\(\int \frac {A+B x^2}{(e x)^{3/2} (a+b x^2)^{3/2}} \, dx\) [812]

Optimal result

Integrand size = 26, antiderivative size = 333 \[ \int \frac {A+B x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/2}} \, dx=-\frac {2 A}{a e \sqrt {e x} \sqrt {a+b x^2}}-\frac {(3 A b-a B) (e x)^{3/2}}{a^2 e^3 \sqrt {a+b x^2}}+\frac {(3 A b-a B) \sqrt {e x} \sqrt {a+b x^2}}{a^2 \sqrt {b} e^2 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {(3 A b-a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{a^{7/4} b^{3/4} e^{3/2} \sqrt {a+b x^2}}+\frac {(3 A b-a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 a^{7/4} b^{3/4} e^{3/2} \sqrt {a+b x^2}} \]

[Out]

-(3*A*b-B*a)*(e*x)^(3/2)/a^2/e^3/(b*x^2+a)^(1/2)-2*A/a/e/(e*x)^(1/2)/(b*x^2+a)^(1/2)+(3*A*b-B*a)*(e*x)^(1/2)*(
b*x^2+a)^(1/2)/a^2/e^2/b^(1/2)/(a^(1/2)+x*b^(1/2))-(3*A*b-B*a)*(cos(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/
2)))^2)^(1/2)/cos(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)))*EllipticE(sin(2*arctan(b^(1/4)*(e*x)^(1/2)/a^
(1/4)/e^(1/2))),1/2*2^(1/2))*(a^(1/2)+x*b^(1/2))*((b*x^2+a)/(a^(1/2)+x*b^(1/2))^2)^(1/2)/a^(7/4)/b^(3/4)/e^(3/
2)/(b*x^2+a)^(1/2)+1/2*(3*A*b-B*a)*(cos(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)))^2)^(1/2)/cos(2*arctan(b
^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)))*EllipticF(sin(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))),1/2*2^(1/2))
*(a^(1/2)+x*b^(1/2))*((b*x^2+a)/(a^(1/2)+x*b^(1/2))^2)^(1/2)/a^(7/4)/b^(3/4)/e^(3/2)/(b*x^2+a)^(1/2)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {464, 296, 335, 311, 226, 1210} \[ \int \frac {A+B x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/2}} \, dx=\frac {\left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (3 A b-a B) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 a^{7/4} b^{3/4} e^{3/2} \sqrt {a+b x^2}}-\frac {\left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} (3 A b-a B) E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{a^{7/4} b^{3/4} e^{3/2} \sqrt {a+b x^2}}-\frac {(e x)^{3/2} (3 A b-a B)}{a^2 e^3 \sqrt {a+b x^2}}+\frac {\sqrt {e x} \sqrt {a+b x^2} (3 A b-a B)}{a^2 \sqrt {b} e^2 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 A}{a e \sqrt {e x} \sqrt {a+b x^2}} \]

[In]

Int[(A + B*x^2)/((e*x)^(3/2)*(a + b*x^2)^(3/2)),x]

[Out]

(-2*A)/(a*e*Sqrt[e*x]*Sqrt[a + b*x^2]) - ((3*A*b - a*B)*(e*x)^(3/2))/(a^2*e^3*Sqrt[a + b*x^2]) + ((3*A*b - a*B
)*Sqrt[e*x]*Sqrt[a + b*x^2])/(a^2*Sqrt[b]*e^2*(Sqrt[a] + Sqrt[b]*x)) - ((3*A*b - a*B)*(Sqrt[a] + Sqrt[b]*x)*Sq
rt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(a^(7
/4)*b^(3/4)*e^(3/2)*Sqrt[a + b*x^2]) + ((3*A*b - a*B)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b
]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(2*a^(7/4)*b^(3/4)*e^(3/2)*Sqrt[a + b
*x^2])

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 296

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-(c*x)^(m + 1))*((a + b*x^n)^(p + 1)/
(a*c*n*(p + 1))), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 311

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, Dist[1/q, 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] && PosQ[b/a]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 464

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

Rule 1210

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a +
c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 A}{a e \sqrt {e x} \sqrt {a+b x^2}}-\frac {(3 A b-a B) \int \frac {\sqrt {e x}}{\left (a+b x^2\right )^{3/2}} \, dx}{a e^2} \\ & = -\frac {2 A}{a e \sqrt {e x} \sqrt {a+b x^2}}-\frac {(3 A b-a B) (e x)^{3/2}}{a^2 e^3 \sqrt {a+b x^2}}+\frac {(3 A b-a B) \int \frac {\sqrt {e x}}{\sqrt {a+b x^2}} \, dx}{2 a^2 e^2} \\ & = -\frac {2 A}{a e \sqrt {e x} \sqrt {a+b x^2}}-\frac {(3 A b-a B) (e x)^{3/2}}{a^2 e^3 \sqrt {a+b x^2}}+\frac {(3 A b-a B) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{a^2 e^3} \\ & = -\frac {2 A}{a e \sqrt {e x} \sqrt {a+b x^2}}-\frac {(3 A b-a B) (e x)^{3/2}}{a^2 e^3 \sqrt {a+b x^2}}+\frac {(3 A b-a B) \text {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{a^{3/2} \sqrt {b} e^2}-\frac {(3 A b-a B) \text {Subst}\left (\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a} e}}{\sqrt {a+\frac {b x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{a^{3/2} \sqrt {b} e^2} \\ & = -\frac {2 A}{a e \sqrt {e x} \sqrt {a+b x^2}}-\frac {(3 A b-a B) (e x)^{3/2}}{a^2 e^3 \sqrt {a+b x^2}}+\frac {(3 A b-a B) \sqrt {e x} \sqrt {a+b x^2}}{a^2 \sqrt {b} e^2 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {(3 A b-a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{a^{7/4} b^{3/4} e^{3/2} \sqrt {a+b x^2}}+\frac {(3 A b-a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{2 a^{7/4} b^{3/4} e^{3/2} \sqrt {a+b x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.23 \[ \int \frac {A+B x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/2}} \, dx=\frac {x \left (-6 a A+2 (-3 A b+a B) x^2 \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\frac {b x^2}{a}\right )\right )}{3 a^2 (e x)^{3/2} \sqrt {a+b x^2}} \]

