\(\int \frac {(2-5 x) (2+5 x+3 x^2)^{3/2}}{x^{15/2}} \, dx\) [1054]

Optimal result

Integrand size = 25, antiderivative size = 256 \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx=-\frac {6907 \sqrt {x} (2+3 x)}{10010 \sqrt {2+5 x+3 x^2}}+\frac {204 \sqrt {2+5 x+3 x^2}}{385 x^{5/2}}-\frac {1231 \sqrt {2+5 x+3 x^2}}{2002 x^{3/2}}+\frac {6907 \sqrt {2+5 x+3 x^2}}{10010 \sqrt {x}}+\frac {(1834+3445 x) \sqrt {2+5 x+3 x^2}}{1001 x^{9/2}}-\frac {4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}+\frac {6907 (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{5005 \sqrt {2} \sqrt {2+5 x+3 x^2}}-\frac {3693 (1+x) \sqrt {\frac {2+3 x}{1+x}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{2002 \sqrt {2} \sqrt {2+5 x+3 x^2}} \]

[Out]

-4/143*(11-25*x)*(3*x^2+5*x+2)^(3/2)/x^(13/2)-6907/10010*(2+3*x)*x^(1/2)/(3*x^2+5*x+2)^(1/2)+6907/10010*(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)-3693/4004*(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)+204/385*(3*x^2+5*x+2)^(1/2)/x^(5/2)-1231/2002*(3*x^2+5*x+2)^(1/2)/x^(3/2)+1/1001
*(1834+3445*x)*(3*x^2+5*x+2)^(1/2)/x^(9/2)+6907/10010*(3*x^2+5*x+2)^(1/2)/x^(1/2)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {824, 848, 853, 1203, 1114, 1150} \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx=-\frac {3693 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{2002 \sqrt {2} \sqrt {3 x^2+5 x+2}}+\frac {6907 (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{5005 \sqrt {2} \sqrt {3 x^2+5 x+2}}+\frac {6907 \sqrt {3 x^2+5 x+2}}{10010 \sqrt {x}}-\frac {6907 \sqrt {x} (3 x+2)}{10010 \sqrt {3 x^2+5 x+2}}-\frac {4 (11-25 x) \left (3 x^2+5 x+2\right )^{3/2}}{143 x^{13/2}}+\frac {(3445 x+1834) \sqrt {3 x^2+5 x+2}}{1001 x^{9/2}}-\frac {1231 \sqrt {3 x^2+5 x+2}}{2002 x^{3/2}}+\frac {204 \sqrt {3 x^2+5 x+2}}{385 x^{5/2}} \]

[In]

Int[((2 - 5*x)*(2 + 5*x + 3*x^2)^(3/2))/x^(15/2),x]

[Out]

