\(\int \frac {\sqrt {a+c x^2}}{x^5 (d+e x)} \, dx\) [325]

Optimal result

Integrand size = 22, antiderivative size = 274 \[ \int \frac {\sqrt {a+c x^2}}{x^5 (d+e x)} \, dx=-\frac {\sqrt {a+c x^2}}{4 d x^4}-\frac {c \sqrt {a+c x^2}}{8 a d x^2}-\frac {e^2 \sqrt {a+c x^2}}{2 d^3 x^2}+\frac {e^3 \sqrt {a+c x^2}}{d^4 x}+\frac {e \left (a+c x^2\right )^{3/2}}{3 a d^2 x^3}+\frac {e^3 \sqrt {c d^2+a e^2} \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^5}+\frac {c^2 \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 a^{3/2} d}-\frac {c e^2 \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 \sqrt {a} d^3}-\frac {\sqrt {a} e^4 \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^5} \]

[Out]

1/3*e*(c*x^2+a)^(3/2)/a/d^2/x^3+1/8*c^2*arctanh((c*x^2+a)^(1/2)/a^(1/2))/a^(3/2)/d-1/2*c*e^2*arctanh((c*x^2+a)
^(1/2)/a^(1/2))/d^3/a^(1/2)-e^4*arctanh((c*x^2+a)^(1/2)/a^(1/2))*a^(1/2)/d^5+e^3*arctanh((-c*d*x+a*e)/(a*e^2+c
*d^2)^(1/2)/(c*x^2+a)^(1/2))*(a*e^2+c*d^2)^(1/2)/d^5-1/4*(c*x^2+a)^(1/2)/d/x^4-1/8*c*(c*x^2+a)^(1/2)/a/d/x^2-1
/2*e^2*(c*x^2+a)^(1/2)/d^3/x^2+e^3*(c*x^2+a)^(1/2)/d^4/x

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {975, 272, 43, 44, 65, 214, 270, 283, 223, 212, 52, 749, 858, 739} \[ \int \frac {\sqrt {a+c x^2}}{x^5 (d+e x)} \, dx=\frac {c^2 \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 a^{3/2} d}-\frac {\sqrt {a} e^4 \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^5}-\frac {c e^2 \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 \sqrt {a} d^3}+\frac {e^3 \sqrt {a e^2+c d^2} \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d^5}+\frac {e^3 \sqrt {a+c x^2}}{d^4 x}-\frac {e^2 \sqrt {a+c x^2}}{2 d^3 x^2}+\frac {e \left (a+c x^2\right )^{3/2}}{3 a d^2 x^3}-\frac {c \sqrt {a+c x^2}}{8 a d x^2}-\frac {\sqrt {a+c x^2}}{4 d x^4} \]

[In]

Int[Sqrt[a + c*x^2]/(x^5*(d + e*x)),x]

[Out]

-1/4*Sqrt[a + c*x^2]/(d*x^4) - (c*Sqrt[a + c*x^2])/(8*a*d*x^2) - (e^2*Sqrt[a + c*x^2])/(2*d^3*x^2) + (e^3*Sqrt
[a + c*x^2])/(d^4*x) + (e*(a + c*x^2)^(3/2))/(3*a*d^2*x^3) + (e^3*Sqrt[c*d^2 + a*e^2]*ArcTanh[(a*e - c*d*x)/(S
qrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/d^5 + (c^2*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(8*a^(3/2)*d) - (c*e^2*ArcT
anh[Sqrt[a + c*x^2]/Sqrt[a]])/(2*Sqrt[a]*d^3) - (Sqrt[a]*e^4*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/d^5

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 214

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

Rule 223

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 283

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1
))), x] - Dist[b*n*(p/(c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 739

