3.11.53 \(\int \frac {(2-5 x) (2+5 x+3 x^2)^{3/2}}{x^{13/2}} \, dx\) [1053]

Optimal. Leaf size=233 \[ \frac {3229 \sqrt {x} (2+3 x)}{1386 \sqrt {2+5 x+3 x^2}}+\frac {1357 \sqrt {2+5 x+3 x^2}}{693 x^{3/2}}-\frac {3229 \sqrt {2+5 x+3 x^2}}{1386 \sqrt {x}}+\frac {(634+1367 x) \sqrt {2+5 x+3 x^2}}{231 x^{7/2}}-\frac {4 (9-20 x) \left (2+5 x+3 x^2\right )^{3/2}}{99 x^{11/2}}-\frac {3229 (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{693 \sqrt {2} \sqrt {2+5 x+3 x^2}}+\frac {1357 (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{231 \sqrt {2} \sqrt {2+5 x+3 x^2}} \]

[Out]

-4/99*(9-20*x)*(3*x^2+5*x+2)^(3/2)/x^(11/2)+3229/1386*(2+3*x)*x^(1/2)/(3*x^2+5*x+2)^(1/2)-3229/1386*(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)+1357/462*(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)+1357/693*(3*x^2+5*x+2)^(1/2)/x^(3/2)+1/231*(634+1367*x)*(3*x^2+5*x+2)^(1/2)/x^(7/2)-3
229/1386*(3*x^2+5*x+2)^(1/2)/x^(1/2)

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Rubi [A]
time = 0.10, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {824, 848, 853, 1203, 1114, 1150} \begin {gather*} \frac {1357 (x+1) \sqrt {\frac {3 x+2}{x+1}} F\left (\text {ArcTan}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{231 \sqrt {2} \sqrt {3 x^2+5 x+2}}-\frac {3229 (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\text {ArcTan}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{693 \sqrt {2} \sqrt {3 x^2+5 x+2}}-\frac {3229 \sqrt {3 x^2+5 x+2}}{1386 \sqrt {x}}+\frac {3229 \sqrt {x} (3 x+2)}{1386 \sqrt {3 x^2+5 x+2}}-\frac {4 (9-20 x) \left (3 x^2+5 x+2\right )^{3/2}}{99 x^{11/2}}+\frac {(1367 x+634) \sqrt {3 x^2+5 x+2}}{231 x^{7/2}}+\frac {1357 \sqrt {3 x^2+5 x+2}}{693 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((2 - 5*x)*(2 + 5*x + 3*x^2)^(3/2))/x^(13/2),x]

[Out]

(3229*Sqrt[x]*(2 + 3*x))/(1386*Sqrt[2 + 5*x + 3*x^2]) + (1357*Sqrt[2 + 5*x + 3*x^2])/(693*x^(3/2)) - (3229*Sqr
t[2 + 5*x + 3*x^2])/(1386*Sqrt[x]) + ((634 + 1367*x)*Sqrt[2 + 5*x + 3*x^2])/(231*x^(7/2)) - (4*(9 - 20*x)*(2 +
5*x + 3*x^2)^(3/2))/(99*x^(11/2)) - (3229*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(
693*Sqrt[2]*Sqrt[2 + 5*x + 3*x^2]) + (1357*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(
231*Sqrt[2]*Sqrt[2 + 5*x + 3*x^2])

Rule 824

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2)
)*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d -
b*e)*(e*f - d*g))*x), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*
x + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m +
1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m +
1) - b*(d*g*(m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*
a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3,
0]

