3.917 \(\int \frac{1}{(e x)^{5/2} (a-b x^2)^2 \sqrt{c-d x^2}} \, dx\)
Optimal. Leaf size=429 \[ \frac{d^{3/4} \sqrt{1-\frac{d x^2}{c}} (7 b c-4 a d) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),-1\right )}{6 a^2 c^{3/4} e^{5/2} \sqrt{c-d x^2} (b c-a d)}+\frac{b \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-9 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)}+\frac{b \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-9 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{c-d x^2} (7 b c-4 a d)}{6 a^2 c e (e x)^{3/2} (b c-a d)}+\frac{b \sqrt{c-d x^2}}{2 a e (e x)^{3/2} \left (a-b x^2\right ) (b c-a d)} \]
[Out]
-((7*b*c - 4*a*d)*Sqrt[c - d*x^2])/(6*a^2*c*(b*c - a*d)*e*(e*x)^(3/2)) + (b*Sqrt[c - d*x^2])/(2*a*(b*c - a*d)*
e*(e*x)^(3/2)*(a - b*x^2)) + (d^(3/4)*(7*b*c - 4*a*d)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])
/(c^(1/4)*Sqrt[e])], -1])/(6*a^2*c^(3/4)*(b*c - a*d)*e^(5/2)*Sqrt[c - d*x^2]) + (b*c^(1/4)*(7*b*c - 9*a*d)*Sqr
t[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])/(4*a^3*d^(1/4)*(b*c - a*d)*e^(5/2)*Sqrt[c - d*x^2]) + (b*c^(1/4)*(7*b*c - 9*a*d)*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])/(4*a^3*d^
(1/4)*(b*c - a*d)*e^(5/2)*Sqrt[c - d*x^2])
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Rubi [A] time = 0.81428, antiderivative size = 429, normalized size of antiderivative = 1.,
number of steps used = 11, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used =
{466, 472, 583, 523, 224, 221, 409, 1219, 1218} \[ \frac{d^{3/4} \sqrt{1-\frac{d x^2}{c}} (7 b c-4 a d) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{6 a^2 c^{3/4} e^{5/2} \sqrt{c-d x^2} (b c-a d)}+\frac{b \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-9 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)}+\frac{b \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-9 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{c-d x^2} (7 b c-4 a d)}{6 a^2 c e (e x)^{3/2} (b c-a d)}+\frac{b \sqrt{c-d x^2}}{2 a e (e x)^{3/2} \left (a-b x^2\right ) (b c-a d)} \]
Antiderivative was successfully verified.
[In]
Int[1/((e*x)^(5/2)*(a - b*x^2)^2*Sqrt[c - d*x^2]),x]
[Out]
-((7*b*c - 4*a*d)*Sqrt[c - d*x^2])/(6*a^2*c*(b*c - a*d)*e*(e*x)^(3/2)) + (b*Sqrt[c - d*x^2])/(2*a*(b*c - a*d)*
e*(e*x)^(3/2)*(a - b*x^2)) + (d^(3/4)*(7*b*c - 4*a*d)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])
/(c^(1/4)*Sqrt[e])], -1])/(6*a^2*c^(3/4)*(b*c - a*d)*e^(5/2)*Sqrt[c - d*x^2]) + (b*c^(1/4)*(7*b*c - 9*a*d)*Sqr
t[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])/(4*a^3*d^(1/4)*(b*c - a*d)*e^(5/2)*Sqrt[c - d*x^2]) + (b*c^(1/4)*(7*b*c - 9*a*d)*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])/(4*a^3*d^
(1/4)*(b*c - a*d)*e^(5/2)*Sqrt[c - d*x^2])
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 472
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*(e*x
)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*e*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d)*(
p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n*(
p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p
, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
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 523
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 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 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 409
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 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]
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]
Rubi steps
\begin{align*} \int \frac{1}{(e x)^{5/2} \left (a-b x^2\right )^2 \sqrt{c-d x^2}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{x^4 \left (a-\frac{b x^4}{e^2}\right )^2 \sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{e}\\ &=\frac{b \sqrt{c-d x^2}}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right )}+\frac{e \operatorname{Subst}\left (\int \frac{\frac{7 b c-4 a d}{e^2}-\frac{5 b d x^4}{e^4}}{x^4 \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 c-a d)}\\ &=-\frac{(7 b c-4 a d) \sqrt{c-d x^2}}{6 a^2 c (b c-a d) e (e x)^{3/2}}+\frac{b \sqrt{c-d x^2}}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right )}-\frac{e \operatorname{Subst}\left (\int \frac{-\frac{21 b^2 c^2-20 a b c d-4 a^2 d^2}{e^4}+\frac{b d (7 b c-4 a d) 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 )}{6 a^2 c (b c-a d)}\\ &=-\frac{(7 b c-4 a d) \sqrt{c-d x^2}}{6 a^2 c (b c-a d) e (e x)^{3/2}}+\frac{b \sqrt{c-d x^2}}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right )}+\frac{(b (7 b c-9 a d)) \operatorname{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 )}{2 a^2 (b c-a d) e^3}+\frac{(d (7 b c-4 a d)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{6 a^2 c (b c-a d) e^3}\\ &=-\frac{(7 b c-4 a d) \sqrt{c-d x^2}}{6 a^2 c (b c-a d) e (e x)^{3/2}}+\frac{b \sqrt{c-d x^2}}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right )}+\frac{(b (7 b c-9 a d)) \operatorname{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 )}{4 a^3 (b c-a d) e^3}+\frac{(b (7 b c-9 a d)) \operatorname{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 )}{4 a^3 (b c-a d) e^3}+\frac{\left (d (7 b c-4 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 )}{6 a^2 c (b c-a d) e^3 \sqrt{c-d x^2}}\\ &=-\frac{(7 b c-4 a d) \sqrt{c-d x^2}}{6 a^2 c (b c-a d) e (e x)^{3/2}}+\frac{b \sqrt{c-d x^2}}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right )}+\frac{d^{3/4} (7 b c-4 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 )}{6 a^2 c^{3/4} (b c-a d) e^{5/2} \sqrt{c-d x^2}}+\frac{\left (b (7 b c-9 a d) \sqrt{1-\frac{d x^2}{c}}\right ) \operatorname{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 )}{4 a^3 (b c-a d) e^3 \sqrt{c-d x^2}}+\frac{\left (b (7 b c-9 a d) \sqrt{1-\frac{d x^2}{c}}\right ) \operatorname{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 )}{4 a^3 (b c-a d) e^3 \sqrt{c-d x^2}}\\ &=-\frac{(7 b c-4 a d) \sqrt{c-d x^2}}{6 a^2 c (b c-a d) e (e x)^{3/2}}+\frac{b \sqrt{c-d x^2}}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right )}+\frac{d^{3/4} (7 b c-4 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 )}{6 a^2 c^{3/4} (b c-a d) e^{5/2} \sqrt{c-d x^2}}+\frac{b \sqrt [4]{c} (7 b c-9 a d) \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 \sqrt [4]{d} (b c-a d) e^{5/2} \sqrt{c-d x^2}}+\frac{b \sqrt [4]{c} (7 b c-9 a d) \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 \sqrt [4]{d} (b c-a d) e^{5/2} \sqrt{c-d x^2}}\\ \end{align*}
Mathematica [C] time = 0.289774, size = 234, normalized size = 0.55 \[ \frac{x \left (5 x^2 \left (a-b x^2\right ) \sqrt{1-\frac{d x^2}{c}} \left (4 a^2 d^2+20 a b c d-21 b^2 c^2\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )-5 a \left (c-d x^2\right ) \left (4 a^2 d-4 a b \left (c+d x^2\right )+7 b^2 c x^2\right )-b d x^4 \left (a-b x^2\right ) \sqrt{1-\frac{d x^2}{c}} (4 a d-7 b c) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}{30 a^3 c (e x)^{5/2} \left (b x^2-a\right ) \sqrt{c-d x^2} (b c-a d)} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[1/((e*x)^(5/2)*(a - b*x^2)^2*Sqrt[c - d*x^2]),x]
[Out]
(x*(-5*a*(c - d*x^2)*(4*a^2*d + 7*b^2*c*x^2 - 4*a*b*(c + d*x^2)) + 5*(-21*b^2*c^2 + 20*a*b*c*d + 4*a^2*d^2)*x^
2*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a] - b*d*(-7*b*c + 4*a*d)*x^4*
(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(30*a^3*c*(b*c - a*d)*(e*x)
^(5/2)*(-a + b*x^2)*Sqrt[c - d*x^2])
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Maple [B] time = 0.035, size = 2622, normalized size = 6.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(e*x)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x)
[Out]
1/24*b*d*(-16*x^2*a^3*d^3*(a*b)^(1/2)+16*a^3*c*d^2*(a*b)^(1/2)+8*2^(1/2)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1
/2))^(1/2),1/2*2^(1/2))*x^3*a^2*b*d^2*(a*b)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c
*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)-14*2^(1/2)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/
2),1/2*2^(1/2))*x*a*b^2*c^2*(a*b)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))
^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)+21*2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^
(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*x*a*b^2*c^2*(a*b)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/
2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)+21*2^(1/2)*EllipticPi(((d*x+(c*
d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*x*a*b^2*c^2*(a*b)^(1/2)*
((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1
/2)+21*2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/
