3.4.1 \(\int \frac {1}{x^5 \sqrt {\frac {e (a+b x^2)}{c+d x^2}}} \, dx\) [301]

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

[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 = 0.17, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1981, 1980, 393, 205, 214} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*((b*c - a*d)^2*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(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 205

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] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

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 393

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 1980

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)))/((c_) + (d_.)*(x_)))^(p_), x_Symbol] :> With[{q = Denominator[p]}
, Dist[q*e*(b*c - a*d), Subst[Int[x^(q*(p + 1) - 1)*(((-a)*e + c*x^q)^m/(b*e - d*x^q)^(m + 2)), x], x, (e*((a
+ b*x)/(c + d*x)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e, m}, x] && FractionQ[p] && IntegerQ[m]

Rule 1981

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Dist[1/n, Sub
st[Int[x^(Simplify[(m + 1)/n] - 1)*(e*((a + b*x)/(c + d*x)))^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n,
p}, x] && 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) \text {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) \text {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)) \text {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 = 1.23, size = 173, normalized size = 0.79 \begin {gather*} \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 )-\left (3 b^2 c^2-2 a b c d-a^2 d^2\right ) x^4 \sqrt {a+b x^2} \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 {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(558\) vs. \(2(194)=388\).
time = 0.09, size = 559, normalized size = 2.56

method	result	size
risch	\(-\frac {\left (b \,x^{2}+a \right ) \left (a d \,x^{2}-3 b c \,x^{2}+2 a c \right )}{8 a^{2} x^{4} c \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}+\frac {\left (\frac {\ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) d^{2}}{16 c \sqrt {a c e}}+\frac {\ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) b d}{8 a \sqrt {a c e}}-\frac {3 c \ln \left (\frac {2 a c e +\left (a d e +b c e \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {d e b \,x^{4}+\left (a d e +b c e \right ) x^{2}+a c e}}{x^{2}}\right ) b^{2}}{16 a^{2} \sqrt {a c e}}\right ) \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{\sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\)	\(326\)
default	\(-\frac {\left (b \,x^{2}+a \right ) \left (2 b \,d^{2} \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, x^{6} a \sqrt {a c}+10 b^{2} d \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, x^{6} c \sqrt {a c}-a^{3} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) d^{2} c \,x^{4}-2 \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) d b \,a^{2} c^{2} x^{4}+3 c^{3} \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) b^{2} a \,x^{4}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, d^{2} a^{2} x^{4} \sqrt {a c}+8 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a b c d \,x^{4}+10 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, b^{2} c^{2} x^{4} \sqrt {a c}-2 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} d a \,x^{2} \sqrt {a c}-10 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} b c \,x^{2} \sqrt {a c}+4 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} a c \sqrt {a c}\right )}{16 \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, a^{3} c^{2} x^{4} \sqrt {a c}}\)	\(559\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)^(1/2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+2*a*
c)/x^2)*d^2*c*x^4-2*ln((a*d*x^2+b*c*x^2+2*(a*c)^(1/2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+2*a*c)/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)^(1/2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+2*a*c)/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)*a*
b*c*d*x^4+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)
*d*a*x^2*(a*c)^(1/2)-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*(a*c)^(1/2))/(e*(b*x^2+a)/(d*x^2+c))^(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 = 0.55, size = 252, normalized size = 1.16 \begin {gather*} \frac {1}{16} \, {\left (\frac {2 \, {\left ({\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d - a^{2} c d^{2}\right )} \left (\frac {b x^{2} + a}{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 {b x^{2} + a}{d x^{2} + c}}\right )}}{a^{4} c - \frac {2 \, {\left (b x^{2} + a\right )} a^{3} c^{2}}{d x^{2} + c} + \frac {{\left (b x^{2} + a\right )}^{2} a^{2} c^{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 {b x^{2} + a}{d x^{2} + c}} - \sqrt {a c}}{c \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} + \sqrt {a c}}\right )}{\sqrt {a c} a^{2} c}\right )} e^{\left (-\frac {1}{2}\right )} \end {gather*}

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

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Fricas [A]
time = 0.66, size = 420, normalized size = 1.93 \begin {gather*} \left [-\frac {{\left ({\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \sqrt {a c} x^{4} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} 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} \sqrt {\frac {b x^{2} + a}{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 x^{2} + a}{d x^{2} + c}}\right )} e^{\left (-\frac {1}{2}\right )}}{32 \, a^{3} c^{2} x^{4}}, \frac {{\left ({\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \sqrt {-a c} x^{4} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {-a c} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}}{2 \, {\left (a b c x^{2} + a^{2} c\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 x^{2} + a}{d x^{2} + c}}\right )} e^{\left (-\frac {1}{2}\right )}}{16 \, a^{3} c^{2} x^{4}}\right ] \end {gather*}

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

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 486 vs. \(2 (194) = 388\).
time = 3.53, size = 486, normalized size = 2.23 \begin {gather*} \frac {{\left (\frac {{\left (3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}\right )} \arctan \left (-\frac {\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}}{\sqrt {-a c}}\right )}{\sqrt {-a c} a^{2} c} - \frac {3 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{3} b^{2} c^{2} - 5 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} a b^{2} c^{3} - 2 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{3} a b c d - 10 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} a^{2} b c^{2} d - {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{3} a^{2} d^{2} - {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )} a^{3} c d^{2} - 8 \, \sqrt {b d} a^{2} b c^{3} - 8 \, {\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{2} \sqrt {b d} a^{2} c d}{{\left ({\left (\sqrt {b d} x^{2} - \sqrt {b d x^{4} + b c x^{2} + a d x^{2} + a c}\right )}^{2} - a c\right )}^{2} a^{2} c}\right )} e^{\left (-\frac {1}{2}\right )}}{8 \, \mathrm {sgn}\left (d x^{2} + c\right )} \end {gather*}

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]

1/8*((3*b^2*c^2 - 2*a*b*c*d - a^2*d^2)*arctan(-(sqrt(b*d)*x^2 - sqrt(b*d*x^4 + b*c*x^2 + a*d*x^2 + a*c))/sqrt(
-a*c))/(sqrt(-a*c)*a^2*c) - (3*(sqrt(b*d)*x^2 - sqrt(b*d*x^4 + b*c*x^2 + a*d*x^2 + a*c))^3*b^2*c^2 - 5*(sqrt(b
*d)*x^2 - sqrt(b*d*x^4 + b*c*x^2 + a*d*x^2 + a*c))*a*b^2*c^3 - 2*(sqrt(b*d)*x^2 - sqrt(b*d*x^4 + b*c*x^2 + a*d
*x^2 + a*c))^3*a*b*c*d - 10*(sqrt(b*d)*x^2 - sqrt(b*d*x^4 + b*c*x^2 + a*d*x^2 + a*c))*a^2*b*c^2*d - (sqrt(b*d)
*x^2 - sqrt(b*d*x^4 + b*c*x^2 + a*d*x^2 + a*c))^3*a^2*d^2 - (sqrt(b*d)*x^2 - sqrt(b*d*x^4 + b*c*x^2 + a*d*x^2
+ a*c))*a^3*c*d^2 - 8*sqrt(b*d)*a^2*b*c^3 - 8*(sqrt(b*d)*x^2 - sqrt(b*d*x^4 + b*c*x^2 + a*d*x^2 + a*c))^2*sqrt
(b*d)*a^2*c*d)/(((sqrt(b*d)*x^2 - sqrt(b*d*x^4 + b*c*x^2 + a*d*x^2 + a*c))^2 - a*c)^2*a^2*c))*e^(-1/2)/sgn(d*x
^2 + c)

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

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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