\(\int \frac {A+B x^2}{\sqrt {x} \sqrt {b x^2+c x^4}} \, dx\) [255]

Optimal result

Integrand size = 28, antiderivative size = 281 \[ \int \frac {A+B x^2}{\sqrt {x} \sqrt {b x^2+c x^4}} \, dx=\frac {2 (b B+A c) x^{3/2} \left (b+c x^2\right )}{b \sqrt {c} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {2 A \sqrt {b x^2+c x^4}}{b x^{3/2}}-\frac {2 (b B+A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{b^{3/4} c^{3/4} \sqrt {b x^2+c x^4}}+\frac {(b B+A c) x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{b^{3/4} c^{3/4} \sqrt {b x^2+c x^4}} \] Output:

2*(A*c+B*b)*x^(3/2)*(c*x^2+b)/b/c^(1/2)/(b^(1/2)+c^(1/2)*x)/(c*x^4+b*x^2)^ 
(1/2)-2*A*(c*x^4+b*x^2)^(1/2)/b/x^(3/2)-2*(A*c+B*b)*x*(b^(1/2)+c^(1/2)*x)* 
((c*x^2+b)/(b^(1/2)+c^(1/2)*x)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/4)*x^( 
1/2)/b^(1/4))),1/2*2^(1/2))/b^(3/4)/c^(3/4)/(c*x^4+b*x^2)^(1/2)+(A*c+B*b)* 
x*(b^(1/2)+c^(1/2)*x)*((c*x^2+b)/(b^(1/2)+c^(1/2)*x)^2)^(1/2)*InverseJacob 
iAM(2*arctan(c^(1/4)*x^(1/2)/b^(1/4)),1/2*2^(1/2))/b^(3/4)/c^(3/4)/(c*x^4+ 
b*x^2)^(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 = 82, normalized size of antiderivative = 0.29 \[ \int \frac {A+B x^2}{\sqrt {x} \sqrt {b x^2+c x^4}} \, dx=\frac {2 \sqrt {x} \left (-3 A \left (b+c x^2\right )+(b B+A c) x^2 \sqrt {1+\frac {c x^2}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^2}{b}\right )\right )}{3 b \sqrt {x^2 \left (b+c x^2\right )}} \] Input:

Integrate[(A + B*x^2)/(Sqrt[x]*Sqrt[b*x^2 + c*x^4]),x]
 

Output:

(2*Sqrt[x]*(-3*A*(b + c*x^2) + (b*B + A*c)*x^2*Sqrt[1 + (c*x^2)/b]*Hyperge 
ometric2F1[1/2, 3/4, 7/4, -((c*x^2)/b)]))/(3*b*Sqrt[x^2*(b + c*x^2)])
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1944, 1431, 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.

\(\displaystyle \int \frac {A+B x^2}{\sqrt {x} \sqrt {b x^2+c x^4}} \, dx\)

\(\Big \downarrow \) 1944

\(\displaystyle \frac {(A c+b B) \int \frac {x^{3/2}}{\sqrt {c x^4+b x^2}}dx}{b}-\frac {2 A \sqrt {b x^2+c x^4}}{b x^{3/2}}\)

\(\Big \downarrow \) 1431

\(\displaystyle \frac {x \sqrt {b+c x^2} (A c+b B) \int \frac {\sqrt {x}}{\sqrt {c x^2+b}}dx}{b \sqrt {b x^2+c x^4}}-\frac {2 A \sqrt {b x^2+c x^4}}{b x^{3/2}}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {2 x \sqrt {b+c x^2} (A c+b B) \int \frac {x}{\sqrt {c x^2+b}}d\sqrt {x}}{b \sqrt {b x^2+c x^4}}-\frac {2 A \sqrt {b x^2+c x^4}}{b x^{3/2}}\)

\(\Big \downarrow \) 834

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 1510

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

Input:

Int[(A + B*x^2)/(Sqrt[x]*Sqrt[b*x^2 + c*x^4]),x]
 

Output:

(-2*A*Sqrt[b*x^2 + c*x^4])/(b*x^(3/2)) + (2*(b*B + A*c)*x*Sqrt[b + c*x^2]* 
(-((-((Sqrt[x]*Sqrt[b + c*x^2])/(Sqrt[b] + Sqrt[c]*x)) + (b^(1/4)*(Sqrt[b] 
 + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan 
[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(c^(1/4)*Sqrt[b + c*x^2]))/Sqrt[c]) + ( 
b^(1/4)*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*El 
lipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(2*c^(3/4)*Sqrt[b + c*x 
^2])))/(b*Sqrt[b*x^2 + c*x^4])
 

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 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 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 1431

