\(\int \frac {(e x)^{7/2} (c-d x^2)^{3/2}}{(a-b x^2)^2} \, dx\) [903]

Optimal result

Integrand size = 30, antiderivative size = 429 \[ \int \frac {(e x)^{7/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\frac {(57 b c-77 a d) e^3 \sqrt {e x} \sqrt {c-d x^2}}{42 b^3}-\frac {11 d e (e x)^{5/2} \sqrt {c-d x^2}}{14 b^2}+\frac {e (e x)^{5/2} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}+\frac {\sqrt [4]{c} \left (48 b^2 c^2-259 a b c d+231 a^2 d^2\right ) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{42 b^4 \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (5 b c-11 a d) (b c-a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b^4 \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (5 b c-11 a d) (b c-a d) e^{7/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b^4 \sqrt [4]{d} \sqrt {c-d x^2}} \]

[Out]

1/2*e*(e*x)^(5/2)*(-d*x^2+c)^(3/2)/b/(-b*x^2+a)-11/14*d*e*(e*x)^(5/2)*(-d*x^2+c)^(1/2)/b^2+1/42*(-77*a*d+57*b*
c)*e^3*(e*x)^(1/2)*(-d*x^2+c)^(1/2)/b^3+1/42*c^(1/4)*(231*a^2*d^2-259*a*b*c*d+48*b^2*c^2)*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^4/d^(1/4)/(-d*x^2+c)^(1/2)-1/4*c^(1/4)*(-11*a*d+5*b*c
)*(-a*d+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^4/d^(1/4)/(-d*x^2+c)^(1/2)-1/4*c^(1/4)*(-11*a*d+5*b*c)*(-a*d+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^4/d^(1/4)/(-d*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {477, 478, 595, 596, 537, 230, 227, 418, 1233, 1232} \[ \int \frac {(e x)^{7/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\frac {\sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} \left (231 a^2 d^2-259 a b c d+48 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{42 b^4 \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} (5 b c-11 a d) (b c-a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b^4 \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} e^{7/2} \sqrt {1-\frac {d x^2}{c}} (5 b c-11 a d) (b c-a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 b^4 \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {e^3 \sqrt {e x} \sqrt {c-d x^2} (57 b c-77 a d)}{42 b^3}+\frac {e (e x)^{5/2} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}-\frac {11 d e (e x)^{5/2} \sqrt {c-d x^2}}{14 b^2} \]

[In]

Int[((e*x)^(7/2)*(c - d*x^2)^(3/2))/(a - b*x^2)^2,x]

[Out]

((57*b*c - 77*a*d)*e^3*Sqrt[e*x]*Sqrt[c - d*x^2])/(42*b^3) - (11*d*e*(e*x)^(5/2)*Sqrt[c - d*x^2])/(14*b^2) + (
e*(e*x)^(5/2)*(c - d*x^2)^(3/2))/(2*b*(a - b*x^2)) + (c^(1/4)*(48*b^2*c^2 - 259*a*b*c*d + 231*a^2*d^2)*e^(7/2)
*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(42*b^4*d^(1/4)*Sqrt[c - d*
x^2]) - (c^(1/4)*(5*b*c - 11*a*d)*(b*c - 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^4*d^(1/4)*Sqrt[c - d*x^2]) - (c^(1/4)*
(5*b*c - 11*a*d)*(b*c - a*d)*e^(7/2)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSi
n[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(4*b^4*d^(1/4)*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 478

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/(b*n*(p + 1))), x] - Dist[e^n/(b*n*(p + 1)), Int[(e*x)^
(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m - n + 1) + d*(m + n*(q - 1) + 1)*x^n, x], x], x] /;
FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 0] && 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 595