(-6907*Sqrt[x]*(2 + 3*x))/(10010*Sqrt[2 + 5*x + 3*x^2]) + (204*Sqrt[2 + 5*x + 3*x^2])/(385*x^(5/2)) - (1231*Sq
rt[2 + 5*x + 3*x^2])/(2002*x^(3/2)) + (6907*Sqrt[2 + 5*x + 3*x^2])/(10010*Sqrt[x]) + ((1834 + 3445*x)*Sqrt[2 +
5*x + 3*x^2])/(1001*x^(9/2)) - (4*(11 - 25*x)*(2 + 5*x + 3*x^2)^(3/2))/(143*x^(13/2)) + (6907*(1 + x)*Sqrt[(2
+ 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(5005*Sqrt[2]*Sqrt[2 + 5*x + 3*x^2]) - (3693*(1 + x)*Sqrt[(
2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(2002*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*} \text {integral}& = -\frac {4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}-\frac {3}{143} \int \frac {(393+465 x) \sqrt {2+5 x+3 x^2}}{x^{11/2}} \, dx \\ & = \frac {(1834+3445 x) \sqrt {2+5 x+3 x^2}}{1001 x^{9/2}}-\frac {4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}+\frac {\int \frac {-7956-\frac {20745 x}{2}}{x^{7/2} \sqrt {2+5 x+3 x^2}} \, dx}{3003} \\ & = \frac {204 \sqrt {2+5 x+3 x^2}}{385 x^{5/2}}+\frac {(1834+3445 x) \sqrt {2+5 x+3 x^2}}{1001 x^{9/2}}-\frac {4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}-\frac {\int \frac {-\frac {55395}{2}-35802 x}{x^{5/2} \sqrt {2+5 x+3 x^2}} \, dx}{15015} \\ & = \frac {204 \sqrt {2+5 x+3 x^2}}{385 x^{5/2}}-\frac {1231 \sqrt {2+5 x+3 x^2}}{2002 x^{3/2}}+\frac {(1834+3445 x) \sqrt {2+5 x+3 x^2}}{1001 x^{9/2}}-\frac {4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}+\frac {\int \frac {-\frac {62163}{2}-\frac {166185 x}{4}}{x^{3/2} \sqrt {2+5 x+3 x^2}} \, dx}{45045} \\ & = \frac {204 \sqrt {2+5 x+3 x^2}}{385 x^{5/2}}-\frac {1231 \sqrt {2+5 x+3 x^2}}{2002 x^{3/2}}+\frac {6907 \sqrt {2+5 x+3 x^2}}{10010 \sqrt {x}}+\frac {(1834+3445 x) \sqrt {2+5 x+3 x^2}}{1001 x^{9/2}}-\frac {4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}-\frac {\int \frac {\frac {166185}{4}+\frac {186489 x}{4}}{\sqrt {x} \sqrt {2+5 x+3 x^2}} \, dx}{45045} \\ & = \frac {204 \sqrt {2+5 x+3 x^2}}{385 x^{5/2}}-\frac {1231 \sqrt {2+5 x+3 x^2}}{2002 x^{3/2}}+\frac {6907 \sqrt {2+5 x+3 x^2}}{10010 \sqrt {x}}+\frac {(1834+3445 x) \sqrt {2+5 x+3 x^2}}{1001 x^{9/2}}-\frac {4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}-\frac {2 \text {Subst}\left (\int \frac {\frac {166185}{4}+\frac {186489 x^2}{4}}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{45045} \\ & = \frac {204 \sqrt {2+5 x+3 x^2}}{385 x^{5/2}}-\frac {1231 \sqrt {2+5 x+3 x^2}}{2002 x^{3/2}}+\frac {6907 \sqrt {2+5 x+3 x^2}}{10010 \sqrt {x}}+\frac {(1834+3445 x) \sqrt {2+5 x+3 x^2}}{1001 x^{9/2}}-\frac {4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}-\frac {3693 \text {Subst}\left (\int \frac {1}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{2002}-\frac {20721 \text {Subst}\left (\int \frac {x^2}{\sqrt {2+5 x^2+3 x^4}} \, dx,x,\sqrt {x}\right )}{10010} \\ & = -\frac {6907 \sqrt {x} (2+3 x)}{10010 \sqrt {2+5 x+3 x^2}}+\frac {204 \sqrt {2+5 x+3 x^2}}{385 x^{5/2}}-\frac {1231 \sqrt {2+5 x+3 x^2}}{2002 x^{3/2}}+\frac {6907 \sqrt {2+5 x+3 x^2}}{10010 \sqrt {x}}+\frac {(1834+3445 x) \sqrt {2+5 x+3 x^2}}{1001 x^{9/2}}-\frac {4 (11-25 x) \left (2+5 x+3 x^2\right )^{3/2}}{143 x^{13/2}}+\frac {6907 (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{5005 \sqrt {2} \sqrt {2+5 x+3 x^2}}-\frac {3693 (1+x) \sqrt {\frac {2+3 x}{1+x}} F\left (\tan ^{-1}\left (\sqrt {x}\right )|-\frac {1}{2}\right )}{2002 \sqrt {2} \sqrt {2+5 x+3 x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 21.17 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.66 \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx=\frac {-24640-67200 x+125440 x^2+654400 x^3+840316 x^4+361120 x^5-29726 x^6-36930 x^7-13814 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{15/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-4651 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{15/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{20020 x^{13/2} \sqrt {2+5 x+3 x^2}} \]

[In]

Integrate[((2 - 5*x)*(2 + 5*x + 3*x^2)^(3/2))/x^(15/2),x]

[Out]

(-24640 - 67200*x + 125440*x^2 + 654400*x^3 + 840316*x^4 + 361120*x^5 - 29726*x^6 - 36930*x^7 - (13814*I)*Sqrt
[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(15/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] - (4651*I)*Sqrt[2]*Sq
rt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(15/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(20020*x^(13/2)*Sqrt[2 + 5
*x + 3*x^2])

