3.449 \(\int (a+\frac{c}{x^2}+\frac{b}{x})^{5/2} \, dx\)

Optimal. Leaf size=204 \[ \frac{5 \left (-48 a^2 c^2-24 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+\frac{2 c}{x}}{2 \sqrt{c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )}{128 c^{3/2}}+\frac{5}{2} a^{3/2} b \tanh ^{-1}\left (\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )-\frac{5 \left (\frac{2 c \left (12 a c+b^2\right )}{x}+b \left (44 a c+b^2\right )\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}{64 c}+x \left (a+\frac{b}{x}+\frac{c}{x^2}\right )^{5/2}-\frac{5}{24} \left (7 b+\frac{6 c}{x}\right ) \left (a+\frac{b}{x}+\frac{c}{x^2}\right )^{3/2} \]

[Out]

(-5*(a + c/x^2 + b/x)^(3/2)*(7*b + (6*c)/x))/24 - (5*Sqrt[a + c/x^2 + b/x]*(b*(b^2 + 44*a*c) + (2*c*(b^2 + 12*
a*c))/x))/(64*c) + (a + c/x^2 + b/x)^(5/2)*x + (5*a^(3/2)*b*ArcTanh[(2*a + b/x)/(2*Sqrt[a]*Sqrt[a + c/x^2 + b/
x])])/2 + (5*(b^4 - 24*a*b^2*c - 48*a^2*c^2)*ArcTanh[(b + (2*c)/x)/(2*Sqrt[c]*Sqrt[a + c/x^2 + b/x])])/(128*c^
(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.230679, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {1342, 732, 814, 843, 621, 206, 724} \[ \frac{5 \left (-48 a^2 c^2-24 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+\frac{2 c}{x}}{2 \sqrt{c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )}{128 c^{3/2}}+\frac{5}{2} a^{3/2} b \tanh ^{-1}\left (\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )-\frac{5 \left (\frac{2 c \left (12 a c+b^2\right )}{x}+b \left (44 a c+b^2\right )\right ) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}{64 c}+x \left (a+\frac{b}{x}+\frac{c}{x^2}\right )^{5/2}-\frac{5}{24} \left (7 b+\frac{6 c}{x}\right ) \left (a+\frac{b}{x}+\frac{c}{x^2}\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + c/x^2 + b/x)^(5/2),x]

[Out]

(-5*(a + c/x^2 + b/x)^(3/2)*(7*b + (6*c)/x))/24 - (5*Sqrt[a + c/x^2 + b/x]*(b*(b^2 + 44*a*c) + (2*c*(b^2 + 12*
a*c))/x))/(64*c) + (a + c/x^2 + b/x)^(5/2)*x + (5*a^(3/2)*b*ArcTanh[(2*a + b/x)/(2*Sqrt[a]*Sqrt[a + c/x^2 + b/
x])])/2 + (5*(b^4 - 24*a*b^2*c - 48*a^2*c^2)*ArcTanh[(b + (2*c)/x)/(2*Sqrt[c]*Sqrt[a + c/x^2 + b/x])])/(128*c^
(3/2))

Rule 1342

Int[((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n + c/x^(2*n))^p/x^2,
x], x, 1/x] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && ILtQ[n, 0]

