3.312 \(\int \frac{1}{x^5 (\frac{e (a+b x^2)}{c+d x^2})^{3/2}} \, dx\)

Optimal. Leaf size=255 \[ -\frac{(7 b c-3 a d) (b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{8 a^3 \left (a e^2-\frac{c e^2 \left (a+b x^2\right )}{c+d x^2}\right )}-\frac{3 (5 b c-a d) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right )}{8 a^{7/2} \sqrt{c} e^{3/2}}-\frac{(b c-a d)^2 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{4 a^2 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^2}+\frac{b (b c-a d)}{a^3 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}} \]

[Out]

(b*(b*c - a*d))/(a^3*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]) - ((b*c - a*d)^2*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/
(4*a^2*(a*e - (c*e*(a + b*x^2))/(c + d*x^2))^2) - ((7*b*c - 3*a*d)*(b*c - a*d)*Sqrt[(e*(a + b*x^2))/(c + d*x^2
)])/(8*a^3*(a*e^2 - (c*e^2*(a + b*x^2))/(c + d*x^2))) - (3*(b*c - a*d)*(5*b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[(e*
(a + b*x^2))/(c + d*x^2)])/(Sqrt[a]*Sqrt[e])])/(8*a^(7/2)*Sqrt[c]*e^(3/2))

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Rubi [A]  time = 0.22828, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {1960, 456, 453, 208} \[ -\frac{(7 b c-3 a d) (b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{8 a^3 \left (a e^2-\frac{c e^2 \left (a+b x^2\right )}{c+d x^2}\right )}-\frac{3 (5 b c-a d) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt{a} \sqrt{e}}\right )}{8 a^{7/2} \sqrt{c} e^{3/2}}-\frac{(b c-a d)^2 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{4 a^2 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^2}+\frac{b (b c-a d)}{a^3 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*((e*(a + b*x^2))/(c + d*x^2))^(3/2)),x]

[Out]

(b*(b*c - a*d))/(a^3*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]) - ((b*c - a*d)^2*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/
(4*a^2*(a*e - (c*e*(a + b*x^2))/(c + d*x^2))^2) - ((7*b*c - 3*a*d)*(b*c - a*d)*Sqrt[(e*(a + b*x^2))/(c + d*x^2
)])/(8*a^3*(a*e^2 - (c*e^2*(a + b*x^2))/(c + d*x^2))) - (3*(b*c - a*d)*(5*b*c - a*d)*ArcTanh[(Sqrt[c]*Sqrt[(e*
(a + b*x^2))/(c + d*x^2)])/(Sqrt[a]*Sqrt[e])])/(8*a^(7/2)*Sqrt[c]*e^(3/2))

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 456

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

Rule 453

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

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 \left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx &=((b c-a d) e) \operatorname{Subst}\left (\int \frac{b e-d x^2}{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 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{4 a^2 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac{1}{4} ((b c-a d) e) \operatorname{Subst}\left (\int \frac{\frac{4 b}{a}+\frac{3 (b c-a d) x^2}{a^2 e}}{x^2 \left (-a e+c x^2\right )^2} \, dx,x,\sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}\right )\\ &=-\frac{(b c-a d)^2 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{4 a^2 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac{(7 b c-3 a d) (b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{8 a^3 \left (a e^2-\frac{c e^2 \left (a+b x^2\right )}{c+d x^2}\right )}+\frac{1}{8} ((b c-a d) e) \operatorname{Subst}\left (\int \frac{\frac{8 b}{a^2 e}+\frac{(7 b c-3 a d) x^2}{a^3 e^2}}{x^2 \left (-a e+c x^2\right )} \, dx,x,\sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}\right )\\ &=\frac{b (b c-a d)}{a^3 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{(b c-a d)^2 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{4 a^2 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac{(7 b c-3 a d) (b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{8 a^3 \left (a e^2-\frac{c e^2 \left (a+b x^2\right )}{c+d x^2}\right )}+\frac{(3 (b c-a d) (5 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^3 e}\\ &=\frac{b (b c-a d)}{a^3 e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}-\frac{(b c-a d)^2 \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{4 a^2 \left (a e-\frac{c e \left (a+b x^2\right )}{c+d x^2}\right )^2}-\frac{(7 b c-3 a d) (b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{8 a^3 \left (a e^2-\frac{c e^2 \left (a+b x^2\right )}{c+d x^2}\right )}-\frac{3 (b c-a d) (5 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^{7/2} \sqrt{c} e^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.120786, size = 189, normalized size = 0.74 \[ \frac{\sqrt{a} \sqrt{c} \sqrt{c+d x^2} \left (-a^2 \left (2 c+5 d x^2\right )+a b x^2 \left (5 c-13 d x^2\right )+15 b^2 c x^4\right )-3 x^4 \sqrt{a+b x^2} \left (a^2 d^2-6 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^2}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{8 a^{7/2} \sqrt{c} e x^4 \sqrt{c+d x^2} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*((e*(a + b*x^2))/(c + d*x^2))^(3/2)),x]

[Out]

(Sqrt[a]*Sqrt[c]*Sqrt[c + d*x^2]*(15*b^2*c*x^4 + a*b*x^2*(5*c - 13*d*x^2) - a^2*(2*c + 5*d*x^2)) - 3*(5*b^2*c^
2 - 6*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^(7/2)*Sqrt[c]*e*x^4*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[c + d*x^2])

