\(\int \frac {A+B x^2}{(e x)^{7/2} (a+b x^2)^{3/2}} \, dx\) [323]

Optimal result

Integrand size = 26, antiderivative size = 379 \[ \int \frac {A+B x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/2}} \, dx=-\frac {2 A}{5 a e (e x)^{5/2} \sqrt {a+b x^2}}-\frac {7 A b-5 a B}{5 a^2 e^3 \sqrt {e x} \sqrt {a+b x^2}}+\frac {3 (7 A b-5 a B) \sqrt {a+b x^2}}{5 a^3 e^3 \sqrt {e x}}-\frac {3 \sqrt {b} (7 A b-5 a B) \sqrt {e x} \sqrt {a+b x^2}}{5 a^3 e^4 \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {3 \sqrt [4]{b} (7 A b-5 a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 a^{11/4} e^{7/2} \sqrt {a+b x^2}}-\frac {3 \sqrt [4]{b} (7 A b-5 a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{10 a^{11/4} e^{7/2} \sqrt {a+b x^2}} \] Output:

-2/5*A/a/e/(e*x)^(5/2)/(b*x^2+a)^(1/2)-1/5*(7*A*b-5*B*a)/a^2/e^3/(e*x)^(1/ 
2)/(b*x^2+a)^(1/2)+3/5*(7*A*b-5*B*a)*(b*x^2+a)^(1/2)/a^3/e^3/(e*x)^(1/2)-3 
/5*b^(1/2)*(7*A*b-5*B*a)*(e*x)^(1/2)*(b*x^2+a)^(1/2)/a^3/e^4/(a^(1/2)+b^(1 
/2)*x)+3/5*b^(1/4)*(7*A*b-5*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)/e^(7/2)/(b*x^2+a)^(1/2)-3/10*b^(1/4)*(7*A*b-5* 
B*a)*(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2)*InverseJa 
cobiAM(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)),1/2*2^(1/2))/a^(11/4) 
/e^(7/2)/(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 = 10.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.21 \[ \int \frac {A+B x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/2}} \, dx=\frac {x \left (-2 a A+2 (7 A b-5 a B) x^2 \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {3}{2},\frac {3}{4},-\frac {b x^2}{a}\right )\right )}{5 a^2 (e x)^{7/2} \sqrt {a+b x^2}} \] Input:

Integrate[(A + B*x^2)/((e*x)^(7/2)*(a + b*x^2)^(3/2)),x]
 

Output:

(x*(-2*a*A + 2*(7*A*b - 5*a*B)*x^2*Sqrt[1 + (b*x^2)/a]*Hypergeometric2F1[- 
1/4, 3/2, 3/4, -((b*x^2)/a)]))/(5*a^2*(e*x)^(7/2)*Sqrt[a + b*x^2])
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 391, 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, 264, 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.

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

\(\Big \downarrow \) 359

\(\displaystyle -\frac {(7 A b-5 a B) \int \frac {1}{(e x)^{3/2} \left (b x^2+a\right )^{3/2}}dx}{5 a e^2}-\frac {2 A}{5 a e (e x)^{5/2} \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 253

\(\displaystyle -\frac {(7 A b-5 a B) \left (\frac {3 \int \frac {1}{(e x)^{3/2} \sqrt {b x^2+a}}dx}{2 a}+\frac {1}{a e \sqrt {e x} \sqrt {a+b x^2}}\right )}{5 a e^2}-\frac {2 A}{5 a e (e x)^{5/2} \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 264

\(\displaystyle -\frac {(7 A b-5 a B) \left (\frac {3 \left (\frac {b \int \frac {\sqrt {e x}}{\sqrt {b x^2+a}}dx}{a e^2}-\frac {2 \sqrt {a+b x^2}}{a e \sqrt {e x}}\right )}{2 a}+\frac {1}{a e \sqrt {e x} \sqrt {a+b x^2}}\right )}{5 a e^2}-\frac {2 A}{5 a e (e x)^{5/2} \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 266

