\(\int \frac {A+B x}{(d x)^{5/2} \sqrt {a+b x^2}} \, dx\) [1369]

Optimal result

Integrand size = 24, antiderivative size = 319 \[ \int \frac {A+B x}{(d x)^{5/2} \sqrt {a+b x^2}} \, dx=-\frac {2 A \sqrt {a+b x^2}}{3 a d (d x)^{3/2}}-\frac {2 B \sqrt {a+b x^2}}{a d^2 \sqrt {d x}}+\frac {2 \sqrt {b} B \sqrt {d x} \sqrt {a+b x^2}}{a d^3 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {2 \sqrt [4]{b} B \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )|\frac {1}{2}\right )}{a^{3/4} d^{5/2} \sqrt {a+b x^2}}-\frac {\sqrt [4]{b} \left (A \sqrt {b}-3 \sqrt {a} B\right ) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right ),\frac {1}{2}\right )}{3 a^{5/4} d^{5/2} \sqrt {a+b x^2}} \] Output:

-2/3*A*(b*x^2+a)^(1/2)/a/d/(d*x)^(3/2)-2*B*(b*x^2+a)^(1/2)/a/d^2/(d*x)^(1/ 
2)+2*b^(1/2)*B*(d*x)^(1/2)*(b*x^2+a)^(1/2)/a/d^3/(a^(1/2)+b^(1/2)*x)-2*b^( 
1/4)*B*(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2)*Ellipti 
cE(sin(2*arctan(b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))),1/2*2^(1/2))/a^(3/4) 
/d^(5/2)/(b*x^2+a)^(1/2)-1/3*b^(1/4)*(A*b^(1/2)-3*a^(1/2)*B)*(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)*(d*x)^(1/2)/a^(1/4)/d^(1/2)),1/2*2^(1/2))/a^(5/4)/d^(5/2)/(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 = 10.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.26 \[ \int \frac {A+B x}{(d x)^{5/2} \sqrt {a+b x^2}} \, dx=-\frac {2 x \sqrt {1+\frac {b x^2}{a}} \left (A \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{2},\frac {1}{4},-\frac {b x^2}{a}\right )+3 B x \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\frac {b x^2}{a}\right )\right )}{3 (d x)^{5/2} \sqrt {a+b x^2}} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.76 (sec) , antiderivative size = 314, normalized size of antiderivative = 0.98, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {553, 27, 553, 27, 556, 555, 1512, 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.

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

\(\Big \downarrow \) 553

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 553

\(\displaystyle \frac {-\frac {2 \int \frac {a b (A-3 B x)}{2 \sqrt {d x} \sqrt {b x^2+a}}dx}{a d}-\frac {6 B \sqrt {a+b x^2}}{d \sqrt {d x}}}{3 a d}-\frac {2 A \sqrt {a+b x^2}}{3 a d (d x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {b \int \frac {A-3 B x}{\sqrt {d x} \sqrt {b x^2+a}}dx}{d}-\frac {6 B \sqrt {a+b x^2}}{d \sqrt {d x}}}{3 a d}-\frac {2 A \sqrt {a+b x^2}}{3 a d (d x)^{3/2}}\)

\(\Big \downarrow \) 556

\(\displaystyle \frac {-\frac {b \sqrt {x} \int \frac {A-3 B x}{\sqrt {x} \sqrt {b x^2+a}}dx}{d \sqrt {d x}}-\frac {6 B \sqrt {a+b x^2}}{d \sqrt {d x}}}{3 a d}-\frac {2 A \sqrt {a+b x^2}}{3 a d (d x)^{3/2}}\)

\(\Big \downarrow \) 555

\(\displaystyle \frac {-\frac {2 b \sqrt {x} \int \frac {A-3 B x}{\sqrt {b x^2+a}}d\sqrt {x}}{d \sqrt {d x}}-\frac {6 B \sqrt {a+b x^2}}{d \sqrt {d x}}}{3 a d}-\frac {2 A \sqrt {a+b x^2}}{3 a d (d x)^{3/2}}\)

\(\Big \downarrow \) 1512

\(\displaystyle \frac {-\frac {2 b \sqrt {x} \left (\left (A-\frac {3 \sqrt {a} B}{\sqrt {b}}\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {x}+\frac {3 \sqrt {a} B \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {a} \sqrt {b x^2+a}}d\sqrt {x}}{\sqrt {b}}\right )}{d \sqrt {d x}}-\frac {6 B \sqrt {a+b x^2}}{d \sqrt {d x}}}{3 a d}-\frac {2 A \sqrt {a+b x^2}}{3 a d (d x)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {2 b \sqrt {x} \left (\left (A-\frac {3 \sqrt {a} B}{\sqrt {b}}\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {x}+\frac {3 B \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {b x^2+a}}d\sqrt {x}}{\sqrt {b}}\right )}{d \sqrt {d x}}-\frac {6 B \sqrt {a+b x^2}}{d \sqrt {d x}}}{3 a d}-\frac {2 A \sqrt {a+b x^2}}{3 a d (d x)^{3/2}}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {-\frac {2 b \sqrt {x} \left (\frac {3 B \int \frac {\sqrt {a}-\sqrt {b} x}{\sqrt {b x^2+a}}d\sqrt {x}}{\sqrt {b}}+\frac {\left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \left (A-\frac {3 \sqrt {a} B}{\sqrt {b}}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a+b x^2}}\right )}{d \sqrt {d x}}-\frac {6 B \sqrt {a+b x^2}}{d \sqrt {d x}}}{3 a d}-\frac {2 A \sqrt {a+b x^2}}{3 a d (d x)^{3/2}}\)

