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

Optimal. Leaf size=485 \[ \frac {2 c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \left (15 a^2 d^2-21 a b c d+4 b^2 c^2\right ) F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{15 b^3 d^{3/4} \sqrt {c-d x^2}}-\frac {2 c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \left (15 a^2 d^2-21 a b c d+4 b^2 c^2\right ) E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{15 b^3 d^{3/4} \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d)^2 \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^{7/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d)^2 \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^{7/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {2 e (e x)^{3/2} \sqrt {c-d x^2} (11 b c-9 a d)}{45 b^2}+\frac {2 d (e x)^{7/2} \sqrt {c-d x^2}}{9 b e} \]

[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]  time = 1.11, antiderivative size = 485, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {466, 477, 582, 584, 307, 224, 221, 1200, 1199, 424, 490, 1219, 1218} \[ \frac {2 c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \left (15 a^2 d^2-21 a b c d+4 b^2 c^2\right ) F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{15 b^3 d^{3/4} \sqrt {c-d x^2}}-\frac {2 c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \left (15 a^2 d^2-21 a b c d+4 b^2 c^2\right ) E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{15 b^3 d^{3/4} \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d)^2 \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^{7/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d)^2 \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^{7/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {2 e (e x)^{3/2} \sqrt {c-d x^2} (11 b c-9 a d)}{45 b^2}+\frac {2 d (e x)^{7/2} \sqrt {c-d x^2}}{9 b e} \]

Antiderivative was successfully verified.

[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 221

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 224

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 307

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 424

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

Rule 466

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 477

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 490

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), In
t[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 582

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 584

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 1199

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 1200

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 1218

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

Rule 1219

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

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Mathematica [C]  time = 0.22, size = 183, normalized size = 0.38 \[ -\frac {2 e (e x)^{3/2} \left (-3 x^2 \sqrt {1-\frac {d x^2}{c}} \left (15 a^2 d^2-21 a b c d+4 b^2 c^2\right ) F_1\left (\frac {7}{4};\frac {1}{2},1;\frac {11}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )+7 a c \sqrt {1-\frac {d x^2}{c}} (9 a d-11 b c) F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )-7 a \left (c-d x^2\right ) \left (9 a d-11 b c+5 b d x^2\right )\right )}{315 a b^2 \sqrt {c-d x^2}} \]

Warning: Unable to verify antiderivative.

[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])

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {5}{2}}}{b x^{2} - a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 0.05, size = 2183, normalized size = 4.50 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/90*e^2*(e*x)^(1/2)*(-d*x^2+c)^(1/2)*(20*x^6*a*b^3*d^4-20*x^6*b^4*c*d^3+36*x^4*a^2*b^2*d^4+64*x^4*b^4*c^2*d^
2-44*x^2*b^4*c^3*d+90*2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)/((c*d)^(1/2)*b+(a*b
)^(1/2)*d)*b,1/2*2^(1/2))*a^2*b*c*d^2*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*(a*b)^
(1/2)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)-90*2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^
(1/2),(c*d)^(1/2)/((c*d)^(1/2)*b-(a*b)^(1/2)*d)*b,1/2*2^(1/2))*a^2*b*c*d^2*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1
/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*(a*b)^(1/2)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)-100*x^4*a*b^3*c*d
^3-216*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*
x)^(1/2)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^2*b^2*c^2*d^2+150*((d*x+(c*d)^(1/2))/(
c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*EllipticF(((d*x+(c
*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b^3*c^3*d+45*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x
+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(
c*d)^(1/2)/((c*d)^(1/2)*b+(a*b)^(1/2)*d)*b,1/2*2^(1/2))*a*b^3*c^3*d+45*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2
^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(
1/2))^(1/2),(c*d)^(1/2)/((c*d)^(1/2)*b-(a*b)^(1/2)*d)*b,1/2*2^(1/2))*a*b^3*c^3*d-180*((d*x+(c*d)^(1/2))/(c*d)^
(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*EllipticE(((d*x+(c*d)^(
1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^3*b*c*d^3+432*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*
d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*EllipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^
(1/2))*a^2*b^2*c^2*d^2-300*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2
)*(-1/(c*d)^(1/2)*d*x)^(1/2)*EllipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b^3*c^3*d+90*((d*x
+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*Ell
ipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^3*b*c*d^3-36*x^2*a^2*b^2*c*d^3+80*x^2*a*b^3*c^2*d^
2-45*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)
^(1/2)*(a*b)^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)/((c*d)^(1/2)*b+(a*b)^(1/2)*d)*
b,1/2*2^(1/2))*(c*d)^(1/2)*a*b^2*c^2*d+45*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c
*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*(a*b)^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)
^(1/2)/((c*d)^(1/2)*b-(a*b)^(1/2)*d)*b,1/2*2^(1/2))*(c*d)^(1/2)*a*b^2*c^2*d-24*((d*x+(c*d)^(1/2))/(c*d)^(1/2))
^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*EllipticF(((d*x+(c*d)^(1/2))/
(c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^4*c^4+48*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(
c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*EllipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^4*
c^4+45*2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)/((c*d)^(1/2)*b+(a*b)^(1/2)*d)*b,1/
2*2^(1/2))*a^3*b*c*d^3*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*((d*x+(c*d)^(1/2))/(c
*d)^(1/2))^(1/2)-45*2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)/((c*d)^(1/2)*b+(a*b)^
(1/2)*d)*b,1/2*2^(1/2))*a^3*d^3*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*(a*b)^(1/2)*
(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)-90*2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),
(c*d)^(1/2)/((c*d)^(1/2)*b+(a*b)^(1/2)*d)*b,1/2*2^(1/2))*a^2*b^2*c^2*d^2*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2
)*(-1/(c*d)^(1/2)*d*x)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)+45*2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c
*d)^(1/2))^(1/2),(c*d)^(1/2)/((c*d)^(1/2)*b-(a*b)^(1/2)*d)*b,1/2*2^(1/2))*a^3*b*c*d^3*((-d*x+(c*d)^(1/2))/(c*d
)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)+45*2^(1/2)*EllipticPi(((d*x+(c
*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)/((c*d)^(1/2)*b-(a*b)^(1/2)*d)*b,1/2*2^(1/2))*a^3*d^3*((-d*x+(c*d)^(1
/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*(a*b)^(1/2)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/
2)-90*2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)/((c*d)^(1/2)*b-(a*b)^(1/2)*d)*b,1/2
*2^(1/2))*a^2*b^2*c^2*d^2*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*((d*x+(c*d)^(1/2))
/(c*d)^(1/2))^(1/2))/x/b^3/(d*x^2-c)/((c*d)^(1/2)*b+(a*b)^(1/2)*d)/((c*d)^(1/2)*b-(a*b)^(1/2)*d)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {5}{2}}}{b x^{2} - a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e\,x\right )}^{5/2}\,{\left (c-d\,x^2\right )}^{3/2}}{a-b\,x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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