\(\int \frac {\sqrt {e x} (A+B x^2)}{(a+b x^2)^{5/2}} \, dx\) [330]

Optimal result

Integrand size = 26, antiderivative size = 344 \[ \int \frac {\sqrt {e x} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {(A b-a B) (e x)^{3/2}}{3 a b e \left (a+b x^2\right )^{3/2}}+\frac {(A b+a B) (e x)^{3/2}}{2 a^2 b e \sqrt {a+b x^2}}-\frac {(A b+a B) \sqrt {e x} \sqrt {a+b x^2}}{2 a^2 b^{3/2} \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {(A b+a B) \sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{2 a^{7/4} b^{7/4} \sqrt {a+b x^2}}-\frac {(A b+a B) \sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 a^{7/4} b^{7/4} \sqrt {a+b x^2}} \] Output:

1/3*(A*b-B*a)*(e*x)^(3/2)/a/b/e/(b*x^2+a)^(3/2)+1/2*(A*b+B*a)*(e*x)^(3/2)/ 
a^2/b/e/(b*x^2+a)^(1/2)-1/2*(A*b+B*a)*(e*x)^(1/2)*(b*x^2+a)^(1/2)/a^2/b^(3 
/2)/(a^(1/2)+b^(1/2)*x)+1/2*(A*b+B*a)*e^(1/2)*(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^(7/4)/b^(7/4)/(b*x^2+a)^(1/2)-1/4*(A*b+B 
*a)*e^(1/2)*(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2)*In 
verseJacobiAM(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)),1/2*2^(1/2))/a 
^(7/4)/b^(7/4)/(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 = 10.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.24 \[ \int \frac {\sqrt {e x} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {2 x \sqrt {e x} \left (-a^2 B+(A b+a B) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{2},\frac {7}{4},-\frac {b x^2}{a}\right )\right )}{3 a^2 b \left (a+b x^2\right )^{3/2}} \] Input:

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

Output:

(2*x*Sqrt[e*x]*(-(a^2*B) + (A*b + a*B)*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Hyp 
ergeometric2F1[3/4, 5/2, 7/4, -((b*x^2)/a)]))/(3*a^2*b*(a + b*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {362, 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.

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

\(\Big \downarrow \) 362

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

\(\Big \downarrow \) 253

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 834

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 1510

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

Input:

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

Output:

((A*b - a*B)*(e*x)^(3/2))/(3*a*b*e*(a + b*x^2)^(3/2)) + ((A*b + a*B)*((e*x 
)^(3/2)/(a*e*Sqrt[a + b*x^2]) - (-((-((e^2*Sqrt[e*x]*Sqrt[a + b*x^2])/(Sqr 
t[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)/(Sq 
rt[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*b)
 

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 362

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*b*e 
*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(2*a*b*(p + 1))   I 
nt[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && N 
eQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || 
  !RationalQ[m] || (ILtQ[p + 1/2, 0] && LeQ[-1, m, -2*(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 = 1.35 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.82

method	result	size
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (\frac {x \left (A b -B a \right ) \sqrt {b e \,x^{3}+a e x}}{3 a \,b^{3} \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {e \,x^{2} \left (A b +B a \right )}{2 b \,a^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}-\frac {e \left (A b +B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{4 b^{2} a^{2} \sqrt {b e \,x^{3}+a e x}}\right )}{e x \sqrt {b \,x^{2}+a}}\)	\(281\)
default	\(-\frac {\left (6 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}-3 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}+6 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 -3 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}-6 A \,x^{4} b^{3}-6 B \,x^{4} a \,b^{2}-10 a A \,b^{2} x^{2}-2 B \,a^{2} b \,x^{2}\right ) \sqrt {e x}}{12 b^{2} a^{2} x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\)	\(764\)

Input:

int((e*x)^(1/2)*(B*x^2+A)/(b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/e/x*(e*x)^(1/2)/(b*x^2+a)^(1/2)*((b*x^2+a)*e*x)^(1/2)*(1/3/a/b^3*x*(A*b- 
B*a)*(b*e*x^3+a*e*x)^(1/2)/(x^2+a/b)^2+1/2/b*e*x^2/a^2*(A*b+B*a)/((x^2+a/b 
)*b*e*x)^(1/2)-1/4/b^2/a^2*e*(A*b+B*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)*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 = 0.08 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.44 \[ \int \frac {\sqrt {e x} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {3 \, {\left ({\left (B a b^{2} + A b^{3}\right )} x^{4} + B a^{3} + A a^{2} b + 2 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2}\right )} \sqrt {b e} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (3 \, {\left (B a b^{2} + A b^{3}\right )} x^{3} + {\left (B a^{2} b + 5 \, A a b^{2}\right )} x\right )} \sqrt {b x^{2} + a} \sqrt {e x}}{6 \, {\left (a^{2} b^{4} x^{4} + 2 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )}} \] Input:

integrate((e*x)^(1/2)*(B*x^2+A)/(b*x^2+a)^(5/2),x, algorithm="fricas")
 

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 32.52 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.27 \[ \int \frac {\sqrt {e x} \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx=\frac {A \sqrt {e} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {B \sqrt {e} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, \frac {5}{2} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} \Gamma \left (\frac {11}{4}\right )} \] Input:

integrate((e*x)**(1/2)*(B*x**2+A)/(b*x**2+a)**(5/2),x)
 

Output:

A*sqrt(e)*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)*gamma(7/4)) + B*sqrt(e)*x**(7/2)*gamma(7/4)*hyper((7/4 
, 5/2), (11/4,), b*x**2*exp_polar(I*pi)/a)/(2*a**(5/2)*gamma(11/4))
 

Maxima [F]

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

integrate((e*x)^(1/2)*(B*x^2+A)/(b*x^2+a)^(5/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((e*x)^(1/2)*(B*x^2+A)/(b*x^2+a)^(5/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(((A + B*x^2)*(e*x)^(1/2))/(a + b*x^2)^(5/2),x)
 

Output:

int(((A + B*x^2)*(e*x)^(1/2))/(a + b*x^2)^(5/2), x)
 

Reduce [F]

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

int((e*x)^(1/2)*(B*x^2+A)/(b*x^2+a)^(5/2),x)
 

Output:

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