\(\int \frac {(e x)^{5/2} (c-d x^2)^{3/2}}{a-b x^2} \, dx\) [873]

Optimal result

Integrand size = 30, antiderivative size = 485 \[ \int \frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{a-b x^2} \, dx=-\frac {2 (11 b c-9 a d) e (e x)^{3/2} \sqrt {c-d x^2}}{45 b^2}+\frac {2 d (e x)^{7/2} \sqrt {c-d x^2}}{9 b e}-\frac {2 c^{3/4} \left (4 b^2 c^2-21 a b c d+15 a^2 d^2\right ) e^{5/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{15 b^3 d^{3/4} \sqrt {c-d x^2}}+\frac {2 c^{3/4} \left (4 b^2 c^2-21 a b c d+15 a^2 d^2\right ) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{15 b^3 d^{3/4} \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} (b c-a d)^2 e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^{7/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} (b c-a d)^2 e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^{7/2} \sqrt [4]{d} \sqrt {c-d x^2}} \]

[Out]

-2/45*(-9*a*d+11*b*c)*e*(e*x)^(3/2)*(-d*x^2+c)^(1/2)/b^2+2/9*d*(e*x)^(7/2)*(-d*x^2+c)^(1/2)/b/e-2/15*c^(3/4)*(
15*a^2*d^2-21*a*b*c*d+4*b^2*c^2)*e^(5/2)*EllipticE(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)*(1-d*x^2/c)^(1/2)/b^
3/d^(3/4)/(-d*x^2+c)^(1/2)+2/15*c^(3/4)*(15*a^2*d^2-21*a*b*c*d+4*b^2*c^2)*e^(5/2)*EllipticF(d^(1/4)*(e*x)^(1/2
)/c^(1/4)/e^(1/2),I)*(1-d*x^2/c)^(1/2)/b^3/d^(3/4)/(-d*x^2+c)^(1/2)-c^(1/4)*(-a*d+b*c)^2*e^(5/2)*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)*a^(1/2)*(1-d*x^2/c)^(1/2)/b^(7/2)/d^(1/4
)/(-d*x^2+c)^(1/2)+c^(1/4)*(-a*d+b*c)^2*e^(5/2)*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)*a^(1/2)*(1-d*x^2/c)^(1/2)/b^(7/2)/d^(1/4)/(-d*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {477, 488, 596, 598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \[ \int \frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{a-b x^2} \, dx=\frac {2 c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \left (15 a^2 d^2-21 a b c d+4 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{15 b^3 d^{3/4} \sqrt {c-d x^2}}-\frac {2 c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \left (15 a^2 d^2-21 a b c d+4 b^2 c^2\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{15 b^3 d^{3/4} \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d)^2 \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^{7/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d)^2 \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^{7/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {2 e (e x)^{3/2} \sqrt {c-d x^2} (11 b c-9 a d)}{45 b^2}+\frac {2 d (e x)^{7/2} \sqrt {c-d x^2}}{9 b e} \]

[In]

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

[Out]

(-2*(11*b*c - 9*a*d)*e*(e*x)^(3/2)*Sqrt[c - d*x^2])/(45*b^2) + (2*d*(e*x)^(7/2)*Sqrt[c - d*x^2])/(9*b*e) - (2*
c^(3/4)*(4*b^2*c^2 - 21*a*b*c*d + 15*a^2*d^2)*e^(5/2)*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])
/(c^(1/4)*Sqrt[e])], -1])/(15*b^3*d^(3/4)*Sqrt[c - d*x^2]) + (2*c^(3/4)*(4*b^2*c^2 - 21*a*b*c*d + 15*a^2*d^2)*
e^(5/2)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(15*b^3*d^(3/4)*Sqrt
[c - d*x^2]) - (Sqrt[a]*c^(1/4)*(b*c - a*d)^2*e^(5/2)*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])/(b^(7/2)*d^(1/4)*Sqrt[c - d*x^2]) + (Sqrt[a
]*c^(1/4)*(b*c - a*d)^2*e^(5/2)*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])/(b^(7/2)*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 313

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Dist[-q^(-1), Int[1/Sqrt[a + b*x^4]
, x], x] + Dist[1/q, Int[(1 + q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 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 488

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

Rule 504

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

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 598

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy
mbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e,
f, g, m, p}, x] && IGtQ[n, 0]

