\(\int \frac {\sqrt {a+b x^2} (c+d x^2)^{3/2}}{e+f x^2} \, dx\) [93]

Optimal result

Integrand size = 32, antiderivative size = 402 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{e+f x^2} \, dx=-\frac {(3 b d e-4 b c f-a d f) x \sqrt {c+d x^2}}{3 f^2 \sqrt {a+b x^2}}+\frac {d x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 f}+\frac {\sqrt {a} (3 b d e-4 b c f-a d f) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 \sqrt {b} f^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} d (3 d e-5 c f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 \sqrt {b} c f^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} (d e-c f)^2 \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e f^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

-1/3*(-a*d*f-4*b*c*f+3*b*d*e)*x*(d*x^2+c)^(1/2)/f^2/(b*x^2+a)^(1/2)+1/3*d* 
x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/f+1/3*a^(1/2)*(-a*d*f-4*b*c*f+3*b*d*e)*( 
d*x^2+c)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^( 
1/2))/b^(1/2)/f^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/3*a^(3 
/2)*d*(-5*c*f+3*d*e)*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1 
/2)),(1-a*d/b/c)^(1/2))/b^(1/2)/c/f^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^ 
2+a))^(1/2)+a^(3/2)*(-c*f+d*e)^2*(d*x^2+c)^(1/2)*EllipticPi(b^(1/2)*x/a^(1 
/2)/(1+b*x^2/a)^(1/2),1-a*f/b/e,(1-a*d/b/c)^(1/2))/b^(1/2)/c/e/f^2/(b*x^2+ 
a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.57 (sec) , antiderivative size = 742, normalized size of antiderivative = 1.85 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{e+f x^2} \, dx=\frac {a \sqrt {\frac {b}{a}} c d e f^2 x+b \sqrt {\frac {b}{a}} c d e f^2 x^3+a \sqrt {\frac {b}{a}} d^2 e f^2 x^3+b \sqrt {\frac {b}{a}} d^2 e f^2 x^5-i c e f (-3 b d e+4 b c f+a d f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i e \left (a d f (-3 d e+4 c f)+b \left (3 d^2 e^2-3 c d e f-c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+3 i b d^2 e^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )-6 i b c d e^2 f \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )-3 i a d^2 e^2 f \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+3 i b c^2 e f^2 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+6 i a c d e f^2 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )-3 i a c^2 f^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 \sqrt {\frac {b}{a}} e f^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2))/(e + f*x^2),x]
 

Output:

(a*Sqrt[b/a]*c*d*e*f^2*x + b*Sqrt[b/a]*c*d*e*f^2*x^3 + a*Sqrt[b/a]*d^2*e*f 
^2*x^3 + b*Sqrt[b/a]*d^2*e*f^2*x^5 - I*c*e*f*(-3*b*d*e + 4*b*c*f + a*d*f)* 
Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], 
(a*d)/(b*c)] - I*e*(a*d*f*(-3*d*e + 4*c*f) + b*(3*d^2*e^2 - 3*c*d*e*f - c^ 
2*f^2))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b 
/a]*x], (a*d)/(b*c)] + (3*I)*b*d^2*e^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2 
)/c]*EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (6*I)* 
b*c*d*e^2*f*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[(a*f)/(b*e) 
, I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (3*I)*a*d^2*e^2*f*Sqrt[1 + (b*x^2 
)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], ( 
a*d)/(b*c)] + (3*I)*b*c^2*e*f^2*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*El 
lipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (6*I)*a*c*d*e 
*f^2*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[(a*f)/(b*e), I*Arc 
Sinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (3*I)*a*c^2*f^3*Sqrt[1 + (b*x^2)/a]*Sqrt 
[1 + (d*x^2)/c]*EllipticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c 
)])/(3*Sqrt[b/a]*e*f^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.95, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {418, 25, 403, 27, 406, 320, 388, 313, 414}

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.

