\(\int \frac {(e x)^{3/2} (c-d x^2)^{3/2}}{a-b x^2} \, dx\) [874]

   Optimal result
   Rubi [A] (verified)
   Mathematica [C] (verified)
   Maple [A] (verified)
   Fricas [F(-1)]
   Sympy [F]
   Maxima [F]
   Giac [F]
   Mupad [F(-1)]

Optimal result

Integrand size = 30, antiderivative size = 372 \[ \int \frac {(e x)^{3/2} \left (c-d x^2\right )^{3/2}}{a-b x^2} \, dx=-\frac {2 (9 b c-7 a d) e \sqrt {e x} \sqrt {c-d x^2}}{21 b^2}+\frac {2 d (e x)^{5/2} \sqrt {c-d x^2}}{7 b e}-\frac {2 \sqrt [4]{c} \left (12 b^2 c^2-35 a b c d+21 a^2 d^2\right ) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{21 b^3 \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d)^2 e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^3 \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d)^2 e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^3 \sqrt [4]{d} \sqrt {c-d x^2}} \]

[Out]

2/7*d*(e*x)^(5/2)*(-d*x^2+c)^(1/2)/b/e-2/21*(-7*a*d+9*b*c)*e*(e*x)^(1/2)*(-d*x^2+c)^(1/2)/b^2-2/21*c^(1/4)*(21
*a^2*d^2-35*a*b*c*d+12*b^2*c^2)*e^(3/2)*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^(1/4)/(-d*x^2+c)^(1/2)+c^(1/4)*(-a*d+b*c)^2*e^(3/2)*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^3/d^(1/4)/(-d*x^2+c)^(1/2)+c^(1/4)*(-a*d+b*c)^2*e^(3/2)*Ellipt
icPi(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^3/d^(1/4)/(-d*
x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {477, 488, 596, 537, 230, 227, 418, 1233, 1232} \[ \int \frac {(e x)^{3/2} \left (c-d x^2\right )^{3/2}}{a-b x^2} \, dx=-\frac {2 \sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} \left (21 a^2 d^2-35 a b c d+12 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{21 b^3 \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d)^2 \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^3 \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d)^2 \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^3 \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {2 e \sqrt {e x} \sqrt {c-d x^2} (9 b c-7 a d)}{21 b^2}+\frac {2 d (e x)^{5/2} \sqrt {c-d x^2}}{7 b e} \]

[In]

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

[Out]

