\(\int \frac {(e x)^{3/2}}{(a-b x^2)^2 (c-d x^2)^{5/2}} \, dx\) [929]

Optimal result

Integrand size = 30, antiderivative size = 447 \[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {5 d e \sqrt {e x}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {e \sqrt {e x}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d (14 b c+a d) e \sqrt {e x}}{6 c (b c-a d)^3 \sqrt {c-d x^2}}+\frac {d^{3/4} (14 b c+a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{6 c^{3/4} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {b \sqrt [4]{c} (b c+9 a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {b \sqrt [4]{c} (b c+9 a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}} \]

[Out]

5/6*d*e*(e*x)^(1/2)/(-a*d+b*c)^2/(-d*x^2+c)^(3/2)+1/2*e*(e*x)^(1/2)/(-a*d+b*c)/(-b*x^2+a)/(-d*x^2+c)^(3/2)+1/6
*d*(a*d+14*b*c)*e*(e*x)^(1/2)/c/(-a*d+b*c)^3/(-d*x^2+c)^(1/2)+1/6*d^(3/4)*(a*d+14*b*c)*e^(3/2)*EllipticF(d^(1/
4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)*(1-d*x^2/c)^(1/2)/c^(3/4)/(-a*d+b*c)^3/(-d*x^2+c)^(1/2)-1/4*b*c^(1/4)*(9*a*d
+b*c)*e^(3/2)*EllipticPi(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),-b^(1/2)*c^(1/2)/a^(1/2)/d^(1/2),I)*(1-d*x^2/c)^(
1/2)/a/d^(1/4)/(-a*d+b*c)^3/(-d*x^2+c)^(1/2)-1/4*b*c^(1/4)*(9*a*d+b*c)*e^(3/2)*EllipticPi(d^(1/4)*(e*x)^(1/2)/
c^(1/4)/e^(1/2),b^(1/2)*c^(1/2)/a^(1/2)/d^(1/2),I)*(1-d*x^2/c)^(1/2)/a/d^(1/4)/(-a*d+b*c)^3/(-d*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {477, 482, 541, 537, 230, 227, 418, 1233, 1232} \[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {d^{3/4} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (a d+14 b c) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{6 c^{3/4} \sqrt {c-d x^2} (b c-a d)^3}-\frac {b \sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (9 a d+b c) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)^3}-\frac {b \sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (9 a d+b c) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)^3}+\frac {d e \sqrt {e x} (a d+14 b c)}{6 c \sqrt {c-d x^2} (b c-a d)^3}+\frac {5 d e \sqrt {e x}}{6 \left (c-d x^2\right )^{3/2} (b c-a d)^2}+\frac {e \sqrt {e x}}{2 \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)} \]

[In]

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

[Out]

(5*d*e*Sqrt[e*x])/(6*(b*c - a*d)^2*(c - d*x^2)^(3/2)) + (e*Sqrt[e*x])/(2*(b*c - a*d)*(a - b*x^2)*(c - d*x^2)^(
3/2)) + (d*(14*b*c + a*d)*e*Sqrt[e*x])/(6*c*(b*c - a*d)^3*Sqrt[c - d*x^2]) + (d^(3/4)*(14*b*c + a*d)*e^(3/2)*S
qrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(6*c^(3/4)*(b*c - a*d)^3*Sqrt
[c - d*x^2]) - (b*c^(1/4)*(b*c + 9*a*d)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sq
rt[d])), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(4*a*d^(1/4)*(b*c - a*d)^3*Sqrt[c - d*x^2]) - (b*
c^(1/4)*(b*c + 9*a*d)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1
/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(4*a*d^(1/4)*(b*c - a*d)^3*Sqrt[c - d*x^2])

Rule 227

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[Rt[-b, 4]*(x/Rt[a, 4])], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 230

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + b*(x^4/a)]/Sqrt[a + b*x^4], Int[1/Sqrt[1 + b*(x^4/
a)], x], x] /; FreeQ[{a, b}, x] && NegQ[b/a] &&  !GtQ[a, 0]

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1
- Rt[-d/c, 2]*x^2)), x], x] + Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-d/c, 2]*x^2)), x], x] /; FreeQ[{a,
b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 482

