\(\int \frac {(e x)^{7/2}}{(a-b x^2)^2 \sqrt {c-d x^2}} \, dx\) [911]

Optimal result

Integrand size = 30, antiderivative size = 376 \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\frac {a e^3 \sqrt {e x} \sqrt {c-d x^2}}{2 b (b c-a d) \left (a-b x^2\right )}+\frac {\sqrt [4]{c} (4 b c-3 a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 b^2 \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (5 b c-3 a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b^2 \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (5 b c-3 a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b^2 \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}} \]

[Out]

1/2*a*e^3*(e*x)^(1/2)*(-d*x^2+c)^(1/2)/b/(-a*d+b*c)/(-b*x^2+a)+1/2*c^(1/4)*(-3*a*d+4*b*c)*e^(7/2)*EllipticF(d^
(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)*(1-d*x^2/c)^(1/2)/b^2/d^(1/4)/(-a*d+b*c)/(-d*x^2+c)^(1/2)-1/4*c^(1/4)*(-3
*a*d+5*b*c)*e^(7/2)*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)/b^2/d^(1/4)/(-a*d+b*c)/(-d*x^2+c)^(1/2)-1/4*c^(1/4)*(-3*a*d+5*b*c)*e^(7/2)*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)/b^2/d^(1/4)/(-a*d+b*c)/(-d*x^2+c)^
(1/2)

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {477, 481, 537, 230, 227, 418, 1233, 1232} \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\frac {\sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} (4 b c-3 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{2 b^2 \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)}-\frac {\sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} (5 b c-3 a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b^2 \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)}-\frac {\sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} (5 b c-3 a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b^2 \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)}+\frac {a e^3 \sqrt {e x} \sqrt {c-d x^2}}{2 b \left (a-b x^2\right ) (b c-a d)} \]

[In]

Int[(e*x)^(7/2)/((a - b*x^2)^2*Sqrt[c - d*x^2]),x]

[Out]

(a*e^3*Sqrt[e*x]*Sqrt[c - d*x^2])/(2*b*(b*c - a*d)*(a - b*x^2)) + (c^(1/4)*(4*b*c - 3*a*d)*e^(7/2)*Sqrt[1 - (d
*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*b^2*d^(1/4)*(b*c - a*d)*Sqrt[c - d*x
^2]) - (c^(1/4)*(5*b*c - 3*a*d)*e^(7/2)*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*b^2*d^(1/4)*(b*c - a*d)*Sqrt[c - d*x^2]) - (c^(1/4)*(5
*b*c - 3*a*d)*e^(7/2)*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*b^2*d^(1/4)*(b*c - a*d)*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 481

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-a)*e^(
2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Dist[e^
(2*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1)
+ (a*d*(m - n + n*q + 1) + b*c*n*(p + 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] && GtQ[m - n + 1, n] && 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 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^8}{\left (a-\frac {b x^4}{e^2}\right )^2 \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{e} \\ & = \frac {a e^3 \sqrt {e x} \sqrt {c-d x^2}}{2 b (b c-a d) \left (a-b x^2\right )}-\frac {e^3 \text {Subst}\left (\int \frac {a c+\frac {(4 b c-3 a d) x^4}{e^2}}{\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 (b c-a d)} \\ & = \frac {a e^3 \sqrt {e x} \sqrt {c-d x^2}}{2 b (b c-a d) \left (a-b x^2\right )}+\frac {\left ((4 b c-3 a d) e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 b^2 (b c-a d)}-\frac {\left (a (5 b c-3 a d) e^3\right ) \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^2 (b c-a d)} \\ & = \frac {a e^3 \sqrt {e x} \sqrt {c-d x^2}}{2 b (b c-a d) \left (a-b x^2\right )}-\frac {\left ((5 b c-3 a d) e^3\right ) \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 b^2 (b c-a d)}-\frac {\left ((5 b c-3 a d) e^3\right ) \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 b^2 (b c-a d)}+\frac {\left ((4 b c-3 a d) e^3 \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 )}{2 b^2 (b c-a d) \sqrt {c-d x^2}} \\ & = \frac {a e^3 \sqrt {e x} \sqrt {c-d x^2}}{2 b (b c-a d) \left (a-b x^2\right )}+\frac {\sqrt [4]{c} (4 b c-3 a d) e^{7/2} \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 )}{2 b^2 \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}-\frac {\left ((5 b c-3 a d) e^3 \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 b^2 (b c-a d) \sqrt {c-d x^2}}-\frac {\left ((5 b c-3 a d) e^3 \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 b^2 (b c-a d) \sqrt {c-d x^2}} \\ & = \frac {a e^3 \sqrt {e x} \sqrt {c-d x^2}}{2 b (b c-a d) \left (a-b x^2\right )}+\frac {\sqrt [4]{c} (4 b c-3 a d) e^{7/2} \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 )}{2 b^2 \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (5 b c-3 a d) e^{7/2} \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 b^2 \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (5 b c-3 a d) e^{7/2} \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 b^2 \sqrt [4]{d} (b c-a d) \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 = 11.17 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.49 \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\frac {e^3 \sqrt {e x} \left (-5 a^2 \left (c-d x^2\right )+5 a c \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )-(-4 b c+3 a d) x^2 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{10 a b (-b c+a d) \left (a-b x^2\right ) \sqrt {c-d x^2}} \]

