3.288 \(\int \frac{(\frac{e (a+b x^2)}{c+d x^2})^{3/2}}{x^6} \, dx\)

Optimal. Leaf size=480 \[ -\frac{e \left (c+d x^2\right ) \left (16 a^2 d^2-16 a b c d+b^2 c^2\right ) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{5 a c^4 x}+\frac{d e x \left (16 a^2 d^2-16 a b c d+b^2 c^2\right ) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{5 a c^4}-\frac{\sqrt{d} e \left (16 a^2 d^2-16 a b c d+b^2 c^2\right ) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{5 a c^{7/2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{e \left (c+d x^2\right ) (7 b c-8 a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{5 c^3 x^3}+\frac{e \left (c+d x^2\right ) (5 b c-6 a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{5 c^2 d x^5}-\frac{b \sqrt{d} e (7 b c-8 a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{5 a c^{5/2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{e (b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c d x^5} \]

[Out]

-(((b*c - a*d)*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(c*d*x^5)) + (d*(b^2*c^2 - 16*a*b*c*d + 16*a^2*d^2)*e*x*Sq
rt[(e*(a + b*x^2))/(c + d*x^2)])/(5*a*c^4) + ((5*b*c - 6*a*d)*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(c + d*x^2))
/(5*c^2*d*x^5) - ((7*b*c - 8*a*d)*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(c + d*x^2))/(5*c^3*x^3) - ((b^2*c^2 - 1
6*a*b*c*d + 16*a^2*d^2)*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(c + d*x^2))/(5*a*c^4*x) - (Sqrt[d]*(b^2*c^2 - 16*
a*b*c*d + 16*a^2*d^2)*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*
d)])/(5*a*c^(7/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]) - (b*Sqrt[d]*(7*b*c - 8*a*d)*e*Sqrt[(e*(a + b*x^2))/(
c + d*x^2)]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(5*a*c^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c +
d*x^2))])

________________________________________________________________________________________

Rubi [A]  time = 0.808583, antiderivative size = 480, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {6719, 468, 583, 531, 418, 492, 411} \[ -\frac{e \left (c+d x^2\right ) \left (16 a^2 d^2-16 a b c d+b^2 c^2\right ) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{5 a c^4 x}+\frac{d e x \left (16 a^2 d^2-16 a b c d+b^2 c^2\right ) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{5 a c^4}-\frac{\sqrt{d} e \left (16 a^2 d^2-16 a b c d+b^2 c^2\right ) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{5 a c^{7/2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{e \left (c+d x^2\right ) (7 b c-8 a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{5 c^3 x^3}+\frac{e \left (c+d x^2\right ) (5 b c-6 a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{5 c^2 d x^5}-\frac{b \sqrt{d} e (7 b c-8 a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{5 a c^{5/2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{e (b c-a d) \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c d x^5} \]

Antiderivative was successfully verified.

[In]

Int[((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^6,x]

[Out]

-(((b*c - a*d)*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(c*d*x^5)) + (d*(b^2*c^2 - 16*a*b*c*d + 16*a^2*d^2)*e*x*Sq
rt[(e*(a + b*x^2))/(c + d*x^2)])/(5*a*c^4) + ((5*b*c - 6*a*d)*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(c + d*x^2))
/(5*c^2*d*x^5) - ((7*b*c - 8*a*d)*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(c + d*x^2))/(5*c^3*x^3) - ((b^2*c^2 - 1
6*a*b*c*d + 16*a^2*d^2)*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(c + d*x^2))/(5*a*c^4*x) - (Sqrt[d]*(b^2*c^2 - 16*
a*b*c*d + 16*a^2*d^2)*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*
d)])/(5*a*c^(7/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]) - (b*Sqrt[d]*(7*b*c - 8*a*d)*e*Sqrt[(e*(a + b*x^2))/(
c + d*x^2)]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(5*a*c^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c +
d*x^2))])

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rule 468

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

Rule 583

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

Rule 531

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

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticF[ArcT
an[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 492

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(x*Sqrt[a + b*x^2])/(b*Sqr
t[c + d*x^2]), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rule 411

