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

Optimal. Leaf size=366 \[ \frac {3 \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (a d+b c) (b c-a d) \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^2 b^2 \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2}}+\frac {3 \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (a d+b c) (b c-a d) \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^2 b^2 \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} d^{3/4} \sqrt {1-\frac {d x^2}{c}} (3 a d+b c) F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a b^2 \sqrt {e} \sqrt {c-d x^2}}+\frac {\sqrt {e x} \sqrt {c-d x^2} (b c-a d)}{2 a b e \left (a-b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.60, antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {466, 413, 523, 224, 221, 409, 1219, 1218} \[ \frac {3 \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (a d+b c) (b c-a d) \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^2 b^2 \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2}}+\frac {3 \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (a d+b c) (b c-a d) \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^2 b^2 \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} d^{3/4} \sqrt {1-\frac {d x^2}{c}} (3 a d+b c) F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a b^2 \sqrt {e} \sqrt {c-d x^2}}+\frac {\sqrt {e x} \sqrt {c-d x^2} (b c-a d)}{2 a b e \left (a-b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*c - a*d)*Sqrt[e*x]*Sqrt[c - d*x^2])/(2*a*b*e*(a - b*x^2)) + (c^(1/4)*d^(3/4)*(b*c + 3*a*d)*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[e]*Sqrt[c - d*x^2]) + (3*c^(1
/4)*(b*c - a*d)*(b*c + 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^2*b^2*d^(1/4)*Sqrt[e]*Sqrt[c - d*x^2]) + (3*c^(1/4)*(b*c - a*d)*
(b*c + 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^2*b^2*d^(1/4)*Sqrt[e]*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 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 413

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

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

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Mathematica [C]  time = 0.20, size = 187, normalized size = 0.51 \[ \frac {5 c x \left (b x^2-a\right ) \sqrt {1-\frac {d x^2}{c}} (a d+3 b c) F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )+d x^3 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} (3 a d+b c) F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )+5 a x \left (c-d x^2\right ) (a d-b c)}{10 a^2 b \sqrt {e x} \left (b x^2-a\right ) \sqrt {c-d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(5*a*(-(b*c) + a*d)*x*(c - d*x^2) + 5*c*(3*b*c + a*d)*x*(-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] + d*(b*c + 3*a*d)*x^3*(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])/(10*a^2*b*Sqrt[e*x]*(-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((-d*x^2+c)^(3/2)/(e*x)^(1/2)/(-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 \[ \int \frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} - a\right )}^{2} \sqrt {e x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.04, size = 2531, normalized size = 6.92 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(-d*x^2+c)^(1/2)/b*d*(-4*2^(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*
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*(a*b)^(1/2)*a^2*b*c*d^2*x-8*(a*b)^(1/2)*a*b^2*c^2*d*x+4*x*b^3*c^3*(a*b)^(1/2)+8*(a*b)^(1
/2)*a*b^2*c*d^2*x^3-6*2^(1/2)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),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)+3*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)-3*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)-2*2^(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^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)+2*2^(1/2)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b^2
*c^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)-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+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+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+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)*(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+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)*(a*b)^(1/2)*E
llipticPi(((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^2-3*((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)*(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^2-3*((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)*(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^2-4*(a*b)^(1/2)*a^2*b*d^3
*x^3-4*(a*b)^(1/2)*b^3*c^2*d*x^3-3*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)-3*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)+6
*2^(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*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
)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),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)-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^3-3*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)+3*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)+3*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)+3*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))/a/(e*x)^(1/2)/(d*x^2-c)/(b*x^2
-a)/(a*b)^(1/2)/((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 \[ \int \frac {{\left (-d x^{2} + c\right )}^{\frac {3}{2}}}{{\left (b x^{2} - a\right )}^{2} \sqrt {e x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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