3.879 \(\int \frac {(c-d x^2)^{3/2}}{(e x)^{7/2} (a-b x^2)} \, dx\)

Optimal. Leaf size=459 \[ -\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (b c-a d)^2 \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a^{5/2} \sqrt {b} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (b c-a d)^2 \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a^{5/2} \sqrt {b} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}}+\frac {2 c^{3/4} \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} (5 b c-7 a d) F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 a^2 e^{7/2} \sqrt {c-d x^2}}-\frac {2 c^{3/4} \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} (5 b c-7 a d) E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 a^2 e^{7/2} \sqrt {c-d x^2}}-\frac {2 \sqrt {c-d x^2} (5 b c-7 a d)}{5 a^2 e^3 \sqrt {e x}}-\frac {2 c \sqrt {c-d x^2}}{5 a e (e x)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 1.06, antiderivative size = 459, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {466, 474, 583, 584, 307, 224, 221, 1200, 1199, 424, 490, 1219, 1218} \[ \frac {2 c^{3/4} \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} (5 b c-7 a d) F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 a^2 e^{7/2} \sqrt {c-d x^2}}-\frac {2 c^{3/4} \sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} (5 b c-7 a d) E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 a^2 e^{7/2} \sqrt {c-d x^2}}-\frac {2 \sqrt {c-d x^2} (5 b c-7 a d)}{5 a^2 e^3 \sqrt {e x}}-\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (b c-a d)^2 \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a^{5/2} \sqrt {b} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (b c-a d)^2 \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a^{5/2} \sqrt {b} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}}-\frac {2 c \sqrt {c-d x^2}}{5 a e (e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(c - d*x^2)^(3/2)/((e*x)^(7/2)*(a - b*x^2)),x]

[Out]

(-2*c*Sqrt[c - d*x^2])/(5*a*e*(e*x)^(5/2)) - (2*(5*b*c - 7*a*d)*Sqrt[c - d*x^2])/(5*a^2*e^3*Sqrt[e*x]) - (2*c^
(3/4)*d^(1/4)*(5*b*c - 7*a*d)*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1]
)/(5*a^2*e^(7/2)*Sqrt[c - d*x^2]) + (2*c^(3/4)*d^(1/4)*(5*b*c - 7*a*d)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d
^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(5*a^2*e^(7/2)*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^(5/2)*Sqrt[b]*d^(1/4)*e^(7/2)*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^(5/2)*Sqrt[b]*d^(1/
4)*e^(7/2)*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 474

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(c*(e*x)^
(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1))/(a*e*(m + 1)), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)
*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*(c*b - a*d)*(m + 1) + c*n*(b*c*(p + 1) + a*d*(q - 1)) + d*((c*b - a*
d)*(m + 1) + c*b*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0]
&& GtQ[q, 1] && LtQ[m, -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 583