Rule 848

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*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] && LtQ[m, -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) \left (2+5 x+3 x^2\right )^{3/2}}{x^{13/2}} \, dx &=-\frac {4 (9-20 x) \left (2+5 x+3 x^2\right )^{3/2}}{99 x^{11/2}}-\frac {1}{33} \int \frac {(317+375 x) \sqrt {2+5 x+3 x^2}}{x^{9/2}} \, dx\\ &=\frac {(634+1367 x) \sqrt {2+5 x+3 x^2}}{231 x^{7/2}}-\frac {4 (9-20 x) \left (2+5 x+3 x^2\right )^{3/2}}{99 x^{11/2}}+\frac {\int \frac {-6785-\frac {17235 x}{2}}{x^{5/2} \sqrt {2+5 x+3 x^2}} \, dx}{1155}\\ &=\frac {1357 \sqrt {2+5 x+3 x^2}}{693 x^{3/2}}+\frac {(634+1367 x) \sqrt {2+5 x+3 x^2}}{231 x^{7/2}}-\frac {4 (9-20 x) \left (2+5 x+3 x^2\right )^{3/2}}{99 x^{11/2}}-\frac {\int \frac {-\frac {16145}{2}-\frac {20355 x}{2}}{x^{3/2} \sqrt {2+5 x+3 x^2}} \, dx}{3465}\\ &=\frac {1357 \sqrt {2+5 x+3 x^2}}{693 x^{3/2}}-\frac {3229 \sqrt {2+5 x+3 x^2}}{1386 \sqrt {x}}+\frac {(634+1367 x) \sqrt {2+5 x+3 x^2}}{231 x^{7/2}}-\frac {4 (9-20 x) \left (2+5 x+3 x^2\right )^{3/2}}{99 x^{11/2}}+\frac {\int \frac {\frac {20355}{2}+\frac {48435 x}{4}}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx}{3465}\\ &=\frac {1357 \sqrt {2+5 x+3 x^2}}{693 x^{3/2}}-\frac {3229 \sqrt {2+5 x+3 x^2}}{1386 \sqrt {x}}+\frac {(634+1367 x) \sqrt {2+5 x+3 x^2}}{231 x^{7/2}}-\frac {4 (9-20 x) \left (2+5 x+3 x^2\right )^{3/2}}{99 x^{11/2}}+\frac {2 \text {Subst}\left (\int \frac {\frac {20355}{2}+\frac {48435 x^2}{4}}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{3465}\\ &=\frac {1357 \sqrt {2+5 x+3 x^2}}{693 x^{3/2}}-\frac {3229 \sqrt {2+5 x+3 x^2}}{1386 \sqrt {x}}+\frac {(634+1367 x) \sqrt {2+5 x+3 x^2}}{231 x^{7/2}}-\frac {4 (9-20 x) \left (2+5 x+3 x^2\right )^{3/2}}{99 x^{11/2}}+\frac {1357}{231} \text {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )+\frac {3229}{462} \text {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )\\ &=\frac {3229 \sqrt {x} (2+3 x)}{1386 \sqrt {2+5 x+3 x^2}}+\frac {1357 \sqrt {2+5 x+3 x^2}}{693 x^{3/2}}-\frac {3229 \sqrt {2+5 x+3 x^2}}{1386 \sqrt {x}}+\frac {(634+1367 x) \sqrt {2+5 x+3 x^2}}{231 x^{7/2}}-\frac {4 (9-20 x) \left (2+5 x+3 x^2\right )^{3/2}}{99 x^{11/2}}-\frac {3229 (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{693 \sqrt {2} \sqrt {2+5 x+3 x^2}}+\frac {1357 (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{231 \sqrt {2} \sqrt {2+5 x+3 x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 20.13, size = 165, normalized size = 0.71 \begin {gather*} \frac {-2016-5600 x+11360 x^2+61744 x^3+86914 x^4+48256 x^5+8142 x^6+3229 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{13/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )+842 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{13/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )}{1386 x^{11/2} \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^(13/2),x]

[Out]

