3.2.78 \(\int \sqrt {2+b x^2} \sqrt {3+d x^2} \, dx\) [178]

3.2.78.1 Optimal result

Integrand size = 23, antiderivative size = 235 \[ \int \sqrt {2+b x^2} \sqrt {3+d x^2} \, dx=\frac {(3 b+2 d) x \sqrt {2+b x^2}}{3 b \sqrt {3+d x^2}}+\frac {1}{3} x \sqrt {2+b x^2} \sqrt {3+d x^2}-\frac {\sqrt {2} (3 b+2 d) \sqrt {2+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{3 b \sqrt {d} \sqrt {\frac {2+b x^2}{3+d x^2}} \sqrt {3+d x^2}}+\frac {2 \sqrt {2} \sqrt {2+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {3}}\right ),1-\frac {3 b}{2 d}\right )}{\sqrt {d} \sqrt {\frac {2+b x^2}{3+d x^2}} \sqrt {3+d x^2}} \]

output

1/3*(3*b+2*d)*x*(b*x^2+2)^(1/2)/b/(d*x^2+3)^(1/2)-1/3*(3*b+2*d)*(1/(3*d*x^ 
2+9))^(1/2)*(3*d*x^2+9)^(1/2)*EllipticE(x*d^(1/2)*3^(1/2)/(3*d*x^2+9)^(1/2 
),1/2*(4-6*b/d)^(1/2))*2^(1/2)*(b*x^2+2)^(1/2)/b/d^(1/2)/((b*x^2+2)/(d*x^2 
+3))^(1/2)/(d*x^2+3)^(1/2)+2*(1/(3*d*x^2+9))^(1/2)*(3*d*x^2+9)^(1/2)*Ellip 
ticF(x*d^(1/2)*3^(1/2)/(3*d*x^2+9)^(1/2),1/2*(4-6*b/d)^(1/2))*2^(1/2)*(b*x 
^2+2)^(1/2)/d^(1/2)/((b*x^2+2)/(d*x^2+3))^(1/2)/(d*x^2+3)^(1/2)+1/3*x*(b*x 
^2+2)^(1/2)*(d*x^2+3)^(1/2)
 

3.2.78.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.88 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.54 \[ \int \sqrt {2+b x^2} \sqrt {3+d x^2} \, dx=\frac {\sqrt {b} d x \sqrt {2+b x^2} \sqrt {3+d x^2}-i \sqrt {3} (3 b+2 d) E\left (i \text {arcsinh}\left (\frac {\sqrt {b} x}{\sqrt {2}}\right )|\frac {2 d}{3 b}\right )+i \sqrt {3} (3 b-2 d) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {b} x}{\sqrt {2}}\right ),\frac {2 d}{3 b}\right )}{3 \sqrt {b} d} \]

input

Integrate[Sqrt[2 + b*x^2]*Sqrt[3 + d*x^2],x]
 

output

(Sqrt[b]*d*x*Sqrt[2 + b*x^2]*Sqrt[3 + d*x^2] - I*Sqrt[3]*(3*b + 2*d)*Ellip 
ticE[I*ArcSinh[(Sqrt[b]*x)/Sqrt[2]], (2*d)/(3*b)] + I*Sqrt[3]*(3*b - 2*d)* 
EllipticF[I*ArcSinh[(Sqrt[b]*x)/Sqrt[2]], (2*d)/(3*b)])/(3*Sqrt[b]*d)
 

3.2.78.3 Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {319, 27, 406, 320, 388, 313}

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.

