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

Optimal. Leaf size=474 \[ -\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-5 a^2 d^2+4 a b c d+b^2 c^2\right ) \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^{3/2} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-5 a^2 d^2+4 a b c d+b^2 c^2\right ) \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^{3/2} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {c^{3/4} \sqrt [4]{d} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (b c-5 a d) F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a b^2 \sqrt {c-d x^2}}-\frac {c^{3/4} \sqrt [4]{d} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (b c-5 a d) E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a b^2 \sqrt {c-d x^2}}+\frac {(e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}{2 a b e \left (a-b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.84, antiderivative size = 474, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {466, 468, 584, 307, 224, 221, 1200, 1199, 424, 490, 1219, 1218} \[ -\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-5 a^2 d^2+4 a b c d+b^2 c^2\right ) \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^{3/2} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} \left (-5 a^2 d^2+4 a b c d+b^2 c^2\right ) \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^{3/2} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {c^{3/4} \sqrt [4]{d} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (b c-5 a d) F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a b^2 \sqrt {c-d x^2}}-\frac {c^{3/4} \sqrt [4]{d} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (b c-5 a d) E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a b^2 \sqrt {c-d x^2}}+\frac {(e x)^{3/2} \sqrt {c-d x^2} (b c-a d)}{2 a b e \left (a-b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*c - a*d)*(e*x)^(3/2)*Sqrt[c - d*x^2])/(2*a*b*e*(a - b*x^2)) - (c^(3/4)*d^(1/4)*(b*c - 5*a*d)*Sqrt[e]*Sqrt[
1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*a*b^2*Sqrt[c - d*x^2]) + (c^(3
/4)*d^(1/4)*(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])/(2*a*b^2*Sqrt[c - d*x^2]) - (c^(1/4)*(b^2*c^2 + 4*a*b*c*d - 5*a^2*d^2)*Sqrt[e]*Sqrt[1 - (d*x^2)/c]*Ellip
ticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(4*a^(3/2)*b
^(5/2)*d^(1/4)*Sqrt[c - d*x^2]) + (c^(1/4)*(b^2*c^2 + 4*a*b*c*d - 5*a^2*d^2)*Sqrt[e]*Sqrt[1 - (d*x^2)/c]*Ellip
ticPi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(4*a^(3/2)*b^(5
/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 468

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[((c*b -
a*d)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1))/(a*b*e*n*(p + 1)), x] + Dist[1/(a*b*n*(p + 1)), I
nt[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c*b - a*d)*(m + 1)) + d*(c*b*n*(p
+ 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[n, 0] && LtQ[p, -1] && 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 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 {\sqrt {e x} \left (c-d x^2\right )^{3/2}}{\left (a-b x^2\right )^2} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {x^2 \left (c-\frac {d x^4}{e^2}\right )^{3/2}}{\left (a-\frac {b x^4}{e^2}\right )^2} \, dx,x,\sqrt {e x}\right )}{e}\\ &=\frac {(b c-a d) (e x)^{3/2} \sqrt {c-d x^2}}{2 a b e \left (a-b x^2\right )}+\frac {e \operatorname {Subst}\left (\int \frac {x^2 \left (\frac {c (b c+3 a d)}{e^2}+\frac {d (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 )}{2 a b}\\ &=\frac {(b c-a d) (e x)^{3/2} \sqrt {c-d x^2}}{2 a b e \left (a-b x^2\right )}+\frac {e \operatorname {Subst}\left (\int \left (-\frac {d (b c-5 a d) x^2}{b e^2 \sqrt {c-\frac {d x^4}{e^2}}}+\frac {\left (b^2 c^2+4 a b c d-5 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 )}{2 a b}\\ &=\frac {(b c-a d) (e x)^{3/2} \sqrt {c-d x^2}}{2 a b e \left (a-b x^2\right )}-\frac {(d (b c-5 a d)) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a b^2 e}+\frac {((b c-a d) (b c+5 a d)) \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 )}{2 a b^2 e}\\ &=\frac {(b c-a d) (e x)^{3/2} \sqrt {c-d x^2}}{2 a b e \left (a-b x^2\right )}+\frac {\left (\sqrt {c} \sqrt {d} (b c-5 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a b^2}-\frac {\left (\sqrt {c} \sqrt {d} (b c-5 a d)\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 )}{2 a b^2}+\frac {((b c-a d) (b c+5 a d) e) \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 )}{4 a b^{5/2}}-\frac {((b c-a d) (b c+5 a d) e) \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 )}{4 a b^{5/2}}\\ &=\frac {(b c-a d) (e x)^{3/2} \sqrt {c-d x^2}}{2 a b e \left (a-b x^2\right )}+\frac {\left (\sqrt {c} \sqrt {d} (b c-5 a d) \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 )}{2 a b^2 \sqrt {c-d x^2}}-\frac {\left (\sqrt {c} \sqrt {d} (b c-5 a d) \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 )}{2 a b^2 \sqrt {c-d x^2}}+\frac {\left ((b c-a d) (b c+5 a d) e \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 )}{4 a b^{5/2} \sqrt {c-d x^2}}-\frac {\left ((b c-a d) (b c+5 a d) e \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 )}{4 a b^{5/2} \sqrt {c-d x^2}}\\ &=\frac {(b c-a d) (e x)^{3/2} \sqrt {c-d x^2}}{2 a b e \left (a-b x^2\right )}+\frac {c^{3/4} \sqrt [4]{d} (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 )}{2 a b^2 \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-a d) (b c+5 a d) \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 )}{4 a^{3/2} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d) (b c+5 a d) \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 )}{4 a^{3/2} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\left (\sqrt {c} \sqrt {d} (b c-5 a d) \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 )}{2 a b^2 \sqrt {c-d x^2}}\\ &=\frac {(b c-a d) (e x)^{3/2} \sqrt {c-d x^2}}{2 a b e \left (a-b x^2\right )}-\frac {c^{3/4} \sqrt [4]{d} (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 )}{2 