Integrand size = 32, antiderivative size = 665 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {b^2 x}{3 a (b c-a d) (b e-a f) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}+\frac {b^2 \left (2 b^2 c e-6 a b d e-5 a b c f+9 a^2 d f\right ) x}{3 a^2 (b c-a d)^2 (b e-a f)^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {d} \left (6 a^3 b d^3 e f-3 a^4 d^3 f^2+2 b^4 c^2 e (d e-c f)-a b^3 c \left (7 d^2 e^2-2 c d e f-5 c^2 f^2\right )-a^2 b^2 d \left (3 d^2 e^2-10 c d e f+10 c^2 f^2\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c} (b c-a d)^3 (b e-a f)^2 (d e-c f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {d} \left (3 a^3 d^3 f^2-3 a b^2 d^2 e (3 d e-4 c f)+6 a^2 b d^2 f (d e-2 c f)+b^3 c (d e-c f)^2\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a^2 (b c-a d)^3 (b e-a f) (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {c^{3/2} f^4 \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e (b e-a f)^2 (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
1/3*b^2*x/a/(-a*d+b*c)/(-a*f+b*e)/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)+1/3*b^2* (9*a^2*d*f-5*a*b*c*f-6*a*b*d*e+2*b^2*c*e)*x/a^2/(-a*d+b*c)^2/(-a*f+b*e)^2/ (b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)+1/3*d^(1/2)*(6*a^3*b*d^3*e*f-3*a^4*d^3*f^2 +2*b^4*c^2*e*(-c*f+d*e)-a*b^3*c*(-5*c^2*f^2-2*c*d*e*f+7*d^2*e^2)-a^2*b^2*d *(10*c^2*f^2-10*c*d*e*f+3*d^2*e^2))*(b*x^2+a)^(1/2)*EllipticE(d^(1/2)*x/c^ (1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))/a^2/c^(1/2)/(-a*d+b*c)^3/(-a*f+ b*e)^2/(-c*f+d*e)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/3*c^(1 /2)*d^(1/2)*(3*a^3*d^3*f^2-3*a*b^2*d^2*e*(-4*c*f+3*d*e)+6*a^2*b*d^2*f*(-2* c*f+d*e)+b^3*c*(-c*f+d*e)^2)*(b*x^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2 )*x/c^(1/2)),(1-b*c/a/d)^(1/2))/a^2/(-a*d+b*c)^3/(-a*f+b*e)/(-c*f+d*e)^2/( c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+c^(3/2)*f^4*(b*x^2+a)^(1/2) *EllipticPi(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),1-c*f/d/e,(1-b*c/a/d)^(1/2 ))/a/d^(1/2)/e/(-a*f+b*e)^2/(-c*f+d*e)^2/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/( d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 11.85 (sec) , antiderivative size = 1662, normalized size of antiderivative = 2.50 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx =\text {Too large to display} \] Input:
Integrate[1/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)*(e + f*x^2)),x]
Output:
((-I)*b*c*e*(-6*a^3*b*d^3*e*f + 3*a^4*d^3*f^2 + 2*b^4*c^2*e*(-(d*e) + c*f) + a*b^3*c*(7*d^2*e^2 - 2*c*d*e*f - 5*c^2*f^2) + a^2*b^2*d*(3*d^2*e^2 - 10 *c*d*e*f + 10*c^2*f^2))*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c ]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (Sqrt[b/a]*(3*a*b^6*c^3 *d*e^3*x - 8*a^2*b^5*c^2*d^2*e^3*x - 3*a^4*b^3*d^4*e^3*x - 3*a*b^6*c^4*e^2 *f*x + 2*a^2*b^5*c^3*d*e^2*f*x + 11*a^3*b^4*c^2*d^2*e^2*f*x + 6*a^5*b^2*d^ 4*e^2*f*x + 6*a^2*b^5*c^4*e*f^2*x - 11*a^3*b^4*c^3*d*e*f^2*x - 3*a^6*b*d^4 *e*f^2*x + 2*b^7*c^3*d*e^3*x^3 - 4*a*b^6*c^2*d^2*e^3*x^3 - 8*a^2*b^5*c*d^3 *e^3*x^3 - 6*a^3*b^4*d^4*e^3*x^3 - 2*b^7*c^4*e^2*f*x^3 - a*b^6*c^3*d*e^2*f *x^3 + 12*a^2*b^5*c^2*d^2*e^2*f*x^3 + 11*a^3*b^4*c*d^3*e^2*f*x^3 + 12*a^4* b^3*d^4*e^2*f*x^3 + 5*a*b^6*c^4*e*f^2*x^3 - 4*a^2*b^5*c^3*d*e*f^2*x^3 - 11 *a^3*b^4*c^2*d^2*e*f^2*x^3 - 6*a^5*b^2*d^4*e*f^2*x^3 + 2*b^7*c^2*d^2*e^3*x ^5 - 7*a*b^6*c*d^3*e^3*x^5 - 3*a^2*b^5*d^4*e^3*x^5 - 2*b^7*c^3*d*e^2*f*x^5 + 2*a*b^6*c^2*d^2*e^2*f*x^5 + 10*a^2*b^5*c*d^3*e^2*f*x^5 + 6*a^3*b^4*d^4* e^2*f*x^5 + 5*a*b^6*c^3*d*e*f^2*x^5 - 10*a^2*b^5*c^2*d^2*e*f^2*x^5 - 3*a^4 *b^3*d^4*e*f^2*x^5 - I*a*b^2*Sqrt[b/a]*c*(-(b*c) + a*d)*e*(-(d*e) + c*f)*( 2*b^2*c*e + 9*a^2*d*f - a*b*(6*d*e + 5*c*f))*(a + b*x^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)*a^4*b^3*Sqrt[b/a]*c^4*f^3*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Elli pticPi[(a*f)/(b*e), I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - (9*I)*a^7*(b...
