\(\int \frac {A+B x^2}{(e x)^{3/2} (a+b x^2)^{5/2}} \, dx\) [331]

Optimal result

Integrand size = 26, antiderivative size = 377 \[ \int \frac {A+B x^2}{(e x)^{3/2} \left (a+b x^2\right )^{5/2}} \, dx=-\frac {2 A}{a e \sqrt {e x} \left (a+b x^2\right )^{3/2}}-\frac {(7 A b-a B) (e x)^{3/2}}{3 a^2 e^3 \left (a+b x^2\right )^{3/2}}-\frac {(7 A b-a B) (e x)^{3/2}}{2 a^3 e^3 \sqrt {a+b x^2}}+\frac {(7 A b-a B) \sqrt {e x} \sqrt {a+b x^2}}{2 a^3 \sqrt {b} e^2 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {(7 A b-a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{2 a^{11/4} b^{3/4} e^{3/2} \sqrt {a+b x^2}}+\frac {(7 A b-a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 a^{11/4} b^{3/4} e^{3/2} \sqrt {a+b x^2}} \] Output:

-2*A/a/e/(e*x)^(1/2)/(b*x^2+a)^(3/2)-1/3*(7*A*b-B*a)*(e*x)^(3/2)/a^2/e^3/( 
b*x^2+a)^(3/2)-1/2*(7*A*b-B*a)*(e*x)^(3/2)/a^3/e^3/(b*x^2+a)^(1/2)+1/2*(7* 
A*b-B*a)*(e*x)^(1/2)*(b*x^2+a)^(1/2)/a^3/b^(1/2)/e^2/(a^(1/2)+b^(1/2)*x)-1 
/2*(7*A*b-B*a)*(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^(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^(11/4)/b^(3/4)/e^(3/2)/(b*x^2+a)^(1/2)+1/4*(7*A*b-B*a)*(a^(1/2)+b^(1/2) 
*x)*((b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2)*InverseJacobiAM(2*arctan(b^(1/ 
4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)),1/2*2^(1/2))/a^(11/4)/b^(3/4)/e^(3/2)/(b*x 
^2+a)^(1/2)
 

Mathematica [C] (verified)

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

Time = 10.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.23 \[ \int \frac {A+B x^2}{(e x)^{3/2} \left (a+b x^2\right )^{5/2}} \, dx=\frac {x \left (-6 a^2 A+2 (-7 A b+a B) x^2 \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{2},\frac {7}{4},-\frac {b x^2}{a}\right )\right )}{3 a^3 (e x)^{3/2} \left (a+b x^2\right )^{3/2}} \] Input:

Integrate[(A + B*x^2)/((e*x)^(3/2)*(a + b*x^2)^(5/2)),x]
 

Output:

(x*(-6*a^2*A + 2*(-7*A*b + a*B)*x^2*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Hyperg 
eometric2F1[3/4, 5/2, 7/4, -((b*x^2)/a)]))/(3*a^3*(e*x)^(3/2)*(a + b*x^2)^ 
(3/2))
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {359, 253, 253, 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.

\(\displaystyle \int \frac {A+B x^2}{(e x)^{3/2} \left (a+b x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 359

\(\displaystyle -\frac {(7 A b-a B) \int \frac {\sqrt {e x}}{\left (b x^2+a\right )^{5/2}}dx}{a e^2}-\frac {2 A}{a e \sqrt {e x} \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 253

\(\displaystyle -\frac {(7 A b-a B) \left (\frac {\int \frac {\sqrt {e x}}{\left (b x^2+a\right )^{3/2}}dx}{2 a}+\frac {(e x)^{3/2}}{3 a e \left (a+b x^2\right )^{3/2}}\right )}{a e^2}-\frac {2 A}{a e \sqrt {e x} \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 253

\(\displaystyle -\frac {(7 A b-a B) \left (\frac {\frac {(e x)^{3/2}}{a e \sqrt {a+b x^2}}-\frac {\int \frac {\sqrt {e x}}{\sqrt {b x^2+a}}dx}{2 a}}{2 a}+\frac {(e x)^{3/2}}{3 a e \left (a+b x^2\right )^{3/2}}\right )}{a e^2}-\frac {2 A}{a e \sqrt {e x} \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 266

