3.11.79 \(\int \frac {2-5 x}{\sqrt {x} (2+5 x+3 x^2)^{5/2}} \, dx\) [1079]

Optimal. Leaf size=185 \[ \frac {2 \sqrt {x} (38+45 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {715 \sqrt {x} (2+3 x)}{3 \sqrt {2+5 x+3 x^2}}-\frac {5 \sqrt {x} (361+429 x)}{3 \sqrt {2+5 x+3 x^2}}-\frac {715 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {2+5 x+3 x^2}}+\frac {295 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{\sqrt {2+5 x+3 x^2}} \]

[Out]

2/3*(38+45*x)*x^(1/2)/(3*x^2+5*x+2)^(3/2)+715/3*(2+3*x)*x^(1/2)/(3*x^2+5*x+2)^(1/2)-5/3*(361+429*x)*x^(1/2)/(3
*x^2+5*x+2)^(1/2)-715/3*(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)+295*(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)

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Rubi [A]
time = 0.08, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {836, 853, 1203, 1114, 1150} \begin {gather*} \frac {295 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\text {ArcTan}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{\sqrt {3 x^2+5 x+2}}-\frac {715 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\text {ArcTan}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}+\frac {715 \sqrt {x} (3 x+2)}{3 \sqrt {3 x^2+5 x+2}}-\frac {5 \sqrt {x} (429 x+361)}{3 \sqrt {3 x^2+5 x+2}}+\frac {2 \sqrt {x} (45 x+38)}{3 \left (3 x^2+5 x+2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(38 + 45*x))/(3*(2 + 5*x + 3*x^2)^(3/2)) + (715*Sqrt[x]*(2 + 3*x))/(3*Sqrt[2 + 5*x + 3*x^2]) - (5*S
qrt[x]*(361 + 429*x))/(3*Sqrt[2 + 5*x + 3*x^2]) - (715*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTa
n[Sqrt[x]], -1/2])/(3*Sqrt[2 + 5*x + 3*x^2]) + (295*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[S
qrt[x]], -1/2])/Sqrt[2 + 5*x + 3*x^2]

Rule 836

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*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] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

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 \frac {2-5 x}{\sqrt {x} \left (2+5 x+3 x^2\right )^{5/2}} \, dx &=\frac {2 \sqrt {x} (38+45 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {1}{3} \int \frac {35-135 x}{\sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=\frac {2 \sqrt {x} (38+45 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {5 \sqrt {x} (361+429 x)}{3 \sqrt {2+5 x+3 x^2}}+\frac {1}{3} \int \frac {885+\frac {2145 x}{2}}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {2 \sqrt {x} (38+45 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {5 \sqrt {x} (361+429 x)}{3 \sqrt {2+5 x+3 x^2}}+\frac {2}{3} \text {Subst}\left (\int \frac {885+\frac {2145 x^2}{2}}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 \sqrt {x} (38+45 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {5 \sqrt {x} (361+429 x)}{3 \sqrt {2+5 x+3 x^2}}+590 \text {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )+715 \text {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 \sqrt {x} (38+45 x)}{3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {715 \sqrt {x} (2+3 x)}{3 \sqrt {2+5 x+3 x^2}}-\frac {5 \sqrt {x} (361+429 x)}{3 \sqrt {2+5 x+3 x^2}}-\frac {715 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {2+5 x+3 x^2}}+\frac {295 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{\sqrt {2+5 x+3 x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 20.18, size = 167, normalized size = 0.90 \begin {gather*} \frac {2 \left (1430+5383 x+6615 x^2+2655 x^3\right )+715 i \sqrt {2+\frac {2}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \left (2+5 x+3 x^2\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+170 i \sqrt {2+\frac {2}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \left (2+5 x+3 x^2\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )}{3 \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(1430 + 5383*x + 6615*x^2 + 2655*x^3) + (715*I)*Sqrt[2 + 2/x]*Sqrt[3 + 2/x]*x^(3/2)*(2 + 5*x + 3*x^2)*Ellip
ticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] + (170*I)*Sqrt[2 + 2/x]*Sqrt[3 + 2/x]*x^(3/2)*(2 + 5*x + 3*x^2)*Ellipt
icF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(3*Sqrt[x]*(2 + 5*x + 3*x^2)^(3/2))

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Maple [A]
time = 0.70, size = 297, normalized size = 1.61

method	result	size
elliptic	\(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (\frac {\left (\frac {76}{27}+\frac {10 x}{3}\right ) \sqrt {3 x^{3}+5 x^{2}+2 x}}{\left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )^{2}}-\frac {2 x \left (\frac {1805}{18}+\frac {715 x}{6}\right ) \sqrt {3}}{\sqrt {x \left (x^{2}+\frac {5}{3} x +\frac {2}{3}\right )}}+\frac {295 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {715 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{6 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(214\)
default	\(-\frac {\left (1125 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{2}-2145 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{2}+1875 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x -3575 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x +750 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-1430 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+38610 x^{4}+96840 x^{3}+79350 x^{2}+21204 x \right ) \sqrt {3 x^{2}+5 x +2}}{18 \sqrt {x}\, \left (x +1\right )^{2} \left (2+3 x \right )^{2}}\)	\(297\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*(1125*(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))*x^2-2145*(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))*x^2+1875*(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))*x-3575*(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))*x+750*(6*x+4)^(1/2)*(3*x+3)^(1/2)*6^(1/2)*(-x)^(1/2)*Ellipt
icF(1/2*(6*x+4)^(1/2),I*2^(1/2))-1430*(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))+38610*x^4+96840*x^3+79350*x^2+21204*x)*(3*x^2+5*x+2)^(1/2)/x^(1/2)/(x+1)^2/(2+3*x)^2

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.62, size = 122, normalized size = 0.66 \begin {gather*} \frac {1735 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 6435 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - 9 \, {\left (6435 \, x^{3} + 16140 \, x^{2} + 13225 \, x + 3534\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}}{27 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*(1735*sqrt(3)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*weierstrassPInverse(28/27, 80/729, x + 5/9) - 6435*sqr
t(3)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*weierstrassZeta(28/27, 80/729, weierstrassPInverse(28/27, 80/729, x
+ 5/9)) - 9*(6435*x^3 + 16140*x^2 + 13225*x + 3534)*sqrt(3*x^2 + 5*x + 2)*sqrt(x))/(9*x^4 + 30*x^3 + 37*x^2 +
20*x + 4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {5 \sqrt {x}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {2}{9 x^{\frac {9}{2}} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{\frac {7}{2}} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{\frac {5}{2}} \sqrt {3 x^{2} + 5 x + 2} + 20 x^{\frac {3}{2}} \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {x} \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(5*sqrt(x)/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*
x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-2/(9*x**(9/2)*sqrt(3*x**2 + 5
*x + 2) + 30*x**(7/2)*sqrt(3*x**2 + 5*x + 2) + 37*x**(5/2)*sqrt(3*x**2 + 5*x + 2) + 20*x**(3/2)*sqrt(3*x**2 +
5*x + 2) + 4*sqrt(x)*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)/(3*x^2+5*x+2)^(5/2)/x^(1/2),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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