\(\displaystyle \int \sqrt {b x^2+2} \sqrt {d x^2+3} \, dx\)

\(\Big \downarrow \) 319

\(\displaystyle \frac {2}{3} \int \frac {(3 b+2 d) x^2+12}{2 \sqrt {b x^2+2} \sqrt {d x^2+3}}dx+\frac {1}{3} x \sqrt {b x^2+2} \sqrt {d x^2+3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \int \frac {(3 b+2 d) x^2+12}{\sqrt {b x^2+2} \sqrt {d x^2+3}}dx+\frac {1}{3} x \sqrt {b x^2+2} \sqrt {d x^2+3}\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {1}{3} \left (12 \int \frac {1}{\sqrt {b x^2+2} \sqrt {d x^2+3}}dx+(3 b+2 d) \int \frac {x^2}{\sqrt {b x^2+2} \sqrt {d x^2+3}}dx\right )+\frac {1}{3} x \sqrt {b x^2+2} \sqrt {d x^2+3}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {1}{3} \left ((3 b+2 d) \int \frac {x^2}{\sqrt {b x^2+2} \sqrt {d x^2+3}}dx+\frac {6 \sqrt {2} \sqrt {b x^2+2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {3}}\right ),1-\frac {3 b}{2 d}\right )}{\sqrt {d} \sqrt {d x^2+3} \sqrt {\frac {b x^2+2}{d x^2+3}}}\right )+\frac {1}{3} x \sqrt {b x^2+2} \sqrt {d x^2+3}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {1}{3} \left ((3 b+2 d) \left (\frac {x \sqrt {b x^2+2}}{b \sqrt {d x^2+3}}-\frac {3 \int \frac {\sqrt {b x^2+2}}{\left (d x^2+3\right )^{3/2}}dx}{b}\right )+\frac {6 \sqrt {2} \sqrt {b x^2+2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {3}}\right ),1-\frac {3 b}{2 d}\right )}{\sqrt {d} \sqrt {d x^2+3} \sqrt {\frac {b x^2+2}{d x^2+3}}}\right )+\frac {1}{3} x \sqrt {b x^2+2} \sqrt {d x^2+3}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {1}{3} \left (\frac {6 \sqrt {2} \sqrt {b x^2+2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {3}}\right ),1-\frac {3 b}{2 d}\right )}{\sqrt {d} \sqrt {d x^2+3} \sqrt {\frac {b x^2+2}{d x^2+3}}}+(3 b+2 d) \left (\frac {x \sqrt {b x^2+2}}{b \sqrt {d x^2+3}}-\frac {\sqrt {2} \sqrt {b x^2+2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{b \sqrt {d} \sqrt {d x^2+3} \sqrt {\frac {b x^2+2}{d x^2+3}}}\right )\right )+\frac {1}{3} x \sqrt {b x^2+2} \sqrt {d x^2+3}\)

input

Int[Sqrt[2 + b*x^2]*Sqrt[3 + d*x^2],x]
 

output

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

3.2.78.3.1 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 313

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim 
p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c 
+ d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ 
[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
 

rule 319

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[x*(a + b*x^2)^p*((c + d*x^2)^q/(2*(p + q) + 1)), x] + Simp[2/(2*(p + q) + 
 1)   Int[(a + b*x^2)^(p - 1)*(c + d*x^2)^(q - 1)*Simp[a*c*(p + q) + (q*(b* 
c - a*d) + a*d*(p + q))*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b* 
c - a*d, 0] && GtQ[q, 0] && GtQ[p, 0] && IntBinomialQ[a, b, c, d, 2, p, q, 
x]
 

rule 320

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( 
c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre 
eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 388

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b   Int[Sqrt[ 
a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 406

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[e   Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim 
p[f   Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, 
f, p, q}, x]
 