Rule 732

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

Rule 814

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 843

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

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{5/2} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (a+b x+c x^2\right )^{5/2}}{x^2} \, dx,x,\frac{1}{x}\right )\\ &=\left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{5/2} x-\frac{5}{2} \operatorname{Subst}\left (\int \frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{5}{24} \left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{3/2} \left (7 b+\frac{6 c}{x}\right )+\left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{5/2} x+\frac{5 \operatorname{Subst}\left (\int \frac{\left (-8 a b c-c \left (b^2+12 a c\right ) x\right ) \sqrt{a+b x+c x^2}}{x} \, dx,x,\frac{1}{x}\right )}{16 c}\\ &=-\frac{5}{24} \left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{3/2} \left (7 b+\frac{6 c}{x}\right )-\frac{5 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} \left (b \left (b^2+44 a c\right )+\frac{2 c \left (b^2+12 a c\right )}{x}\right )}{64 c}+\left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{5/2} x-\frac{5 \operatorname{Subst}\left (\int \frac{32 a^2 b c^2-\frac{1}{2} c \left (b^4-24 a b^2 c-48 a^2 c^2\right ) x}{x \sqrt{a+b x+c x^2}} \, dx,x,\frac{1}{x}\right )}{64 c^2}\\ &=-\frac{5}{24} \left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{3/2} \left (7 b+\frac{6 c}{x}\right )-\frac{5 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} \left (b \left (b^2+44 a c\right )+\frac{2 c \left (b^2+12 a c\right )}{x}\right )}{64 c}+\left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{5/2} x-\frac{1}{2} \left (5 a^2 b\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx,x,\frac{1}{x}\right )+\frac{\left (5 \left (b^4-24 a b^2 c-48 a^2 c^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,\frac{1}{x}\right )}{128 c}\\ &=-\frac{5}{24} \left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{3/2} \left (7 b+\frac{6 c}{x}\right )-\frac{5 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} \left (b \left (b^2+44 a c\right )+\frac{2 c \left (b^2+12 a c\right )}{x}\right )}{64 c}+\left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{5/2} x+\left (5 a^2 b\right ) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+\frac{b}{x}}{\sqrt{a+\frac{c}{x^2}+\frac{b}{x}}}\right )+\frac{\left (5 \left (b^4-24 a b^2 c-48 a^2 c^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+\frac{2 c}{x}}{\sqrt{a+\frac{c}{x^2}+\frac{b}{x}}}\right )}{64 c}\\ &=-\frac{5}{24} \left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{3/2} \left (7 b+\frac{6 c}{x}\right )-\frac{5 \sqrt{a+\frac{c}{x^2}+\frac{b}{x}} \left (b \left (b^2+44 a c\right )+\frac{2 c \left (b^2+12 a c\right )}{x}\right )}{64 c}+\left (a+\frac{c}{x^2}+\frac{b}{x}\right )^{5/2} x+\frac{5}{2} a^{3/2} b \tanh ^{-1}\left (\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}}}\right )+\frac{5 \left (b^4-24 a b^2 c-48 a^2 c^2\right ) \tanh ^{-1}\left (\frac{b+\frac{2 c}{x}}{2 \sqrt{c} \sqrt{a+\frac{c}{x^2}+\frac{b}{x}}}\right )}{128 c^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.538654, size = 213, normalized size = 1.04 \[ \frac{\sqrt{a+\frac{b x+c}{x^2}} \left (-2 \sqrt{c} \sqrt{x (a x+b)+c} \left (2 c x^2 \left (-96 a^2 x^2+278 a b x+59 b^2\right )+8 c^2 x (27 a x+17 b)+15 b^3 x^3+48 c^3\right )+15 x^4 \left (-48 a^2 c^2-24 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b x+2 c}{2 \sqrt{c} \sqrt{x (a x+b)+c}}\right )+960 a^{3/2} b c^{3/2} x^4 \tanh ^{-1}\left (\frac{2 a x+b}{2 \sqrt{a} \sqrt{x (a x+b)+c}}\right )\right )}{384 c^{3/2} x^3 \sqrt{x (a x+b)+c}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c/x^2 + b/x)^(5/2),x]

[Out]

(Sqrt[a + (c + b*x)/x^2]*(-2*Sqrt[c]*Sqrt[c + x*(b + a*x)]*(48*c^3 + 15*b^3*x^3 + 8*c^2*x*(17*b + 27*a*x) + 2*
c*x^2*(59*b^2 + 278*a*b*x - 96*a^2*x^2)) + 960*a^(3/2)*b*c^(3/2)*x^4*ArcTanh[(b + 2*a*x)/(2*Sqrt[a]*Sqrt[c + x
*(b + a*x)])] + 15*(b^4 - 24*a*b^2*c - 48*a^2*c^2)*x^4*ArcTanh[(2*c + b*x)/(2*Sqrt[c]*Sqrt[c + x*(b + a*x)])])
)/(384*c^(3/2)*x^3*Sqrt[c + x*(b + a*x)])