Time = 1.06 (sec) , antiderivative size = 786, normalized size of antiderivative = 1.18, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {421, 25, 402, 25, 402, 25, 27, 400, 313, 320, 421, 25, 400, 313, 320, 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.
| \(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}-\frac {b \int -\frac {-b f x^2+b e-2 a f}{\left (b x^2+a\right )^{5/2} \left (d x^2+c\right )^{3/2}}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}+\frac {b \int \frac {-b f x^2+b e-2 a f}{\left (b x^2+a\right )^{5/2} \left (d x^2+c\right )^{3/2}}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}-\frac {\int -\frac {6 d f a^2-b (3 d e+5 c f) a+3 b d (b e-a f) x^2+2 b^2 c e}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}dx}{3 a (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {\int \frac {6 d f a^2-3 b d e a-5 b c f a+3 b d (b e-a f) x^2+2 b^2 c e}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{3/2}}dx}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b \left (\frac {\frac {b x \left (9 a^2 d f-5 a b c f-6 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}-\frac {\int -\frac {d \left (b \left (9 d f a^2-6 b d e a-5 b c f a+2 b^2 c e\right ) x^2+a \left (-6 d f a^2+b (3 d e+2 c f) a+b^2 c e\right )\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{a (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {\frac {\int \frac {d \left (b \left (9 d f a^2-6 b d e a-5 b c f a+2 b^2 c e\right ) x^2+a \left (-6 d f a^2+b (3 d e+2 c f) a+b^2 c e\right )\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{a (b c-a d)}+\frac {b x \left (9 a^2 d f-5 a b c f-6 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {\frac {d \int \frac {b \left (9 d f a^2-6 b d e a-5 b c f a+2 b^2 c e\right ) x^2+a \left (-6 d f a^2+b (3 d e+2 c f) a+b^2 c e\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{a (b c-a d)}+\frac {b x \left (9 a^2 d f-5 a b c f-6 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {b \left (\frac {\frac {d \left (\frac {\left (6 a^3 d^2 f-a^2 b d (3 d e-7 c f)-a b^2 c (5 c f+7 d e)+2 b^3 c^2 e\right ) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}-\frac {a b \left (15 a^2 d f-7 a b c f-9 a b d e+b^2 c e\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}\right )}{a (b c-a d)}+\frac {b x \left (9 a^2 d f-5 a b c f-6 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b \left (\frac {\frac {d \left (\frac {\sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (3 d e-7 c f)-a b^2 c (5 c f+7 d e)+2 b^3 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {a b \left (15 a^2 d f-7 a b c f-9 a b d e+b^2 c e\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}\right )}{a (b c-a d)}+\frac {b x \left (9 a^2 d f-5 a b c f-6 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {f^2 \int \frac {1}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2} \left (f x^2+e\right )}dx}{(b e-a f)^2}+\frac {b \left (\frac {\frac {b x \left (9 a^2 d f-5 a b c f-6 