3.1035 \(\int (2-5 x) x^{5/2} \sqrt{2+5 x+3 x^2} \, dx\)

Optimal. Leaf size=228 \[ -\frac{13016 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{x}\right ),-\frac{1}{2}\right )}{56133 \sqrt{3 x^2+5 x+2}}-\frac{10}{33} \left (3 x^2+5 x+2\right )^{3/2} x^{5/2}+\frac{532}{891} \left (3 x^2+5 x+2\right )^{3/2} x^{3/2}-\frac{4420 \left (3 x^2+5 x+2\right )^{3/2} \sqrt{x}}{6237}+\frac{8 (74313 x+57860) \sqrt{3 x^2+5 x+2} \sqrt{x}}{280665}-\frac{261784 (3 x+2) \sqrt{x}}{841995 \sqrt{3 x^2+5 x+2}}+\frac{261784 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{841995 \sqrt{3 x^2+5 x+2}} \]

[Out]

(-261784*Sqrt[x]*(2 + 3*x))/(841995*Sqrt[2 + 5*x + 3*x^2]) + (8*Sqrt[x]*(57860 + 74313*x)*Sqrt[2 + 5*x + 3*x^2
])/280665 - (4420*Sqrt[x]*(2 + 5*x + 3*x^2)^(3/2))/6237 + (532*x^(3/2)*(2 + 5*x + 3*x^2)^(3/2))/891 - (10*x^(5
/2)*(2 + 5*x + 3*x^2)^(3/2))/33 + (261784*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -
1/2])/(841995*Sqrt[2 + 5*x + 3*x^2]) - (13016*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]
], -1/2])/(56133*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.173006, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {832, 814, 839, 1189, 1100, 1136} \[ -\frac{10}{33} \left (3 x^2+5 x+2\right )^{3/2} x^{5/2}+\frac{532}{891} \left (3 x^2+5 x+2\right )^{3/2} x^{3/2}-\frac{4420 \left (3 x^2+5 x+2\right )^{3/2} \sqrt{x}}{6237}+\frac{8 (74313 x+57860) \sqrt{3 x^2+5 x+2} \sqrt{x}}{280665}-\frac{261784 (3 x+2) \sqrt{x}}{841995 \sqrt{3 x^2+5 x+2}}-\frac{13016 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{56133 \sqrt{3 x^2+5 x+2}}+\frac{261784 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{841995 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-261784*Sqrt[x]*(2 + 3*x))/(841995*Sqrt[2 + 5*x + 3*x^2]) + (8*Sqrt[x]*(57860 + 74313*x)*Sqrt[2 + 5*x + 3*x^2
])/280665 - (4420*Sqrt[x]*(2 + 5*x + 3*x^2)^(3/2))/6237 + (532*x^(3/2)*(2 + 5*x + 3*x^2)^(3/2))/891 - (10*x^(5
/2)*(2 + 5*x + 3*x^2)^(3/2))/33 + (261784*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -
1/2])/(841995*Sqrt[2 + 5*x + 3*x^2]) - (13016*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]
], -1/2])/(56133*Sqrt[2 + 5*x + 3*x^2])

Rule 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 814

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*(a + b*x + c*x^
2)^p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 839

Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2, Subst[Int[(f +
g*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x, Sqrt[x]], x] /; FreeQ[{a, b, c, f, g}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1189

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}
, Dist[d, Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[e, Int[x^2/Sqrt[a + b*x^2 + c*x^4], x], x] /; PosQ[(b +
q)/a] || PosQ[(b - q)/a]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]

Rule 1100

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[((2*a + (b -
q)*x^2)*Sqrt[(2*a + (b + q)*x^2)/(2*a + (b - q)*x^2)]*EllipticF[ArcTan[Rt[(b - q)/(2*a), 2]*x], (-2*q)/(b - q)
])/(2*a*Rt[(b - q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]), x] /; PosQ[(b - q)/a]] /; FreeQ[{a, b, c}, x] && GtQ[b^
2 - 4*a*c, 0]

Rule 1136

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(x*(b -
q + 2*c*x^2))/(2*c*Sqrt[a + b*x^2 + c*x^4]), x] - Simp[(Rt[(b - q)/(2*a), 2]*(2*a + (b - q)*x^2)*Sqrt[(2*a + (
b + q)*x^2)/(2*a + (b - q)*x^2)]*EllipticE[ArcTan[Rt[(b - q)/(2*a), 2]*x], (-2*q)/(b - q)])/(2*c*Sqrt[a + b*x^
2 + c*x^4]), x] /; PosQ[(b - q)/a]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]