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

Rule 749

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

Rule 858

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

Rule 975

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (ILtQ[m, 0] && ILtQ[n, 0])) &&  !(IGtQ[m, 0] || IGtQ[n, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt {a+c x^2}}{d x^5}-\frac {e \sqrt {a+c x^2}}{d^2 x^4}+\frac {e^2 \sqrt {a+c x^2}}{d^3 x^3}-\frac {e^3 \sqrt {a+c x^2}}{d^4 x^2}+\frac {e^4 \sqrt {a+c x^2}}{d^5 x}-\frac {e^5 \sqrt {a+c x^2}}{d^5 (d+e x)}\right ) \, dx \\ & = \frac {\int \frac {\sqrt {a+c x^2}}{x^5} \, dx}{d}-\frac {e \int \frac {\sqrt {a+c x^2}}{x^4} \, dx}{d^2}+\frac {e^2 \int \frac {\sqrt {a+c x^2}}{x^3} \, dx}{d^3}-\frac {e^3 \int \frac {\sqrt {a+c x^2}}{x^2} \, dx}{d^4}+\frac {e^4 \int \frac {\sqrt {a+c x^2}}{x} \, dx}{d^5}-\frac {e^5 \int \frac {\sqrt {a+c x^2}}{d+e x} \, dx}{d^5} \\ & = -\frac {e^4 \sqrt {a+c x^2}}{d^5}+\frac {e^3 \sqrt {a+c x^2}}{d^4 x}+\frac {e \left (a+c x^2\right )^{3/2}}{3 a d^2 x^3}+\frac {\text {Subst}\left (\int \frac {\sqrt {a+c x}}{x^3} \, dx,x,x^2\right )}{2 d}+\frac {e^2 \text {Subst}\left (\int \frac {\sqrt {a+c x}}{x^2} \, dx,x,x^2\right )}{2 d^3}-\frac {\left (c e^3\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{d^4}+\frac {e^4 \text {Subst}\left (\int \frac {\sqrt {a+c x}}{x} \, dx,x,x^2\right )}{2 d^5}-\frac {e^4 \int \frac {a e-c d x}{(d+e x) \sqrt {a+c x^2}} \, dx}{d^5} \\ & = -\frac {\sqrt {a+c x^2}}{4 d x^4}-\frac {e^2 \sqrt {a+c x^2}}{2 d^3 x^2}+\frac {e^3 \sqrt {a+c x^2}}{d^4 x}+\frac {e \left (a+c x^2\right )^{3/2}}{3 a d^2 x^3}+\frac {c \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+c x}} \, dx,x,x^2\right )}{8 d}+\frac {\left (c e^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{4 d^3}+\frac {\left (c e^3\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{d^4}-\frac {\left (c e^3\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{d^4}+\frac {\left (a e^4\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{2 d^5}-\frac {\left (e^3 \left (c d^2+a e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{d^5} \\ & = -\frac {\sqrt {a+c x^2}}{4 d x^4}-\frac {c \sqrt {a+c x^2}}{8 a d x^2}-\frac {e^2 \sqrt {a+c x^2}}{2 d^3 x^2}+\frac {e^3 \sqrt {a+c x^2}}{d^4 x}+\frac {e \left (a+c x^2\right )^{3/2}}{3 a d^2 x^3}-\frac {\sqrt {c} e^3 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{d^4}-\frac {c^2 \text {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{16 a d}+\frac {e^2 \text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{2 d^3}+\frac {\left (c e^3\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{d^4}+\frac {\left (a e^4\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{c d^5}+\frac {\left (e^3 \left (c d^2+a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{c d^2+a e^2-x^2} \, dx,x,\frac {a e-c d x}{\sqrt {a+c x^2}}\right )}{d^5} \\ & = -\frac {\sqrt {a+c x^2}}{4 d x^4}-\frac {c \sqrt {a+c x^2}}{8 a d x^2}-\frac {e^2 \sqrt {a+c x^2}}{2 d^3 x^2}+\frac {e^3 \sqrt {a+c x^2}}{d^4 x}+\frac {e \left (a+c x^2\right )^{3/2}}{3 a d^2 x^3}+\frac {e^3 \sqrt {c d^2+a e^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^5}-\frac {c e^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 \sqrt {a} d^3}-\frac {\sqrt {a} e^4 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^5}-\frac {c \text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{8 a d} \\ & = -\frac {\sqrt {a+c x^2}}{4 d x^4}-\frac {c \sqrt {a+c x^2}}{8 a d x^2}-\frac {e^2 \sqrt {a+c x^2}}{2 d^3 x^2}+\frac {e^3 \sqrt {a+c x^2}}{d^4 x}+\frac {e \left (a+c x^2\right )^{3/2}}{3 a d^2 x^3}+\frac {e^3 \sqrt {c d^2+a e^2} \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^5}+\frac {c^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{8 a^{3/2} d}-\frac {c e^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 \sqrt {a} d^3}-\frac {\sqrt {a} e^4 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{d^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {a+c x^2}}{x^5 (d+e x)} \, dx=\frac {\sqrt {a} \left (d \sqrt {a+c x^2} \left (c d^2 x^2 (-3 d+8 e x)+a \left (-6 d^3+8 d^2 e x-12 d e^2 x^2+24 e^3 x^3\right )\right )-48 a e^3 \sqrt {-c d^2-a e^2} x^4 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )\right )-6 \left (c^2 d^4-4 a c d^2 e^2-8 a^2 e^4\right ) x^4 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+c x^2}}{\sqrt {a}}\right )}{24 a^{3/2} d^5 x^4} \]

