4 x -1\right ) \sqrt {2 x^{2}-x +3}}{524288}+\frac {16851295 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{2097152}\)	\(102\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/16*x*(2*x^2-x+3)^(7/2)+141/448*(2*x^2-x+3)^(7/2)+277/3072*(4*x-1)*(2*x^2-x+3)^(5/2)+31855/98304*(4*x-1)*(2*x
^2-x+3)^(3/2)+732665/524288*(4*x-1)*(2*x^2-x+3)^(1/2)[phone]/2097152*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))

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Maxima [A]
time = 0.50, size = 133, normalized size = 1.04 \begin {gather*} \frac {5}{16} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x + \frac {141}{448} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} + \frac {277}{768} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x - \frac {277}{3072} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {31855}{24576} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {31855}{98304} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {732665}{131072} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {16851295}{2097152} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {732665}{524288} \, \sqrt {2 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/16*(2*x^2 - x + 3)^(7/2)*x + 141/448*(2*x^2 - x + 3)^(7/2) + 277/768*(2*x^2 - x + 3)^(5/2)*x - 277/3072*(2*x
^2 - x + 3)^(5/2) + 31855/24576*(2*x^2 - x + 3)^(3/2)*x - 31855/98304*(2*x^2 - x + 3)^(3/2) + 732665/131072*sq
rt(2*x^2 - x + 3)*x + 16851295/2097152*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 732665/524288*sqrt(2*x^2 - x
+ 3)

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Fricas [A]
time = 3.21, size = 88, normalized size = 0.69 \begin {gather*} \frac {1}{11010048} \, {\left (27525120 \, x^{7} - 13565952 \, x^{6} + 118808576 \, x^{5} - 1619968 \, x^{4} + 172684416 \, x^{3} + 67272352 \, x^{2} + 148957444 \, x + 58536675\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {16851295}{4194304} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11010048*(27525120*x^7 - 13565952*x^6 + 118808576*x^5 - 1619968*x^4 + 172684416*x^3 + 67272352*x^2 + 1489574
44*x + 58536675)*sqrt(2*x^2 - x + 3) + 16851295/4194304*sqrt(2)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) -
32*x^2 + 16*x - 25)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 3.20, size = 83, normalized size = 0.65 \begin {gather*} \frac {1}{11010048} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (4 \, {\left (24 \, {\left (140 \, x - 69\right )} x + 14503\right )} x - 791\right )} x + 1349097\right )} x + 2102261\right )} x + 37239361\right )} x + 58536675\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {16851295}{2097152} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11010048*(4*(8*(4*(16*(4*(24*(140*x - 69)*x + 14503)*x - 791)*x + 1349097)*x + 2102261)*x + 37239361)*x + 58
536675)*sqrt(2*x^2 - x + 3) - 16851295/2097152*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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