\(\Big \downarrow \) 1510

\(\displaystyle \frac {-\frac {2 b \sqrt {x} \left (\frac {\left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \left (A-\frac {3 \sqrt {a} B}{\sqrt {b}}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a+b x^2}}+\frac {3 B \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {\sqrt {x} \sqrt {a+b x^2}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {b}}\right )}{d \sqrt {d x}}-\frac {6 B \sqrt {a+b x^2}}{d \sqrt {d x}}}{3 a d}-\frac {2 A \sqrt {a+b x^2}}{3 a d (d x)^{3/2}}\)

Input:

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

Output:

(-2*A*Sqrt[a + b*x^2])/(3*a*d*(d*x)^(3/2)) + ((-6*B*Sqrt[a + b*x^2])/(d*Sq 
rt[d*x]) - (2*b*Sqrt[x]*((3*B*(-((Sqrt[x]*Sqrt[a + b*x^2])/(Sqrt[a] + Sqrt 
[b]*x)) + (a^(1/4)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[ 
b]*x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(b^(1/4)*Sqr 
t[a + b*x^2])))/Sqrt[b] + ((A - (3*Sqrt[a]*B)/Sqrt[b])*(Sqrt[a] + Sqrt[b]* 
x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*S 
qrt[x])/a^(1/4)], 1/2])/(2*a^(1/4)*b^(1/4)*Sqrt[a + b*x^2])))/(d*Sqrt[d*x] 
))/(3*a*d)
 

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 553

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

rule 555

Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> 
Simp[2   Subst[Int[(f + g*x^2)/Sqrt[a + c*x^4], x], x, Sqrt[x]], x] /; Free 
Q[{a, c, f, g}, x]
 

rule 556

Int[((c_) + (d_.)*(x_))/(Sqrt[(e_)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symb 
ol] :> Simp[Sqrt[x]/Sqrt[e*x]   Int[(c + d*x)/(Sqrt[x]*Sqrt[a + b*x^2]), x] 
, x] /; FreeQ[{a, b, c, d, e}, x]
 

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 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]
 

rule 1512

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + c*x^4], x], x] - Simp[e/q 
 Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, c 
, d, e}, x] && PosQ[c/a]
 

Maple [A] (verified)

Time = 1.20 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.96

method	result	size
default	\(-\frac {A \sqrt {2}\, \sqrt {-a b}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) x +3 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a x -6 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a x +6 B b \,x^{3}+2 A b \,x^{2}+6 B a x +2 A a}{3 x \sqrt {b \,x^{2}+a}\, a \,d^{2} \sqrt {d x}}\)	\(306\)
risch	\(-\frac {2 \sqrt {b \,x^{2}+a}\, \left (3 B x +A \right )}{3 a x \,d^{2} \sqrt {d x}}-\frac {b \left (\frac {A \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b d \,x^{3}+a d x}}-\frac {3 B \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {b d \,x^{3}+a d x}}\right ) \sqrt {d \left (b \,x^{2}+a \right ) x}}{3 a \,d^{2} \sqrt {d x}\, \sqrt {b \,x^{2}+a}}\)	\(340\)
elliptic	\(\frac {\sqrt {d \left (b \,x^{2}+a \right ) x}\, \left (-\frac {2 A \sqrt {b d \,x^{3}+a d x}}{3 d^{3} a \,x^{2}}-\frac {2 \left (b d \,x^{2}+a d \right ) B}{d^{3} a \sqrt {x \left (b d \,x^{2}+a d \right )}}-\frac {A \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{3 a \,d^{2} \sqrt {b d \,x^{3}+a d x}}+\frac {B \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{a \,d^{2} \sqrt {b d \,x^{3}+a d x}}\right )}{\sqrt {d x}\, \sqrt {b \,x^{2}+a}}\)	\(364\)

Input:

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

Output:

-1/3/x*(A*2^(1/2)*(-a*b)^(1/2)*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*((- 
b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-1/(-a*b)^(1/2)*b*x)^(1/2)*Elliptic 
F(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*x+3*B*2^(1/2)*((b*x 
+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2 
)*(-1/(-a*b)^(1/2)*b*x)^(1/2)*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^ 
(1/2),1/2*2^(1/2))*a*x-6*B*2^(1/2)*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2) 
*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-1/(-a*b)^(1/2)*b*x)^(1/2)*Elli 
pticE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a*x+6*B*b*x^3+2 
*A*b*x^2+6*B*a*x+2*A*a)/(b*x^2+a)^(1/2)/a/d^2/(d*x)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.25 \[ \int \frac {A+B x}{(d x)^{5/2} \sqrt {a+b x^2}} \, dx=-\frac {2 \, {\left (\sqrt {b d} A x^{2} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + 3 \, \sqrt {b d} B x^{2} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + \sqrt {b x^{2} + a} {\left (3 \, B x + A\right )} \sqrt {d x}\right )}}{3 \, a d^{3} x^{2}} \] Input:

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

Output:

-2/3*(sqrt(b*d)*A*x^2*weierstrassPInverse(-4*a/b, 0, x) + 3*sqrt(b*d)*B*x^ 
2*weierstrassZeta(-4*a/b, 0, weierstrassPInverse(-4*a/b, 0, x)) + sqrt(b*x 
^2 + a)*(3*B*x + A)*sqrt(d*x))/(a*d^3*x^2)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.91 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.31 \[ \int \frac {A+B x}{(d x)^{5/2} \sqrt {a+b x^2}} \, dx=\frac {A \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} d^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {B \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} d^{\frac {5}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(d)*(int((sqrt(x)*sqrt(a + b*x**2))/(a*x**3 + b*x**5),x)*a + int((sqr 
t(x)*sqrt(a + b*x**2))/(a*x**2 + b*x**4),x)*b))/d**3