\(\int \frac {\sqrt {e x} (c-d x^2)^{3/2}}{a-b x^2} \, dx\) [875]

Optimal result

Integrand size = 30, antiderivative size = 421 \[ \int \frac {\sqrt {e x} \left (c-d x^2\right )^{3/2}}{a-b x^2} \, dx=\frac {2 d (e x)^{3/2} \sqrt {c-d x^2}}{5 b e}+\frac {2 c^{3/4} \sqrt [4]{d} (7 b c-5 a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 b^2 \sqrt {c-d x^2}}-\frac {2 c^{3/4} \sqrt [4]{d} (7 b c-5 a d) \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{5 b^2 \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-a d)^2 \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {a} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d)^2 \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {a} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {477, 488, 598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \[ \int \frac {\sqrt {e x} \left (c-d x^2\right )^{3/2}}{a-b x^2} \, dx=-\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (b c-a d)^2 \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {a} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (b c-a d)^2 \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {a} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {2 c^{3/4} \sqrt [4]{d} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (7 b c-5 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{5 b^2 \sqrt {c-d x^2}}+\frac {2 c^{3/4} \sqrt [4]{d} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (7 b c-5 a d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 b^2 \sqrt {c-d x^2}}+\frac {2 d (e x)^{3/2} \sqrt {c-d x^2}}{5 b e} \]

[In]

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

[Out]

(2*d*(e*x)^(3/2)*Sqrt[c - d*x^2])/(5*b*e) + (2*c^(3/4)*d^(1/4)*(7*b*c - 5*a*d)*Sqrt[e]*Sqrt[1 - (d*x^2)/c]*Ell
ipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(5*b^2*Sqrt[c - d*x^2]) - (2*c^(3/4)*d^(1/4)*(7*b*c
- 5*a*d)*Sqrt[e]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(5*b^2*Sqr
t[c - d*x^2]) - (c^(1/4)*(b*c - a*d)^2*Sqrt[e]*Sqrt[1 - (d*x^2)/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqr
t[d])), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(Sqrt[a]*b^(5/2)*d^(1/4)*Sqrt[c - d*x^2]) + (c^(1/
4)*(b*c - a*d)^2*Sqrt[e]*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*S
qrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(Sqrt[a]*b^(5/2)*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 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^2 \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)^{3/2} \sqrt {c-d x^2}}{5 b e}-\frac {(2 e) \text {Subst}\left (\int \frac {x^2 \left (-\frac {c (5 b c-3 a d)}{e^2}+\frac {d (7 b c-5 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 )}{5 b} \\ & = \frac {2 d (e x)^{3/2} \sqrt {c-d x^2}}{5 b e}-\frac {(2 e) \text {Subst}\left (\int \left (-\frac {d (7 b c-5 a d) x^2}{b e^2 \sqrt {c-\frac {d x^4}{e^2}}}-\frac {5 \left (b^2 c^2-2 a b c d+a^2 d^2\right ) x^2}{b e^2 \left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}}\right ) \, dx,x,\sqrt {e x}\right )}{5 b} \\ & = \frac {2 d (e x)^{3/2} \sqrt {c-d x^2}}{5 b e}+\frac {(2 d (7 b c-5 a d)) \text {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 b^2 e}+\frac {\left (2 (b c-a d)^2\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^2 e} \\ & = \frac {2 d (e x)^{3/2} \sqrt {c-d x^2}}{5 b e}-\frac {\left (2 \sqrt {c} \sqrt {d} (7 b c-5 a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 b^2}+\frac {\left (2 \sqrt {c} \sqrt {d} (7 b c-5 a d)\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 )}{5 b^2}+\frac {\left ((b c-a d)^2 e\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^{5/2}}-\frac {\left ((b c-a d)^2 e\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^{5/2}} \\ & = \frac {2 d (e x)^{3/2} \sqrt {c-d x^2}}{5 b e}-\frac {\left (2 \sqrt {c} \sqrt {d} (7 b c-5 a d) \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 )}{5 b^2 \sqrt {c-d x^2}}+\frac {\left (2 \sqrt {c} \sqrt {d} (7 b c-5 a d) \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 )}{5 b^2 \sqrt {c-d x^2}}+\frac {\left ((b c-a d)^2 e \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^{5/2} \sqrt {c-d x^2}}-\frac {\left ((b c-a d)^2 e \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^{5/2} \sqrt {c-d x^2}} \\ & = \frac {2 d (e x)^{3/2} \sqrt {c-d x^2}}{5 b e}-\frac {2 c^{3/4} \sqrt [4]{d} (7 b c-5 a d) \sqrt {e} \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 )}{5 b^2 \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-a d)^2 \sqrt {e} \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 )}{\sqrt {a} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d)^2 \sqrt {e} \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 )}{\sqrt {a} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\left (2 \sqrt {c} \sqrt {d} (7 b c-5 a d) \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 )}{5 b^2 \sqrt {c-d x^2}} \\ & = \frac {2 d (e x)^{3/2} \sqrt {c-d x^2}}{5 b e}+\frac {2 c^{3/4} \sqrt [4]{d} (7 b c-5 a d) \sqrt {e} \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 )}{5 b^2 \sqrt {c-d x^2}}-\frac {2 c^{3/4} \sqrt [4]{d} (7 b c-5 a d) \sqrt {e} \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 )}{5 b^2 \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-a d)^2 \sqrt {e} \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 )}{\sqrt {a} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d)^2 \sqrt {e} \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 )}{\sqrt {a} b^{5/2} \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 = 155, normalized size of antiderivative = 0.37 \[ \int \frac {\sqrt {e x} \left (c-d x^2\right )^{3/2}}{a-b x^2} \, dx=\frac {2 x \sqrt {e x} \left (7 c (5 b c-3 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+3 d \left (7 a \left (c-d x^2\right )+(-7 b c+5 a d) x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )\right )}{105 a b \sqrt {c-d x^2}} \]

