3.11.48 \(\int \frac {(2-5 x) (2+5 x+3 x^2)^{3/2}}{x^{3/2}} \, dx\) [1048]

Optimal. Leaf size=187 \[ \frac {5848 \sqrt {x} (2+3 x)}{315 \sqrt {2+5 x+3 x^2}}+\frac {2}{105} \sqrt {x} (1045+531 x) \sqrt {2+5 x+3 x^2}-\frac {2 (14+5 x) \left (2+5 x+3 x^2\right )^{3/2}}{7 \sqrt {x}}-\frac {5848 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{315 \sqrt {2+5 x+3 x^2}}+\frac {482 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{21 \sqrt {2+5 x+3 x^2}} \]

[Out]

-2/7*(14+5*x)*(3*x^2+5*x+2)^(3/2)/x^(1/2)+5848/315*(2+3*x)*x^(1/2)/(3*x^2+5*x+2)^(1/2)-5848/315*(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)+4
82/21*(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)+2/105*(1045+531*x)*x^(1/2)*(3*x^2+5*x+2)^(1/2)

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Rubi [A]
time = 0.08, antiderivative size = 187, 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 = {826, 828, 853, 1203, 1114, 1150} \begin {gather*} \frac {482 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\text {ArcTan}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{21 \sqrt {3 x^2+5 x+2}}-\frac {5848 \sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\text {ArcTan}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{315 \sqrt {3 x^2+5 x+2}}-\frac {2 (5 x+14) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt {x}}+\frac {2}{105} \sqrt {x} (531 x+1045) \sqrt {3 x^2+5 x+2}+\frac {5848 \sqrt {x} (3 x+2)}{315 \sqrt {3 x^2+5 x+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5848*Sqrt[x]*(2 + 3*x))/(315*Sqrt[2 + 5*x + 3*x^2]) + (2*Sqrt[x]*(1045 + 531*x)*Sqrt[2 + 5*x + 3*x^2])/105 -
(2*(14 + 5*x)*(2 + 5*x + 3*x^2)^(3/2))/(7*Sqrt[x]) - (5848*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[A
rcTan[Sqrt[x]], -1/2])/(315*Sqrt[2 + 5*x + 3*x^2]) + (482*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[Ar
cTan[Sqrt[x]], -1/2])/(21*Sqrt[2 + 5*x + 3*x^2])

Rule 826

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

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 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) \left (2+5 x+3 x^2\right )^{3/2}}{x^{3/2}} \, dx &=-\frac {2 (14+5 x) \left (2+5 x+3 x^2\right )^{3/2}}{7 \sqrt {x}}-\frac {6}{7} \int \frac {\left (-25-\frac {59 x}{2}\right ) \sqrt {2+5 x+3 x^2}}{\sqrt {x}} \, dx\\ &=\frac {2}{105} \sqrt {x} (1045+531 x) \sqrt {2+5 x+3 x^2}-\frac {2 (14+5 x) \left (2+5 x+3 x^2\right )^{3/2}}{7 \sqrt {x}}+\frac {4}{105} \int \frac {\frac {1205}{2}+731 x}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {2}{105} \sqrt {x} (1045+531 x) \sqrt {2+5 x+3 x^2}-\frac {2 (14+5 x) \left (2+5 x+3 x^2\right )^{3/2}}{7 \sqrt {x}}+\frac {8}{105} \text {Subst}\left (\int \frac {\frac {1205}{2}+731 x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2}{105} \sqrt {x} (1045+531 x) \sqrt {2+5 x+3 x^2}-\frac {2 (14+5 x) \left (2+5 x+3 x^2\right )^{3/2}}{7 \sqrt {x}}+\frac {964}{21} \text {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )+\frac {5848}{105} \text {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=\frac {5848 \sqrt {x} (2+3 x)}{315 \sqrt {2+5 x+3 x^2}}+\frac {2}{105} \sqrt {x} (1045+531 x) \sqrt {2+5 x+3 x^2}-\frac {2 (14+5 x) \left (2+5 x+3 x^2\right )^{3/2}}{7 \sqrt {x}}-\frac {5848 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{315 \sqrt {2+5 x+3 x^2}}+\frac {482 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{21 \sqrt {2+5 x+3 x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 20.15, size = 163, normalized size = 0.87 \begin {gather*} \frac {-2 \left (-3328-7390 x+177 x^2+9855 x^3+7641 x^4+2025 x^5\right )+5848 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 )+1382 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 )}{315 \sqrt {x} \sqrt {2+5 x+3 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(-3328 - 7390*x + 177*x^2 + 9855*x^3 + 7641*x^4 + 2025*x^5) + (5848*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2
/x]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] + (1382*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(
3/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(315*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]
time = 0.74, size = 123, normalized size = 0.66

method	result	size
default	\(-\frac {2 \left (6075 x^{5}+771 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-1462 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )+22923 x^{4}+29565 x^{3}+26847 x^{2}+21690 x +7560\right )}{945 \sqrt {3 x^{2}+5 x +2}\, \sqrt {x}}\)	\(123\)
risch	\(-\frac {2 \left (675 x^{5}+2547 x^{4}+3285 x^{3}+2983 x^{2}+2410 x +840\right )}{105 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}-\frac {\left (-\frac {482 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{63 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {2924 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{315 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right ) \sqrt {x \left (3 x^{2}+5 x +2\right )}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(203\)
elliptic	\(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {8 \left (3 x^{2}+5 x +2\right )}{\sqrt {x \left (3 x^{2}+5 x +2\right )}}-\frac {30 x^{2} \sqrt {3 x^{3}+5 x^{2}+2 x}}{7}-\frac {316 x \sqrt {3 x^{3}+5 x^{2}+2 x}}{35}-\frac {62 \sqrt {3 x^{3}+5 x^{2}+2 x}}{21}+\frac {482 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{63 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {2924 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{315 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(243\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/945*(6075*x^5+771*(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))-146
2*(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))+22923*x^4+29565*x^3+26
847*x^2+21690*x+7560)/(3*x^2+5*x+2)^(1/2)/x^(1/2)

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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)^(3/2)/x^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.61, size = 65, normalized size = 0.35 \begin {gather*} \frac {2 \, {\left (7070 \, \sqrt {3} x {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 26316 \, \sqrt {3} x {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - 27 \, {\left (225 \, x^{3} + 474 \, x^{2} + 155 \, x + 420\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}\right )}}{2835 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/2835*(7070*sqrt(3)*x*weierstrassPInverse(28/27, 80/729, x + 5/9) - 26316*sqrt(3)*x*weierstrassZeta(28/27, 80
/729, weierstrassPInverse(28/27, 80/729, x + 5/9)) - 27*(225*x^3 + 474*x^2 + 155*x + 420)*sqrt(3*x^2 + 5*x + 2
)*sqrt(x))/x

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {4 \sqrt {3 x^{2} + 5 x + 2}}{x^{\frac {3}{2}}}\right )\, dx - \int 19 \sqrt {x} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int 15 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)*(3*x**2+5*x+2)**(3/2)/x**(3/2),x)

[Out]

-Integral(-4*sqrt(3*x**2 + 5*x + 2)/x**(3/2), x) - Integral(19*sqrt(x)*sqrt(3*x**2 + 5*x + 2), x) - Integral(1
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)*(3*x^2+5*x+2)^(3/2)/x^(3/2),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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