________________________________________________________________________________________

Maple [B]  time = 0.013, size = 701, normalized size = 3.4 \begin{align*}{\frac{x}{384\,{c}^{4}} \left ({\frac{a{x}^{2}+bx+c}{{x}^{2}}} \right ) ^{{\frac{5}{2}}} \left ( -96\, \left ( a{x}^{2}+bx+c \right ) ^{7/2}{c}^{3}{a}^{3/2}-30\,{a}^{3/2}\sqrt{a{x}^{2}+bx+c}{x}^{4}{b}^{4}{c}^{2}+4\,{a}^{3/2} \left ( a{x}^{2}+bx+c \right ) ^{7/2}{x}^{2}{b}^{2}c-10\,{a}^{3/2} \left ( a{x}^{2}+bx+c \right ) ^{3/2}{x}^{4}{b}^{4}c+16\,{a}^{3/2} \left ( a{x}^{2}+bx+c \right ) ^{7/2}xb{c}^{2}+660\,{a}^{5/2}\sqrt{a{x}^{2}+bx+c}{x}^{4}{b}^{2}{c}^{3}+600\,{a}^{7/2}\sqrt{a{x}^{2}+bx+c}{x}^{5}b{c}^{3}-30\,{a}^{5/2}\sqrt{a{x}^{2}+bx+c}{x}^{5}{b}^{3}{c}^{2}+960\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx+c}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){a}^{3}{x}^{4}b{c}^{4}+15\,\ln \left ({\frac{2\,c+bx+2\,\sqrt{c}\sqrt{a{x}^{2}+bx+c}}{x}} \right ){c}^{5/2}{a}^{3/2}{x}^{4}{b}^{4}+260\,{a}^{5/2} \left ( a{x}^{2}+bx+c \right ) ^{3/2}{x}^{4}{b}^{2}{c}^{2}+280\,{a}^{7/2} \left ( a{x}^{2}+bx+c \right ) ^{3/2}{x}^{5}b{c}^{2}-10\,{a}^{5/2} \left ( a{x}^{2}+bx+c \right ) ^{3/2}{x}^{5}{b}^{3}c-152\,{a}^{5/2} \left ( a{x}^{2}+bx+c \right ) ^{7/2}{x}^{3}bc+148\,{a}^{5/2} \left ( a{x}^{2}+bx+c \right ) ^{5/2}{x}^{4}{b}^{2}c+152\,{a}^{7/2} \left ( a{x}^{2}+bx+c \right ) ^{5/2}{x}^{5}bc-360\,\ln \left ({\frac{2\,c+bx+2\,\sqrt{c}\sqrt{a{x}^{2}+bx+c}}{x}} \right ){c}^{7/2}{a}^{5/2}{x}^{4}{b}^{2}+720\,{a}^{7/2}\sqrt{a{x}^{2}+bx+c}{x}^{4}{c}^{4}-6\,{a}^{5/2} \left ( a{x}^{2}+bx+c \right ) ^{5/2}{x}^{5}{b}^{3}+144\,{a}^{7/2} \left ( a{x}^{2}+bx+c \right ) ^{5/2}{x}^{4}{c}^{2}-144\,{a}^{5/2} \left ( a{x}^{2}+bx+c \right ) ^{7/2}{x}^{2}{c}^{2}+240\,{a}^{7/2} \left ( a{x}^{2}+bx+c \right ) ^{3/2}{x}^{4}{c}^{3}+6\,{a}^{3/2} \left ( a{x}^{2}+bx+c \right ) ^{7/2}{x}^{3}{b}^{3}-6\,{a}^{3/2} \left ( a{x}^{2}+bx+c \right ) ^{5/2}{x}^{4}{b}^{4}-720\,\ln \left ({\frac{2\,c+bx+2\,\sqrt{c}\sqrt{a{x}^{2}+bx+c}}{x}} \right ){c}^{9/2}{a}^{7/2}{x}^{4} \right ) \left ( a{x}^{2}+bx+c \right ) ^{-{\frac{5}{2}}}{a}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+c/x^2+b/x)^(5/2),x)