\(\displaystyle \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{e+f x^2} \, dx\)

\(\Big \downarrow \) 418

\(\displaystyle \frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}+\frac {d \int -\frac {\sqrt {b x^2+a} \left (-d f x^2+d e-2 c f\right )}{\sqrt {d x^2+c}}dx}{f^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}-\frac {d \int \frac {\sqrt {b x^2+a} \left (-d f x^2+d e-2 c f\right )}{\sqrt {d x^2+c}}dx}{f^2}\)

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

\(\displaystyle \frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}-\frac {d \left (\frac {1}{3} \left ((-a d f-4 b c f+3 b d e) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {\sqrt {c} \sqrt {a+b x^2} (3 d e-5 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )-\frac {1}{3} f x \sqrt {a+b x^2} \sqrt {c+d x^2}\right )}{f^2}\)

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

\(\displaystyle \frac {(d e-c f)^2 \int \frac {\sqrt {b x^2+a}}{\sqrt {d x^2+c} \left (f x^2+e\right )}dx}{f^2}-\frac {d \left (\frac {1}{3} \left (\frac {\sqrt {c} \sqrt {a+b x^2} (3 d e-5 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+(-a d f-4 b c f+3 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )-\frac {1}{3} f x \sqrt {a+b x^2} \sqrt {c+d x^2}\right )}{f^2}\)

\(\Big \downarrow \) 414

\(\displaystyle \frac {a^{3/2} \sqrt {c+d x^2} (d e-c f)^2 \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e f^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {d \left (\frac {1}{3} \left (\frac {\sqrt {c} \sqrt {a+b x^2} (3 d e-5 c f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+(-a d f-4 b c f+3 b d e) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )-\frac {1}{3} f x \sqrt {a+b x^2} \sqrt {c+d x^2}\right )}{f^2}\)

Input:

Int[(Sqrt[a + b*x^2]*(c + d*x^2)^(3/2))/(e + f*x^2),x]
 

Output:

-((d*(-1/3*(f*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]) + ((3*b*d*e - 4*b*c*f - a 
*d*f)*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]* 
EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt[( 
c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (Sqrt[c]*(3*d*e - 5*c* 
f)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)] 
)/(Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/3))/f^2 
) + (a^(3/2)*(d*e - c*f)^2*Sqrt[c + d*x^2]*EllipticPi[1 - (a*f)/(b*e), Arc 
Tan[(Sqrt[b]*x)/Sqrt[a]], 1 - (a*d)/(b*c)])/(Sqrt[b]*c*e*f^2*Sqrt[a + b*x^ 
2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 403

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

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]
 

rule 414

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_) 
^2]), x_Symbol] :> Simp[c*(Sqrt[e + f*x^2]/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]* 
Sqrt[c*((e + f*x^2)/(e*(c + d*x^2)))]))*EllipticPi[1 - b*(c/(a*d)), ArcTan[ 
Rt[d/c, 2]*x], 1 - c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ 
[d/c]
 

rule 418

Int[(((c_) + (d_.)*(x_)^2)^(3/2)*Sqrt[(e_) + (f_.)*(x_)^2])/((a_) + (b_.)*( 
x_)^2), x_Symbol] :> Simp[(b*c - a*d)^2/b^2   Int[Sqrt[e + f*x^2]/((a + b*x 
^2)*Sqrt[c + d*x^2]), x], x] + Simp[d/b^2   Int[(2*b*c - a*d + b*d*x^2)*(Sq 
rt[e + f*x^2]/Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && P 
osQ[d/c] && PosQ[f/e]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(830\) vs. \(2(372)=744\).