Rubi steps

\begin{align*} \int (2-5 x) x^{5/2} \sqrt{2+5 x+3 x^2} \, dx &=-\frac{10}{33} x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac{2}{33} \int x^{3/2} (25+133 x) \sqrt{2+5 x+3 x^2} \, dx\\ &=\frac{532}{891} x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac{10}{33} x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac{4}{891} \int \left (-399-\frac{3315 x}{2}\right ) \sqrt{x} \sqrt{2+5 x+3 x^2} \, dx\\ &=-\frac{4420 \sqrt{x} \left (2+5 x+3 x^2\right )^{3/2}}{6237}+\frac{532}{891} x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac{10}{33} x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac{8 \int \frac{\left (\frac{3315}{2}+\frac{24771 x}{2}\right ) \sqrt{2+5 x+3 x^2}}{\sqrt{x}} \, dx}{18711}\\ &=\frac{8 \sqrt{x} (57860+74313 x) \sqrt{2+5 x+3 x^2}}{280665}-\frac{4420 \sqrt{x} \left (2+5 x+3 x^2\right )^{3/2}}{6237}+\frac{532}{891} x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac{10}{33} x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac{16 \int \frac{\frac{24405}{2}+\frac{98169 x}{4}}{\sqrt{x} \sqrt{2+5 x+3 x^2}} \, dx}{841995}\\ &=\frac{8 \sqrt{x} (57860+74313 x) \sqrt{2+5 x+3 x^2}}{280665}-\frac{4420 \sqrt{x} \left (2+5 x+3 x^2\right )^{3/2}}{6237}+\frac{532}{891} x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac{10}{33} x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac{32 \operatorname{Subst}\left (\int \frac{\frac{24405}{2}+\frac{98169 x^2}{4}}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )}{841995}\\ &=\frac{8 \sqrt{x} (57860+74313 x) \sqrt{2+5 x+3 x^2}}{280665}-\frac{4420 \sqrt{x} \left (2+5 x+3 x^2\right )^{3/2}}{6237}+\frac{532}{891} x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac{10}{33} x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac{26032 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )}{56133}-\frac{261784 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{2+5 x^2+3 x^4}} \, dx,x,\sqrt{x}\right )}{280665}\\ &=-\frac{261784 \sqrt{x} (2+3 x)}{841995 \sqrt{2+5 x+3 x^2}}+\frac{8 \sqrt{x} (57860+74313 x) \sqrt{2+5 x+3 x^2}}{280665}-\frac{4420 \sqrt{x} \left (2+5 x+3 x^2\right )^{3/2}}{6237}+\frac{532}{891} x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac{10}{33} x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}+\frac{261784 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{841995 \sqrt{2+5 x+3 x^2}}-\frac{13016 \sqrt{2} (1+x) \sqrt{\frac{2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{56133 \sqrt{2+5 x+3 x^2}}\\ \end{align*}

Mathematica [C]  time = 0.18365, size = 170, normalized size = 0.75 \[ \frac{66544 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right ),\frac{3}{2}\right )-2296350 x^7-3129840 x^6+271350 x^5+947916 x^4+39780 x^3-198168 x^2-261784 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-918440 x-523568}{841995 \sqrt{x} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-523568 - 918440*x - 198168*x^2 + 39780*x^3 + 947916*x^4 + 271350*x^5 - 3129840*x^6 - 2296350*x^7 - (261784*I
)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] + (66544*I)*Sqrt
[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(841995*Sqrt[x]*Sqrt[
2 + 5*x + 3*x^2])

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Maple [A]  time = 0.02, size = 132, normalized size = 0.6 \begin{align*}{\frac{2}{2525985} \left ( -3444525\,{x}^{7}-4694760\,{x}^{6}+407025\,{x}^{5}+98718\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{6}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -65446\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{6}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) +1421874\,{x}^{4}+59670\,{x}^{3}+880776\,{x}^{2}+585720\,x \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/2525985/x^(1/2)/(3*x^2+5*x+2)^(1/2)*(-3444525*x^7-4694760*x^6+407025*x^5+98718*(6*x+4)^(1/2)*(3+3*x)^(1/2)*6
^(1/2)*(-x)^(1/2)*EllipticF(1/2*(6*x+4)^(1/2),I*2^(1/2))-65446*(6*x+4)^(1/2)*(3+3*x)^(1/2)*6^(1/2)*(-x)^(1/2)*
EllipticE(1/2*(6*x+4)^(1/2),I*2^(1/2))+1421874*x^4+59670*x^3+880776*x^2+585720*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (5 \, x^{3} - 2 \, x^{2}\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(5*x^3 - 2*x^2)*sqrt(3*x^2 + 5*x + 2)*sqrt(x), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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