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]

-((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])

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Rubi [A]  time = 0.135421, antiderivative size = 218, normalized size of antiderivative = 1., 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 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]]

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 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]

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*}

Mathematica [A]  time = 0.156722, 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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Maple [B]  time = 0.013, size = 558, normalized size = 2.6 \begin{align*}{\frac{b{x}^{2}+a}{16\,{a}^{3}{c}^{2}{x}^{4}} \left ( -2\,b{d}^{2}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}{x}^{6}a\sqrt{ac}-10\,{b}^{2}d\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}{x}^{6}c\sqrt{ac}+{a}^{3}\ln \left ({\frac{1}{{x}^{2}} \left ( ad{x}^{2}+bc{x}^{2}+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}+2\,ac \right ) } \right ){d}^{2}c{x}^{4}+2\,\ln \left ({\frac{ad{x}^{2}+bc{x}^{2}+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}+2\,ac}{{x}^{2}}} \right ) db{a}^{2}{c}^{2}{x}^{4}-3\,{c}^{3}\ln \left ({\frac{ad{x}^{2}+bc{x}^{2}+2\,\sqrt{ac}\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}+2\,ac}{{x}^{2}}} \right ){b}^{2}a{x}^{4}-2\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}{d}^{2}{a}^{2}{x}^{4}\sqrt{ac}-8\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}\sqrt{ac}{x}^{4}abcd-10\,\sqrt{bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac}{b}^{2}{c}^{2}{x}^{4}\sqrt{ac}+2\, \left ( bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac \right ) ^{3/2}da{x}^{2}\sqrt{ac}+10\, \left ( bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac \right ) ^{3/2}bc{x}^{2}\sqrt{ac}-4\, \left ( bd{x}^{4}+ad{x}^{2}+bc{x}^{2}+ac \right ) ^{3/2}ac\sqrt{ac} \right ){\frac{1}{\sqrt{{\frac{e \left ( b{x}^{2}+a \right ) }{d{x}^{2}+c}}}}}{\frac{1}{\sqrt{ \left ( d{x}^{2}+c \right ) \left ( b{x}^{2}+a \right ) }}}{\frac{1}{\sqrt{ac}}}} \end{align*}

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)

[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)*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)
*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/(a*c)^(1/2)/c^2/x^4

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 8.21647, size = 926, normalized size = 4.25 \begin{align*} \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 ] \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\frac{{\left (b x^{2} + a\right )} e}{d x^{2} + c}} x^{5}}\,{d x} \end{align*}

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]

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