3.5.49 \(\int (a+\frac {c}{x^2}+\frac {b}{x})^{5/2} \, dx\) [449]

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

[Out]

-5/24*(a+c/x^2+b/x)^(3/2)*(7*b+6*c/x)+(a+c/x^2+b/x)^(5/2)*x+5/2*a^(3/2)*b*arctanh(1/2*(2*a+b/x)/a^(1/2)/(a+c/x
^2+b/x)^(1/2))+5/128*(-48*a^2*c^2-24*a*b^2*c+b^4)*arctanh(1/2*(b+2*c/x)/c^(1/2)/(a+c/x^2+b/x)^(1/2))/c^(3/2)-5
/64*(b*(44*a*c+b^2)+2*c*(12*a*c+b^2)/x)*(a+c/x^2+b/x)^(1/2)/c

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Rubi [A]
time = 0.16, antiderivative size = 204, normalized size of antiderivative = 1.00, 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 = {1356, 746, 828, 857, 635, 212, 738} \begin {gather*} \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 (-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 \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} \end {gather*}

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 212

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

Rule 635

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 738

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]

Rule 746

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 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 857

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 1356

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]

Rubi steps

\begin {align*} \int \left (a+\frac {c}{x^2}+\frac {b}{x}\right )^{5/2} \, dx &=-\text {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} \text {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 \text {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 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 1.24, size = 208, normalized size = 1.02 \begin {gather*} -\frac {\sqrt {a+\frac {c+b x}{x^2}} \left (15 \left (b^4-24 a b^2 c-48 a^2 c^2\right ) x^4 \tanh ^{-1}\left (\frac {\sqrt {a} x-\sqrt {c+x (b+a x)}}{\sqrt {c}}\right )+\sqrt {c} \left (\sqrt {c+x (b+a x)} \left (48 c^3+15 b^3 x^3+8 c^2 x (17 b+27 a x)+2 c x^2 \left (59 b^2+278 a b x-96 a^2 x^2\right )\right )+480 a^{3/2} b c x^4 \log \left (b+2 a x-2 \sqrt {a} \sqrt {c+x (b+a x)}\right )\right )\right )}{192 c^{3/2} x^3 \sqrt {c+x (b+a x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c/x^2 + b/x)^(5/2),x]

[Out]

-1/192*(Sqrt[a + (c + b*x)/x^2]*(15*(b^4 - 24*a*b^2*c - 48*a^2*c^2)*x^4*ArcTanh[(Sqrt[a]*x - Sqrt[c + x*(b + a
*x)])/Sqrt[c]] + 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)) + 480*a^(3/2)*b*c*x^4*Log[b + 2*a*x - 2*Sqrt[a]*Sqrt[c + x*(b + a*x)]])))/(c^(3/
2)*x^3*Sqrt[c + x*(b + a*x)])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(700\) vs. \(2(174)=348\).
time = 0.06, size = 701, normalized size = 3.44

method	result	size
risch	\(-\frac {\left (556 a b c \,x^{3}+15 b^{3} x^{3}+216 a \,c^{2} x^{2}+118 b^{2} c \,x^{2}+136 b \,c^{2} x +48 c^{3}\right ) \sqrt {\frac {a \,x^{2}+b x +c}{x^{2}}}}{192 x^{3} c}+\frac {\left (a^{2} \sqrt {a \,x^{2}+b x +c}+\frac {5 a^{\frac {3}{2}} b \ln \left (\frac {\frac {b}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+b x +c}\right )}{2}-\frac {15 \sqrt {c}\, \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right ) a^{2}}{8}-\frac {15 \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right ) a \,b^{2}}{16 \sqrt {c}}+\frac {5 \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right ) b^{4}}{128 c^{\frac {3}{2}}}\right ) \sqrt {\frac {a \,x^{2}+b x +c}{x^{2}}}\, x}{\sqrt {a \,x^{2}+b x +c}}\)	\(263\)
default	\(\frac {\left (\frac {a \,x^{2}+b x +c}{x^{2}}\right )^{\frac {5}{2}} x \left (-6 a^{\frac {3}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}} b^{4} x^{4}-96 \left (a \,x^{2}+b x +c \right )^{\frac {7}{2}} c^{3} a^{\frac {3}{2}}-360 \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right ) a^{\frac {5}{2}} c^{\frac {7}{2}} b^{2} x^{4}+152 a^{\frac {7}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}} b c \,x^{5}-152 a^{\frac {5}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {7}{2}} b c \,x^{3}+148 a^{\frac {5}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}} b^{2} c \,x^{4}+280 a^{\frac {7}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} b \,c^{2} x^{5}-10 a^{\frac {5}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} b^{3} c \,x^{5}+6 a^{\frac {3}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {7}{2}} b^{3} x^{3}-144 a^{\frac {5}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {7}{2}} c^{2} x^{2}+4 a^{\frac {3}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {7}{2}} b^{2} c \,x^{2}+240 a^{\frac {7}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} c^{3} x^{4}-10 a^{\frac {3}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} b^{4} c \,x^{4}+16 a^{\frac {3}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {7}{2}} b \,c^{2} x +720 a^{\frac {7}{2}} \sqrt {a \,x^{2}+b x +c}\, c^{4} x^{4}-30 a^{\frac {3}{2}} \sqrt {a \,x^{2}+b x +c}\, b^{4} c^{2} x^{4}-6 a^{\frac {5}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}} b^{3} x^{5}+144 a^{\frac {7}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}} c^{2} x^{4}-720 \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right ) a^{\frac {7}{2}} c^{\frac {9}{2}} x^{4}+15 \ln \left (\frac {2 c +b x +2 \sqrt {c}\, \sqrt {a \,x^{2}+b x +c}}{x}\right ) a^{\frac {3}{2}} c^{\frac {5}{2}} b^{4} x^{4}+260 a^{\frac {5}{2}} \left (a \,x^{2}+b x +c \right )^{\frac {3}{2}} b^{2} c^{2} x^{4}+600 a^{\frac {7}{2}} \sqrt {a \,x^{2}+b x +c}\, b \,c^{3} x^{5}-30 a^{\frac {5}{2}} \sqrt {a \,x^{2}+b x +c}\, b^{3} c^{2} x^{5}+960 \ln \left (\frac {2 \sqrt {a \,x^{2}+b x +c}\, \sqrt {a}+2 a x +b}{2 \sqrt {a}}\right ) a^{3} b \,c^{4} x^{4}+660 a^{\frac {5}{2}} \sqrt {a \,x^{2}+b x +c}\, b^{2} c^{3} x^{4}\right )}{384 \left (a \,x^{2}+b x +c \right )^{\frac {5}{2}} c^{4} a^{\frac {3}{2}}}\)	\(701\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate((a + b/x + c/x^2)^(5/2), x)

