\(\int \frac {\sqrt {a+b x^2} (A+B x^2)}{(e x)^{7/2}} \, dx\) [296]

Optimal result

Integrand size = 26, antiderivative size = 338 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{(e x)^{7/2}} \, dx=-\frac {2 (A b+5 a B) \sqrt {a+b x^2}}{5 a e^3 \sqrt {e x}}+\frac {4 \sqrt {b} (A b+5 a B) \sqrt {e x} \sqrt {a+b x^2}}{5 a e^4 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 A \left (a+b x^2\right )^{3/2}}{5 a e (e x)^{5/2}}-\frac {4 \sqrt [4]{b} (A b+5 a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 a^{3/4} e^{7/2} \sqrt {a+b x^2}}+\frac {2 \sqrt [4]{b} (A b+5 a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{5 a^{3/4} e^{7/2} \sqrt {a+b x^2}} \] Output:

-2/5*(A*b+5*B*a)*(b*x^2+a)^(1/2)/a/e^3/(e*x)^(1/2)+4/5*b^(1/2)*(A*b+5*B*a) 
*(e*x)^(1/2)*(b*x^2+a)^(1/2)/a/e^4/(a^(1/2)+b^(1/2)*x)-2/5*A*(b*x^2+a)^(3/ 
2)/a/e/(e*x)^(5/2)-4/5*b^(1/4)*(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^(3/4)/e^(7/2)/(b*x^2+a)^(1/2)+2/5*b^(1/4)*( 
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)*Inv 
erseJacobiAM(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)),1/2*2^(1/2))/a^ 
(3/4)/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 = 95, normalized size of antiderivative = 0.28 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{(e x)^{7/2}} \, dx=-\frac {2 x \sqrt {a+b x^2} \left (A \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}}+(A b+5 a B) x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{4},\frac {3}{4},-\frac {b x^2}{a}\right )\right )}{5 a (e x)^{7/2} \sqrt {1+\frac {b x^2}{a}}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {359, 247, 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.

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

\(\Big \downarrow \) 359

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

\(\Big \downarrow \) 247

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

\(\Big \downarrow \) 266

\(\displaystyle \frac {(5 a B+A b) \left (\frac {4 b \int \frac {e x}{\sqrt {b x^2+a}}d\sqrt {e x}}{e^3}-\frac {2 \sqrt {a+b x^2}}{e \sqrt {e x}}\right )}{5 a e^2}-\frac {2 A \left (a+b x^2\right )^{3/2}}{5 a e (e x)^{5/2}}\)

\(\Big \downarrow \) 834

\(\displaystyle \frac {(5 a B+A b) \left (\frac {4 b \left (\frac {\sqrt {a} e \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}-\frac {\sqrt {a} e \int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {a} e \sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}\right )}{e^3}-\frac {2 \sqrt {a+b x^2}}{e \sqrt {e x}}\right )}{5 a e^2}-\frac {2 A \left (a+b x^2\right )^{3/2}}{5 a e (e x)^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(5 a B+A b) \left (\frac {4 b \left (\frac {\sqrt {a} e \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}-\frac {\int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}\right )}{e^3}-\frac {2 \sqrt {a+b x^2}}{e \sqrt {e x}}\right )}{5 a e^2}-\frac {2 A \left (a+b x^2\right )^{3/2}}{5 a e (e x)^{5/2}}\)

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 1510

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

Input:

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

Output:

(-2*A*(a + b*x^2)^(3/2))/(5*a*e*(e*x)^(5/2)) + ((A*b + 5*a*B)*((-2*Sqrt[a 
+ b*x^2])/(e*Sqrt[e*x]) + (4*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]*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)/(Sqr 
t[a]*e + Sqrt[b]*e*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*S 
qrt[e])], 1/2])/(2*b^(3/4)*Sqrt[a + b*x^2])))/e^3))/(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 247

