\(\int \frac {(e x)^{3/2} \sqrt {c-d x^2}}{a-b x^2} \, dx\) [867]

Optimal result

Integrand size = 30, antiderivative size = 315 \[ \int \frac {(e x)^{3/2} \sqrt {c-d x^2}}{a-b x^2} \, dx=-\frac {2 e \sqrt {e x} \sqrt {c-d x^2}}{3 b}-\frac {2 \sqrt [4]{c} (2 b c-3 a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{3 b^2 \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^2 \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d) e^{3/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^2 \sqrt [4]{d} \sqrt {c-d x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 315, 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, 489, 537, 230, 227, 418, 1233, 1232} \[ \int \frac {(e x)^{3/2} \sqrt {c-d x^2}}{a-b x^2} \, dx=-\frac {2 \sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (2 b c-3 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{3 b^2 \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^2 \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} e^{3/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^2 \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {2 e \sqrt {e x} \sqrt {c-d x^2}}{3 b} \]

[In]

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

[Out]

(-2*e*Sqrt[e*x]*Sqrt[c - d*x^2])/(3*b) - (2*c^(1/4)*(2*b*c - 3*a*d)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcS
in[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(3*b^2*d^(1/4)*Sqrt[c - d*x^2]) + (c^(1/4)*(b*c - a*d)*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)*Sq
rt[e])], -1])/(b^2*d^(1/4)*Sqrt[c - d*x^2]) + (c^(1/4)*(b*c - a*d)*e^(3/2)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqr
t[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(b^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 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 489

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[e^(n - 1
)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(b*(m + n*(p + q) + 1))), x] - Dist[e^n/(b*(m + n*(p +
q) + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[a*c*(m - n + 1) + (a*d*(m - n + 1) - n*q*(b
*c - a*d))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 0] &&
GtQ[m - n + 1, 0] && 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 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 \sqrt {c-\frac {d x^4}{e^2}}}{a-\frac {b x^4}{e^2}} \, dx,x,\sqrt {e x}\right )}{e} \\ & = -\frac {2 e \sqrt {e x} \sqrt {c-d x^2}}{3 b}+\frac {(2 e) \text {Subst}\left (\int \frac {a c+\frac {(2 b c-3 a d) x^4}{e^2}}{\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 e \sqrt {e x} \sqrt {c-d x^2}}{3 b}-\frac {(2 (2 b c-3 a d) e) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{3 b^2}+\frac {(2 a (b c-a d) e) \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} \\ & = -\frac {2 e \sqrt {e x} \sqrt {c-d x^2}}{3 b}+\frac {((b c-a d) e) \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^2}+\frac {((b c-a d) e) \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^2}-\frac {\left (2 (2 b c-3 a d) 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 )}{3 b^2 \sqrt {c-d x^2}} \\ & = -\frac {2 e \sqrt {e x} \sqrt {c-d x^2}}{3 b}-\frac {2 \sqrt [4]{c} (2 b c-3 a d) 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 )}{3 b^2 \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\left ((b c-a d) 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^2 \sqrt {c-d x^2}}+\frac {\left ((b c-a d) 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^2 \sqrt {c-d x^2}} \\ & = -\frac {2 e \sqrt {e x} \sqrt {c-d x^2}}{3 b}-\frac {2 \sqrt [4]{c} (2 b c-3 a d) 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 )}{3 b^2 \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d) 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^2 \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d) 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^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 = 11.12 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.45 \[ \int \frac {(e x)^{3/2} \sqrt {c-d x^2}}{a-b x^2} \, dx=\frac {2 e \sqrt {e x} \left (-5 a \left (c-d x^2\right )+5 a c \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+(2 b c-3 a d) x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{15 a b \sqrt {c-d x^2}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(483\) vs. \(2(237)=474\).

Time = 3.93 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.54

method	result	size
risch	\(-\frac {2 \sqrt {-d \,x^{2}+c}\, x \,e^{2}}{3 b \sqrt {e x}}+\frac {\left (\frac {\left (3 a d -2 b c \right ) \sqrt {c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{b d \sqrt {-d e \,x^{3}+c e x}}+\frac {3 \left (a d -b c \right ) a \left (\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{b}\right ) e^{2} \sqrt {\left (-d \,x^{2}+c \right ) e x}}{3 b \sqrt {e x}\, \sqrt {-d \,x^{2}+c}}\)	\(484\)
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (-\frac {2 e \sqrt {-d e \,x^{3}+c e x}}{3 b}+\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}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) e^{2} a}{\sqrt {-d e \,x^{3}+c e x}\, b^{2}}-\frac {2 \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}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) e^{2} c}{3 d \sqrt {-d e \,x^{3}+c e x}\, b}+\frac {a^{2} e^{2} \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^{2} \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {a \,e^{2} \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 ) c}{2 b \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 {a^{2} e^{2} \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^{2} \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}+\frac {a \,e^{2} \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 ) c}{2 b \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 )}{e x \sqrt {-d \,x^{2}+c}}\)	\(851\)
default	\(\text {Expression too large to display}\)	\(1275\)

[In]

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

[Out]

-2/3*(-d*x^2+c)^(1/2)*x/b*e^2/(e*x)^(1/2)+1/3/b*((3*a*d-2*b*c)/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)*Ellipt
icF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+3*(a*d-b*c)*a/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. \[ \int \frac {(e x)^{3/2} \sqrt {c-d x^2}}{a-b x^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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