\(\displaystyle -\frac {(7 A b-a B) \left (\frac {\frac {(e x)^{3/2}}{a e \sqrt {a+b x^2}}-\frac {\int \frac {e x}{\sqrt {b x^2+a}}d\sqrt {e x}}{a e}}{2 a}+\frac {(e x)^{3/2}}{3 a e \left (a+b x^2\right )^{3/2}}\right )}{a e^2}-\frac {2 A}{a e \sqrt {e x} \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 834

\(\displaystyle -\frac {(7 A b-a B) \left (\frac {\frac {(e x)^{3/2}}{a e \sqrt {a+b x^2}}-\frac {\frac {\sqrt {a} e \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}-\frac {\sqrt {a} e \int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {a} e \sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}}{a e}}{2 a}+\frac {(e x)^{3/2}}{3 a e \left (a+b x^2\right )^{3/2}}\right )}{a e^2}-\frac {2 A}{a e \sqrt {e x} \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {(7 A b-a B) \left (\frac {\frac {(e x)^{3/2}}{a e \sqrt {a+b x^2}}-\frac {\frac {\sqrt {a} e \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}-\frac {\int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}}{a e}}{2 a}+\frac {(e x)^{3/2}}{3 a e \left (a+b x^2\right )^{3/2}}\right )}{a e^2}-\frac {2 A}{a e \sqrt {e x} \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 761

\(\displaystyle -\frac {(7 A b-a B) \left (\frac {\frac {(e x)^{3/2}}{a e \sqrt {a+b x^2}}-\frac {\frac {\sqrt [4]{a} \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^2}}-\frac {\int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}}{a e}}{2 a}+\frac {(e x)^{3/2}}{3 a e \left (a+b x^2\right )^{3/2}}\right )}{a e^2}-\frac {2 A}{a e \sqrt {e x} \left (a+b x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1510

\(\displaystyle -\frac {(7 A b-a B) \left (\frac {\frac {(e x)^{3/2}}{a e \sqrt {a+b x^2}}-\frac {\frac {\sqrt [4]{a} \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^2}}-\frac {\frac {\sqrt [4]{a} \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {e^2 \sqrt {e x} \sqrt {a+b x^2}}{\sqrt {a} e+\sqrt {b} e x}}{\sqrt {b}}}{a e}}{2 a}+\frac {(e x)^{3/2}}{3 a e \left (a+b x^2\right )^{3/2}}\right )}{a e^2}-\frac {2 A}{a e \sqrt {e x} \left (a+b x^2\right )^{3/2}}\)

Input:

Int[(A + B*x^2)/((e*x)^(3/2)*(a + b*x^2)^(5/2)),x]
 

Output:

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

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 253

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x 
)^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 
2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m 
}, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, 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]
 

Maple [A] (verified)

Time = 2.73 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.87

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (-\frac {x \left (A b -B a \right ) \sqrt {b e \,x^{3}+a e x}}{3 a^{2} e^{2} b^{2} \left (x^{2}+\frac {a}{b}\right )^{2}}-\frac {x^{2} \left (3 A b -B a \right )}{2 e \,a^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}-\frac {2 \left (b e \,x^{2}+a e \right ) A}{a^{3} e^{2} \sqrt {x \left (b e \,x^{2}+a e \right )}}+\frac {\left (\frac {3 A b -B a}{4 a^{3} e}+\frac {b A}{a^{3} e}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(327\)
risch	\(-\frac {2 A \sqrt {b \,x^{2}+a}}{a^{3} 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 {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\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}\, \operatorname {EllipticF}\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^{2} \left (A b -B a \right ) \left (\frac {x \sqrt {b e \,x^{3}+a e x}}{3 a e \,b^{2} \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {x^{2}}{2 a^{2} \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 {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\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}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{4 a^{2} b \sqrt {b e \,x^{3}+a e x}}\right )-a b A \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 {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\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}\, \operatorname {EllipticF}\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^{3} e \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(643\)
default	\(\frac {42 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a \,b^{2} x^{2}-21 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a \,b^{2} x^{2}-6 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b \,x^{2}+3 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b \,x^{2}+42 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b -21 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b -6 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{3}+3 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{3}-42 A \,x^{4} b^{3}+6 B \,x^{4} a \,b^{2}-70 a A \,b^{2} x^{2}+10 B \,a^{2} b \,x^{2}-24 a^{2} b A}{12 b \,a^{3} e \sqrt {e x}\, \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\)	\(771\)