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}+\frac {d \left (\frac {\sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (3 d e-7 c f)-a b^2 c (5 c f+7 d e)+2 b^3 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} \left (15 a^2 d f-7 a b c f-9 a b d e+b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {f^2 \left (\frac {f^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(d e-c f)^2}-\frac {d \int -\frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{(d e-c f)^2}\right )}{(b e-a f)^2}+\frac {b \left (\frac {\frac {b x \left (9 a^2 d f-5 a b c f-6 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}+\frac {d \left (\frac {\sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (3 d e-7 c f)-a b^2 c (5 c f+7 d e)+2 b^3 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} \left (15 a^2 d f-7 a b c f-9 a b d e+b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {f^2 \left (\frac {f^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \int \frac {-d f x^2+d e-2 c f}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{(d e-c f)^2}\right )}{(b e-a f)^2}+\frac {b \left (\frac {\frac {b x \left (9 a^2 d f-5 a b c f-6 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}+\frac {d \left (\frac {\sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (3 d e-7 c f)-a b^2 c (5 c f+7 d e)+2 b^3 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} \left (15 a^2 d f-7 a b c f-9 a b d e+b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {f^2 \left (\frac {f^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {(a d f-2 b c f+b d e) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}-\frac {d (d e-c f) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}\right )}{(d e-c f)^2}\right )}{(b e-a f)^2}+\frac {b \left (\frac {\frac {b x \left (9 a^2 d f-5 a b c f-6 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}+\frac {d \left (\frac {\sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (3 d e-7 c f)-a b^2 c (5 c f+7 d e)+2 b^3 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} \left (15 a^2 d f-7 a b c f-9 a b d e+b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {f^2 \left (\frac {d \left (\frac {(a d f-2 b c f+b d e) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}-\frac {\sqrt {d} \sqrt {a+b x^2} (d e-c f) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(d e-c f)^2}+\frac {f^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(d e-c f)^2}\right )}{(b e-a f)^2}+\frac {b \left (\frac {\frac {b x \left (9 a^2 d f-5 a b c f-6 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}+\frac {d \left (\frac {\sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (3 d e-7 c f)-a b^2 c (5 c f+7 d e)+2 b^3 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} \left (15 a^2 d f-7 a b c f-9 a b d e+b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {f^2 \left (\frac {f^2 \int \frac {\sqrt {d x^2+c}}{\sqrt {b x^2+a} \left (f x^2+e\right )}dx}{(d e-c f)^2}+\frac {d \left (\frac {\sqrt {c} \sqrt {a+b x^2} (a d f-2 b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {d} \sqrt {a+b x^2} (d e-c f) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(d e-c f)^2}\right )}{(b e-a f)^2}+\frac {b \left (\frac {\frac {b x \left (9 a^2 d f-5 a b c f-6 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}+\frac {d \left (\frac {\sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (3 d e-7 c f)-a b^2 c (5 c f+7 d e)+2 b^3 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} \left (15 a^2 d f-7 a b c f-9 a b d e+b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\right )}{(b e-a f)^2}\) |