[In]

Integrate[(A + B*x^2)/((e*x)^(3/2)*(a + b*x^2)^(3/2)),x]

[Out]

(x*(-6*a*A + 2*(-3*A*b + a*B)*x^2*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, -((b*x^2)/a)]))/(3*a^2*
(e*x)^(3/2)*Sqrt[a + b*x^2])

Maple [A] (verified)

Time = 3.56 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.84

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (-\frac {2 \left (b e \,x^{2}+a e \right ) A}{a^{2} e^{2} \sqrt {x \left (b e \,x^{2}+a e \right )}}-\frac {x^{2} \left (A b -B a \right )}{e \,a^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {\left (\frac {b A}{a^{2} e}+\frac {A b -B a}{2 a^{2} e}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(281\)
default	\(\frac {6 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a b -3 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a b -2 B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2}+B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2}-6 A \,b^{2} x^{2}+2 B a b \,x^{2}-4 a b A}{2 \sqrt {b \,x^{2}+a}\, b e \sqrt {e x}\, a^{2}}\)	\(386\)
risch	\(-\frac {2 A \sqrt {b \,x^{2}+a}}{a^{2} e \sqrt {e x}}+\frac {\left (\frac {A \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{\sqrt {b e \,x^{3}+a e x}}-a \left (A b -B a \right ) \left (\frac {x^{2}}{a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}-\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{2 a b \sqrt {b e \,x^{3}+a e x}}\right )\right ) \sqrt {\left (b \,x^{2}+a \right ) e x}}{a^{2} e \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(413\)

[In]

int((B*x^2+A)/(e*x)^(3/2)/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

((b*x^2+a)*e*x)^(1/2)/(e*x)^(1/2)/(b*x^2+a)^(1/2)*(-2*(b*e*x^2+a*e)/a^2/e^2*A/(x*(b*e*x^2+a*e))^(1/2)-1/e*x^2/
a^2*(A*b-B*a)/((x^2+a/b)*b*e*x)^(1/2)+(b/a^2/e*A+1/2/a^2*(A*b-B*a)/e)*(-a*b)^(1/2)/b*((x+(-a*b)^(1/2)/b)/(-a*b
)^(1/2)*b)^(1/2)*(-2*(x-(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2)*(-x/(-a*b)^(1/2)*b)^(1/2)/(b*e*x^3+a*e*x)^(1/2)*
(-2*(-a*b)^(1/2)/b*EllipticE(((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2),1/2*2^(1/2))+(-a*b)^(1/2)/b*EllipticF((
(x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2),1/2*2^(1/2))))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.08 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.35 \[ \int \frac {A+B x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (B a b - 3 \, A b^{2}\right )} x^{3} + {\left (B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {b e} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) - {\left (2 \, A a b - {\left (B a b - 3 \, A b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {e x}}{a^{2} b^{2} e^{2} x^{3} + a^{3} b e^{2} x} \]

[In]

integrate((B*x^2+A)/(e*x)^(3/2)/(b*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

(((B*a*b - 3*A*b^2)*x^3 + (B*a^2 - 3*A*a*b)*x)*sqrt(b*e)*weierstrassZeta(-4*a/b, 0, weierstrassPInverse(-4*a/b
, 0, x)) - (2*A*a*b - (B*a*b - 3*A*b^2)*x^2)*sqrt(b*x^2 + a)*sqrt(e*x))/(a^2*b^2*e^2*x^3 + a^3*b*e^2*x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 12.75 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.29 \[ \int \frac {A+B x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/2}} \, dx=\frac {A \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {B x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} \]

[In]

integrate((B*x**2+A)/(e*x)**(3/2)/(b*x**2+a)**(3/2),x)

[Out]

A*gamma(-1/4)*hyper((-1/4, 3/2), (3/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(3/2)*e**(3/2)*sqrt(x)*gamma(3/4)) +
B*x**(3/2)*gamma(3/4)*hyper((3/4, 3/2), (7/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(3/2)*e**(3/2)*gamma(7/4))

Maxima [F]

\[ \int \frac {A+B x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate((B*x^2+A)/(e*x)^(3/2)/(b*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*(e*x)^(3/2)), x)

Giac [F]

\[ \int \frac {A+B x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate((B*x^2+A)/(e*x)^(3/2)/(b*x^2+a)^(3/2),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*(e*x)^(3/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {B\,x^2+A}{{\left (e\,x\right )}^{3/2}\,{\left (b\,x^2+a\right )}^{3/2}} \,d x \]

[In]

int((A + B*x^2)/((e*x)^(3/2)*(a + b*x^2)^(3/2)),x)

[Out]

int((A + B*x^2)/((e*x)^(3/2)*(a + b*x^2)^(3/2)), x)