Input:

int((B*x^2+A)/(e*x)^(3/2)/(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.47 \[ \int \frac {A+B x^2}{(e x)^{3/2} \left (a+b x^2\right )^{5/2}} \, dx=\frac {3 \, {\left ({\left (B a b^{2} - 7 \, A b^{3}\right )} x^{5} + 2 \, {\left (B a^{2} b - 7 \, A a b^{2}\right )} x^{3} + {\left (B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt {b e} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (3 \, {\left (B a b^{2} - 7 \, A b^{3}\right )} x^{4} - 12 \, A a^{2} b + 5 \, {\left (B a^{2} b - 7 \, A a b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {e x}}{6 \, {\left (a^{3} b^{3} e^{2} x^{5} + 2 \, a^{4} b^{2} e^{2} x^{3} + a^{5} b e^{2} x\right )}} \] Input:

integrate((B*x^2+A)/(e*x)^(3/2)/(b*x^2+a)^(5/2),x, algorithm="fricas")
 

Output:

1/6*(3*((B*a*b^2 - 7*A*b^3)*x^5 + 2*(B*a^2*b - 7*A*a*b^2)*x^3 + (B*a^3 - 7 
*A*a^2*b)*x)*sqrt(b*e)*weierstrassZeta(-4*a/b, 0, weierstrassPInverse(-4*a 
/b, 0, x)) + (3*(B*a*b^2 - 7*A*b^3)*x^4 - 12*A*a^2*b + 5*(B*a^2*b - 7*A*a* 
b^2)*x^2)*sqrt(b*x^2 + a)*sqrt(e*x))/(a^3*b^3*e^2*x^5 + 2*a^4*b^2*e^2*x^3 
+ a^5*b*e^2*x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 72.83 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.26 \[ \int \frac {A+B x^2}{(e x)^{3/2} \left (a+b x^2\right )^{5/2}} \, dx=\frac {A \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {5}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {B x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} e^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} \] Input:

integrate((B*x**2+A)/(e*x)**(3/2)/(b*x**2+a)**(5/2),x)
 

Output:

A*gamma(-1/4)*hyper((-1/4, 5/2), (3/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**( 
5/2)*e**(3/2)*sqrt(x)*gamma(3/4)) + B*x**(3/2)*gamma(3/4)*hyper((3/4, 5/2) 
, (7/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(5/2)*e**(3/2)*gamma(7/4))
 

Maxima [F]

\[ \int \frac {A+B x^2}{(e x)^{3/2} \left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((B*x^2+A)/(e*x)^(3/2)/(b*x^2+a)^(5/2),x, algorithm="maxima")
 

Output:

integrate((B*x^2 + A)/((b*x^2 + a)^(5/2)*(e*x)^(3/2)), x)
 

Giac [F]

\[ \int \frac {A+B x^2}{(e x)^{3/2} \left (a+b x^2\right )^{5/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((B*x^2+A)/(e*x)^(3/2)/(b*x^2+a)^(5/2),x, algorithm="giac")
 

Output:

integrate((B*x^2 + A)/((b*x^2 + a)^(5/2)*(e*x)^(3/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2}{(e x)^{3/2} \left (a+b x^2\right )^{5/2}} \, dx=\int \frac {B\,x^2+A}{{\left (e\,x\right )}^{3/2}\,{\left (b\,x^2+a\right )}^{5/2}} \,d x \] Input:

int((A + B*x^2)/((e*x)^(3/2)*(a + b*x^2)^(5/2)),x)
 

Output:

int((A + B*x^2)/((e*x)^(3/2)*(a + b*x^2)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {A+B x^2}{(e x)^{3/2} \left (a+b x^2\right )^{5/2}} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{2} x^{6}+2 a b \,x^{4}+a^{2} x^{2}}d x \right )}{e^{2}} \] Input:

int((B*x^2+A)/(e*x)^(3/2)/(b*x^2+a)^(5/2),x)
 

Output:

(sqrt(e)*int((sqrt(x)*sqrt(a + b*x**2))/(a**2*x**2 + 2*a*b*x**4 + b**2*x** 
6),x))/e**2