\(\displaystyle -\frac {(7 A b-5 a B) \left (\frac {3 \left (\frac {2 b \int \frac {e x}{\sqrt {b x^2+a}}d\sqrt {e x}}{a e^3}-\frac {2 \sqrt {a+b x^2}}{a e \sqrt {e x}}\right )}{2 a}+\frac {1}{a e \sqrt {e x} \sqrt {a+b x^2}}\right )}{5 a e^2}-\frac {2 A}{5 a e (e x)^{5/2} \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 834

\(\displaystyle -\frac {(7 A b-5 a B) \left (\frac {3 \left (\frac {2 b \left (\frac {\sqrt {a} e \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}-\frac {\sqrt {a} e \int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {a} e \sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}\right )}{a e^3}-\frac {2 \sqrt {a+b x^2}}{a e \sqrt {e x}}\right )}{2 a}+\frac {1}{a e \sqrt {e x} \sqrt {a+b x^2}}\right )}{5 a e^2}-\frac {2 A}{5 a e (e x)^{5/2} \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {(7 A b-5 a B) \left (\frac {3 \left (\frac {2 b \left (\frac {\sqrt {a} e \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}-\frac {\int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}\right )}{a e^3}-\frac {2 \sqrt {a+b x^2}}{a e \sqrt {e x}}\right )}{2 a}+\frac {1}{a e \sqrt {e x} \sqrt {a+b x^2}}\right )}{5 a e^2}-\frac {2 A}{5 a e (e x)^{5/2} \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 761

\(\displaystyle -\frac {(7 A b-5 a B) \left (\frac {3 \left (\frac {2 b \left (\frac {\sqrt [4]{a} \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^2}}-\frac {\int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}\right )}{a e^3}-\frac {2 \sqrt {a+b x^2}}{a e \sqrt {e x}}\right )}{2 a}+\frac {1}{a e \sqrt {e x} \sqrt {a+b x^2}}\right )}{5 a e^2}-\frac {2 A}{5 a e (e x)^{5/2} \sqrt {a+b x^2}}\)

\(\Big \downarrow \) 1510

\(\displaystyle -\frac {(7 A b-5 a B) \left (\frac {3 \left (\frac {2 b \left (\frac {\sqrt [4]{a} \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^2}}-\frac {\frac {\sqrt [4]{a} \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {e^2 \sqrt {e x} \sqrt {a+b x^2}}{\sqrt {a} e+\sqrt {b} e x}}{\sqrt {b}}\right )}{a e^3}-\frac {2 \sqrt {a+b x^2}}{a e \sqrt {e x}}\right )}{2 a}+\frac {1}{a e \sqrt {e x} \sqrt {a+b x^2}}\right )}{5 a e^2}-\frac {2 A}{5 a e (e x)^{5/2} \sqrt {a+b x^2}}\)

Input:

Int[(A + B*x^2)/((e*x)^(7/2)*(a + b*x^2)^(3/2)),x]
 

Output:

(-2*A)/(5*a*e*(e*x)^(5/2)*Sqrt[a + b*x^2]) - ((7*A*b - 5*a*B)*(1/(a*e*Sqrt 
[e*x]*Sqrt[a + b*x^2]) + (3*((-2*Sqrt[a + b*x^2])/(a*e*Sqrt[e*x]) + (2*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 + S 
qrt[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]*Ellipti 
cF[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)))/(2*a)))/(5*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 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -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 = 2.09 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.85