(-2016 - 5600*x + 11360*x^2 + 61744*x^3 + 86914*x^4 + 48256*x^5 + 8142*x^6 + (3229*I)*Sqrt[2]*Sqrt[1 + x^(-1)]
*Sqrt[3 + 2/x]*x^(13/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] + (842*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3
+ 2/x]*x^(13/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(1386*x^(11/2)*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]
time = 0.75, size = 139, normalized size = 0.60

method	result	size
default	\(-\frac {1545 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{5}-3229 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {6}\, \sqrt {-x}\, \EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{5}+58122 x^{7}+48018 x^{6}-250788 x^{5}-521484 x^{4}-370464 x^{3}-68160 x^{2}+33600 x +12096}{8316 \sqrt {3 x^{2}+5 x +2}\, x^{\frac {11}{2}}}\)	\(139\)
risch	\(-\frac {9687 x^{7}+8003 x^{6}-41798 x^{5}-86914 x^{4}-61744 x^{3}-11360 x^{2}+5600 x +2016}{1386 x^{\frac {11}{2}} \sqrt {3 x^{2}+5 x +2}}-\frac {\left (-\frac {3229 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{2772 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {1357 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{1386 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right ) \sqrt {x \left (3 x^{2}+5 x +2\right )}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(213\)
elliptic	\(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {8 \sqrt {3 x^{3}+5 x^{2}+2 x}}{11 x^{6}}-\frac {20 \sqrt {3 x^{3}+5 x^{2}+2 x}}{99 x^{5}}+\frac {3946 \sqrt {3 x^{3}+5 x^{2}+2 x}}{693 x^{4}}+\frac {1927 \sqrt {3 x^{3}+5 x^{2}+2 x}}{231 x^{3}}+\frac {1357 \sqrt {3 x^{3}+5 x^{2}+2 x}}{693 x^{2}}-\frac {3229 \left (3 x^{2}+5 x +2\right )}{1386 \sqrt {x \left (3 x^{2}+5 x +2\right )}}+\frac {1357 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{1386 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {3229 \sqrt {6 x +4}\, \sqrt {3 x +3}\, \sqrt {-6 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-\EllipticF \left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{2772 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(290\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2-5*x)*(3*x^2+5*x+2)^(3/2)/x^(13/2),x,method=_RETURNVERBOSE)

[Out]

-1/8316*(1545*(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^5-3229*(
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^5+58122*x^7+48018*x^6-2
50788*x^5-521484*x^4-370464*x^3-68160*x^2+33600*x+12096)/(3*x^2+5*x+2)^(1/2)/x^(11/2)

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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^(13/2),x, algorithm="maxima")

[Out]

-integrate((3*x^2 + 5*x + 2)^(3/2)*(5*x - 2)/x^(13/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.36, size = 79, normalized size = 0.34 \begin {gather*} \frac {8281 \, \sqrt {3} x^{6} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 29061 \, \sqrt {3} x^{6} {\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 (3229 \, x^{5} - 2714 \, x^{4} - 11562 \, x^{3} - 7892 \, x^{2} + 280 \, x + 1008\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}}{12474 \, x^{6}} \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^(13/2),x, algorithm="fricas")

[Out]

1/12474*(8281*sqrt(3)*x^6*weierstrassPInverse(28/27, 80/729, x + 5/9) - 29061*sqrt(3)*x^6*weierstrassZeta(28/2
7, 80/729, weierstrassPInverse(28/27, 80/729, x + 5/9)) - 9*(3229*x^5 - 2714*x^4 - 11562*x^3 - 7892*x^2 + 280*
x + 1008)*sqrt(3*x^2 + 5*x + 2)*sqrt(x))/x^6

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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**(13/2),x)

[Out]

Timed out

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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^(13/2),x, algorithm="giac")

[Out]

integrate(-(3*x^2 + 5*x + 2)^(3/2)*(5*x - 2)/x^(13/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {\left (5\,x-2\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{x^{13/2}} \,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^(13/2),x)

[Out]

int(-((5*x - 2)*(5*x + 3*x^2 + 2)^(3/2))/x^(13/2), x)

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