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticE[ArcTan
[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rubi steps

\begin{align*} \int \frac{\left (\frac{e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^6} \, dx &=\frac{\left (e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}\right ) \int \frac{\left (a+b x^2\right )^{3/2}}{x^6 \left (c+d x^2\right )^{3/2}} \, dx}{\sqrt{a+b x^2}}\\ &=-\frac{(b c-a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c d x^5}-\frac{\left (e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}\right ) \int \frac{a (5 b c-6 a d)+b (4 b c-5 a d) x^2}{x^6 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{c d \sqrt{a+b x^2}}\\ &=-\frac{(b c-a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c d x^5}+\frac{(5 b c-6 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^2 d x^5}+\frac{\left (e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}\right ) \int \frac{3 a^2 d (7 b c-8 a d)+3 a b d (5 b c-6 a d) x^2}{x^4 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{5 a c^2 d \sqrt{a+b x^2}}\\ &=-\frac{(b c-a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c d x^5}+\frac{(5 b c-6 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^2 d x^5}-\frac{(7 b c-8 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^3 x^3}-\frac{\left (e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}\right ) \int \frac{-3 a^2 d \left (b^2 c^2-16 a b c d+16 a^2 d^2\right )+3 a^2 b d^2 (7 b c-8 a d) x^2}{x^2 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{15 a^2 c^3 d \sqrt{a+b x^2}}\\ &=-\frac{(b c-a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c d x^5}+\frac{(5 b c-6 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^2 d x^5}-\frac{(7 b c-8 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^3 x^3}-\frac{\left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 a c^4 x}+\frac{\left (e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}\right ) \int \frac{-3 a^3 b c d^2 (7 b c-8 a d)+3 a^2 b d^2 \left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{15 a^3 c^4 d \sqrt{a+b x^2}}\\ &=-\frac{(b c-a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c d x^5}+\frac{(5 b c-6 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^2 d x^5}-\frac{(7 b c-8 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^3 x^3}-\frac{\left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 a c^4 x}-\frac{\left (b d (7 b c-8 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}\right ) \int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{5 c^3 \sqrt{a+b x^2}}+\frac{\left (b d \left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}\right ) \int \frac{x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx}{5 a c^4 \sqrt{a+b x^2}}\\ &=-\frac{(b c-a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c d x^5}+\frac{d \left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) e x \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{5 a c^4}+\frac{(5 b c-6 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^2 d x^5}-\frac{(7 b c-8 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^3 x^3}-\frac{\left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 a c^4 x}-\frac{b \sqrt{d} (7 b c-8 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{5 a c^{5/2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{\left (d \left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \sqrt{c+d x^2}\right ) \int \frac{\sqrt{a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{5 a c^3 \sqrt{a+b x^2}}\\ &=-\frac{(b c-a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{c d x^5}+\frac{d \left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) e x \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}}}{5 a c^4}+\frac{(5 b c-6 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^2 d x^5}-\frac{(7 b c-8 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^3 x^3}-\frac{\left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 a c^4 x}-\frac{\sqrt{d} \left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{5 a c^{7/2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{b \sqrt{d} (7 b c-8 a d) e \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{5 a c^{5/2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\\ \end{align*}

Mathematica [C]  time = 0.658236, size = 357, normalized size = 0.74 \[ -\frac{e \sqrt{\frac{b}{a}} \sqrt{\frac{e \left (a+b x^2\right )}{c+d x^2}} \left (\sqrt{\frac{b}{a}} \left (a^2 b x^2 \left (-11 c^2 d x^2+3 c^3-8 c d^2 x^4+16 d^3 x^6\right )+a^3 \left (-2 c^2 d x^2+c^3+8 c d^2 x^4+16 d^3 x^6\right )+a b^2 c x^4 \left (3 c^2-8 c d x^2-16 d^2 x^4\right )+b^3 c^2 x^6 \left (c+d x^2\right )\right )-i b c x^5 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (8 a^2 d^2-9 a b c d+b^2 c^2\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+i b c x^5 \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (16 a^2 d^2-16 a b c d+b^2 c^2\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )\right )}{5 b c^4 x^5 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[((e*(a + b*x^2))/(c + d*x^2))^(3/2)/x^6,x]

[Out]

-(Sqrt[b/a]*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[b/a]*(b^3*c^2*x^6*(c + d*x^2) + a*b^2*c*x^4*(3*c^2 - 8*c
*d*x^2 - 16*d^2*x^4) + a^2*b*x^2*(3*c^3 - 11*c^2*d*x^2 - 8*c*d^2*x^4 + 16*d^3*x^6) + a^3*(c^3 - 2*c^2*d*x^2 +
8*c*d^2*x^4 + 16*d^3*x^6)) + I*b*c*(b^2*c^2 - 16*a*b*c*d + 16*a^2*d^2)*x^5*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2
)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*b*c*(b^2*c^2 - 9*a*b*c*d + 8*a^2*d^2)*x^5*Sqrt[1 + (b*
x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(5*b*c^4*x^5*(a + b*x^2))