3.2.78.4 Maple [A] (verified)

Time = 3.40 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.07

method	result	size
risch	\(\frac {x \sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}}{3}+\frac {\left (\frac {2 \sqrt {3 d \,x^{2}+9}\, \sqrt {2 b \,x^{2}+4}\, F\left (\frac {x \sqrt {-3 d}}{3}, \frac {\sqrt {-4+\frac {6 b +4 d}{d}}}{2}\right )}{\sqrt {-3 d}\, \sqrt {b d \,x^{4}+3 b \,x^{2}+2 d \,x^{2}+6}}-\frac {\left (3 b +2 d \right ) \sqrt {3 d \,x^{2}+9}\, \sqrt {2 b \,x^{2}+4}\, \left (F\left (\frac {x \sqrt {-3 d}}{3}, \frac {\sqrt {-4+\frac {6 b +4 d}{d}}}{2}\right )-E\left (\frac {x \sqrt {-3 d}}{3}, \frac {\sqrt {-4+\frac {6 b +4 d}{d}}}{2}\right )\right )}{3 \sqrt {-3 d}\, \sqrt {b d \,x^{4}+3 b \,x^{2}+2 d \,x^{2}+6}\, b}\right ) \sqrt {\left (b \,x^{2}+2\right ) \left (d \,x^{2}+3\right )}}{\sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}}\)	\(252\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+2\right ) \left (d \,x^{2}+3\right )}\, \left (\frac {x \sqrt {b d \,x^{4}+3 b \,x^{2}+2 d \,x^{2}+6}}{3}+\frac {2 \sqrt {3 d \,x^{2}+9}\, \sqrt {2 b \,x^{2}+4}\, F\left (\frac {x \sqrt {-3 d}}{3}, \frac {\sqrt {-4+\frac {6 b +4 d}{d}}}{2}\right )}{\sqrt {-3 d}\, \sqrt {b d \,x^{4}+3 b \,x^{2}+2 d \,x^{2}+6}}-\frac {\left (b +\frac {2 d}{3}\right ) \sqrt {3 d \,x^{2}+9}\, \sqrt {2 b \,x^{2}+4}\, \left (F\left (\frac {x \sqrt {-3 d}}{3}, \frac {\sqrt {-4+\frac {6 b +4 d}{d}}}{2}\right )-E\left (\frac {x \sqrt {-3 d}}{3}, \frac {\sqrt {-4+\frac {6 b +4 d}{d}}}{2}\right )\right )}{\sqrt {-3 d}\, \sqrt {b d \,x^{4}+3 b \,x^{2}+2 d \,x^{2}+6}\, b}\right )}{\sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}}\)	\(253\)
default	\(\frac {\sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}\, \left (b^{2} d \,x^{5} \sqrt {-d}+3 b^{2} x^{3} \sqrt {-d}+2 b d \,x^{3} \sqrt {-d}+3 \sqrt {2}\, F\left (\frac {x \sqrt {3}\, \sqrt {-d}}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {b}{d}}}{2}\right ) b \sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}-2 \sqrt {2}\, F\left (\frac {x \sqrt {3}\, \sqrt {-d}}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {b}{d}}}{2}\right ) d \sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}+3 \sqrt {2}\, E\left (\frac {x \sqrt {3}\, \sqrt {-d}}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {b}{d}}}{2}\right ) b \sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}+2 \sqrt {2}\, E\left (\frac {x \sqrt {3}\, \sqrt {-d}}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {b}{d}}}{2}\right ) d \sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}+6 b x \sqrt {-d}\right )}{3 \left (b d \,x^{4}+3 b \,x^{2}+2 d \,x^{2}+6\right ) \sqrt {-d}\, b}\)	\(303\)

input

int((b*x^2+2)^(1/2)*(d*x^2+3)^(1/2),x,method=_RETURNVERBOSE)
 

output

1/3*x*(b*x^2+2)^(1/2)*(d*x^2+3)^(1/2)+(2/(-3*d)^(1/2)*(3*d*x^2+9)^(1/2)*(2 
*b*x^2+4)^(1/2)/(b*d*x^4+3*b*x^2+2*d*x^2+6)^(1/2)*EllipticF(1/3*x*(-3*d)^( 
1/2),1/2*(-4+2*(3*b+2*d)/d)^(1/2))-1/3*(3*b+2*d)/(-3*d)^(1/2)*(3*d*x^2+9)^ 
(1/2)*(2*b*x^2+4)^(1/2)/(b*d*x^4+3*b*x^2+2*d*x^2+6)^(1/2)/b*(EllipticF(1/3 
*x*(-3*d)^(1/2),1/2*(-4+2*(3*b+2*d)/d)^(1/2))-EllipticE(1/3*x*(-3*d)^(1/2) 
,1/2*(-4+2*(3*b+2*d)/d)^(1/2))))*((b*x^2+2)*(d*x^2+3))^(1/2)/(b*x^2+2)^(1/ 
2)/(d*x^2+3)^(1/2)
 

