3.11.37 \(\int (2-5 x) \sqrt {x} \sqrt {2+5 x+3 x^2} \, dx\) [1037]

Optimal. Leaf size=182 \[ -\frac {2476 \sqrt {x} (2+3 x)}{2835 \sqrt {2+5 x+3 x^2}}+\frac {4}{945} \sqrt {x} (430+639 x) \sqrt {2+5 x+3 x^2}-\frac {10}{21} \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}+\frac {2476 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{2835 \sqrt {2+5 x+3 x^2}}-\frac {164 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{189 \sqrt {2+5 x+3 x^2}} \]

[Out]

-10/21*(3*x^2+5*x+2)^(3/2)*x^(1/2)-2476/2835*(2+3*x)*x^(1/2)/(3*x^2+5*x+2)^(1/2)+2476/2835*(1+x)^(3/2)*(1/(1+x
))^(1/2)*EllipticE(x^(1/2)/(1+x)^(1/2),1/2*I*2^(1/2))*2^(1/2)*((2+3*x)/(1+x))^(1/2)/(3*x^2+5*x+2)^(1/2)-164/18
9*(1+x)^(3/2)*(1/(1+x))^(1/2)*EllipticF(x^(1/2)/(1+x)^(1/2),1/2*I*2^(1/2))*2^(1/2)*((2+3*x)/(1+x))^(1/2)/(3*x^
2+5*x+2)^(1/2)+4/945*(430+639*x)*x^(1/2)*(3*x^2+5*x+2)^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {846, 828, 853, 1203, 1114, 1150} \begin {gather*} -\frac {164 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\text {ArcTan}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{189 \sqrt {3 x^2+5 x+2}}+\frac {2476 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\text {ArcTan}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{2835 \sqrt {3 x^2+5 x+2}}-\frac {10}{21} \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}+\frac {4}{945} \sqrt {x} (639 x+430) \sqrt {3 x^2+5 x+2}-\frac {2476 \sqrt {x} (3 x+2)}{2835 \sqrt {3 x^2+5 x+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2476*Sqrt[x]*(2 + 3*x))/(2835*Sqrt[2 + 5*x + 3*x^2]) + (4*Sqrt[x]*(430 + 639*x)*Sqrt[2 + 5*x + 3*x^2])/945 -
(10*Sqrt[x]*(2 + 5*x + 3*x^2)^(3/2))/21 + (2476*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt
[x]], -1/2])/(2835*Sqrt[2 + 5*x + 3*x^2]) - (164*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt
[x]], -1/2])/(189*Sqrt[2 + 5*x + 3*x^2])

Rule 828

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 846

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 853

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 1114

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)]/(2*a*Rt[(b - q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]))*Elli
pticF[ArcTan[Rt[(b - q)/(2*a), 2]*x], -2*(q/(b - q))], x] /; PosQ[(b - q)/a]] /; FreeQ[{a, b, c}, x] && GtQ[b^
2 - 4*a*c, 0]

Rule 1150

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)]/(2*c*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[ArcTan[Rt[(b - q)/(2*a), 2]*x], -2*(
q/(b - q))], x] /; PosQ[(b - q)/a]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]

Rule 1203

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]

Rubi steps

\begin {align*} \int (2-5 x) \sqrt {x} \sqrt {2+5 x+3 x^2} \, dx &=-\frac {10}{21} \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}+\frac {2}{21} \int \frac {(5+71 x) \sqrt {2+5 x+3 x^2}}{\sqrt {x}} \, dx\\ &=\frac {4}{945} \sqrt {x} (430+639 x) \sqrt {2+5 x+3 x^2}-\frac {10}{21} \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}-\frac {4}{945} \int \frac {205+\frac {619 x}{2}}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {4}{945} \sqrt {x} (430+639 x) \sqrt {2+5 x+3 x^2}-\frac {10}{21} \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}-\frac {8}{945} \text {Subst}\left (\int \frac {205+\frac {619 x^2}{2}}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=\frac {4}{945} \sqrt {x} (430+639 x) \sqrt {2+5 x+3 x^2}-\frac {10}{21} \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}-\frac {328}{189} \text {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )-\frac {2476}{945} \text {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2476 \sqrt {x} (2+3 x)}{2835 \sqrt {2+5 x+3 x^2}}+\frac {4}{945} \sqrt {x} (430+639 x) \sqrt {2+5 x+3 x^2}-\frac {10}{21} \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}+\frac {2476 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{2835 \sqrt {2+5 x+3 x^2}}-\frac {164 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{189 \sqrt {2+5 x+3 x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 20.14, size = 163, normalized size = 0.90 \begin {gather*} \frac {-2 \left (2476+3730 x-3354 x^2-1935 x^3+8748 x^4+6075 x^5\right )-2476 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+16 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )}{2835 \sqrt {x} \sqrt {2+5 x+3 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(2476 + 3730*x - 3354*x^2 - 1935*x^3 + 8748*x^4 + 6075*x^5) - (2476*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2
/x]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] + (16*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/
2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(2835*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]
time = 0.73, size = 122, normalized size = 0.67

method	result	size
default	\(\frac {-\frac {30 x^{5}}{7}+\frac {418 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{2835}-\frac {1238 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{8505}-\frac {216 x^{4}}{35}+\frac {86 x^{3}}{63}+\frac {4712 x^{2}}{945}+\frac {328 x}{189}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(122\)
risch	\(-\frac {2 \left (675 x^{2}-153 x -410\right ) \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{945}-\frac {\left (\frac {164 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{567 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {1238 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{2835 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right ) \sqrt {x \left (3 x^{2}+5 x +2\right )}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(188\)
elliptic	\(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {10 x^{2} \sqrt {3 x^{3}+5 x^{2}+2 x}}{7}+\frac {34 x \sqrt {3 x^{3}+5 x^{2}+2 x}}{105}+\frac {164 \sqrt {3 x^{3}+5 x^{2}+2 x}}{189}-\frac {164 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{567 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {1238 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{2835 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(217\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/8505/x^(1/2)/(3*x^2+5*x+2)^(1/2)*(-18225*x^5+627*(6*x+4)^(1/2)*(3*x+3)^(1/2)*6^(1/2)*(-x)^(1/2)*EllipticF(1/
2*(6*x+4)^(1/2),I*2^(1/2))-619*(6*x+4)^(1/2)*(3*x+3)^(1/2)*6^(1/2)*(-x)^(1/2)*EllipticE(1/2*(6*x+4)^(1/2),I*2^
(1/2))-26244*x^4+5805*x^3+21204*x^2+7380*x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.86, size = 53, normalized size = 0.29 \begin {gather*} -\frac {2}{945} \, {\left (675 \, x^{2} - 153 \, x - 410\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x} - \frac {68}{729} \, \sqrt {3} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) + \frac {2476}{2835} \, \sqrt {3} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/945*(675*x^2 - 153*x - 410)*sqrt(3*x^2 + 5*x + 2)*sqrt(x) - 68/729*sqrt(3)*weierstrassPInverse(28/27, 80/72
9, x + 5/9) + 2476/2835*sqrt(3)*weierstrassZeta(28/27, 80/729, weierstrassPInverse(28/27, 80/729, x + 5/9))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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