(-2*(9*b*c - 7*a*d)*e*Sqrt[e*x]*Sqrt[c - d*x^2])/(21*b^2) + (2*d*(e*x)^(5/2)*Sqrt[c - d*x^2])/(7*b*e) - (2*c^(
1/4)*(12*b^2*c^2 - 35*a*b*c*d + 21*a^2*d^2)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(
c^(1/4)*Sqrt[e])], -1])/(21*b^3*d^(1/4)*Sqrt[c - d*x^2]) + (c^(1/4)*(b*c - a*d)^2*e^(3/2)*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^3*d^
(1/4)*Sqrt[c - d*x^2]) + (c^(1/4)*(b*c - a*d)^2*e^(3/2)*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^3*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 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 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 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^4 \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)^{5/2} \sqrt {c-d x^2}}{7 b e}-\frac {(2 e) \text {Subst}\left (\int \frac {x^4 \left (-\frac {c (7 b c-5 a d)}{e^2}+\frac {d (9 b c-7 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 )}{7 b} \\ & = -\frac {2 (9 b c-7 a d) e \sqrt {e x} \sqrt {c-d x^2}}{21 b^2}+\frac {2 d (e x)^{5/2} \sqrt {c-d x^2}}{7 b e}+\frac {\left (2 e^5\right ) \text {Subst}\left (\int \frac {\frac {a c d (9 b c-7 a d)}{e^4}+\frac {d \left (12 b^2 c^2-35 a b c d+21 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 )}{21 b^2 d} \\ & = -\frac {2 (9 b c-7 a d) e \sqrt {e x} \sqrt {c-d x^2}}{21 b^2}+\frac {2 d (e x)^{5/2} \sqrt {c-d x^2}}{7 b e}+\frac {\left (2 a (b c-a d)^2 e\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^3}-\frac {\left (2 \left (12 b^2 c^2-35 a b c d+21 a^2 d^2\right ) e\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{21 b^3} \\ & = -\frac {2 (9 b c-7 a d) e \sqrt {e x} \sqrt {c-d x^2}}{21 b^2}+\frac {2 d (e x)^{5/2} \sqrt {c-d x^2}}{7 b e}+\frac {\left ((b c-a d)^2 e\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 )}{b^3}+\frac {\left ((b c-a d)^2 e\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 )}{b^3}-\frac {\left (2 \left (12 b^2 c^2-35 a b c d+21 a^2 d^2\right ) e \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 )}{21 b^3 \sqrt {c-d x^2}} \\ & = -\frac {2 (9 b c-7 a d) e \sqrt {e x} \sqrt {c-d x^2}}{21 b^2}+\frac {2 d (e x)^{5/2} \sqrt {c-d x^2}}{7 b e}-\frac {2 \sqrt [4]{c} \left (12 b^2 c^2-35 a b c d+21 a^2 d^2\right ) e^{3/2} \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 )}{21 b^3 \sqrt [4]{d} \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 (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 )}{b^3 \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 (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 )}{b^3 \sqrt {c-d x^2}} \\ & = -\frac {2 (9 b c-7 a d) e \sqrt {e x} \sqrt {c-d x^2}}{21 b^2}+\frac {2 d (e x)^{5/2} \sqrt {c-d x^2}}{7 b e}-\frac {2 \sqrt [4]{c} \left (12 b^2 c^2-35 a b c d+21 a^2 d^2\right ) e^{3/2} \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 )}{21 b^3 \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d)^2 e^{3/2} \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^3 \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d)^2 e^{3/2} \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^3 \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 = 11.17 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.49 \[ \int \frac {(e x)^{3/2} \left (c-d x^2\right )^{3/2}}{a-b x^2} \, dx=-\frac {2 e \sqrt {e x} \left (-5 a \left (c-d x^2\right ) \left (-9 b c+7 a d+3 b d x^2\right )+5 a c (-9 b c+7 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+\left (-12 b^2 c^2+35 a b c d-21 a^2 d^2\right ) x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{105 a b^2 \sqrt {c-d x^2}} \]

[In]

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

[Out]

(-2*e*Sqrt[e*x]*(-5*a*(c - d*x^2)*(-9*b*c + 7*a*d + 3*b*d*x^2) + 5*a*c*(-9*b*c + 7*a*d)*Sqrt[1 - (d*x^2)/c]*Ap
pellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] + (-12*b^2*c^2 + 35*a*b*c*d - 21*a^2*d^2)*x^2*Sqrt[1 - (d*x^2)/
c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(105*a*b^2*Sqrt[c - d*x^2])

Maple [A] (verified)

Time = 4.05 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.42

method result size
risch \(\frac {2 \left (3 b d \,x^{2}+7 a d -9 b c \right ) \sqrt {-d \,x^{2}+c}\, x \,e^{2}}{21 b^{2} \sqrt {e x}}-\frac {\left (\frac {\left (21 a^{2} d^{2}-35 a b c d +12 b^{2} c^{2}\right ) \sqrt {c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{b d \sqrt {-d e \,x^{3}+c e x}}+\frac {21 a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{b}\right ) e^{2} \sqrt {\left (-d \,x^{2}+c \right ) e x}}{21 b^{2} \sqrt {e x}\, \sqrt {-d \,x^{2}+c}}\) \(527\)
elliptic \(\text {Expression too large to display}\) \(1313\)
default \(\text {Expression too large to display}\) \(1909\)

[In]

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

[Out]

2/21*(3*b*d*x^2+7*a*d-9*b*c)*(-d*x^2+c)^(1/2)*x/b^2*e^2/(e*x)^(1/2)-1/21/b^2*((21*a^2*d^2-35*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))+2
1*a*(a^2*d^2-2*a*b*c*d+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))*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^2*((-d*x^2+c)*e*x)^(1/2)/(e*x)^(1/2)/(-d*x^2+c)^(
1/2)

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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