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

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 {\left (c-d x^2\right )^{3/2}}{(e x)^{7/2} \left (a-b x^2\right )} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {\left (c-\frac {d x^4}{e^2}\right )^{3/2}}{x^6 \left (a-\frac {b x^4}{e^2}\right )} \, dx,x,\sqrt {e x}\right )}{e}\\ &=-\frac {2 c \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}+\frac {2 \operatorname {Subst}\left (\int \frac {\frac {c (5 b c-7 a d)}{e^2}-\frac {d (3 b c-5 a d) x^4}{e^4}}{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 )}{5 a e}\\ &=-\frac {2 c \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}-\frac {2 (5 b c-7 a d) \sqrt {c-d x^2}}{5 a^2 e^3 \sqrt {e x}}-\frac {2 \operatorname {Subst}\left (\int \frac {x^2 \left (-\frac {c \left (5 b^2 c^2-15 a b c d+12 a^2 d^2\right )}{e^4}-\frac {b c d (5 b c-7 a d) x^4}{e^6}\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 a^2 c e}\\ &=-\frac {2 c \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}-\frac {2 (5 b c-7 a d) \sqrt {c-d x^2}}{5 a^2 e^3 \sqrt {e x}}-\frac {2 \operatorname {Subst}\left (\int \left (\frac {c d (5 b c-7 a d) x^2}{e^4 \sqrt {c-\frac {d x^4}{e^2}}}-\frac {5 \left (b^2 c^3-2 a b c^2 d+a^2 c d^2\right ) x^2}{e^4 \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 a^2 c e}\\ &=-\frac {2 c \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}-\frac {2 (5 b c-7 a d) \sqrt {c-d x^2}}{5 a^2 e^3 \sqrt {e x}}-\frac {(2 d (5 b c-7 a d)) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 a^2 e^5}+\frac {\left (2 (b c-a d)^2\right ) \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 )}{a^2 e^5}\\ &=-\frac {2 c \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}-\frac {2 (5 b c-7 a d) \sqrt {c-d x^2}}{5 a^2 e^3 \sqrt {e x}}+\frac {\left (2 \sqrt {c} \sqrt {d} (5 b c-7 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 a^2 e^4}-\frac {\left (2 \sqrt {c} \sqrt {d} (5 b c-7 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 )}{5 a^2 e^4}+\frac {(b c-a d)^2 \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 )}{a^2 \sqrt {b} e^3}-\frac {(b c-a d)^2 \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 )}{a^2 \sqrt {b} e^3}\\ &=-\frac {2 c \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}-\frac {2 (5 b c-7 a d) \sqrt {c-d x^2}}{5 a^2 e^3 \sqrt {e x}}+\frac {\left (2 \sqrt {c} \sqrt {d} (5 b c-7 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 )}{5 a^2 e^4 \sqrt {c-d x^2}}-\frac {\left (2 \sqrt {c} \sqrt {d} (5 b c-7 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 )}{5 a^2 e^4 \sqrt {c-d x^2}}+\frac {\left ((b c-a d)^2 \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 )}{a^2 \sqrt {b} e^3 \sqrt {c-d x^2}}-\frac {\left ((b c-a d)^2 \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 )}{a^2 \sqrt {b} e^3 \sqrt {c-d x^2}}\\ &=-\frac {2 c \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}-\frac {2 (5 b c-7 a d) \sqrt {c-d x^2}}{5 a^2 e^3 \sqrt {e x}}+\frac {2 c^{3/4} \sqrt [4]{d} (5 b c-7 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 )}{5 a^2 e^{7/2} \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^{5/2} \sqrt {b} \sqrt [4]{d} e^{7/2} \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^{5/2} \sqrt {b} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}}-\frac {\left (2 \sqrt {c} \sqrt {d} (5 b c-7 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 )}{5 a^2 e^4 \sqrt {c-d x^2}}\\ &=-\frac {2 c \sqrt {c-d x^2}}{5 a e (e x)^{5/2}}-\frac {2 (5 b c-7 a d) \sqrt {c-d x^2}}{5 a^2 e^3 \sqrt {e x}}-\frac {2 c^{3/4} \sqrt [4]{d} (5 b c-7 a d) \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 a^2 e^{7/2} \sqrt {c-d x^2}}+\frac {2 c^{3/4} \sqrt [4]{d} (5 b c-7 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 )}{5 a^2 e^{7/2} \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^{5/2} \sqrt {b} \sqrt [4]{d} e^{7/2} \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^{5/2} \sqrt {b} \sqrt [4]{d} e^{7/2} \sqrt {c-d x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.26, size = 187, normalized size = 0.41 \[ \frac {x \left (14 x^4 \sqrt {1-\frac {d x^2}{c}} \left (12 a^2 d^2-15 a b c d+5 b^2 c^2\right ) F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )-6 \left (b d x^6 \sqrt {1-\frac {d x^2}{c}} (7 a d-5 b c) F_1\left (\frac {7}{4};\frac {1}{2},1;\frac {11}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )+7 a \left (c-d x^2\right ) \left (a c-7 a d x^2+5 b c x^2\right )\right )\right )}{105 a^3 (e x)^{7/2} \sqrt {c-d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c - d*x^2)^(3/2)/((e*x)^(7/2)*(a - b*x^2)),x]