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

Rule 596

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[f*g^(n - 1)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q +
1) + 1))), x] - Dist[g^n/(b*d*(m + n*(p + q + 1) + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*
f*c*(m - n + 1) + (a*f*d*(m + n*q + 1) + b*(f*c*(m + n*p + 1) - e*d*(m + n*(p + q + 1) + 1)))*x^n, x], x], x]
/; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && GtQ[m, n - 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^8 \left (c-\frac {d x^4}{e^2}\right )^{3/2}}{\left (a-\frac {b x^4}{e^2}\right )^2} \, dx,x,\sqrt {e x}\right )}{e} \\ & = \frac {e (e x)^{5/2} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}-\frac {e \text {Subst}\left (\int \frac {x^4 \left (5 c-\frac {11 d x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}}{a-\frac {b x^4}{e^2}} \, dx,x,\sqrt {e x}\right )}{2 b} \\ & = -\frac {11 d e (e x)^{5/2} \sqrt {c-d x^2}}{14 b^2}+\frac {e (e x)^{5/2} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}+\frac {e^3 \text {Subst}\left (\int \frac {x^4 \left (-\frac {5 c (7 b c-11 a d)}{e^2}+\frac {d (57 b c-77 a d) x^4}{e^4}\right )}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{14 b^2} \\ & = \frac {(57 b c-77 a d) e^3 \sqrt {e x} \sqrt {c-d x^2}}{42 b^3}-\frac {11 d e (e x)^{5/2} \sqrt {c-d x^2}}{14 b^2}+\frac {e (e x)^{5/2} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}-\frac {e^7 \text {Subst}\left (\int \frac {\frac {a c d (57 b c-77 a d)}{e^4}+\frac {d \left (48 b^2 c^2-259 a b c d+231 a^2 d^2\right ) 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 )}{42 b^3 d} \\ & = \frac {(57 b c-77 a d) e^3 \sqrt {e x} \sqrt {c-d x^2}}{42 b^3}-\frac {11 d e (e x)^{5/2} \sqrt {c-d x^2}}{14 b^2}+\frac {e (e x)^{5/2} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}-\frac {\left (a (5 b c-11 a d) (b c-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^4}+\frac {\left (\left (48 b^2 c^2-259 a b c d+231 a^2 d^2\right ) e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{42 b^4} \\ & = \frac {(57 b c-77 a d) e^3 \sqrt {e x} \sqrt {c-d x^2}}{42 b^3}-\frac {11 d e (e x)^{5/2} \sqrt {c-d x^2}}{14 b^2}+\frac {e (e x)^{5/2} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}-\frac {\left ((5 b c-11 a d) (b c-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^4}-\frac {\left ((5 b c-11 a d) (b c-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^4}+\frac {\left (\left (48 b^2 c^2-259 a b c d+231 a^2 d^2\right ) 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 )}{42 b^4 \sqrt {c-d x^2}} \\ & = \frac {(57 b c-77 a d) e^3 \sqrt {e x} \sqrt {c-d x^2}}{42 b^3}-\frac {11 d e (e x)^{5/2} \sqrt {c-d x^2}}{14 b^2}+\frac {e (e x)^{5/2} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}+\frac {\sqrt [4]{c} \left (48 b^2 c^2-259 a b c d+231 a^2 d^2\right ) 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 )}{42 b^4 \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\left ((5 b c-11 a d) (b c-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^4 \sqrt {c-d x^2}}-\frac {\left ((5 b c-11 a d) (b c-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^4 \sqrt {c-d x^2}} \\ & = \frac {(57 b c-77 a d) e^3 \sqrt {e x} \sqrt {c-d x^2}}{42 b^3}-\frac {11 d e (e x)^{5/2} \sqrt {c-d x^2}}{14 b^2}+\frac {e (e x)^{5/2} \left (c-d x^2\right )^{3/2}}{2 b \left (a-b x^2\right )}+\frac {\sqrt [4]{c} \left (48 b^2 c^2-259 a b c d+231 a^2 d^2\right ) 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 )}{42 b^4 \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (5 b c-11 a d) (b c-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^4 \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (5 b c-11 a d) (b c-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^4 \sqrt [4]{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 = 11.22 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.54 \[ \int \frac {(e x)^{7/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\frac {e^3 \sqrt {e x} \left (5 a \left (c-d x^2\right ) \left (77 a^2 d-12 b^2 x^2 \left (-3 c+d x^2\right )-a b \left (57 c+44 d x^2\right )\right )-5 a c (-57 b c+77 a d) \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+\left (48 b^2 c^2-259 a b c d+231 a^2 d^2\right ) x^2 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{210 a b^3 \left (-a+b x^2\right ) \sqrt {c-d x^2}} \]