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.56

method	result	size
default	\(\frac {2256 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{6}-6907 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right ) x^{6}+124326 x^{8}+96420 x^{7}-6294 x^{6}+1083360 x^{5}+2520948 x^{4}+1963200 x^{3}+376320 x^{2}-201600 x -73920}{60060 \sqrt {3 x^{2}+5 x +2}\, x^{\frac {13}{2}}}\)	\(144\)
risch	\(\frac {20721 x^{8}+16070 x^{7}-1049 x^{6}+180560 x^{5}+420158 x^{4}+327200 x^{3}+62720 x^{2}-33600 x -12320}{10010 x^{\frac {13}{2}} \sqrt {3 x^{2}+5 x +2}}-\frac {\left (\frac {1231 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{4004 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {6907 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{20020 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right ) \sqrt {x \left (3 x^{2}+5 x +2\right )}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(218\)
elliptic	\(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {8 \sqrt {3 x^{3}+5 x^{2}+2 x}}{13 x^{7}}-\frac {20 \sqrt {3 x^{3}+5 x^{2}+2 x}}{143 x^{6}}+\frac {630 \sqrt {3 x^{3}+5 x^{2}+2 x}}{143 x^{5}}+\frac {5545 \sqrt {3 x^{3}+5 x^{2}+2 x}}{1001 x^{4}}+\frac {204 \sqrt {3 x^{3}+5 x^{2}+2 x}}{385 x^{3}}-\frac {1231 \sqrt {3 x^{3}+5 x^{2}+2 x}}{2002 x^{2}}+\frac {\frac {20721}{10010} x^{2}+\frac {6907}{2002} x +\frac {6907}{5005}}{\sqrt {x \left (3 x^{2}+5 x +2\right )}}-\frac {1231 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{4004 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {6907 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{20020 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\)	\(311\)

[In]

int((2-5*x)*(3*x^2+5*x+2)^(3/2)/x^(15/2),x,method=_RETURNVERBOSE)

[Out]

1/60060*(2256*(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))*x^6-6907*(
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))*x^6+124326*x^8+96420*x^7-
6294*x^6+1083360*x^5+2520948*x^4+1963200*x^3+376320*x^2-201600*x-73920)/(3*x^2+5*x+2)^(1/2)/x^(13/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.18 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.33 \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx=-\frac {20860 \, \sqrt {3} x^{7} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 62163 \, \sqrt {3} x^{7} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) - 9 \, {\left (6907 \, x^{6} - 6155 \, x^{5} + 5304 \, x^{4} + 55450 \, x^{3} + 44100 \, x^{2} - 1400 \, x - 6160\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}}{90090 \, x^{7}} \]

[In]

integrate((2-5*x)*(3*x^2+5*x+2)^(3/2)/x^(15/2),x, algorithm="fricas")

[Out]

-1/90090*(20860*sqrt(3)*x^7*weierstrassPInverse(28/27, 80/729, x + 5/9) - 62163*sqrt(3)*x^7*weierstrassZeta(28
/27, 80/729, weierstrassPInverse(28/27, 80/729, x + 5/9)) - 9*(6907*x^6 - 6155*x^5 + 5304*x^4 + 55450*x^3 + 44
100*x^2 - 1400*x - 6160)*sqrt(3*x^2 + 5*x + 2)*sqrt(x))/x^7

Sympy [F(-1)]

Timed out. \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx=\text {Timed out} \]

[In]

integrate((2-5*x)*(3*x**2+5*x+2)**(3/2)/x**(15/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (5 \, x - 2\right )}}{x^{\frac {15}{2}}} \,d x } \]

[In]

integrate((2-5*x)*(3*x^2+5*x+2)^(3/2)/x^(15/2),x, algorithm="maxima")

[Out]

-integrate((3*x^2 + 5*x + 2)^(3/2)*(5*x - 2)/x^(15/2), x)

Giac [F]

\[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx=\int { -\frac {{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} {\left (5 \, x - 2\right )}}{x^{\frac {15}{2}}} \,d x } \]

[In]

integrate((2-5*x)*(3*x^2+5*x+2)^(3/2)/x^(15/2),x, algorithm="giac")

[Out]

integrate(-(3*x^2 + 5*x + 2)^(3/2)*(5*x - 2)/x^(15/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{15/2}} \, dx=\int -\frac {\left (5\,x-2\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{x^{15/2}} \,d x \]

[In]

int(-((5*x - 2)*(5*x + 3*x^2 + 2)^(3/2))/x^(15/2),x)

[Out]

int(-((5*x - 2)*(5*x + 3*x^2 + 2)^(3/2))/x^(15/2), x)