3.9.4 \(\int \frac {A+B x^2}{(e x)^{3/2} \sqrt {a+b x^2}} \, dx\) [804]

3.9.4.1 Optimal result

Integrand size = 26, antiderivative size = 290 \[ \int \frac {A+B x^2}{(e x)^{3/2} \sqrt {a+b x^2}} \, dx=-\frac {2 A \sqrt {a+b x^2}}{a e \sqrt {e x}}+\frac {2 (A b+a B) \sqrt {e x} \sqrt {a+b x^2}}{a \sqrt {b} e^2 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 (A b+a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{a^{3/4} b^{3/4} e^{3/2} \sqrt {a+b x^2}}+\frac {(A b+a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{a^{3/4} b^{3/4} e^{3/2} \sqrt {a+b x^2}} \]

output

-2*A*(b*x^2+a)^(1/2)/a/e/(e*x)^(1/2)+2*(A*b+B*a)*(e*x)^(1/2)*(b*x^2+a)^(1/ 
2)/a/e^2/b^(1/2)/(a^(1/2)+x*b^(1/2))-2*(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 
^(3/4)/b^(3/4)/e^(3/2)/(b*x^2+a)^(1/2)+(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 
^(3/4)/b^(3/4)/e^(3/2)/(b*x^2+a)^(1/2)
 

3.9.4.2 Mathematica [C] (verified)

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

Time = 10.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.28 \[ \int \frac {A+B x^2}{(e x)^{3/2} \sqrt {a+b x^2}} \, dx=\frac {x \left (-6 A \left (a+b x^2\right )+2 (A b+a B) x^2 \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {b x^2}{a}\right )\right )}{3 a (e x)^{3/2} \sqrt {a+b x^2}} \]

input

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

output

(x*(-6*A*(a + b*x^2) + 2*(A*b + a*B)*x^2*Sqrt[1 + (b*x^2)/a]*Hypergeometri 
c2F1[1/2, 3/4, 7/4, -((b*x^2)/a)]))/(3*a*(e*x)^(3/2)*Sqrt[a + b*x^2])
 

3.9.4.3 Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {359, 266, 834, 27, 761, 1510}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 359

\(\displaystyle \frac {(a B+A b) \int \frac {\sqrt {e x}}{\sqrt {b x^2+a}}dx}{a e^2}-\frac {2 A \sqrt {a+b x^2}}{a e \sqrt {e x}}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {2 (a B+A b) \int \frac {e x}{\sqrt {b x^2+a}}d\sqrt {e x}}{a e^3}-\frac {2 A \sqrt {a+b x^2}}{a e \sqrt {e x}}\)

\(\Big \downarrow \) 834

\(\displaystyle \frac {2 (a B+A b) \left (\frac {\sqrt {a} e \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}-\frac {\sqrt {a} e \int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {a} e \sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}\right )}{a e^3}-\frac {2 A \sqrt {a+b x^2}}{a e \sqrt {e x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (a B+A b) \left (\frac {\sqrt {a} e \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}-\frac {\int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}\right )}{a e^3}-\frac {2 A \sqrt {a+b x^2}}{a e \sqrt {e x}}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {2 (a B+A b) \left (\frac {\sqrt [4]{a} \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^2}}-\frac {\int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}\right )}{a e^3}-\frac {2 A \sqrt {a+b x^2}}{a e \sqrt {e x}}\)

\(\Big \downarrow \) 1510

\(\displaystyle \frac {2 (a B+A b) \left (\frac {\sqrt [4]{a} \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^2}}-\frac {\frac {\sqrt [4]{a} \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {e^2 \sqrt {e x} \sqrt {a+b x^2}}{\sqrt {a} e+\sqrt {b} e x}}{\sqrt {b}}\right )}{a e^3}-\frac {2 A \sqrt {a+b x^2}}{a e \sqrt {e x}}\)

input

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

output

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

3.9.4.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 359

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + 
Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1))   Int[(e*x)^(m + 2)* 
(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] 
&& LtQ[m, -1] &&  !ILtQ[p, -1]
 

rule 761

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 834

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

rule 1510

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]))*E 
llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e 
}, x] && PosQ[c/a]
 

3.9.4.4 Maple [A] (verified)

Time = 3.06 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.77

method	result	size
risch	\(-\frac {2 A \sqrt {b \,x^{2}+a}}{a e \sqrt {e x}}+\frac {\left (A b +B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right ) \sqrt {\left (b \,x^{2}+a \right ) e x}}{a b \sqrt {b e \,x^{3}+a e x}\, e \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(223\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (-\frac {2 \left (b e \,x^{2}+a e \right ) A}{e^{2} a \sqrt {x \left (b e \,x^{2}+a e \right )}}+\frac {\left (\frac {B}{e}+\frac {b A}{a e}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(236\)
default	\(\frac {2 A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a b -A \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a b +2 B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2}-B \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, a^{2}-2 A \,b^{2} x^{2}-2 a b A}{\sqrt {b \,x^{2}+a}\, b e \sqrt {e x}\, a}\)	\(378\)

input

int((B*x^2+A)/(e*x)^(3/2)/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

output

-2*A*(b*x^2+a)^(1/2)/a/e/(e*x)^(1/2)+(A*b+B*a)/a*(-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*Elli 
pticE(((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)))/e*((b* 
x^2+a)*e*x)^(1/2)/(e*x)^(1/2)/(b*x^2+a)^(1/2)
 

3.9.4.5 Fricas [C] (verification not implemented)

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

Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.22 \[ \int \frac {A+B x^2}{(e x)^{3/2} \sqrt {a+b x^2}} \, dx=-\frac {2 \, {\left ({\left (B a + A b\right )} \sqrt {b e} x {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + \sqrt {b x^{2} + a} \sqrt {e x} A b\right )}}{a b e^{2} x} \]

input

integrate((B*x^2+A)/(e*x)^(3/2)/(b*x^2+a)^(1/2),x, algorithm="fricas")
 

output

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

3.9.4.6 Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.43 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.33 \[ \int \frac {A+B x^2}{(e x)^{3/2} \sqrt {a+b x^2}} \, dx=\frac {A \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {B x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} e^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} \]

input

integrate((B*x**2+A)/(e*x)**(3/2)/(b*x**2+a)**(1/2),x)
 

output

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

3.9.4.7 Maxima [F]

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

input

integrate((B*x^2+A)/(e*x)^(3/2)/(b*x^2+a)^(1/2),x, algorithm="maxima")
 

output

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

3.9.4.8 Giac [F]

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

input

integrate((B*x^2+A)/(e*x)^(3/2)/(b*x^2+a)^(1/2),x, algorithm="giac")
 

output

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

3.9.4.9 Mupad [F(-1)]

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

input

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

output

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