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1)))   Int[ 
(c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 
0] && LtQ[m, -1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && 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 = 0.73 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.72

method	result	size
risch	\(-\frac {2 \sqrt {b \,x^{2}+a}\, \left (2 A b \,x^{2}+5 B a \,x^{2}+A a \right )}{5 x^{2} a \,e^{3} \sqrt {e x}}+\frac {2 \left (A b +5 B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right ) \sqrt {\left (b \,x^{2}+a \right ) e x}}{5 a \sqrt {b e \,x^{3}+a e x}\, e^{3} \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(242\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (-\frac {2 A \sqrt {b e \,x^{3}+a e x}}{5 e^{4} x^{3}}-\frac {2 \left (b e \,x^{2}+a e \right ) \left (2 A b +5 B a \right )}{5 e^{4} a \sqrt {x \left (b e \,x^{2}+a e \right )}}+\frac {\left (\frac {B b}{e^{3}}+\frac {b \left (2 A b +5 B a \right )}{5 a \,e^{3}}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {b \,x^{2}+a}}\)	\(276\)
default	\(\frac {\frac {4 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,x^{2}}{5}-\frac {2 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,x^{2}}{5}+4 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} x^{2}-2 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} x^{2}-\frac {4 A \,b^{2} x^{4}}{5}-2 B a b \,x^{4}-\frac {6 a A b \,x^{2}}{5}-2 B \,a^{2} x^{2}-\frac {2 a^{2} A}{5}}{x^{2} \sqrt {b \,x^{2}+a}\, e^{3} \sqrt {e x}\, a}\)	\(417\)

Input:

int((b*x^2+a)^(1/2)*(B*x^2+A)/(e*x)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-2/5*(b*x^2+a)^(1/2)*(2*A*b*x^2+5*B*a*x^2+A*a)/x^2/a/e^3/(e*x)^(1/2)+2/5*( 
A*b+5*B*a)/a*(-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)))/e^3*((b*x^2+a)*e*x)^(1/2)/(e*x)^(1 
/2)/(b*x^2+a)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.23 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{(e x)^{7/2}} \, dx=-\frac {2 \, {\left (2 \, {\left (5 \, B a + A b\right )} \sqrt {b e} x^{3} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left ({\left (5 \, B a + 2 \, A b\right )} x^{2} + A a\right )} \sqrt {b x^{2} + a} \sqrt {e x}\right )}}{5 \, a e^{4} x^{3}} \] Input:

integrate((b*x^2+a)^(1/2)*(B*x^2+A)/(e*x)^(7/2),x, algorithm="fricas")
 

Output:

-2/5*(2*(5*B*a + A*b)*sqrt(b*e)*x^3*weierstrassZeta(-4*a/b, 0, weierstrass 
PInverse(-4*a/b, 0, x)) + ((5*B*a + 2*A*b)*x^2 + A*a)*sqrt(b*x^2 + a)*sqrt 
(e*x))/(a*e^4*x^3)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 11.57 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.32 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x^2\right )}{(e x)^{7/2}} \, dx=\frac {A \sqrt {a} \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {1}{2} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {7}{2}} x^{\frac {5}{2}} \Gamma \left (- \frac {1}{4}\right )} + \frac {B \sqrt {a} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {7}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} \] Input:

integrate((b*x**2+a)**(1/2)*(B*x**2+A)/(e*x)**(7/2),x)
 

Output:

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

Maxima [F]

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

integrate((b*x^2+a)^(1/2)*(B*x^2+A)/(e*x)^(7/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((b*x^2+a)^(1/2)*(B*x^2+A)/(e*x)^(7/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(((A + B*x^2)*(a + b*x^2)^(1/2))/(e*x)^(7/2),x)
 

Output:

int(((A + B*x^2)*(a + b*x^2)^(1/2))/(e*x)^(7/2), x)
 

Reduce [F]

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

int((b*x^2+a)^(1/2)*(B*x^2+A)/(e*x)^(7/2),x)
 

Output:

(2*sqrt(e)*( - sqrt(a + b*x**2)*a + sqrt(a + b*x**2)*b*x**2 - 2*sqrt(x)*in 
t((sqrt(x)*sqrt(a + b*x**2))/(a*x**4 + b*x**6),x)*a**2*x**2))/(sqrt(x)*e** 
4*x**2)