________________________________________________________________________________________

Maple [B]  time = 0.017, size = 1197, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*(b*x^2+a)/(d*x^2+c))^(3/2)/x^6,x)

[Out]

-1/5*(e*(b*x^2+a)/(d*x^2+c))^(3/2)*(d*x^2+c)*(5*(-b/a)^(1/2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*x^8*a^2*b*d^3
-5*(-b/a)^(1/2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*x^8*a*b^2*c*d^2+11*(-b/a)^(1/2)*((d*x^2+c)*(b*x^2+a))^(1/2
)*x^8*a^2*b*d^3-11*(-b/a)^(1/2)*((d*x^2+c)*(b*x^2+a))^(1/2)*x^8*a*b^2*c*d^2+(-b/a)^(1/2)*((d*x^2+c)*(b*x^2+a))
^(1/2)*x^8*b^3*c^2*d+8*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*((d*x
^2+c)*(b*x^2+a))^(1/2)*x^5*a^2*b*c*d^2-9*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d
/b/c)^(1/2))*((d*x^2+c)*(b*x^2+a))^(1/2)*x^5*a*b^2*c^2*d+((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(
-b/a)^(1/2),(a*d/b/c)^(1/2))*((d*x^2+c)*(b*x^2+a))^(1/2)*x^5*b^3*c^3-16*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2
)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*((d*x^2+c)*(b*x^2+a))^(1/2)*x^5*a^2*b*c*d^2+16*((b*x^2+a)/a)^(1/2)
*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*((d*x^2+c)*(b*x^2+a))^(1/2)*x^5*a*b^2*c^2*d-((b
*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*((d*x^2+c)*(b*x^2+a))^(1/2)*x^5
*b^3*c^3+5*(-b/a)^(1/2)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*x^6*a^3*d^3-5*(-b/a)^(1/2)*(b*d*x^4+a*d*x^2+b*c*x^
2+a*c)^(1/2)*x^6*a^2*b*c*d^2+11*(-b/a)^(1/2)*((d*x^2+c)*(b*x^2+a))^(1/2)*x^6*a^3*d^3-3*(-b/a)^(1/2)*((d*x^2+c)
*(b*x^2+a))^(1/2)*x^6*a^2*b*c*d^2-8*(-b/a)^(1/2)*((d*x^2+c)*(b*x^2+a))^(1/2)*x^6*a*b^2*c^2*d+(-b/a)^(1/2)*((d*
x^2+c)*(b*x^2+a))^(1/2)*x^6*b^3*c^3+8*(-b/a)^(1/2)*((d*x^2+c)*(b*x^2+a))^(1/2)*x^4*a^3*c*d^2-11*(-b/a)^(1/2)*(
(d*x^2+c)*(b*x^2+a))^(1/2)*x^4*a^2*b*c^2*d+3*(-b/a)^(1/2)*((d*x^2+c)*(b*x^2+a))^(1/2)*x^4*a*b^2*c^3-2*(-b/a)^(
1/2)*((d*x^2+c)*(b*x^2+a))^(1/2)*x^2*a^3*c^2*d+3*(-b/a)^(1/2)*((d*x^2+c)*(b*x^2+a))^(1/2)*x^2*a^2*b*c^3+(-b/a)
^(1/2)*((d*x^2+c)*(b*x^2+a))^(1/2)*a^3*c^3)/a/(b*x^2+a)^2/c^4/(-b/a)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)
/x^5

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac{3}{2}}}{x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*(b*x^2+a)/(d*x^2+c))^(3/2)/x^6,x, algorithm="maxima")

[Out]

integrate(((b*x^2 + a)*e/(d*x^2 + c))^(3/2)/x^6, x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b e x^{2} + a e\right )} \sqrt{\frac{b e x^{2} + a e}{d x^{2} + c}}}{d x^{8} + c x^{6}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*(b*x^2+a)/(d*x^2+c))^(3/2)/x^6,x, algorithm="fricas")

[Out]

integral((b*e*x^2 + a*e)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c))/(d*x^8 + c*x^6), x)

________________________________________________________________________________________

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((e*(b*x**2+a)/(d*x**2+c))**(3/2)/x**6,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac{3}{2}}}{x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*(b*x^2+a)/(d*x^2+c))^(3/2)/x^6,x, algorithm="giac")

[Out]

integrate(((b*x^2 + a)*e/(d*x^2 + c))^(3/2)/x^6, x)