Rule 1213

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

Rule 1214

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

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^6 \left (c-\frac {d x^4}{e^2}\right )^{3/2}}{a-\frac {b x^4}{e^2}} \, dx,x,\sqrt {e x}\right )}{e} \\ & = \frac {2 d (e x)^{7/2} \sqrt {c-d x^2}}{9 b e}-\frac {(2 e) \text {Subst}\left (\int \frac {x^6 \left (-\frac {c (9 b c-7 a d)}{e^2}+\frac {d (11 b c-9 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 )}{9 b} \\ & = -\frac {2 (11 b c-9 a d) e (e x)^{3/2} \sqrt {c-d x^2}}{45 b^2}+\frac {2 d (e x)^{7/2} \sqrt {c-d x^2}}{9 b e}+\frac {\left (2 e^5\right ) \text {Subst}\left (\int \frac {x^2 \left (\frac {3 a c d (11 b c-9 a d)}{e^4}+\frac {3 d \left (4 b^2 c^2-21 a b c d+15 a^2 d^2\right ) x^4}{e^6}\right )}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{45 b^2 d} \\ & = -\frac {2 (11 b c-9 a d) e (e x)^{3/2} \sqrt {c-d x^2}}{45 b^2}+\frac {2 d (e x)^{7/2} \sqrt {c-d x^2}}{9 b e}+\frac {\left (2 e^5\right ) \text {Subst}\left (\int \left (-\frac {3 d \left (4 b^2 c^2-21 a b c d+15 a^2 d^2\right ) x^2}{b e^4 \sqrt {c-\frac {d x^4}{e^2}}}+\frac {45 \left (a b^2 c^2 d-2 a^2 b c d^2+a^3 d^3\right ) x^2}{b e^4 \left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}}\right ) \, dx,x,\sqrt {e x}\right )}{45 b^2 d} \\ & = -\frac {2 (11 b c-9 a d) e (e x)^{3/2} \sqrt {c-d x^2}}{45 b^2}+\frac {2 d (e x)^{7/2} \sqrt {c-d x^2}}{9 b e}+\frac {\left (2 a (b c-a d)^2 e\right ) \text {Subst}\left (\int \frac {x^2}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b^3}-\frac {\left (2 \left (4 b^2 c^2-21 a b c d+15 a^2 d^2\right ) e\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{15 b^3} \\ & = -\frac {2 (11 b c-9 a d) e (e x)^{3/2} \sqrt {c-d x^2}}{45 b^2}+\frac {2 d (e x)^{7/2} \sqrt {c-d x^2}}{9 b e}+\frac {\left (2 \sqrt {c} \left (4 b^2 c^2-21 a b c d+15 a^2 d^2\right ) e^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{15 b^3 \sqrt {d}}-\frac {\left (2 \sqrt {c} \left (4 b^2 c^2-21 a b c d+15 a^2 d^2\right ) e^2\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{15 b^3 \sqrt {d}}+\frac {\left (a (b c-a d)^2 e^3\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} x^2\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b^{7/2}}-\frac {\left (a (b c-a d)^2 e^3\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e+\sqrt {b} x^2\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b^{7/2}} \\ & = -\frac {2 (11 b c-9 a d) e (e x)^{3/2} \sqrt {c-d x^2}}{45 b^2}+\frac {2 d (e x)^{7/2} \sqrt {c-d x^2}}{9 b e}+\frac {\left (2 \sqrt {c} \left (4 b^2 c^2-21 a b c d+15 a^2 d^2\right ) e^2 \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 )}{15 b^3 \sqrt {d} \sqrt {c-d x^2}}-\frac {\left (2 \sqrt {c} \left (4 b^2 c^2-21 a b c d+15 a^2 d^2\right ) e^2 \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{15 b^3 \sqrt {d} \sqrt {c-d x^2}}+\frac {\left (a (b c-a d)^2 e^3 \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} x^2\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{b^{7/2} \sqrt {c-d x^2}}-\frac {\left (a (b c-a d)^2 e^3 \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e+\sqrt {b} x^2\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{b^{7/2} \sqrt {c-d x^2}} \\ & = -\frac {2 (11 b c-9 a d) e (e x)^{3/2} \sqrt {c-d x^2}}{45 b^2}+\frac {2 d (e x)^{7/2} \sqrt {c-d x^2}}{9 b e}+\frac {2 c^{3/4} \left (4 b^2 c^2-21 a b c d+15 a^2 d^2\right ) e^{5/2} \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 )}{15 b^3 d^{3/4} \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} (b c-a d)^2 e^{5/2} \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 )}{b^{7/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} (b c-a d)^2 e^{5/2} \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 )}{b^{7/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\left (2 \sqrt {c} \left (4 b^2 c^2-21 a b c d+15 a^2 d^2\right ) e^2 \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {\sqrt {d} x^2}{\sqrt {c} e}}}{\sqrt {1-\frac {\sqrt {d} x^2}{\sqrt {c} e}}} \, dx,x,\sqrt {e x}\right )}{15 b^3 \sqrt {d} \sqrt {c-d x^2}} \\ & = -\frac {2 (11 b c-9 a d) e (e x)^{3/2} \sqrt {c-d x^2}}{45 b^2}+\frac {2 d (e x)^{7/2} \sqrt {c-d x^2}}{9 b e}-\frac {2 c^{3/4} \left (4 b^2 c^2-21 a b c d+15 a^2 d^2\right ) e^{5/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{15 b^3 d^{3/4} \sqrt {c-d x^2}}+\frac {2 c^{3/4} \left (4 b^2 c^2-21 a b c d+15 a^2 d^2\right ) e^{5/2} \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 )}{15 b^3 d^{3/4} \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} (b c-a d)^2 e^{5/2} \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 )}{b^{7/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} (b c-a d)^2 e^{5/2} \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 )}{b^{7/2} \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 = 11.18 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.38 \[ \int \frac {(e x)^{5/2} \left (c-d x^2\right )^{3/2}}{a-b x^2} \, dx=-\frac {2 e (e x)^{3/2} \left (-7 a \left (c-d x^2\right ) \left (-11 b c+9 a d+5 b d x^2\right )+7 a c (-11 b c+9 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )-3 \left (4 b^2 c^2-21 a b c d+15 a^2 d^2\right ) x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{315 a b^2 \sqrt {c-d x^2}} \]