[In]

Integrate[((e*x)^(7/2)*(c - d*x^2)^(3/2))/(a - b*x^2)^2,x]

[Out]

(e^3*Sqrt[e*x]*(5*a*(c - d*x^2)*(77*a^2*d - 12*b^2*x^2*(-3*c + d*x^2) - a*b*(57*c + 44*d*x^2)) - 5*a*c*(-57*b*
c + 77*a*d)*(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] + (48*b^2*c^2 - 2
59*a*b*c*d + 231*a^2*d^2)*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]
))/(210*a*b^3*(-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. \(1325\) vs. \(2(335)=670\).

Time = 5.86 (sec) , antiderivative size = 1326, normalized size of antiderivative = 3.09

method	result	size
risch	\(\text {Expression too large to display}\)	\(1326\)
elliptic	\(\text {Expression too large to display}\)	\(1357\)
default	\(\text {Expression too large to display}\)	\(3778\)

[In]

int((e*x)^(7/2)*(-d*x^2+c)^(3/2)/(-b*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

-2/21*(3*b*d*x^2+14*a*d-9*b*c)*(-d*x^2+c)^(1/2)*x/b^3*e^4/(e*x)^(1/2)+1/21/b^3*((63*a^2*d^2-70*a*b*c*d+12*b^2*
c^2)/b/d*(c*d)^(1/2)*((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2)*(-2*(x-1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(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))
+21*a^2*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b*(1/2*b/(a*d-b*c)/a/e*(-d*e*x^3+c*e*x)^(1/2)/(b*x^2-a)-1/4/(a*d-b*c)/a*(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))-5/8/(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))+3/8/(a*d-b*c)/a/(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))*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))*
b*c+5/8/(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))-3/8/(a*d-b*c)/a/(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))*b*c)+42*a/b*(2*a^2*d^2-3*a*b*c*d+b^2*c^2)*(1/2/(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))-1/2/(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))*Ellipti
cPi(((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))
))*e^4*((-d*x^2+c)*e*x)^(1/2)/(e*x)^(1/2)/(-d*x^2+c)^(1/2)

Fricas [F(-1)]

Timed out. \[ \int \frac {(e x)^{7/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\text {Timed out} \]

[In]

integrate((e*x)^(7/2)*(-d*x^2+c)^(3/2)/(-b*x^2+a)^2,x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {(e x)^{7/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\text {Timed out} \]

[In]

integrate((e*x)**(7/2)*(-d*x**2+c)**(3/2)/(-b*x**2+a)**2,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(e x)^{7/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\int { \frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} - a\right )}^{2}} \,d x } \]

[In]

integrate((e*x)^(7/2)*(-d*x^2+c)^(3/2)/(-b*x^2+a)^2,x, algorithm="maxima")

[Out]

integrate((-d*x^2 + c)^(3/2)*(e*x)^(7/2)/(b*x^2 - a)^2, x)

Giac [F]

\[ \int \frac {(e x)^{7/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\int { \frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {7}{2}}}{{\left (b x^{2} - a\right )}^{2}} \,d x } \]

[In]

integrate((e*x)^(7/2)*(-d*x^2+c)^(3/2)/(-b*x^2+a)^2,x, algorithm="giac")

[Out]

integrate((-d*x^2 + c)^(3/2)*(e*x)^(7/2)/(b*x^2 - a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{7/2} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx=\int \frac {{\left (e\,x\right )}^{7/2}\,{\left (c-d\,x^2\right )}^{3/2}}{{\left (a-b\,x^2\right )}^2} \,d x \]

[In]

int(((e*x)^(7/2)*(c - d*x^2)^(3/2))/(a - b*x^2)^2,x)

[Out]

int(((e*x)^(7/2)*(c - d*x^2)^(3/2))/(a - b*x^2)^2, x)