3.2.78.5 Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.63 \[ \int \sqrt {2+b x^2} \sqrt {3+d x^2} \, dx=-\frac {3 \, \sqrt {3} \sqrt {b d} {\left (3 \, b + 2 \, d\right )} x \sqrt {-\frac {1}{d}} E(\arcsin \left (\frac {\sqrt {3} \sqrt {-\frac {1}{d}}}{x}\right )\,|\,\frac {2 \, d}{3 \, b}) - \sqrt {3} \sqrt {b d} {\left (4 \, d^{2} + 9 \, b + 6 \, d\right )} x \sqrt {-\frac {1}{d}} F(\arcsin \left (\frac {\sqrt {3} \sqrt {-\frac {1}{d}}}{x}\right )\,|\,\frac {2 \, d}{3 \, b}) - {\left (b d^{2} x^{2} + 3 \, b d + 2 \, d^{2}\right )} \sqrt {b x^{2} + 2} \sqrt {d x^{2} + 3}}{3 \, b d^{2} x} \]

input

integrate((b*x^2+2)^(1/2)*(d*x^2+3)^(1/2),x, algorithm="fricas")
 

output

-1/3*(3*sqrt(3)*sqrt(b*d)*(3*b + 2*d)*x*sqrt(-1/d)*elliptic_e(arcsin(sqrt( 
3)*sqrt(-1/d)/x), 2/3*d/b) - sqrt(3)*sqrt(b*d)*(4*d^2 + 9*b + 6*d)*x*sqrt( 
-1/d)*elliptic_f(arcsin(sqrt(3)*sqrt(-1/d)/x), 2/3*d/b) - (b*d^2*x^2 + 3*b 
*d + 2*d^2)*sqrt(b*x^2 + 2)*sqrt(d*x^2 + 3))/(b*d^2*x)
 

3.2.78.6 Sympy [F]

\[ \int \sqrt {2+b x^2} \sqrt {3+d x^2} \, dx=\int \sqrt {b x^{2} + 2} \sqrt {d x^{2} + 3}\, dx \]

input

integrate((b*x**2+2)**(1/2)*(d*x**2+3)**(1/2),x)
 

output

Integral(sqrt(b*x**2 + 2)*sqrt(d*x**2 + 3), x)
 

3.2.78.7 Maxima [F]

\[ \int \sqrt {2+b x^2} \sqrt {3+d x^2} \, dx=\int { \sqrt {b x^{2} + 2} \sqrt {d x^{2} + 3} \,d x } \]

input

integrate((b*x^2+2)^(1/2)*(d*x^2+3)^(1/2),x, algorithm="maxima")
 

output

integrate(sqrt(b*x^2 + 2)*sqrt(d*x^2 + 3), x)
 

3.2.78.8 Giac [F]

\[ \int \sqrt {2+b x^2} \sqrt {3+d x^2} \, dx=\int { \sqrt {b x^{2} + 2} \sqrt {d x^{2} + 3} \,d x } \]

input

integrate((b*x^2+2)^(1/2)*(d*x^2+3)^(1/2),x, algorithm="giac")
 

output

integrate(sqrt(b*x^2 + 2)*sqrt(d*x^2 + 3), x)
 

3.2.78.9 Mupad [F(-1)]

Timed out. \[ \int \sqrt {2+b x^2} \sqrt {3+d x^2} \, dx=\int \sqrt {b\,x^2+2}\,\sqrt {d\,x^2+3} \,d x \]

input

int((b*x^2 + 2)^(1/2)*(d*x^2 + 3)^(1/2),x)
 

output

int((b*x^2 + 2)^(1/2)*(d*x^2 + 3)^(1/2), x)