\(\int \frac {x^4 \sqrt {a+b x^2} (c+d x^2)^{3/2}}{e+f x^2} \, dx\) [97]

Optimal result

Integrand size = 35, antiderivative size = 790 \[ \int \frac {x^4 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{e+f x^2} \, dx=\frac {\left (8 a^3 d^3 f^3+a^2 b d^2 f^2 (14 d e-19 c f)+a b^2 d f \left (35 d^2 e^2-49 c d e f+9 c^2 f^2\right )-b^3 \left (105 d^3 e^3-140 c d^2 e^2 f+21 c^2 d e f^2+6 c^3 f^3\right )\right ) x \sqrt {c+d x^2}}{105 b^2 d^2 f^4 \sqrt {a+b x^2}}-\frac {\left (\frac {4 a^2 d^2 f}{b}+a d (7 d e-6 c f)+b \left (21 c d e-\frac {35 d^2 e^2}{f}+6 c^2 f\right )\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{105 b d f^2}-\frac {(7 b d e+2 b c f-a d f) x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{35 b d f^2}+\frac {x \sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}}{7 d f}-\frac {\sqrt {a} \left (8 a^3 d^3 f^3+a^2 b d^2 f^2 (14 d e-19 c f)+a b^2 d f \left (35 d^2 e^2-49 c d e f+9 c^2 f^2\right )-b^3 \left (105 d^3 e^3-140 c d^2 e^2 f+21 c^2 d e f^2+6 c^3 f^3\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{105 b^{5/2} d^2 f^4 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} \left (4 a^2 c d^2 f^3+a b c d f^2 (7 d e-9 c f)-b^2 \left (105 d^3 e^3-175 c d^2 e^2 f+63 c^2 d e f^2+3 c^3 f^3\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{105 b^{5/2} c d f^4 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} e (d e-c f)^2 \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c f^4 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

1/105*(8*a^3*d^3*f^3+a^2*b*d^2*f^2*(-19*c*f+14*d*e)+a*b^2*d*f*(9*c^2*f^2-4 
9*c*d*e*f+35*d^2*e^2)-b^3*(6*c^3*f^3+21*c^2*d*e*f^2-140*c*d^2*e^2*f+105*d^ 
3*e^3))*x*(d*x^2+c)^(1/2)/b^2/d^2/f^4/(b*x^2+a)^(1/2)-1/105*(4*a^2*d^2*f/b 
+a*d*(-6*c*f+7*d*e)+b*(21*c*d*e-35*d^2*e^2/f+6*c^2*f))*x*(b*x^2+a)^(1/2)*( 
d*x^2+c)^(1/2)/b/d/f^2-1/35*(-a*d*f+2*b*c*f+7*b*d*e)*x*(b*x^2+a)^(1/2)*(d* 
x^2+c)^(3/2)/b/d/f^2+1/7*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(5/2)/d/f-1/105*a^(1/ 
2)*(8*a^3*d^3*f^3+a^2*b*d^2*f^2*(-19*c*f+14*d*e)+a*b^2*d*f*(9*c^2*f^2-49*c 
*d*e*f+35*d^2*e^2)-b^3*(6*c^3*f^3+21*c^2*d*e*f^2-140*c*d^2*e^2*f+105*d^3*e 
^3))*(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^(5/2)/d^2/f^4/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2 
)+1/105*a^(3/2)*(4*a^2*c*d^2*f^3+a*b*c*d*f^2*(-9*c*f+7*d*e)-b^2*(3*c^3*f^3 
+63*c^2*d*e*f^2-175*c*d^2*e^2*f+105*d^3*e^3))*(d*x^2+c)^(1/2)*InverseJacob 
iAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(5/2)/c/d/f^4/(b*x^2+a) 
^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+a^(3/2)*e*(-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/f^4/(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 = 6.07 (sec) , antiderivative size = 609, normalized size of antiderivative = 0.77 \[ \int \frac {x^4 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{e+f x^2} \, dx=\frac {-i c f \left (8 a^3 d^3 f^3+a^2 b d^2 f^2 (14 d e-19 c f)+a b^2 d f \left (35 d^2 e^2-49 c d e f+9 c^2 f^2\right )-b^3 \left (105 d^3 e^3-140 c d^2 e^2 f+21 c^2 d e f^2+6 c^3 f^3\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i \left (4 a^3 c d^3 f^4+a^2 b c d^2 f^3 (7 d e-10 c f)+a b^2 d f \left (105 d^3 e^3-140 c d^2 e^2 f+14 c^2 d e f^2+12 c^3 f^3\right )+b^3 \left (-105 d^4 e^4+105 c d^3 e^3 f+35 c^2 d^2 e^2 f^2-21 c^3 d e f^3-6 c^4 f^4\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )+d \left (-\sqrt {\frac {b}{a}} f^2 x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 a^2 d^2 f^2+a b d f \left (7 d e-9 c f-3 d f x^2\right )-b^2 \left (3 c^2 f^2+6 c d f \left (-7 e+4 f x^2\right )+d^2 \left (35 e^2-21 e f x^2+15 f^2 x^4\right )\right )\right )-105 i b^2 d e (-b e+a f) (d e-c f)^2 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{105 a^2 \left (\frac {b}{a}\right )^{5/2} d^2 f^5 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[(x^4*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2))/(e + f*x^2),x]
 