[Out]

(x*(14*(5*b^2*c^2 - 15*a*b*c*d + 12*a^2*d^2)*x^4*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*
x^2)/a] - 6*(7*a*(c - d*x^2)*(a*c + 5*b*c*x^2 - 7*a*d*x^2) + b*d*(-5*b*c + 7*a*d)*x^6*Sqrt[1 - (d*x^2)/c]*Appe
llF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a])))/(105*a^3*(e*x)^(7/2)*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((-d*x^2+c)^(3/2)/(e*x)^(7/2)/(-b*x^2+a),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} - a\right )} \left (e x\right )^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-d*x^2+c)^(3/2)/(e*x)^(7/2)/(-b*x^2+a),x, algorithm="giac")

[Out]

integrate(-(-d*x^2 + c)^(3/2)/((b*x^2 - a)*(e*x)^(7/2)), x)

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maple [B]  time = 0.03, size = 2028, normalized size = 4.42 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-d*x^2+c)^(3/2)/(e*x)^(7/2)/(-b*x^2+a),x)

[Out]

-1/10*(-d*x^2+c)^(1/2)*d*(28*x^4*a^2*b*d^3+20*x^4*b^3*c^2*d+4*a^2*b*c^2*d-48*x^4*a*b^2*c*d^2+28*((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*c*d^2-48*((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*b^2*c^2*d-14*((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*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)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a*b^2*c^2*d-5*((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)*Ell
ipticPi(((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*a^2*d^2-5*((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^2*c^2+5*((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*a^2*d^2+5*((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)*Ellip
ticPi(((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^2*c^2+5*((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*a^2*b*c*d^2-10*((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)*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^2*d+5*((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*a^2*b*c*d^2-10*((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)*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^2*d-20*x^2*b^3*c^3+52*x^2*a*b^2*c
^2*d-32*x^2*a^2*b*c*d^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))*x^2*b^3*c^3+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)*(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*a*b*c*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)*(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*a*b*c*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*b^3*c^3+5*((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^3*c^3+5*((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^3*c^3-4*a*b^2*c^3)/x^2/e^3/(e*x)^(1/2)/(d*x^2-c)/a^2/((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 \[ -\int \frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} - a\right )} \left (e x\right )^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-d*x^2+c)^(3/2)/(e*x)^(7/2)/(-b*x^2+a),x, algorithm="maxima")

[Out]

-integrate((-d*x^2 + c)^(3/2)/((b*x^2 - a)*(e*x)^(7/2)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c-d\,x^2\right )}^{3/2}}{{\left (e\,x\right )}^{7/2}\,\left (a-b\,x^2\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c - d*x^2)^(3/2)/((e*x)^(7/2)*(a - b*x^2)),x)

[Out]

int((c - d*x^2)^(3/2)/((e*x)^(7/2)*(a - b*x^2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {c \sqrt {c - d x^{2}}}{- a \left (e x\right )^{\frac {7}{2}} + b x^{2} \left (e x\right )^{\frac {7}{2}}}\, dx - \int \left (- \frac {d x^{2} \sqrt {c - d x^{2}}}{- a \left (e x\right )^{\frac {7}{2}} + b x^{2} \left (e x\right )^{\frac {7}{2}}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-d*x**2+c)**(3/2)/(e*x)**(7/2)/(-b*x**2+a),x)

[Out]

-Integral(c*sqrt(c - d*x**2)/(-a*(e*x)**(7/2) + b*x**2*(e*x)**(7/2)), x) - Integral(-d*x**2*sqrt(c - d*x**2)/(
-a*(e*x)**(7/2) + b*x**2*(e*x)**(7/2)), x)

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