3.4.10 \(\int \frac {1}{x (\frac {e (a+b x^2)}{c+d x^2})^{3/2}} \, dx\) [310]

Optimal. Leaf size=152 \[ \frac {b c-a d}{a b e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}-\frac {c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{a^{3/2} e^{3/2}}+\frac {d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{b^{3/2} e^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.20, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1981, 1980, 491, 536, 214} \begin {gather*} -\frac {c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{a^{3/2} e^{3/2}}+\frac {d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{b^{3/2} e^{3/2}}+\frac {b c-a d}{a b e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*((e*(a + b*x^2))/(c + d*x^2))^(3/2)),x]

[Out]

(b*c - a*d)/(a*b*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]) - (c^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[(e*(a + b*x^2))/(c + d*
x^2)])/(Sqrt[a]*Sqrt[e])])/(a^(3/2)*e^(3/2)) + (d^(3/2)*ArcTanh[(Sqrt[d]*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(S
qrt[b]*Sqrt[e])])/(b^(3/2)*e^(3/2))

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 491

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*e*(m + 1))), x] - Dist[1/(a*c*e^n*(m + 1)), Int[(e*x)^(m +
n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*
x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntBino
mialQ[a, b, c, d, e, m, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

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

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Mathematica [A]
time = 1.70, size = 189, normalized size = 1.24 \begin {gather*} \frac {-b^{3/2} c^{3/2} \sqrt {a+b x^2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )+\sqrt {a} \left (\sqrt {b} (b c-a d) \sqrt {c+d x^2}+a d^{3/2} \sqrt {a+b x^2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )\right )}{a^{3/2} b^{3/2} e \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*((e*(a + b*x^2))/(c + d*x^2))^(3/2)),x]

[Out]

(-(b^(3/2)*c^(3/2)*Sqrt[a + b*x^2]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])]) + Sqrt[a]*(Sq
rt[b]*(b*c - a*d)*Sqrt[c + d*x^2] + a*d^(3/2)*Sqrt[a + b*x^2]*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^2])/(Sqrt[b]*Sqrt[
c + d*x^2])]))/(a^(3/2)*b^(3/2)*e*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. \(400\) vs. \(2(122)=244\).
time = 0.07, size = 401, normalized size = 2.64

method	result	size
default	\(\frac {\left (\ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a b \,d^{2} x^{2}-\sqrt {b d}\, \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) b^{2} c^{2} x^{2}+\ln \left (\frac {2 b d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) \sqrt {a c}\, a^{2} d^{2}-\sqrt {b d}\, \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) a b \,c^{2}-2 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, a d +2 \sqrt {b d}\, \sqrt {a c}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, b c \right ) \left (b \,x^{2}+a \right )}{2 a b \sqrt {a c}\, \sqrt {b d}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \left (d \,x^{2}+c \right ) \left (\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}\right )^{\frac {3}{2}}}\)	\(401\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(e*(b*x^2+a)/(d*x^2+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.52, size = 185, normalized size = 1.22 \begin {gather*} \frac {1}{2} \, {\left (\frac {c^{2} \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} - \frac {d^{2} \log \left (\frac {d \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} - \sqrt {b d}}{d \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} + \sqrt {b d}}\right )}{\sqrt {b d} b} + \frac {2 \, {\left (b c - a d\right )}}{a b \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}}\right )} e^{\left (-\frac {3}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(e*(b*x^2+a)/(d*x^2+c))^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (110) = 220\).
time = 0.87, size = 1177, normalized size = 7.74 \begin {gather*} \left [\frac {{\left ({\left (a b d x^{2} + a^{2} d\right )} \sqrt {\frac {d}{b}} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + a b c d + {\left (3 \, b^{2} c d + a b d^{2}\right )} x^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} \sqrt {\frac {d}{b}}\right ) + {\left (b^{2} c x^{2} + a b c\right )} \sqrt {\frac {c}{a}} \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 (a b c d + a^{2} d^{2}\right )} x^{4} + 2 \, a^{2} c^{2} + {\left (a b c^{2} + 3 \, a^{2} c d\right )} x^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} \sqrt {\frac {c}{a}}}{x^{4}}\right ) + 4 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}\right )} e^{\left (-\frac {3}{2}\right )}}{4 \, {\left (a b^{2} x^{2} + a^{2} b\right )}}, -\frac {{\left (2 \, {\left (a b d x^{2} + a^{2} d\right )} \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d x^{2} + a d\right )}}\right ) - {\left (b^{2} c x^{2} + a b c\right )} \sqrt {\frac {c}{a}} \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 (a b c d + a^{2} d^{2}\right )} x^{4} + 2 \, a^{2} c^{2} + {\left (a b c^{2} + 3 \, a^{2} c d\right )} x^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} \sqrt {\frac {c}{a}}}{x^{4}}\right ) - 4 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}\right )} e^{\left (-\frac {3}{2}\right )}}{4 \, {\left (a b^{2} x^{2} + a^{2} b\right )}}, \frac {{\left (2 \, {\left (b^{2} c x^{2} + a b c\right )} \sqrt {-\frac {c}{a}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} \sqrt {-\frac {c}{a}}}{2 \, {\left (b c x^{2} + a c\right )}}\right ) + {\left (a b d x^{2} + a^{2} d\right )} \sqrt {\frac {d}{b}} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + a b c d + {\left (3 \, b^{2} c d + a b d^{2}\right )} x^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} \sqrt {\frac {d}{b}}\right ) + 4 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}\right )} e^{\left (-\frac {3}{2}\right )}}{4 \, {\left (a b^{2} x^{2} + a^{2} b\right )}}, \frac {{\left ({\left (b^{2} c x^{2} + a b c\right )} \sqrt {-\frac {c}{a}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} \sqrt {-\frac {c}{a}}}{2 \, {\left (b c x^{2} + a c\right )}}\right ) - {\left (a b d x^{2} + a^{2} d\right )} \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d x^{2} + a d\right )}}\right ) + 2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )} \sqrt {\frac {b x^{2} + a}{d x^{2} + c}}\right )} e^{\left (-\frac {3}{2}\right )}}{2 \, {\left (a b^{2} x^{2} + a^{2} b\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(e*(b*x^2+a)/(d*x^2+c))^(3/2),x, algorithm="fricas")

[Out]

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

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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/(e*(b*x**2+a)/(d*x**2+c))**(3/2),x)

[Out]

Timed out

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(e*(b*x^2+a)/(d*x^2+c))^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x\,{\left (\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*((e*(a + b*x^2))/(c + d*x^2))^(3/2)),x)

[Out]

int(1/(x*((e*(a + b*x^2))/(c + d*x^2))^(3/2)), x)

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