3.301 \(\int \frac {1}{x^5 \sqrt {\frac {e (a+b x^2)}{c+d x^2}}} \, dx\)

Optimal. Leaf size=218 \[ -\frac {(a d+3 b c) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{8 a^{5/2} c^{3/2} \sqrt {e}}-\frac {(a d+3 b c) (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 a^2 c \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {e (b c-a d)^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 a c \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.14, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1960, 385, 199, 208} \[ -\frac {(a d+3 b c) (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{8 a^{5/2} c^{3/2} \sqrt {e}}-\frac {(a d+3 b c) (b c-a d) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{8 a^2 c \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {e (b c-a d)^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 a c \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]),x]

[Out]

-((b*c - a*d)^2*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(4*a*c*(a*e - (c*e*(a + b*x^2))/(c + d*x^2))^2) - ((b*c -
a*d)*(3*b*c + a*d)*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(8*a^2*c*(a*e - (c*e*(a + b*x^2))/(c + d*x^2))) - ((b*c
- a*d)*(3*b*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(Sqrt[a]*Sqrt[e])])/(8*a^(5/2)*c^(3/
2)*Sqrt[e])

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 208

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

Rule 385

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

Rule 1960

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> With[{q = Den
ominator[p]}, Dist[(q*e*(b*c - a*d))/n, Subst[Int[(x^(q*(p + 1) - 1)*(-(a*e) + c*x^q)^(Simplify[(m + 1)/n] - 1
))/(b*e - d*x^q)^(Simplify[(m + 1)/n] + 1), x], x, ((e*(a + b*x^n))/(c + d*x^n))^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, m, n}, x] && FractionQ[p] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

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

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Mathematica [A]  time = 0.16, size = 173, normalized size = 0.79 \[ \frac {\sqrt {a} \sqrt {c} \left (a+b x^2\right ) \sqrt {c+d x^2} \left (3 b c x^2-a \left (2 c+d x^2\right )\right )-x^4 \sqrt {a+b x^2} \left (-a^2 d^2-2 a b c d+3 b^2 c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{8 a^{5/2} c^{3/2} x^4 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]),x]

[Out]

(Sqrt[a]*Sqrt[c]*(a + b*x^2)*Sqrt[c + d*x^2]*(3*b*c*x^2 - a*(2*c + d*x^2)) - (3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)
*x^4*Sqrt[a + b*x^2]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/(8*a^(5/2)*c^(3/2)*x^4*Sqrt
[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[c + d*x^2])

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fricas [A]  time = 1.32, size = 443, normalized size = 2.03 \[ \left [-\frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \sqrt {a c e} x^{4} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} e x^{4} + 8 \, a^{2} c^{2} e + 8 \, {\left (a b c^{2} + a^{2} c d\right )} e x^{2} + 4 \, {\left ({\left (b c d + a d^{2}\right )} x^{4} + 2 \, a c^{2} + {\left (b c^{2} + 3 \, a c d\right )} x^{2}\right )} \sqrt {a c e} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{x^{4}}\right ) + 4 \, {\left (2 \, a^{2} c^{3} - {\left (3 \, a b c^{2} d - a^{2} c d^{2}\right )} x^{4} - 3 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{32 \, a^{3} c^{2} e x^{4}}, \frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \sqrt {-a c e} x^{4} \arctan \left (\frac {\sqrt {-a c e} {\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{2 \, {\left (a b c e x^{2} + a^{2} c e\right )}}\right ) - 2 \, {\left (2 \, a^{2} c^{3} - {\left (3 \, a b c^{2} d - a^{2} c d^{2}\right )} x^{4} - 3 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{16 \, a^{3} c^{2} e x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/32*((3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*sqrt(a*c*e)*x^4*log(((b^2*c^2 + 6*a*b*c*d + a^2*d^2)*e*x^4 + 8*a^2*c
^2*e + 8*(a*b*c^2 + a^2*c*d)*e*x^2 + 4*((b*c*d + a*d^2)*x^4 + 2*a*c^2 + (b*c^2 + 3*a*c*d)*x^2)*sqrt(a*c*e)*sqr
t((b*e*x^2 + a*e)/(d*x^2 + c)))/x^4) + 4*(2*a^2*c^3 - (3*a*b*c^2*d - a^2*c*d^2)*x^4 - 3*(a*b*c^3 - a^2*c^2*d)*
x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(a^3*c^2*e*x^4), 1/16*((3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*sqrt(-a*c*e)*
x^4*arctan(1/2*sqrt(-a*c*e)*((b*c + a*d)*x^2 + 2*a*c)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c))/(a*b*c*e*x^2 + a^2*c*e
)) - 2*(2*a^2*c^3 - (3*a*b*c^2*d - a^2*c*d^2)*x^4 - 3*(a*b*c^3 - a^2*c^2*d)*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 +
c)))/(a^3*c^2*e*x^4)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep*d+c)]Unable to divide, perhaps due to rounding error%%%{%%%{1,[4,1,4]%%%},[2,7,0]%%%}+%%%{%%%{-4,[3
,2,4]%%%},[2,6,1]%%%}+%%%{%%%{6,[2,3,4]%%%},[2,5,2]%%%}+%%%{%%%{-4,[1,4,4]%%%},[2,4,3]%%%}+%%%{%%%{1,[0,5,4]%%
%},[2,3,4]%%%}+%%%{%%{[%%%{-2,[4,0,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,8,0]%%%}+%%%{%%{[%%%{8,[3,1,4]%%%}
,0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,7,1]%%%}+%%%{%%{[%%%{-12,[2,2,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,6,2]
%%%}+%%%{%%{[%%%{8,[1,3,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,5,3]%%%}+%%%{%%{[%%%{-2,[0,4,4]%%%},0]:[1,0,%
%%{-1,[1,1,1]%%%}]%%},[1,4,4]%%%}+%%%{%%%{1,[5,0,5]%%%},[0,9,0]%%%}+%%%{%%%{-4,[4,1,5]%%%},[0,8,1]%%%}+%%%{%%%
{6,[3,2,5]%%%},[0,7,2]%%%}+%%%{%%%{-4,[2,3,5]%%%},[0,6,3]%%%}+%%%{%%%{1,[1,4,5]%%%},[0,5,4]%%%} / %%%{%%%{1,[0
,2,0]%%%},[2,0,0]%%%}+%%%{%%{[%%%{-2,[0,1,0]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[1,1,0]%%%}+%%%{%%%{1,[1,1,1]%
%%},[0,2,0]%%%} Error: Bad Argument Value