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

Rule 537

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

Rule 541

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

Rule 1232

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[
a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]

Rule 1233

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4]
, Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a)]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] &&  !GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {x^4}{\left (a-\frac {b x^4}{e^2}\right )^2 \left (c-\frac {d x^4}{e^2}\right )^{5/2}} \, dx,x,\sqrt {e x}\right )}{e} \\ & = \frac {e \sqrt {e x}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}-\frac {e \text {Subst}\left (\int \frac {c+\frac {9 d x^4}{e^2}}{\left (a-\frac {b x^4}{e^2}\right ) \left (c-\frac {d x^4}{e^2}\right )^{5/2}} \, dx,x,\sqrt {e x}\right )}{2 (b c-a d)} \\ & = \frac {5 d e \sqrt {e x}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {e \sqrt {e x}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {e^3 \text {Subst}\left (\int \frac {-\frac {2 c (3 b c+2 a d)}{e^2}-\frac {50 b c d x^4}{e^4}}{\left (a-\frac {b x^4}{e^2}\right ) \left (c-\frac {d x^4}{e^2}\right )^{3/2}} \, dx,x,\sqrt {e x}\right )}{12 c (b c-a d)^2} \\ & = \frac {5 d e \sqrt {e x}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {e \sqrt {e x}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d (14 b c+a d) e \sqrt {e x}}{6 c (b c-a d)^3 \sqrt {c-d x^2}}-\frac {e^5 \text {Subst}\left (\int \frac {\frac {4 c \left (3 b^2 c^2+13 a b c d-a^2 d^2\right )}{e^4}+\frac {4 b c d (14 b c+a d) x^4}{e^6}}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{24 c^2 (b c-a d)^3} \\ & = \frac {5 d e \sqrt {e x}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {e \sqrt {e x}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d (14 b c+a d) e \sqrt {e x}}{6 c (b c-a d)^3 \sqrt {c-d x^2}}+\frac {(d (14 b c+a d) e) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{6 c (b c-a d)^3}-\frac {(b (b c+9 a d) e) \text {Subst}\left (\int \frac {1}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 (b c-a d)^3} \\ & = \frac {5 d e \sqrt {e x}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {e \sqrt {e x}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d (14 b c+a d) e \sqrt {e x}}{6 c (b c-a d)^3 \sqrt {c-d x^2}}-\frac {(b (b c+9 a d) e) \text {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a (b c-a d)^3}-\frac {(b (b c+9 a d) e) \text {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a (b c-a d)^3}+\frac {\left (d (14 b c+a d) e \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{6 c (b c-a d)^3 \sqrt {c-d x^2}} \\ & = \frac {5 d e \sqrt {e x}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {e \sqrt {e x}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d (14 b c+a d) e \sqrt {e x}}{6 c (b c-a d)^3 \sqrt {c-d x^2}}+\frac {d^{3/4} (14 b c+a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{6 c^{3/4} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\left (b (b c+9 a d) e \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\left (b (b c+9 a d) e \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{4 a (b c-a d)^3 \sqrt {c-d x^2}} \\ & = \frac {5 d e \sqrt {e x}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {e \sqrt {e x}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d (14 b c+a d) e \sqrt {e x}}{6 c (b c-a d)^3 \sqrt {c-d x^2}}+\frac {d^{3/4} (14 b c+a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{6 c^{3/4} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {b \sqrt [4]{c} (b c+9 a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {b \sqrt [4]{c} (b c+9 a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 11.34 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.62 \[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {e \sqrt {e x} \left (5 a \left (a^2 d^2 \left (c+d x^2\right )+b^2 c \left (-3 c^2+19 c d x^2-14 d^2 x^4\right )-a b d \left (13 c^2-10 c d x^2+d^2 x^4\right )\right )-5 \left (-3 b^2 c^2-13 a b c d+a^2 d^2\right ) \left (a-b x^2\right ) \left (c-d x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+b d (14 b c+a d) x^2 \left (a-b x^2\right ) \left (c-d x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{30 a c (b c-a d)^3 \left (-a+b x^2\right ) \left (c-d x^2\right )^{3/2}} \]

[In]

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

[Out]