Int[((d_.)*(x_))^(m_)*((b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp 
[(b*x^2 + c*x^4)^p/((d*x)^(2*p)*(b + c*x^2)^p)   Int[(d*x)^(m + 2*p)*(b + c 
*x^2)^p, x], x] /; FreeQ[{b, c, d, m, p}, x] &&  !IntegerQ[p]
 

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 1944

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + 
(d_.)*(x_)^(n_.)), x_Symbol] :> Simp[c*e^(j - 1)*(e*x)^(m - j + 1)*((a*x^j 
+ b*x^(j + n))^(p + 1)/(a*(m + j*p + 1))), x] + Simp[(a*d*(m + j*p + 1) - b 
*c*(m + n + p*(j + n) + 1))/(a*e^n*(m + j*p + 1))   Int[(e*x)^(m + n)*(a*x^ 
j + b*x^(j + n))^p, x], x] /; FreeQ[{a, b, c, d, e, j, p}, x] && EqQ[jn, j 
+ n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && (LtQ[m + j*p, -1 
] || (IntegersQ[m - 1/2, p - 1/2] && LtQ[p, 0] && LtQ[m, (-n)*p - 1])) && ( 
GtQ[e, 0] || IntegersQ[j, n]) && NeQ[m + j*p + 1, 0] && NeQ[m - n + j*p + 1 
, 0]
 

Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.80

method	result	size
risch	\(-\frac {2 A \left (c \,x^{2}+b \right ) \sqrt {x}}{b \sqrt {x^{2} \left (c \,x^{2}+b \right )}}+\frac {\left (A c +B b \right ) \sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \left (-\frac {2 \sqrt {-b c}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-b c}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right ) \sqrt {x}\, \sqrt {x \left (c \,x^{2}+b \right )}}{b c \sqrt {c \,x^{3}+b x}\, \sqrt {x^{2} \left (c \,x^{2}+b \right )}}\)	\(225\)
default	\(\frac {\sqrt {x}\, \left (2 A \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) b c -A \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) b c +2 B \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) b^{2}-B \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) b^{2}-2 A \,c^{2} x^{2}-2 A b c \right )}{\sqrt {c \,x^{4}+b \,x^{2}}\, c b}\)	\(377\)

Input:

int((B*x^2+A)/x^(1/2)/(c*x^4+b*x^2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.22 \[ \int \frac {A+B x^2}{\sqrt {x} \sqrt {b x^2+c x^4}} \, dx=-\frac {2 \, {\left ({\left (B b + A c\right )} \sqrt {c} x^{2} {\rm weierstrassZeta}\left (-\frac {4 \, b}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right )\right ) + \sqrt {c x^{4} + b x^{2}} A c \sqrt {x}\right )}}{b c x^{2}} \] Input:

integrate((B*x^2+A)/x^(1/2)/(c*x^4+b*x^2)^(1/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate((B*x**2+A)/x**(1/2)/(c*x**4+b*x**2)**(1/2),x)
 

Output:

Integral((A + B*x**2)/(sqrt(x)*sqrt(x**2*(b + c*x**2))), x)
 

Maxima [F]

\[ \int \frac {A+B x^2}{\sqrt {x} \sqrt {b x^2+c x^4}} \, dx=\int { \frac {B x^{2} + A}{\sqrt {c x^{4} + b x^{2}} \sqrt {x}} \,d x } \] Input:

integrate((B*x^2+A)/x^(1/2)/(c*x^4+b*x^2)^(1/2),x, algorithm="maxima")
 

Output:

integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*sqrt(x)), x)
 

Giac [F]

\[ \int \frac {A+B x^2}{\sqrt {x} \sqrt {b x^2+c x^4}} \, dx=\int { \frac {B x^{2} + A}{\sqrt {c x^{4} + b x^{2}} \sqrt {x}} \,d x } \] Input:

integrate((B*x^2+A)/x^(1/2)/(c*x^4+b*x^2)^(1/2),x, algorithm="giac")
 

Output:

integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*sqrt(x)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2}{\sqrt {x} \sqrt {b x^2+c x^4}} \, dx=\int \frac {B\,x^2+A}{\sqrt {x}\,\sqrt {c\,x^4+b\,x^2}} \,d x \] Input:

int((A + B*x^2)/(x^(1/2)*(b*x^2 + c*x^4)^(1/2)),x)
 

Output:

int((A + B*x^2)/(x^(1/2)*(b*x^2 + c*x^4)^(1/2)), x)
 

Reduce [F]

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

int((B*x^2+A)/x^(1/2)/(c*x^4+b*x^2)^(1/2),x)
 

Output:

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