a b^2 \sqrt {c-d x^2}}+\frac {c^{3/4} \sqrt [4]{d} (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 )}{2 a b^2 \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-a d) (b c+5 a d) \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 )}{4 a^{3/2} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d) (b c+5 a d) \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 )}{4 a^{3/2} b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.23, size = 189, normalized size = 0.40 \[ \frac {\sqrt {e x} \left (7 c x \left (b x^2-a\right ) \sqrt {1-\frac {d x^2}{c}} (3 a d+b c) F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )+3 d x^3 \left (b x^2-a\right ) \sqrt {1-\frac {d x^2}{c}} (b c-5 a d) F_1\left (\frac {7}{4};\frac {1}{2},1;\frac {11}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )+21 a x \left (c-d x^2\right ) (a d-b c)\right )}{42 a^2 b \left (b x^2-a\right ) \sqrt {c-d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[e*x]*(21*a*(-(b*c) + a*d)*x*(c - d*x^2) + 7*c*(b*c + 3*a*d)*x*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[
3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + 3*d*(b*c - 5*a*d)*x^3*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4,
1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(42*a^2*b*(-a + b*x^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)^(1/2)*(-d*x^2+c)^(3/2)/(-b*x^2+a)^2,x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.03, size = 3858, normalized size = 8.14 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(e*x)^(1/2)*(-d*x^2+c)^(1/2)*d*(4*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))*x^2*a*b^2*c*d*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/
2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)-4*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))*x^2*a*b^2*c*d*((d*x+(c*d)^(1/2
))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*(c*d)^(1/2)*(a*b)^(1/2
)-4*x^4*a^2*b^2*d^3-4*x^4*b^4*c^2*d+8*x^4*a*b^3*c*d^2+5*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))*x^2*a^2*b^2*c*d^2*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^
(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)-4*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))*x^2*a*b^3*c^2*d*((d*x+(c*d)^(1/2
))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)+5*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))*x^2*a^2*b^2*c*d^
2*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)-4*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)
)*x^2*a*b^3*c^2*d*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)
*d*x)^(1/2)-4*((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))*x^2*b^4*c^3+2*((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))*x^2*b^4*c^3-((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))*x^2*b^4*c^3-((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))*x^2*b^4*c^3+4*((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^3-2*((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*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^3-20*((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))*x^2*a^2*b^2*c*d^2+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)*Ellipti
cE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a*b^3*c^2*d+10*((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))*x^2*a^2*b^2*c*d^2-12*((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))*
x^2*a*b^3*c^2*d+((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)*x^2*b^3*c^2-((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)*x^2*b^3*c^2-((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*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^2-5*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))*x^2*a^2*b*d^2*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(
1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)+5*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))*x^2*a^2*b*d^2*((d*x+(c*d)^(1
/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*(c*d)^(1/2)*(a*b)^(1
/2)-4*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*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/
2)*d*x)^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)+4*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*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2)
)/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)+20*((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^2-24*((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)*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-10*((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^3*b*c*d^2+12*((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+4*x^2*a^2*b^2*c*d^2-8*x^2*a*b^3*c^2*d+4*x^2*b^4*c^3+
((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^3-5*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^2*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*
d)^(1/2)*d*x)^(1/2)+5*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^2*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)
*(-1/(c*d)^(1/2)*d*x)^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)+4*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*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*
((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)-5*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^2*((d*x+(c*d)^(1/2))/(c*d)^(
1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)-5*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^2*((d*x+(c*d)^(1/2))
/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)+
4*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*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/
2)*d*x)^(1/2))/b^2/x/(d*x^2-c)/a/(b*x^2-a)/((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 \[ \int \frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}} \sqrt {e x}}{{\left (b x^{2} - a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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