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maple [B]  time = 0.06, size = 558, normalized size = 2.56 \[ \frac {\left (b \,x^{2}+a \right ) \left (a^{3} c \,d^{2} x^{4} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}{x^{2}}\right )+2 a^{2} b \,c^{2} d \,x^{4} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}{x^{2}}\right )-3 a \,b^{2} c^{3} x^{4} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 a c +2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}{x^{2}}\right )-2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a b \,d^{2} x^{6}-10 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, b^{2} c d \,x^{6}-2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a^{2} d^{2} x^{4}-8 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a b c d \,x^{4}-10 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, b^{2} c^{2} x^{4}+2 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, a d \,x^{2}+10 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, b c \,x^{2}-4 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, a c \right )}{16 \sqrt {\frac {\left (b \,x^{2}+a \right ) e}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {a c}\, a^{3} c^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(b*x^2+a)*(-2*b*d^2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*x^6*a*(a*c)^(1/2)-10*b^2*d*(b*d*x^4+a*d*x^2+b*c*x
^2+a*c)^(1/2)*x^6*c*(a*c)^(1/2)+a^3*ln((a*d*x^2+b*c*x^2+2*a*c+2*(a*c)^(1/2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2
))/x^2)*d^2*c*x^4+2*ln((a*d*x^2+b*c*x^2+2*a*c+2*(a*c)^(1/2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2))/x^2)*d*b*a^2*
c^2*x^4-3*c^3*ln((a*d*x^2+b*c*x^2+2*a*c+2*(a*c)^(1/2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2))/x^2)*b^2*a*x^4-2*(b
*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*d^2*a^2*x^4*(a*c)^(1/2)-8*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(a*c)^(1/2)*x^
4*a*b*c*d-10*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*b^2*c^2*x^4*(a*c)^(1/2)+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(3/2)
*(a*c)^(1/2)*a*d*x^2+10*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(3/2)*b*c*x^2*(a*c)^(1/2)-4*(b*d*x^4+a*d*x^2+b*c*x^2+a*c
)^(3/2)*(a*c)^(1/2)*a*c)/((b*x^2+a)/(d*x^2+c)*e)^(1/2)/((d*x^2+c)*(b*x^2+a))^(1/2)/a^3/c^2/x^4/(a*c)^(1/2)

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maxima [A]  time = 1.76, size = 271, normalized size = 1.24 \[ \frac {1}{16} \, e {\left (\frac {2 \, {\left ({\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2}\right )} \left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}} - {\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}} e\right )}}{a^{4} c e^{3} - \frac {2 \, {\left (b x^{2} + a\right )} a^{3} c^{2} e^{3}}{d x^{2} + c} + \frac {{\left (b x^{2} + a\right )}^{2} a^{2} c^{3} e^{3}}{{\left (d x^{2} + c\right )}^{2}}} + \frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \log \left (\frac {c \sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}} - \sqrt {a c e}}{c \sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}} + \sqrt {a c e}}\right )}{\sqrt {a c e} a^{2} c e}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*e*(2*((3*b^2*c^3 - 2*a*b*c^2*d - a^2*c*d^2)*((b*x^2 + a)*e/(d*x^2 + c))^(3/2) - (5*a*b^2*c^2 - 6*a^2*b*c*
d + a^3*d^2)*sqrt((b*x^2 + a)*e/(d*x^2 + c))*e)/(a^4*c*e^3 - 2*(b*x^2 + a)*a^3*c^2*e^3/(d*x^2 + c) + (b*x^2 +
a)^2*a^2*c^3*e^3/(d*x^2 + c)^2) + (3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*log((c*sqrt((b*x^2 + a)*e/(d*x^2 + c)) - s
qrt(a*c*e))/(c*sqrt((b*x^2 + a)*e/(d*x^2 + c)) + sqrt(a*c*e)))/(sqrt(a*c*e)*a^2*c*e))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^5\,\sqrt {\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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