[Out]

1/384*((a*x^2+b*x+c)/x^2)^(5/2)*x*(-96*(a*x^2+b*x+c)^(7/2)*c^3*a^(3/2)-30*a^(3/2)*(a*x^2+b*x+c)^(1/2)*x^4*b^4*
c^2+4*a^(3/2)*(a*x^2+b*x+c)^(7/2)*x^2*b^2*c-10*a^(3/2)*(a*x^2+b*x+c)^(3/2)*x^4*b^4*c+16*a^(3/2)*(a*x^2+b*x+c)^
(7/2)*x*b*c^2+660*a^(5/2)*(a*x^2+b*x+c)^(1/2)*x^4*b^2*c^3+600*a^(7/2)*(a*x^2+b*x+c)^(1/2)*x^5*b*c^3-30*a^(5/2)
*(a*x^2+b*x+c)^(1/2)*x^5*b^3*c^2+960*ln(1/2*(2*(a*x^2+b*x+c)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^3*x^4*b*c^4+15*
ln((2*c+b*x+2*c^(1/2)*(a*x^2+b*x+c)^(1/2))/x)*c^(5/2)*a^(3/2)*x^4*b^4+260*a^(5/2)*(a*x^2+b*x+c)^(3/2)*x^4*b^2*
c^2+280*a^(7/2)*(a*x^2+b*x+c)^(3/2)*x^5*b*c^2-10*a^(5/2)*(a*x^2+b*x+c)^(3/2)*x^5*b^3*c-152*a^(5/2)*(a*x^2+b*x+
c)^(7/2)*x^3*b*c+148*a^(5/2)*(a*x^2+b*x+c)^(5/2)*x^4*b^2*c+152*a^(7/2)*(a*x^2+b*x+c)^(5/2)*x^5*b*c-360*ln((2*c
+b*x+2*c^(1/2)*(a*x^2+b*x+c)^(1/2))/x)*c^(7/2)*a^(5/2)*x^4*b^2+720*a^(7/2)*(a*x^2+b*x+c)^(1/2)*x^4*c^4-6*a^(5/
2)*(a*x^2+b*x+c)^(5/2)*x^5*b^3+144*a^(7/2)*(a*x^2+b*x+c)^(5/2)*x^4*c^2-144*a^(5/2)*(a*x^2+b*x+c)^(7/2)*x^2*c^2
+240*a^(7/2)*(a*x^2+b*x+c)^(3/2)*x^4*c^3+6*a^(3/2)*(a*x^2+b*x+c)^(7/2)*x^3*b^3-6*a^(3/2)*(a*x^2+b*x+c)^(5/2)*x
^4*b^4-720*ln((2*c+b*x+2*c^(1/2)*(a*x^2+b*x+c)^(1/2))/x)*c^(9/2)*a^(7/2)*x^4)/(a*x^2+b*x+c)^(5/2)/c^4/a^(3/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a + \frac{b}{x} + \frac{c}{x^{2}}\right )}^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+c/x^2+b/x)^(5/2),x, algorithm="maxima")

[Out]

integrate((a + b/x + c/x^2)^(5/2), x)

________________________________________________________________________________________

Fricas [A]  time = 3.94398, size = 2272, normalized size = 11.14 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+c/x^2+b/x)^(5/2),x, algorithm="fricas")

[Out]

