\(\int \frac {(a+b x^2)^{3/2} \sqrt {c+d x^2}}{e+f x^2} \, dx\) [99]

Optimal result

Integrand size = 32, antiderivative size = 421 \[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{e+f x^2} \, dx=-\frac {b (3 b d e-b c f-4 a d f) x \sqrt {c+d x^2}}{3 d f^2 \sqrt {a+b x^2}}+\frac {b x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 f}+\frac {\sqrt {a} \sqrt {b} (3 b d e-b c f-4 a d f) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 d f^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {a^{3/2} (3 b d e-2 b c f-3 a d f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 \sqrt {b} c f^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} (b e-a f) (d e-c f) \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e f^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

-1/3*b*(-4*a*d*f-b*c*f+3*b*d*e)*x*(d*x^2+c)^(1/2)/d/f^2/(b*x^2+a)^(1/2)+1/ 
3*b*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/f+1/3*a^(1/2)*b^(1/2)*(-4*a*d*f-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))/d/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)*(-3*a*d*f-2*b*c*f+3*b*d*e)*(d*x^2+c)^(1/2)*InverseJacobiAM(arcta 
n(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)*(-a*f+b*e)*(-c*f+d*e)*(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 = 4.94 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{e+f x^2} \, dx=\frac {-i b c e f (-3 b d e+b c f+4 a d f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i e \left (3 a^2 d^2 f^2+a b d f (-6 d e+c f)+b^2 \left (3 d^2 e^2-c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+d \left (b \sqrt {\frac {b}{a}} e f^2 x \left (a+b x^2\right ) \left (c+d x^2\right )-3 i (b e-a f)^2 (-d e+c f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 \sqrt {\frac {b}{a}} d e f^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

((-I)*b*c*e*f*(-3*b*d*e + b*c*f + 4*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*(3*a^2*d^2*f 
^2 + a*b*d*f*(-6*d*e + c*f) + b^2*(3*d^2*e^2 - 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)] + d* 
(b*Sqrt[b/a]*e*f^2*x*(a + b*x^2)*(c + d*x^2) - (3*I)*(b*e - a*f)^2*(-(d*e) 
 + c*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*Sqrt[b/a]*d*e*f^3*Sqrt[a + b*x^2]* 
Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 385, normalized size of antiderivative = 0.91, 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.

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

\(\Big \downarrow \) 418

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

\(\Big \downarrow \) 414

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

Input:

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

Output:

-((b*(-1/3*(f*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]) + ((3*b*d*e - b*c*f - 4*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])) + (c^(3/2)*(3*b*e - 5*a* 
f)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)] 
)/(a*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/3))/f 
^2) + (c^(3/2)*(b*e - a*f)^2*Sqrt[a + b*x^2]*EllipticPi[1 - (c*f)/(d*e), A 
rcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*e*f^2*Sqrt[(c*(a 
+ b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*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. \(834\) vs. \(2(391)=782\).

Time = 9.69 (sec) , antiderivative size = 835, normalized size of antiderivative = 1.98

method	result	size
risch	\(\frac {b x \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{3 f}+\frac {\left (\frac {-\frac {f b \left (4 a d f +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}\, d}+\frac {3 a^{2} 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 b^{2} d \,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 b c \,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^{2} c 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 a b 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^{2} c \,f^{3}-a^{2} d e \,f^{2}-2 a b c e \,f^{2}+2 a b d \,e^{2} f +b^{2} c \,e^{2} f -b^{2} d \,e^{3}\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}}\)	\(835\)
default	\(\text {Expression too large to display}\)	\(1059\)
elliptic	\(\text {Expression too large to display}\)	\(1444\)

Input:

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

Output:

1/3*b*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/f+1/3/f*(1/f^2*(-f*b*(4*a*d*f+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)/d*(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*a^2*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*b^2*d*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*b*c*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^2*c*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*a*b*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^2*c*f^3-a^2* 
d*e*f^2-2*a*b*c*e*f^2+2*a*b*d*e^2*f+b^2*c*e^2*f-b^2*d*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 
)*EllipticPi(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. \[ \int \frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{e+f x^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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