2*2^(1/2))*x^3*b^4*c^3*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d
)^(1/2))^(1/2)+16*x^4*a^2*b*d^3*(a*b)^(1/2)-32*a^2*b*c^2*d*(a*b)^(1/2)-21*2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2)
)/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*x*a*b^3*c^3*((d*x+(c*d)^(1/2))/(
c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)+27*2^(1/2)*EllipticPi(((d*x+
(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*x*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)*(-x*d/(c*d)^(1/2))^(1/2)-27*2^(1/2)*El
lipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*x*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)*(-x*d/(c*d)^(1/2))^(1/2
)+14*2^(1/2)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^3*b^3*c^2*(a*b)^(1/2)*((d*x+(c*d)^
(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)-27*2^(1/
2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*x
^3*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)*(-x*d/(c*d)^(1/2))
^(1/2)-28*x^2*b^3*c^3*(a*b)^(1/2)+16*a*b^2*c^3*(a*b)^(1/2)-21*2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2
))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*x^3*b^3*c^2*(a*b)^(1/2)*((d*x+(c*d)^(1/2))/(
c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)+27*2^(1/2)*Ellip
ticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*x^3*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)*(-x*d/(c*d)^(1/2))^(1/2)-2
1*2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(
1/2))*x^3*b^3*c^2*(a*b)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x
*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)-8*2^(1/2)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*x*a^3
*d^2*(a*b)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2
))^(1/2)*(c*d)^(1/2)-44*x^4*a*b^2*c*d^2*(a*b)^(1/2)+28*x^4*b^3*c^2*d*(a*b)^(1/2)+16*x^2*a^2*b*c*d^2*(a*b)^(1/2
)+28*x^2*a*b^2*c^2*d*(a*b)^(1/2)+21*2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c
*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*x*a*b^3*c^3*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/
(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)-21*2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^
(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*x^3*b^4*c^3*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c
*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)+22*2^(1/2)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1
/2),1/2*2^(1/2))*x*a^2*b*c*d*(a*b)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2)
)^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)-27*2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)
^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*x*a^2*b*c*d*(a*b)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1
/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)-27*2^(1/2)*EllipticPi(((d*x+(c
*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*x*a^2*b*c*d*(a*b)^(1/2)
*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(
1/2)+27*2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1
/2*2^(1/2))*x^3*a*b^2*c*d*(a*b)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(
1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)-22*2^(1/2)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/
2))*x^3*a*b^2*c*d*(a*b)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x
*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)+27*2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((
a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*x^3*a*b^2*c*d*(a*b)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*
x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(c*d)^(1/2))*(-d*x^2+c)^(1/2)/x/c/a^2/((c*d)^(1/2)*
b-(a*b)^(1/2)*d)/((a*b)^(1/2)*d+(c*d)^(1/2)*b)/(a*b)^(1/2)/(b*x^2-a)/(a*d-b*c)/(d*x^2-c)/e^2/(e*x)^(1/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} - a\right )}^{2} \sqrt{-d x^{2} + c} \left (e x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x, algorithm="maxima")
[Out]
integrate(1/((b*x^2 - a)^2*sqrt(-d*x^2 + c)*(e*x)^(5/2)), x)
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x, algorithm="fricas")
[Out]
Exception raised: UnboundLocalError
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x)**(5/2)/(-b*x**2+a)**2/(-d*x**2+c)**(1/2),x)
[Out]
Timed out
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x, algorithm="giac")
[Out]
Timed out