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Fricas [A]
time = 0.48, size = 959, normalized size = 4.70 \begin {gather*} \left [\frac {960 \, a^{\frac {3}{2}} b c^{2} x^{3} \log \left (-8 \, a^{2} x^{2} - 8 \, a b x - b^{2} - 4 \, a c - 4 \, {\left (2 \, a x^{2} + b x\right )} \sqrt {a} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}\right ) - 15 \, {\left (b^{4} - 24 \, a b^{2} c - 48 \, a^{2} c^{2}\right )} \sqrt {c} x^{3} \log \left (-\frac {8 \, b c x + {\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, c^{2} - 4 \, {\left (b x^{2} + 2 \, c x\right )} \sqrt {c} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{x^{2}}\right ) + 4 \, {\left (192 \, a^{2} c^{2} x^{4} - 136 \, b c^{3} x - 48 \, c^{4} - {\left (15 \, b^{3} c + 556 \, a b c^{2}\right )} x^{3} - 2 \, {\left (59 \, b^{2} c^{2} + 108 \, a c^{3}\right )} x^{2}\right )} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{768 \, c^{2} x^{3}}, -\frac {1920 \, \sqrt {-a} a b c^{2} x^{3} \arctan \left (\frac {{\left (2 \, a x^{2} + b x\right )} \sqrt {-a} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{2 \, {\left (a^{2} x^{2} + a b x + a c\right )}}\right ) + 15 \, {\left (b^{4} - 24 \, a b^{2} c - 48 \, a^{2} c^{2}\right )} \sqrt {c} x^{3} \log \left (-\frac {8 \, b c x + {\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, c^{2} - 4 \, {\left (b x^{2} + 2 \, c x\right )} \sqrt {c} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{x^{2}}\right ) - 4 \, {\left (192 \, a^{2} c^{2} x^{4} - 136 \, b c^{3} x - 48 \, c^{4} - {\left (15 \, b^{3} c + 556 \, a b c^{2}\right )} x^{3} - 2 \, {\left (59 \, b^{2} c^{2} + 108 \, a c^{3}\right )} x^{2}\right )} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{768 \, c^{2} x^{3}}, \frac {480 \, a^{\frac {3}{2}} b c^{2} x^{3} \log \left (-8 \, a^{2} x^{2} - 8 \, a b x - b^{2} - 4 \, a c - 4 \, {\left (2 \, a x^{2} + b x\right )} \sqrt {a} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}\right ) - 15 \, {\left (b^{4} - 24 \, a b^{2} c - 48 \, a^{2} c^{2}\right )} \sqrt {-c} x^{3} \arctan \left (\frac {{\left (b x^{2} + 2 \, c x\right )} \sqrt {-c} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{2 \, {\left (a c x^{2} + b c x + c^{2}\right )}}\right ) + 2 \, {\left (192 \, a^{2} c^{2} x^{4} - 136 \, b c^{3} x - 48 \, c^{4} - {\left (15 \, b^{3} c + 556 \, a b c^{2}\right )} x^{3} - 2 \, {\left (59 \, b^{2} c^{2} + 108 \, a c^{3}\right )} x^{2}\right )} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{384 \, c^{2} x^{3}}, -\frac {960 \, \sqrt {-a} a b c^{2} x^{3} \arctan \left (\frac {{\left (2 \, a x^{2} + b x\right )} \sqrt {-a} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{2 \, {\left (a^{2} x^{2} + a b x + a c\right )}}\right ) + 15 \, {\left (b^{4} - 24 \, a b^{2} c - 48 \, a^{2} c^{2}\right )} \sqrt {-c} x^{3} \arctan \left (\frac {{\left (b x^{2} + 2 \, c x\right )} \sqrt {-c} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{2 \, {\left (a c x^{2} + b c x + c^{2}\right )}}\right ) - 2 \, {\left (192 \, a^{2} c^{2} x^{4} - 136 \, b c^{3} x - 48 \, c^{4} - {\left (15 \, b^{3} c + 556 \, a b c^{2}\right )} x^{3} - 2 \, {\left (59 \, b^{2} c^{2} + 108 \, a c^{3}\right )} x^{2}\right )} \sqrt {\frac {a x^{2} + b x + c}{x^{2}}}}{384 \, c^{2} x^{3}}\right ] \end {gather*}

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)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + \frac {b}{x} + \frac {c}{x^{2}}\right )^{\frac {5}{2}}\, dx \end {gather*}

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)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x + c/x^2)^(5/2),x)

[Out]

int((a + b/x + c/x^2)^(5/2), x)

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