[In]

Integrate[Sqrt[a + c*x^2]/(x^5*(d + e*x)),x]

[Out]

(Sqrt[a]*(d*Sqrt[a + c*x^2]*(c*d^2*x^2*(-3*d + 8*e*x) + a*(-6*d^3 + 8*d^2*e*x - 12*d*e^2*x^2 + 24*e^3*x^3)) -
48*a*e^3*Sqrt[-(c*d^2) - a*e^2]*x^4*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a + c*x^2])/Sqrt[-(c*d^2) - a*e^2]]) -
6*(c^2*d^4 - 4*a*c*d^2*e^2 - 8*a^2*e^4)*x^4*ArcTanh[(Sqrt[c]*x - Sqrt[a + c*x^2])/Sqrt[a]])/(24*a^(3/2)*d^5*x^
4)

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.03

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+a}\, \left (-24 a \,e^{3} x^{3}-8 c \,d^{2} e \,x^{3}+12 a d \,e^{2} x^{2}+3 c \,d^{3} x^{2}-8 a \,d^{2} e x +6 a \,d^{3}\right )}{24 d^{4} x^{4} a}-\frac {-\frac {\left (-8 a^{2} e^{4}-4 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{d \sqrt {a}}-\frac {8 a \,e^{2} \left (e^{2} a +c \,d^{2}\right ) \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}}{8 d^{4} a}\)	\(282\)
default	\(\frac {-\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4 a \,x^{4}}-\frac {c \left (-\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}}}{2 a \,x^{2}}+\frac {c \left (\sqrt {c \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )\right )}{2 a}\right )}{4 a}}{d}+\frac {e^{4} \left (\sqrt {c \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )\right )}{d^{5}}+\frac {e^{2} \left (-\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}}}{2 a \,x^{2}}+\frac {c \left (\sqrt {c \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )\right )}{2 a}\right )}{d^{3}}-\frac {e^{3} \left (-\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}}}{a x}+\frac {2 c \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{a}\right )}{d^{4}}+\frac {e \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{3 a \,d^{2} x^{3}}-\frac {e^{4} \left (\sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}-\frac {\sqrt {c}\, d \ln \left (\frac {-\frac {c d}{e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}\right )}{e}-\frac {\left (e^{2} a +c \,d^{2}\right ) \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{d^{5}}\)	\(558\)

[In]

int((c*x^2+a)^(1/2)/x^5/(e*x+d),x,method=_RETURNVERBOSE)

[Out]

-1/24*(c*x^2+a)^(1/2)*(-24*a*e^3*x^3-8*c*d^2*e*x^3+12*a*d*e^2*x^2+3*c*d^3*x^2-8*a*d^2*e*x+6*a*d^3)/d^4/x^4/a-1
/8/d^4/a*(-(-8*a^2*e^4-4*a*c*d^2*e^2+c^2*d^4)/d/a^(1/2)*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x)-8*a*e^2*(a*e^2+c
*d^2)/d/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2/e*c*d*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*((x+d/e)
^2*c-2/e*c*d*(x+d/e)+(a*e^2+c*d^2)/e^2)^(1/2))/(x+d/e)))

Fricas [A] (verification not implemented)