[In]

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

[Out]

(-2*e*(e*x)^(3/2)*(-7*a*(c - d*x^2)*(-11*b*c + 9*a*d + 5*b*d*x^2) + 7*a*c*(-11*b*c + 9*a*d)*Sqrt[1 - (d*x^2)/c
]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] - 3*(4*b^2*c^2 - 21*a*b*c*d + 15*a^2*d^2)*x^2*Sqrt[1 - (d*x
^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(315*a*b^2*Sqrt[c - d*x^2])

Maple [A] (verified)

Time = 4.42 (sec) , antiderivative size = 571, normalized size of antiderivative = 1.18

method	result	size
risch	\(\frac {2 x^{2} \left (5 b d \,x^{2}+9 a d -11 b c \right ) \sqrt {-d \,x^{2}+c}\, e^{3}}{45 b^{2} \sqrt {e x}}-\frac {\left (\frac {\left (15 a^{2} d^{2}-21 a b c d +4 b^{2} c^{2}\right ) \sqrt {c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \left (-\frac {2 \sqrt {c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{b d \sqrt {-d e \,x^{3}+c e x}}+\frac {15 a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 b d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 b d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{b}\right ) e^{3} \sqrt {\left (-d \,x^{2}+c \right ) e x}}{15 b^{2} \sqrt {e x}\, \sqrt {-d \,x^{2}+c}}\)	\(571\)
elliptic	\(\text {Expression too large to display}\)	\(1540\)
default	\(\text {Expression too large to display}\)	\(2172\)

[In]

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

[Out]

2/45*x^2*(5*b*d*x^2+9*a*d-11*b*c)*(-d*x^2+c)^(1/2)/b^2*e^3/(e*x)^(1/2)-1/15/b^2*((15*a^2*d^2-21*a*b*c*d+4*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)*(-2/d*(c*d)^(1/2)*EllipticE(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^
(1/2),1/2*2^(1/2))+1/d*(c*d)^(1/2)*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2)))+15*a*(a^2
*d^2-2*a*b*c*d+b^2*c^2)/b*(1/2/b/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/b/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))))*e^3*((-d*x^2+c)*e*x)^(1/2)/(e*x)^(1/2)/(-d*x^2+c)^(1/2)

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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