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Maple [B]  time = 0.018, size = 1042, normalized size = 4.1 \begin{align*} \text{result too large to display} \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))^(3/2),x)

[Out]

-1/16*(-6*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(a*c)^(1/2)*x^8*a*b^2*d^2+18*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)
*(a*c)^(1/2)*x^8*b^3*c*d+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)*x
^6*a^3*b*c*d^2-18*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)*x^6*a^2*b^
2*c^2*d+15*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)*x^6*a*b^3*c^3-12*
(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(a*c)^(1/2)*x^6*a^2*b*d^2+26*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(a*c)^(1/
2)*x^6*a*b^2*c*d+18*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(a*c)^(1/2)*x^6*b^3*c^2+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)*x^4*a^4*c*d^2-18*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)*x^4*a^3*b*c^2*d+15*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)*x^4*a^2*b^2*c^3+6*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(3/2)*(a*c)^(1/2)*x^4*a*b*d-
18*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(3/2)*(a*c)^(1/2)*x^4*b^2*c-6*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(a*c)^(1/2)
*x^4*a^3*d^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^2*b*c*d+18*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^
(1/2)*(a*c)^(1/2)*x^4*a*b^2*c^2+16*(a*c)^(1/2)*((d*x^2+c)*(b*x^2+a))^(1/2)*x^4*a^2*b*c*d-16*(a*c)^(1/2)*((d*x^
2+c)*(b*x^2+a))^(1/2)*x^4*a*b^2*c^2+6*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(3/2)*(a*c)^(1/2)*x^2*a^2*d-14*(b*d*x^4+a*
d*x^2+b*c*x^2+a*c)^(3/2)*(a*c)^(1/2)*x^2*a*b*c+4*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(3/2)*(a*c)^(1/2)*a^2*c)/c*(b*x
^2+a)/x^4/(a*c)^(1/2)/a^4/((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 [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))^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 36.7036, size = 1284, normalized size = 5.04 \begin{align*} \left [\frac{3 \,{\left ({\left (5 \, b^{3} c^{2} - 6 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{6} +{\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{4}\right )} \sqrt{a c e} \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 ({\left (15 \, a b^{2} c^{2} d - 13 \, a^{2} b c d^{2}\right )} x^{6} - 2 \, a^{3} c^{3} +{\left (15 \, a b^{2} c^{3} - 8 \, a^{2} b c^{2} d - 5 \, a^{3} c d^{2}\right )} x^{4} +{\left (5 \, a^{2} b c^{3} - 7 \, a^{3} c^{2} d\right )} x^{2}\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}}}{32 \,{\left (a^{4} b c e^{2} x^{6} + a^{5} c e^{2} x^{4}\right )}}, \frac{3 \,{\left ({\left (5 \, b^{3} c^{2} - 6 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{6} +{\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{4}\right )} \sqrt{-a c e} \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 ({\left (15 \, a b^{2} c^{2} d - 13 \, a^{2} b c d^{2}\right )} x^{6} - 2 \, a^{3} c^{3} +{\left (15 \, a b^{2} c^{3} - 8 \, a^{2} b c^{2} d - 5 \, a^{3} c d^{2}\right )} x^{4} +{\left (5 \, a^{2} b c^{3} - 7 \, a^{3} c^{2} d\right )} x^{2}\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}}}{16 \,{\left (a^{4} b c e^{2} x^{6} + a^{5} c e^{2} x^{4}\right )}}\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))^(3/2),x, algorithm="fricas")

[Out]

[1/32*(3*((5*b^3*c^2 - 6*a*b^2*c*d + a^2*b*d^2)*x^6 + (5*a*b^2*c^2 - 6*a^2*b*c*d + a^3*d^2)*x^4)*sqrt(a*c*e)*l
og(((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)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/x^4) + 4*((15*a*b^2*c^2*d -
13*a^2*b*c*d^2)*x^6 - 2*a^3*c^3 + (15*a*b^2*c^3 - 8*a^2*b*c^2*d - 5*a^3*c*d^2)*x^4 + (5*a^2*b*c^3 - 7*a^3*c^2
*d)*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(a^4*b*c*e^2*x^6 + a^5*c*e^2*x^4), 1/16*(3*((5*b^3*c^2 - 6*a*b^2*c
*d + a^2*b*d^2)*x^6 + (5*a*b^2*c^2 - 6*a^2*b*c*d + a^3*d^2)*x^4)*sqrt(-a*c*e)*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*((15*a*b^2*c^2*d - 13*a^2*b*c
*d^2)*x^6 - 2*a^3*c^3 + (15*a*b^2*c^3 - 8*a^2*b*c^2*d - 5*a^3*c*d^2)*x^4 + (5*a^2*b*c^3 - 7*a^3*c^2*d)*x^2)*sq
rt((b*e*x^2 + a*e)/(d*x^2 + c)))/(a^4*b*c*e^2*x^6 + a^5*c*e^2*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))**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (\frac{{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac{3}{2}} 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))^(3/2),x, algorithm="giac")

[Out]

integrate(1/(((b*x^2 + a)*e/(d*x^2 + c))^(3/2)*x^5), x)