[In]

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

[Out]

(2*x*Sqrt[e*x]*(7*c*(5*b*c - 3*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*d
*(7*a*(c - d*x^2) + (-7*b*c + 5*a*d)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]
)))/(105*a*b*Sqrt[c - d*x^2])

Maple [A] (verified)

Time = 4.33 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.27

method	result	size
risch	\(\frac {2 d \sqrt {-d \,x^{2}+c}\, x^{2} e}{5 b \sqrt {e x}}-\frac {\left (\frac {\left (5 a d -7 b c \right ) \sqrt {c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \left (-\frac {2 \sqrt {c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{b \sqrt {-d e \,x^{3}+c e x}}+\frac {5 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 b d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 b d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{b}\right ) e \sqrt {\left (-d \,x^{2}+c \right ) e x}}{5 b \sqrt {e x}\, \sqrt {-d \,x^{2}+c}}\)	\(534\)
elliptic	\(\text {Expression too large to display}\)	\(1272\)
default	\(\text {Expression too large to display}\)	\(1916\)

[In]

int((-d*x^2+c)^(3/2)*(e*x)^(1/2)/(-b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

2/5*d*(-d*x^2+c)^(1/2)*x^2/b*e/(e*x)^(1/2)-1/5/b*((5*a*d-7*b*c)/b*(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)))+5*(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*((-d*x^2+c)*e*x)^(1/2)/
(e*x)^(1/2)/(-d*x^2+c)^(1/2)

Fricas [F(-1)]

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

[In]

integrate((-d*x^2+c)^(3/2)*(e*x)^(1/2)/(-b*x^2+a),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

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

[In]

integrate((-d*x**2+c)**(3/2)*(e*x)**(1/2)/(-b*x**2+a),x)

[Out]

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

Maxima [F]

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

[In]

integrate((-d*x^2+c)^(3/2)*(e*x)^(1/2)/(-b*x^2+a),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate((-d*x^2+c)^(3/2)*(e*x)^(1/2)/(-b*x^2+a),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int(((e*x)^(1/2)*(c - d*x^2)^(3/2))/(a - b*x^2),x)

[Out]

int(((e*x)^(1/2)*(c - d*x^2)^(3/2))/(a - b*x^2), x)