Time = 9.68 (sec) , antiderivative size = 831, normalized size of antiderivative = 2.07

method	result	size
risch	\(\frac {d x \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{3 f}+\frac {\left (\frac {-\frac {f \left (a d f +4 b c f -3 b d e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {3 b \,c^{2} f^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {3 b \,d^{2} e^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {5 a c d \,f^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {3 a \,d^{2} e f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {6 b c d e f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}}{f^{2}}+\frac {3 \left (a \,c^{2} f^{3}-2 a c d e \,f^{2}+a \,d^{2} e^{2} f -b \,c^{2} e \,f^{2}+2 b c d \,e^{2} f -e^{3} b \,d^{2}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right )}{f^{2} e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 f \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(831\)
default	\(\text {Expression too large to display}\)	\(1028\)
elliptic	\(\text {Expression too large to display}\)	\(1435\)

Input:

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

Output:

1/3*d*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/f+1/3/f*(1/f^2*(-f*(a*d*f+4*b*c*f- 
3*b*d*e)*c/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x 
^2+b*c*x^2+a*c)^(1/2)*(EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))- 
EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2)))+3*b*c^2*f^2/(-b/a)^(1/ 
2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2) 
*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))+3*b*d^2*e^2/(-b/a)^(1/ 
2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2) 
*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))+5*a*c*d*f^2/(-b/a)^(1/ 
2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2) 
*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-3*a*d^2*e*f/(-b/a)^(1/ 
2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2) 
*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-6*b*c*d*e*f/(-b/a)^(1/ 
2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2) 
*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2)))+3*(a*c^2*f^3-2*a*c*d* 
e*f^2+a*d^2*e^2*f-b*c^2*e*f^2+2*b*c*d*e^2*f-b*d^2*e^3)/f^2/e/(-b/a)^(1/2)* 
(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*El 
lipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2)))*((b*x^2+a)*( 
d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{e+f x^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{e+f x^2} \, dx=\int \frac {\sqrt {a + b x^{2}} \left (c + d x^{2}\right )^{\frac {3}{2}}}{e + f x^{2}}\, dx \] Input:

integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(3/2)/(f*x**2+e),x)
 

Output:

Integral(sqrt(a + b*x**2)*(c + d*x**2)**(3/2)/(e + f*x**2), x)
 

Maxima [F]

\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{e+f x^2} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}{f x^{2} + e} \,d x } \] Input:

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

Output:

integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)/(f*x^2 + e), x)
 

Giac [F]

\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{e+f x^2} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}{f x^{2} + e} \,d x } \] Input:

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

Output:

integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)/(f*x^2 + e), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{e+f x^2} \, dx=\int \frac {\sqrt {b\,x^2+a}\,{\left (d\,x^2+c\right )}^{3/2}}{f\,x^2+e} \,d x \] Input:

int(((a + b*x^2)^(1/2)*(c + d*x^2)^(3/2))/(e + f*x^2),x)
 

Output:

int(((a + b*x^2)^(1/2)*(c + d*x^2)^(3/2))/(e + f*x^2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{e+f x^2} \, dx=\frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, d x +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) a \,d^{2} f +4 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) b c d f -3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{4}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) b \,d^{2} e +5 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) a c d f -2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) a \,d^{2} e +3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) b \,c^{2} f -2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) b c d e +3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) a \,c^{2} f -\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) a c d e}{3 f} \] Input:

int((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/(f*x^2+e),x)
 

Output:

(sqrt(c + d*x**2)*sqrt(a + b*x**2)*d*x + int((sqrt(c + d*x**2)*sqrt(a + b* 
x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b 
*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*a*d**2*f + 4*int((sqrt(c + d*x**2) 
*sqrt(a + b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b* 
c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*b*c*d*f - 3*int((sqrt( 
c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d* 
f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*b*d**2*e + 
5*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c*e + a*c*f*x**2 + a*d*e 
*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x) 
*a*c*d*f - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c*e + a*c*f*x 
**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d 
*f*x**6),x)*a*d**2*e + 3*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c 
*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d* 
e*x**4 + b*d*f*x**6),x)*b*c**2*f - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2 
)*x**2)/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f 
*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*b*c*d*e + 3*int((sqrt(c + d*x**2)*sqrt 
(a + b*x**2))/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x**2 + 
 b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*a*c**2*f - int((sqrt(c + d*x**2) 
*sqrt(a + b*x**2))/(a*c*e + a*c*f*x**2 + a*d*e*x**2 + a*d*f*x**4 + b*c*e*x 
**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*a*c*d*e)/(3*f)