none

Time = 0.51 (sec) , antiderivative size = 1007, normalized size of antiderivative = 3.68 \[ \int \frac {\sqrt {a+c x^2}}{x^5 (d+e x)} \, dx=\left [\frac {24 \, \sqrt {c d^{2} + a e^{2}} a^{2} e^{3} x^{4} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 3 \, {\left (c^{2} d^{4} - 4 \, a c d^{2} e^{2} - 8 \, a^{2} e^{4}\right )} \sqrt {a} x^{4} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (8 \, a^{2} d^{3} e x - 6 \, a^{2} d^{4} + 8 \, {\left (a c d^{3} e + 3 \, a^{2} d e^{3}\right )} x^{3} - 3 \, {\left (a c d^{4} + 4 \, a^{2} d^{2} e^{2}\right )} x^{2}\right )} \sqrt {c x^{2} + a}}{48 \, a^{2} d^{5} x^{4}}, \frac {48 \, \sqrt {-c d^{2} - a e^{2}} a^{2} e^{3} x^{4} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) - 3 \, {\left (c^{2} d^{4} - 4 \, a c d^{2} e^{2} - 8 \, a^{2} e^{4}\right )} \sqrt {a} x^{4} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (8 \, a^{2} d^{3} e x - 6 \, a^{2} d^{4} + 8 \, {\left (a c d^{3} e + 3 \, a^{2} d e^{3}\right )} x^{3} - 3 \, {\left (a c d^{4} + 4 \, a^{2} d^{2} e^{2}\right )} x^{2}\right )} \sqrt {c x^{2} + a}}{48 \, a^{2} d^{5} x^{4}}, \frac {12 \, \sqrt {c d^{2} + a e^{2}} a^{2} e^{3} x^{4} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 3 \, {\left (c^{2} d^{4} - 4 \, a c d^{2} e^{2} - 8 \, a^{2} e^{4}\right )} \sqrt {-a} x^{4} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + {\left (8 \, a^{2} d^{3} e x - 6 \, a^{2} d^{4} + 8 \, {\left (a c d^{3} e + 3 \, a^{2} d e^{3}\right )} x^{3} - 3 \, {\left (a c d^{4} + 4 \, a^{2} d^{2} e^{2}\right )} x^{2}\right )} \sqrt {c x^{2} + a}}{24 \, a^{2} d^{5} x^{4}}, \frac {24 \, \sqrt {-c d^{2} - a e^{2}} a^{2} e^{3} x^{4} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) - 3 \, {\left (c^{2} d^{4} - 4 \, a c d^{2} e^{2} - 8 \, a^{2} e^{4}\right )} \sqrt {-a} x^{4} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + {\left (8 \, a^{2} d^{3} e x - 6 \, a^{2} d^{4} + 8 \, {\left (a c d^{3} e + 3 \, a^{2} d e^{3}\right )} x^{3} - 3 \, {\left (a c d^{4} + 4 \, a^{2} d^{2} e^{2}\right )} x^{2}\right )} \sqrt {c x^{2} + a}}{24 \, a^{2} d^{5} x^{4}}\right ] \]

[In]

integrate((c*x^2+a)^(1/2)/x^5/(e*x+d),x, algorithm="fricas")

[Out]

[1/48*(24*sqrt(c*d^2 + a*e^2)*a^2*e^3*x^4*log((2*a*c*d*e*x - a*c*d^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2 +
2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)) - 3*(c^2*d^4 - 4*a*c*d^2*e^2
- 8*a^2*e^4)*sqrt(a)*x^4*log(-(c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(8*a^2*d^3*e*x - 6*a^2*d^4 +
8*(a*c*d^3*e + 3*a^2*d*e^3)*x^3 - 3*(a*c*d^4 + 4*a^2*d^2*e^2)*x^2)*sqrt(c*x^2 + a))/(a^2*d^5*x^4), 1/48*(48*sq
rt(-c*d^2 - a*e^2)*a^2*e^3*x^4*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a)/(a*c*d^2 + a^2*e^2 +
(c^2*d^2 + a*c*e^2)*x^2)) - 3*(c^2*d^4 - 4*a*c*d^2*e^2 - 8*a^2*e^4)*sqrt(a)*x^4*log(-(c*x^2 - 2*sqrt(c*x^2 + a
)*sqrt(a) + 2*a)/x^2) + 2*(8*a^2*d^3*e*x - 6*a^2*d^4 + 8*(a*c*d^3*e + 3*a^2*d*e^3)*x^3 - 3*(a*c*d^4 + 4*a^2*d^
2*e^2)*x^2)*sqrt(c*x^2 + a))/(a^2*d^5*x^4), 1/24*(12*sqrt(c*d^2 + a*e^2)*a^2*e^3*x^4*log((2*a*c*d*e*x - a*c*d^
2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2 + 2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*
d*e*x + d^2)) - 3*(c^2*d^4 - 4*a*c*d^2*e^2 - 8*a^2*e^4)*sqrt(-a)*x^4*arctan(sqrt(-a)/sqrt(c*x^2 + a)) + (8*a^2
*d^3*e*x - 6*a^2*d^4 + 8*(a*c*d^3*e + 3*a^2*d*e^3)*x^3 - 3*(a*c*d^4 + 4*a^2*d^2*e^2)*x^2)*sqrt(c*x^2 + a))/(a^
2*d^5*x^4), 1/24*(24*sqrt(-c*d^2 - a*e^2)*a^2*e^3*x^4*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a
)/(a*c*d^2 + a^2*e^2 + (c^2*d^2 + a*c*e^2)*x^2)) - 3*(c^2*d^4 - 4*a*c*d^2*e^2 - 8*a^2*e^4)*sqrt(-a)*x^4*arctan
(sqrt(-a)/sqrt(c*x^2 + a)) + (8*a^2*d^3*e*x - 6*a^2*d^4 + 8*(a*c*d^3*e + 3*a^2*d*e^3)*x^3 - 3*(a*c*d^4 + 4*a^2
*d^2*e^2)*x^2)*sqrt(c*x^2 + a))/(a^2*d^5*x^4)]