Output:

((-I)*c*f*(8*a^3*d^3*f^3 + a^2*b*d^2*f^2*(14*d*e - 19*c*f) + a*b^2*d*f*(35 
*d^2*e^2 - 49*c*d*e*f + 9*c^2*f^2) - b^3*(105*d^3*e^3 - 140*c*d^2*e^2*f + 
21*c^2*d*e*f^2 + 6*c^3*f^3))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Ellip 
ticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*(4*a^3*c*d^3*f^4 + a^2*b*c*d 
^2*f^3*(7*d*e - 10*c*f) + a*b^2*d*f*(105*d^3*e^3 - 140*c*d^2*e^2*f + 14*c^ 
2*d*e*f^2 + 12*c^3*f^3) + b^3*(-105*d^4*e^4 + 105*c*d^3*e^3*f + 35*c^2*d^2 
*e^2*f^2 - 21*c^3*d*e*f^3 - 6*c^4*f^4))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^ 
2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + d*(-(Sqrt[b/a]*f^2* 
x*(a + b*x^2)*(c + d*x^2)*(4*a^2*d^2*f^2 + a*b*d*f*(7*d*e - 9*c*f - 3*d*f* 
x^2) - b^2*(3*c^2*f^2 + 6*c*d*f*(-7*e + 4*f*x^2) + d^2*(35*e^2 - 21*e*f*x^ 
2 + 15*f^2*x^4)))) - (105*I)*b^2*d*e*(-(b*e) + a*f)*(d*e - c*f)^2*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)]))/(105*a^2*(b/a)^(5/2)*d^2*f^5*Sqrt[a + b*x^2]*Sqrt[c + 
 d*x^2])
 

Rubi [F]

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.

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

\(\Big \downarrow \) 450

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

Input:

Int[(x^4*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2))/(e + f*x^2),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 450

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

Maple [A] (verified)

Time = 18.56 (sec) , antiderivative size = 826, normalized size of antiderivative = 1.05