[1/768*(960*a^(3/2)*b*c^2*x^3*log(-8*a^2*x^2 - 8*a*b*x - b^2 - 4*a*c - 4*(2*a*x^2 + b*x)*sqrt(a)*sqrt((a*x^2 +
b*x + c)/x^2)) - 15*(b^4 - 24*a*b^2*c - 48*a^2*c^2)*sqrt(c)*x^3*log(-(8*b*c*x + (b^2 + 4*a*c)*x^2 + 8*c^2 - 4
*(b*x^2 + 2*c*x)*sqrt(c)*sqrt((a*x^2 + b*x + c)/x^2))/x^2) + 4*(192*a^2*c^2*x^4 - 136*b*c^3*x - 48*c^4 - (15*b
^3*c + 556*a*b*c^2)*x^3 - 2*(59*b^2*c^2 + 108*a*c^3)*x^2)*sqrt((a*x^2 + b*x + c)/x^2))/(c^2*x^3), -1/768*(1920
*sqrt(-a)*a*b*c^2*x^3*arctan(1/2*(2*a*x^2 + b*x)*sqrt(-a)*sqrt((a*x^2 + b*x + c)/x^2)/(a^2*x^2 + a*b*x + a*c))
+ 15*(b^4 - 24*a*b^2*c - 48*a^2*c^2)*sqrt(c)*x^3*log(-(8*b*c*x + (b^2 + 4*a*c)*x^2 + 8*c^2 - 4*(b*x^2 + 2*c*x
)*sqrt(c)*sqrt((a*x^2 + b*x + c)/x^2))/x^2) - 4*(192*a^2*c^2*x^4 - 136*b*c^3*x - 48*c^4 - (15*b^3*c + 556*a*b*
c^2)*x^3 - 2*(59*b^2*c^2 + 108*a*c^3)*x^2)*sqrt((a*x^2 + b*x + c)/x^2))/(c^2*x^3), 1/384*(480*a^(3/2)*b*c^2*x^
3*log(-8*a^2*x^2 - 8*a*b*x - b^2 - 4*a*c - 4*(2*a*x^2 + b*x)*sqrt(a)*sqrt((a*x^2 + b*x + c)/x^2)) - 15*(b^4 -
24*a*b^2*c - 48*a^2*c^2)*sqrt(-c)*x^3*arctan(1/2*(b*x^2 + 2*c*x)*sqrt(-c)*sqrt((a*x^2 + b*x + c)/x^2)/(a*c*x^2
+ b*c*x + c^2)) + 2*(192*a^2*c^2*x^4 - 136*b*c^3*x - 48*c^4 - (15*b^3*c + 556*a*b*c^2)*x^3 - 2*(59*b^2*c^2 +
108*a*c^3)*x^2)*sqrt((a*x^2 + b*x + c)/x^2))/(c^2*x^3), -1/384*(960*sqrt(-a)*a*b*c^2*x^3*arctan(1/2*(2*a*x^2 +
b*x)*sqrt(-a)*sqrt((a*x^2 + b*x + c)/x^2)/(a^2*x^2 + a*b*x + a*c)) + 15*(b^4 - 24*a*b^2*c - 48*a^2*c^2)*sqrt(
-c)*x^3*arctan(1/2*(b*x^2 + 2*c*x)*sqrt(-c)*sqrt((a*x^2 + b*x + c)/x^2)/(a*c*x^2 + b*c*x + c^2)) - 2*(192*a^2*
c^2*x^4 - 136*b*c^3*x - 48*c^4 - (15*b^3*c + 556*a*b*c^2)*x^3 - 2*(59*b^2*c^2 + 108*a*c^3)*x^2)*sqrt((a*x^2 +
b*x + c)/x^2))/(c^2*x^3)]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + \frac{b}{x} + \frac{c}{x^{2}}\right )^{\frac{5}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+c/x**2+b/x)**(5/2),x)

[Out]

Integral((a + b/x + c/x**2)**(5/2), x)

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+c/x^2+b/x)^(5/2),x, algorithm="giac")

[Out]

Timed out