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (-\frac {2 A \sqrt {b e \,x^{3}+a e x}}{5 e^{4} a^{2} x^{3}}+\frac {2 \left (b e \,x^{2}+a e \right ) \left (8 A b -5 B a \right )}{5 e^{4} a^{3} \sqrt {x \left (b e \,x^{2}+a e \right )}}+\frac {b \,x^{2} \left (A b -B a \right )}{e^{3} a^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {\left (-\frac {b \left (8 A b -5 B a \right )}{5 a^{3} e^{3}}-\frac {\left (A b -B a \right ) b}{2 a^{3} e^{3}}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(324\)
default	\(-\frac {42 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,x^{2}-21 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,x^{2}-30 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} x^{2}+15 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} x^{2}-42 A \,b^{2} x^{4}+30 B a b \,x^{4}-28 a A b \,x^{2}+20 B \,a^{2} x^{2}+4 a^{2} A}{10 x^{2} \sqrt {b \,x^{2}+a}\, e^{3} \sqrt {e x}\, a^{3}}\)	\(417\)
risch	\(-\frac {2 \sqrt {b \,x^{2}+a}\, \left (-8 A b \,x^{2}+5 B a \,x^{2}+A a \right )}{5 a^{3} x^{2} e^{3} \sqrt {e x}}-\frac {b^{2} \left (\frac {\left (8 A b -5 B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b^{2} \sqrt {b e \,x^{3}+a e x}}-\frac {5 a \left (A b -B a \right ) \left (\frac {x^{2}}{a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}-\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{2 a b \sqrt {b e \,x^{3}+a e x}}\right )}{b}\right ) \sqrt {\left (b \,x^{2}+a \right ) e x}}{5 a^{3} e^{3} \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(451\)

Input:

int((B*x^2+A)/(e*x)^(7/2)/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

((b*x^2+a)*e*x)^(1/2)/(e*x)^(1/2)/(b*x^2+a)^(1/2)*(-2/5/e^4/a^2*A*(b*e*x^3 
+a*e*x)^(1/2)/x^3+2/5*(b*e*x^2+a*e)/e^4/a^3*(8*A*b-5*B*a)/(x*(b*e*x^2+a*e) 
)^(1/2)+b/e^3*x^2/a^3*(A*b-B*a)/((x^2+a/b)*b*e*x)^(1/2)+(-1/5*b/a^3*(8*A*b 
-5*B*a)/e^3-1/2*(A*b-B*a)/a^3*b/e^3)/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)*El 
lipticF(((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 = 0.12 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.37 \[ \int \frac {A+B x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/2}} \, dx=-\frac {3 \, {\left ({\left (5 \, B a b - 7 \, A b^{2}\right )} x^{5} + {\left (5 \, B a^{2} - 7 \, A a b\right )} x^{3}\right )} \sqrt {b e} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (3 \, {\left (5 \, B a b - 7 \, A b^{2}\right )} x^{4} + 2 \, A a^{2} + 2 \, {\left (5 \, B a^{2} - 7 \, A a b\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {e x}}{5 \, {\left (a^{3} b e^{4} x^{5} + a^{4} e^{4} x^{3}\right )}} \] Input:

integrate((B*x^2+A)/(e*x)^(7/2)/(b*x^2+a)^(3/2),x, algorithm="fricas")
 

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 51.65 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.27 \[ \int \frac {A+B x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/2}} \, dx=\frac {A \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {3}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {7}{2}} x^{\frac {5}{2}} \Gamma \left (- \frac {1}{4}\right )} + \frac {B \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{2}} e^{\frac {7}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} \] Input:

integrate((B*x**2+A)/(e*x)**(7/2)/(b*x**2+a)**(3/2),x)
 

Output:

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

Maxima [F]

\[ \int \frac {A+B x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate((B*x^2+A)/(e*x)^(7/2)/(b*x^2+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*(e*x)^(7/2)), x)
 

Giac [F]

\[ \int \frac {A+B x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate((B*x^2+A)/(e*x)^(7/2)/(b*x^2+a)^(3/2),x, algorithm="giac")
 

Output:

integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*(e*x)^(7/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {B\,x^2+A}{{\left (e\,x\right )}^{7/2}\,{\left (b\,x^2+a\right )}^{3/2}} \,d x \] Input:

int((A + B*x^2)/((e*x)^(7/2)*(a + b*x^2)^(3/2)),x)
 

Output:

int((A + B*x^2)/((e*x)^(7/2)*(a + b*x^2)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {A+B x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/2}} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b \,x^{6}+a \,x^{4}}d x \right )}{e^{4}} \] Input:

int((B*x^2+A)/(e*x)^(7/2)/(b*x^2+a)^(3/2),x)
 

Output:

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