method	result	size
risch	\(-\frac {x \left (-15 f^{2} x^{4} b^{2} d^{2}-3 a b \,d^{2} f^{2} x^{2}-24 b^{2} c d \,f^{2} x^{2}+21 b^{2} d^{2} e f \,x^{2}+4 a^{2} d^{2} f^{2}-9 a b c d \,f^{2}+7 a b \,d^{2} e f -3 b^{2} c^{2} f^{2}+42 b^{2} c d e f -35 b^{2} d^{2} e^{2}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{105 d \,b^{2} f^{3}}+\frac {\left (\frac {\left (4 a^{3} c \,d^{2} f^{4}-9 a^{2} b \,c^{2} d \,f^{4}+7 a^{2} b c \,d^{2} e \,f^{3}-3 a \,b^{2} c^{3} f^{4}-63 a \,b^{2} c^{2} d e \,f^{3}+175 a \,b^{2} c \,d^{2} e^{2} f^{2}-105 a \,b^{2} d^{3} e^{3} f +105 b^{3} c^{2} d \,e^{2} f^{2}-210 b^{3} c \,d^{2} e^{3} f +105 b^{3} d^{3} e^{4}\right ) \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 )}{f^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (8 a^{3} d^{3} f^{3}-19 a^{2} b c \,d^{2} f^{3}+14 a^{2} b \,d^{3} e \,f^{2}+9 a \,b^{2} c^{2} d \,f^{3}-49 a \,b^{2} c \,d^{2} e \,f^{2}+35 a \,b^{2} d^{3} e^{2} f -6 b^{3} c^{3} f^{3}-21 b^{3} c^{2} d e \,f^{2}+140 b^{3} c \,d^{2} e^{2} f -105 b^{3} d^{3} e^{3}\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 )}{f \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {105 e \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 ) d \,b^{2} \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} \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 )}}{105 b^{2} f^{3} d \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(826\)
default	\(\text {Expression too large to display}\)	\(2484\)
elliptic	\(\text {Expression too large to display}\)	\(3042\)

Input:

int(x^4*(b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/(f*x^2+e),x,method=_RETURNVERBOSE)
 

Output:

-1/105*x/d*(-15*b^2*d^2*f^2*x^4-3*a*b*d^2*f^2*x^2-24*b^2*c*d*f^2*x^2+21*b^ 
2*d^2*e*f*x^2+4*a^2*d^2*f^2-9*a*b*c*d*f^2+7*a*b*d^2*e*f-3*b^2*c^2*f^2+42*b 
^2*c*d*e*f-35*b^2*d^2*e^2)*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b^2/f^3+1/105/b 
^2/f^3/d*((4*a^3*c*d^2*f^4-9*a^2*b*c^2*d*f^4+7*a^2*b*c*d^2*e*f^3-3*a*b^2*c 
^3*f^4-63*a*b^2*c^2*d*e*f^3+175*a*b^2*c*d^2*e^2*f^2-105*a*b^2*d^3*e^3*f+10 
5*b^3*c^2*d*e^2*f^2-210*b^3*c*d^2*e^3*f+105*b^3*d^3*e^4)/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)*El 
lipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-1/f*(8*a^3*d^3*f^3-19*a^2 
*b*c*d^2*f^3+14*a^2*b*d^3*e*f^2+9*a*b^2*c^2*d*f^3-49*a*b^2*c*d^2*e*f^2+35* 
a*b^2*d^3*e^2*f-6*b^3*c^3*f^3-21*b^3*c^2*d*e*f^2+140*b^3*c*d^2*e^2*f-105*b 
^3*d^3*e^3)*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)))+105*e*(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)*d*b^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)*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. \[ \int \frac {x^4 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{e+f x^2} \, dx=\text {Timed out} \] Input:

integrate(x^4*(b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/(f*x^2+e),x, algorithm="fric 
as")
 

Output:

Timed out
 

Sympy [F]

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

integrate(x**4*(b*x**2+a)**(1/2)*(d*x**2+c)**(3/2)/(f*x**2+e),x)
 

Output:

Integral(x**4*sqrt(a + b*x**2)*(c + d*x**2)**(3/2)/(e + f*x**2), x)
 

Maxima [F]

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

integrate(x^4*(b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/(f*x^2+e),x, algorithm="maxi 
ma")
 

Output:

integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2)*x^4/(f*x^2 + e), x)
 

Giac [F]

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

integrate(x^4*(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)*x^4/(f*x^2 + e), x)
 

Mupad [F(-1)]

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

int((x^4*(a + b*x^2)^(1/2)*(c + d*x^2)^(3/2))/(e + f*x^2),x)
 

Output:

int((x^4*(a + b*x^2)^(1/2)*(c + d*x^2)^(3/2))/(e + f*x^2), x)
 

Reduce [F]

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

int(x^4*(b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/(f*x^2+e),x)
 

Output:

int(x^4*(b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/(f*x^2+e),x)