[In]

Integrate[(e*x)^(7/2)/((a - b*x^2)^2*Sqrt[c - d*x^2]),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(922\) vs. \(2(294)=588\).

Time = 3.08 (sec) , antiderivative size = 923, normalized size of antiderivative = 2.45

method	result	size
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (-\frac {e^{3} a \sqrt {-d e \,x^{3}+c e x}}{2 \left (a d -b c \right ) b \left (-b \,x^{2}+a \right )}+\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) e^{4}}{d \sqrt {-d e \,x^{3}+c e x}\, b^{2}}-\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) a \,e^{4}}{4 \sqrt {-d e \,x^{3}+c e x}\, b^{2} \left (a d -b c \right )}+\frac {3 a^{2} e^{4} \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{8 b^{2} \left (a d -b c \right ) \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {5 a \,e^{4} \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) c}{8 b \left (a d -b c \right ) \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {3 a^{2} e^{4} \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{8 b^{2} \left (a d -b c \right ) \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}+\frac {5 a \,e^{4} \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) c}{8 b \left (a d -b c \right ) \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{e x \sqrt {-d \,x^{2}+c}}\)	\(923\)
default	\(\text {Expression too large to display}\)	\(2508\)

[In]

int((e*x)^(7/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),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/(a*d-b*c)*e^3*a/b*(-d*e*x^3+c*e*x)^(1/2)/(-b*x
^2+a)+1/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))*e^4/b^2-1/4*(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))*a/b^2*e^4/(a*d-b*c)+3/8*a^2*e^4/b^2/(a*d-b*c)/(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))-5/8*a*e^4/b/(a*d-b*c)/(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-3/8*a^2*e^4/b^2/(a*d-b*c)/(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))*Elliptic
Pi(((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))+
5/8*a*e^4/b/(a*d-b*c)/(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. \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\text {Timed out} \]

[In]

integrate((e*x)^(7/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\text {Timed out} \]

[In]

integrate((e*x)**(7/2)/(-b*x**2+a)**2/(-d*x**2+c)**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\int { \frac {\left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} - a\right )}^{2} \sqrt {-d x^{2} + c}} \,d x } \]

[In]

integrate((e*x)^(7/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x, algorithm="maxima")

[Out]

integrate((e*x)^(7/2)/((b*x^2 - a)^2*sqrt(-d*x^2 + c)), x)

Giac [F]

\[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\int { \frac {\left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} - a\right )}^{2} \sqrt {-d x^{2} + c}} \,d x } \]

[In]

integrate((e*x)^(7/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x, algorithm="giac")

[Out]

integrate((e*x)^(7/2)/((b*x^2 - a)^2*sqrt(-d*x^2 + c)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{7/2}}{\left (a-b x^2\right )^2 \sqrt {c-d x^2}} \, dx=\int \frac {{\left (e\,x\right )}^{7/2}}{{\left (a-b\,x^2\right )}^2\,\sqrt {c-d\,x^2}} \,d x \]

[In]

int((e*x)^(7/2)/((a - b*x^2)^2*(c - d*x^2)^(1/2)),x)

[Out]

int((e*x)^(7/2)/((a - b*x^2)^2*(c - d*x^2)^(1/2)), x)