\(\int \frac {(c-d x^2)^{3/2}}{\sqrt {e x} (a-b x^2)} \, dx\) [876]

Optimal result

Integrand size = 30, antiderivative size = 328 \[ \int \frac {\left (c-d x^2\right )^{3/2}}{\sqrt {e x} \left (a-b x^2\right )} \, dx=\frac {2 d \sqrt {e x} \sqrt {c-d x^2}}{3 b e}+\frac {2 \sqrt [4]{c} d^{3/4} (5 b c-3 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{3 b^2 \sqrt {e} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d)^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a b^2 \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d)^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a b^2 \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 328, 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, 427, 537, 230, 227, 418, 1233, 1232} \[ \int \frac {\left (c-d x^2\right )^{3/2}}{\sqrt {e x} \left (a-b x^2\right )} \, dx=\frac {2 \sqrt [4]{c} d^{3/4} \sqrt {1-\frac {d x^2}{c}} (5 b c-3 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{3 b^2 \sqrt {e} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (b c-a d)^2 \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a b^2 \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (b c-a d)^2 \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{a b^2 \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2}}+\frac {2 d \sqrt {e x} \sqrt {c-d x^2}}{3 b e} \]

[In]

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

[Out]

(2*d*Sqrt[e*x]*Sqrt[c - d*x^2])/(3*b*e) + (2*c^(1/4)*d^(3/4)*(5*b*c - 3*a*d)*Sqrt[1 - (d*x^2)/c]*EllipticF[Arc
Sin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(3*b^2*Sqrt[e]*Sqrt[c - d*x^2]) + (c^(1/4)*(b*c - a*d)^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])/(a*b^2*d^(1/4)*Sqrt[e]*Sqrt[c - d*x^2]) + (c^(1/4)*(b*c - a*d)^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])/(a*b^2*d^(1/4)*Sqrt[e]*Sqr
t[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 427

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c
+ d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

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 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 {\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 \sqrt {e x} \sqrt {c-d x^2}}{3 b e}-\frac {(2 e) \text {Subst}\left (\int \frac {-\frac {c (3 b c-a d)}{e^2}+\frac {d (5 b c-3 a d) x^4}{e^4}}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{3 b} \\ & = \frac {2 d \sqrt {e x} \sqrt {c-d x^2}}{3 b e}+\frac {(2 d (5 b c-3 a d)) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{3 b^2 e}+\frac {\left (2 (b c-a d)^2\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 )}{b^2 e} \\ & = \frac {2 d \sqrt {e x} \sqrt {c-d x^2}}{3 b e}+\frac {(b c-a d)^2 \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 )}{a b^2 e}+\frac {(b c-a d)^2 \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 )}{a b^2 e}+\frac {\left (2 d (5 b c-3 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 )}{3 b^2 e \sqrt {c-d x^2}} \\ & = \frac {2 d \sqrt {e x} \sqrt {c-d x^2}}{3 b e}+\frac {2 \sqrt [4]{c} d^{3/4} (5 b c-3 a d) \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 )}{3 b^2 \sqrt {e} \sqrt {c-d x^2}}+\frac {\left ((b c-a d)^2 \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 )}{a b^2 e \sqrt {c-d x^2}}+\frac {\left ((b c-a d)^2 \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 )}{a b^2 e \sqrt {c-d x^2}} \\ & = \frac {2 d \sqrt {e x} \sqrt {c-d x^2}}{3 b e}+\frac {2 \sqrt [4]{c} d^{3/4} (5 b c-3 a d) \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 )}{3 b^2 \sqrt {e} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d)^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 )}{a b^2 \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d)^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 )}{a b^2 \sqrt [4]{d} \sqrt {e} \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 = 11.13 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.47 \[ \int \frac {\left (c-d x^2\right )^{3/2}}{\sqrt {e x} \left (a-b x^2\right )} \, dx=\frac {2 x \left (5 a d \left (c-d x^2\right )+5 c (3 b c-a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+d (-5 b c+3 a d) x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{15 a b \sqrt {e x} \sqrt {c-d x^2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 4.07 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.49

method	result	size
risch	\(\frac {2 d \sqrt {-d \,x^{2}+c}\, x}{3 b \sqrt {e x}}-\frac {\left (\frac {\left (3 a d -5 b c \right ) \sqrt {c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {-d e \,x^{3}+c e x}}+\frac {\left (3 a^{2} d^{2}-6 a b c d +3 b^{2} c^{2}\right ) \left (\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{b}\right ) \sqrt {\left (-d \,x^{2}+c \right ) e x}}{3 b \sqrt {e x}\, \sqrt {-d \,x^{2}+c}}\)	\(489\)
elliptic	\(\text {Expression too large to display}\)	\(1122\)
default	\(\text {Expression too large to display}\)	\(1710\)

[In]

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

[Out]

2/3*d*(-d*x^2+c)^(1/2)*x/b/(e*x)^(1/2)-1/3/b*((3*a*d-5*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)*EllipticF((
(x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+(3*a^2*d^2-6*a*b*c*d+3*b^2*c^2)/b*(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))
))*((-d*x^2+c)*e*x)^(1/2)/(e*x)^(1/2)/(-d*x^2+c)^(1/2)

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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