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

Optimal. Leaf size=372 \[ -\frac{2 \sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} \left (21 a^2 d^2-35 a b c d+12 b^2 c^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),-1\right )}{21 b^3 \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^3 \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^3 \sqrt [4]{d} \sqrt{c-d x^2}}-\frac{2 e \sqrt{e x} \sqrt{c-d x^2} (9 b c-7 a d)}{21 b^2}+\frac{2 d (e x)^{5/2} \sqrt{c-d x^2}}{7 b e} \]

[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])

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Rubi [A]  time = 0.778367, antiderivative size = 372, 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, 477, 582, 523, 224, 221, 409, 1219, 1218} \[ -\frac{2 \sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} \left (21 a^2 d^2-35 a b c d+12 b^2 c^2\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{21 b^3 \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^3 \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^3 \sqrt [4]{d} \sqrt{c-d x^2}}-\frac{2 e \sqrt{e x} \sqrt{c-d x^2} (9 b c-7 a d)}{21 b^2}+\frac{2 d (e x)^{5/2} \sqrt{c-d x^2}}{7 b e} \]

Antiderivative was successfully verified.

[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 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 477

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 582

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 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{(e x)^{3/2} \left (c-d x^2\right )^{3/2}}{a-b x^2} \, dx &=\frac{2 \operatorname{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) \operatorname{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 ) \operatorname{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 ) \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 )}{b^3}-\frac{\left (2 \left (12 b^2 c^2-35 a b c d+21 a^2 d^2\right ) e\right ) \operatorname{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 ) \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 )}{b^3}+\frac{\left ((b c-a d)^2 e\right ) \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 )}{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 ) \operatorname{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 ) \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 )}{b^3 \sqrt{c-d x^2}}+\frac{\left ((b c-a d)^2 e \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 )}{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]  time = 0.189403, size = 182, normalized size = 0.49 \[ -\frac{2 e \sqrt{e x} \left (x^2 \sqrt{1-\frac{d x^2}{c}} \left (-21 a^2 d^2+35 a b c d-12 b^2 c^2\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+5 a c \sqrt{1-\frac{d x^2}{c}} (7 a d-9 b c) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )-5 a \left (c-d x^2\right ) \left (7 a d-9 b c+3 b d x^2\right )\right )}{105 a b^2 \sqrt{c-d x^2}} \]

Warning: Unable to verify antiderivative.

[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])

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Maple [B]  time = 0.024, size = 1920, normalized size = 5.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/42*e*(e*x)^(1/2)*(-d*x^2+c)^(1/2)/b^2*(21*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))*2^(1/2)*a^3*b*c*d^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*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))*2^(1/2)*a^3*d^3*(a*b)^(1/2)*(c*d)^(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)-42*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))*2^(1/2)*a^2*b^2*c^2*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)*(-x*d/(c*d)^(1/2))^(1/2)-42*EllipticP
i(((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))*2^(1/2)*a^2*b
*c*d^2*(a*b)^(1/2)*(c*d)^(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)+21*((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)
*(-x*d/(c*d)^(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))*a*b^3*c^3*d+21*((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)*(-x*d/(c*d)^(1/2))^(1/2)*(a*b)^(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))*(c*d)^(1/2)*a*b^2*c^2*d-21*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))*2^(1/2)*a^3*b*c*d^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*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))*2^(1/2)*a^3*d^3*(a*b)^(1/2)*(c*
d)^(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)
+42*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))*
2^(1/2)*a^2*b^2*c^2*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)*(-x*d/(c*
d)^(1/2))^(1/2)-42*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))*2^(1/2)*a^2*b*c*d^2*(a*b)^(1/2)*(c*d)^(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)-21*((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)*(-x*d/(c*d)^(1/2))^(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))*a*b^3*c^3*d+21*((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)*(-x*d/(c*d)^(1/2))^(1/2)*(a*b)^(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))*(c*d)^(1/2)*a*b^2*c^2*d-42*EllipticF(
((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a^3*d^3*(a*b)^(1/2)*(c*d)^(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)+112*EllipticF(((d*x+(c*d)^(
1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a^2*b*c*d^2*(a*b)^(1/2)*(c*d)^(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)-94*EllipticF(((d*x+(c*d)^(1/2))/(c*d
)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a*b^2*c^2*d*(a*b)^(1/2)*(c*d)^(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)+24*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(
1/2),1/2*2^(1/2))*2^(1/2)*b^3*c^3*(a*b)^(1/2)*(c*d)^(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)-12*x^5*a*b^2*d^4*(a*b)^(1/2)+12*x^5*b^3*c*d^3*(a*b)^(1/2)-28
*x^3*a^2*b*d^4*(a*b)^(1/2)+76*x^3*a*b^2*c*d^3*(a*b)^(1/2)-48*x^3*b^3*c^2*d^2*(a*b)^(1/2)+28*x*a^2*b*c*d^3*(a*b
)^(1/2)-64*x*a*b^2*c^2*d^2*(a*b)^(1/2)+36*x*b^3*c^3*d*(a*b)^(1/2))/x/(d*x^2-c)/(a*b)^(1/2)/((a*b)^(1/2)*d+(c*d
)^(1/2)*b)/((c*d)^(1/2)*b-(a*b)^(1/2)*d)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (-d x^{2} + c\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{3}{2}}}{b x^{2} - a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{c \left (e x\right )^{\frac{3}{2}} \sqrt{c - d x^{2}}}{- a + b x^{2}}\, dx - \int - \frac{d x^{2} \left (e x\right )^{\frac{3}{2}} \sqrt{c - d x^{2}}}{- a + b x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (-d x^{2} + c\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{3}{2}}}{b x^{2} - a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)