(e*Sqrt[e*x]*(5*a*(a^2*d^2*(c + d*x^2) + b^2*c*(-3*c^2 + 19*c*d*x^2 - 14*d^2*x^4) - a*b*d*(13*c^2 - 10*c*d*x^2
+ d^2*x^4)) - 5*(-3*b^2*c^2 - 13*a*b*c*d + a^2*d^2)*(a - b*x^2)*(c - d*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4,
1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + b*d*(14*b*c + a*d)*x^2*(a - b*x^2)*(c - d*x^2)*Sqrt[1 - (d*x^2)/c]*Appel
lF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(30*a*c*(b*c - a*d)^3*(-a + b*x^2)*(c - d*x^2)^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1191\) vs. \(2(353)=706\).

Time = 3.29 (sec) , antiderivative size = 1192, normalized size of antiderivative = 2.67

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1192\)
default	\(\text {Expression too large to display}\)	\(4391\)

[In]

int((e*x)^(3/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/e/x*(e*x)^(1/2)/(-d*x^2+c)^(1/2)*((-d*x^2+c)*e*x)^(1/2)*(1/2*b^2*d*e/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)*(
-d*e*x^3+c*e*x)^(1/2)/(b*d*x^2-a*d)+1/3*e/(a*d-b*c)^2/d*(-d*e*x^3+c*e*x)^(1/2)/(x^2-c/d)^2-1/6*d*e^2*x/c*(a*d+
11*b*c)/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)/(-(x^2-c/d)*d*e*x)^(1/2)-7/6*(c*d)^(1/2)*(d*x/(c*d)^(1/2)+1)^(1/
2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)*EllipticF(((x+1/d*(c*d)^(1/2))
*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*b*e^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)-1/12*d*(c*d)^(1/2)*(d*x/(c*d)^(
1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)*EllipticF(((x+1/d*(
c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))/c*e^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)*a-9/8*e^2/(a^2*d^2-2
*a*b*c*d+b^2*c^2)/(a*d-b*c)*b/(a*b)^(1/2)*(c*d)^(1/2)*(d*x/(c*d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(
-d*x/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2))*EllipticPi(((x+1/d*(c*d)^(1/
2))*d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2)),1/2*2^(1/2))*a-1/8*e^2/(a^2*d^2-2
*a*b*c*d+b^2*c^2)/(a*d-b*c)*b^2/(a*b)^(1/2)/d*(c*d)^(1/2)*(d*x/(c*d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/
2)*(-d*x/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2))*EllipticPi(((x+1/d*(c*d)
^(1/2))*d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2)),1/2*2^(1/2))*c+9/8*e^2/(a^2*d
^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)*b/(a*b)^(1/2)*(c*d)^(1/2)*(d*x/(c*d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/
2)*(-d*x/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)/(-1/d*(c*d)^(1/2)+1/b*(a*b)^(1/2))*EllipticPi(((x+1/d*(c*d)
^(1/2))*d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/(-1/d*(c*d)^(1/2)+1/b*(a*b)^(1/2)),1/2*2^(1/2))*a+1/8*e^2/(a^2*d
^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)*b^2/(a*b)^(1/2)/d*(c*d)^(1/2)*(d*x/(c*d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)
^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)/(-1/d*(c*d)^(1/2)+1/b*(a*b)^(1/2))*EllipticPi(((x+1/d*(
c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/(-1/d*(c*d)^(1/2)+1/b*(a*b)^(1/2)),1/2*2^(1/2))*c)

Fricas [F(-1)]

Timed out. \[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((e*x)^(3/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int \frac {\left (e x\right )^{\frac {3}{2}}}{\left (- a + b x^{2}\right )^{2} \left (c - d x^{2}\right )^{\frac {5}{2}}}\, dx \]

[In]

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

[Out]

Integral((e*x)**(3/2)/((-a + b*x**2)**2*(c - d*x**2)**(5/2)), x)

Maxima [F]

\[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {3}{2}}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((e*x)^(3/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int { \frac {\left (e x\right )^{\frac {3}{2}}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((e*x)^(3/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{3/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int \frac {{\left (e\,x\right )}^{3/2}}{{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{5/2}} \,d x \]

[In]

int((e*x)^(3/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x)

[Out]

int((e*x)^(3/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)), x)