Sympy [F]

\[ \int \frac {\sqrt {a+c x^2}}{x^5 (d+e x)} \, dx=\int \frac {\sqrt {a + c x^{2}}}{x^{5} \left (d + e x\right )}\, dx \]

[In]

integrate((c*x**2+a)**(1/2)/x**5/(e*x+d),x)

[Out]

Integral(sqrt(a + c*x**2)/(x**5*(d + e*x)), x)

Maxima [F]

\[ \int \frac {\sqrt {a+c x^2}}{x^5 (d+e x)} \, dx=\int { \frac {\sqrt {c x^{2} + a}}{{\left (e x + d\right )} x^{5}} \,d x } \]

[In]

integrate((c*x^2+a)^(1/2)/x^5/(e*x+d),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2 + a)/((e*x + d)*x^5), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 605 vs. \(2 (229) = 458\).

Time = 0.30 (sec) , antiderivative size = 605, normalized size of antiderivative = 2.21 \[ \int \frac {\sqrt {a+c x^2}}{x^5 (d+e x)} \, dx=-\frac {2 \, {\left (c d^{2} e^{3} + a e^{5}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{\sqrt {-c d^{2} - a e^{2}} d^{5}} - \frac {{\left (c^{2} d^{4} - 4 \, a c d^{2} e^{2} - 8 \, a^{2} e^{4}\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a d^{5}} + \frac {3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} c^{2} d^{3} + 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} a c d e^{2} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{6} a c^{\frac {3}{2}} d^{2} e - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{6} a^{2} \sqrt {c} e^{3} + 21 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} a c^{2} d^{3} - 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} a^{2} c d e^{2} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a^{2} c^{\frac {3}{2}} d^{2} e + 72 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} a^{3} \sqrt {c} e^{3} + 21 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{2} c^{2} d^{3} - 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} a^{3} c d e^{2} - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{3} c^{\frac {3}{2}} d^{2} e - 72 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a^{4} \sqrt {c} e^{3} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{3} c^{2} d^{3} + 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a^{4} c d e^{2} + 8 \, a^{4} c^{\frac {3}{2}} d^{2} e + 24 \, a^{5} \sqrt {c} e^{3}}{12 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{4} a d^{4}} \]

[In]

integrate((c*x^2+a)^(1/2)/x^5/(e*x+d),x, algorithm="giac")

[Out]

-2*(c*d^2*e^3 + a*e^5)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2))/(sqrt(-c*d^
2 - a*e^2)*d^5) - 1/4*(c^2*d^4 - 4*a*c*d^2*e^2 - 8*a^2*e^4)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/(s
qrt(-a)*a*d^5) + 1/12*(3*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^2*d^3 + 12*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a*c*d*e^
2 - 24*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a*c^(3/2)*d^2*e - 24*(sqrt(c)*x - sqrt(c*x^2 + a))^6*a^2*sqrt(c)*e^3 +
21*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a*c^2*d^3 - 12*(sqrt(c)*x - sqrt(c*x^2 + a))^5*a^2*c*d*e^2 + 24*(sqrt(c)*x
- sqrt(c*x^2 + a))^4*a^2*c^(3/2)*d^2*e + 72*(sqrt(c)*x - sqrt(c*x^2 + a))^4*a^3*sqrt(c)*e^3 + 21*(sqrt(c)*x -
sqrt(c*x^2 + a))^3*a^2*c^2*d^3 - 12*(sqrt(c)*x - sqrt(c*x^2 + a))^3*a^3*c*d*e^2 - 8*(sqrt(c)*x - sqrt(c*x^2 +
a))^2*a^3*c^(3/2)*d^2*e - 72*(sqrt(c)*x - sqrt(c*x^2 + a))^2*a^4*sqrt(c)*e^3 + 3*(sqrt(c)*x - sqrt(c*x^2 + a))
*a^3*c^2*d^3 + 12*(sqrt(c)*x - sqrt(c*x^2 + a))*a^4*c*d*e^2 + 8*a^4*c^(3/2)*d^2*e + 24*a^5*sqrt(c)*e^3)/(((sqr
t(c)*x - sqrt(c*x^2 + a))^2 - a)^4*a*d^4)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+c x^2}}{x^5 (d+e x)} \, dx=\int \frac {\sqrt {c\,x^2+a}}{x^5\,\left (d+e\,x\right )} \,d x \]

[In]

int((a + c*x^2)^(1/2)/(x^5*(d + e*x)),x)

[Out]

int((a + c*x^2)^(1/2)/(x^5*(d + e*x)), x)