| \(\Big \downarrow \) 414 |
| \(\displaystyle \frac {b \left (\frac {\frac {b x \left (9 a^2 d f-5 a b c f-6 a b d e+2 b^2 c e\right )}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}+\frac {d \left (\frac {\sqrt {a+b x^2} \left (6 a^3 d^2 f-a^2 b d (3 d e-7 c f)-a b^2 c (5 c f+7 d e)+2 b^3 c^2 e\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} \left (15 a^2 d f-7 a b c f-9 a b d e+b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a (b c-a d)}}{3 a (b c-a d)}+\frac {b x (b e-a f)}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)}\right )}{(b e-a f)^2}+\frac {f^2 \left (\frac {c^{3/2} f^2 \sqrt {a+b x^2} \operatorname {EllipticPi}\left (1-\frac {c f}{d e},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} e \sqrt {c+d x^2} (d e-c f)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {d \left (\frac {\sqrt {c} \sqrt {a+b x^2} (a d f-2 b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {\sqrt {d} \sqrt {a+b x^2} (d e-c f) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{(d e-c f)^2}\right )}{(b e-a f)^2}\) |
Input:
Int[1/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)*(e + f*x^2)),x]
Output:
(b*((b*(b*e - a*f)*x)/(3*a*(b*c - a*d)*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2]) + ((b*(2*b^2*c*e - 6*a*b*d*e - 5*a*b*c*f + 9*a^2*d*f)*x)/(a*(b*c - a*d)*Sq rt[a + b*x^2]*Sqrt[c + d*x^2]) + (d*(((2*b^3*c^2*e + 6*a^3*d^2*f - a^2*b*d *(3*d*e - 7*c*f) - a*b^2*c*(7*d*e + 5*c*f))*Sqrt[a + b*x^2]*EllipticE[ArcT an[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[c]*Sqrt[d]*(b*c - a*d)*Sq rt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (b*Sqrt[c]*(b^2*c*e - 9*a*b*d*e - 7*a*b*c*f + 15*a^2*d*f)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(S qrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*(b*c - a*d)*Sqrt[(c*(a + b* x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])))/(a*(b*c - a*d)))/(3*a*(b*c - a*d ))))/(b*e - a*f)^2 + (f^2*((d*(-((Sqrt[d]*(d*e - c*f)*Sqrt[a + b*x^2]*Elli pticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[c]*(b*c - a*d)* Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])) + (Sqrt[c]*(b*d*e - 2*b*c*f + a*d*f)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^ 2))]*Sqrt[c + d*x^2])))/(d*e - c*f)^2 + (c^(3/2)*f^2*Sqrt[a + b*x^2]*Ellip ticPi[1 - (c*f)/(d*e), ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*S qrt[d]*e*(d*e - c*f)^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^ 2])))/(b*e - a*f)^2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^ (3/2)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(Sqrt[a + b*x^2]* Sqrt[c + d*x^2]), x], x] - Simp[(d*e - c*f)/(b*c - a*d) Int[Sqrt[a + b*x^ 2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] & & PosQ[d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
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]
Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[b^2/(b*c - a*d)^2 Int[(c + d*x^2)^(q + 2)*((e + f*x^2)^r/(a + b*x^2)), x], x] - Simp[d/(b*c - a*d)^2 Int[(c + d*x^2)^q*( e + f*x^2)^r*(2*b*c - a*d + b*d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, r} , x] && LtQ[q, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(2126\) vs. \(2(635)=1270\).
Time = 20.29 (sec) , antiderivative size = 2127, normalized size of antiderivative = 3.20
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(2127\) |
| default | \(\text {Expression too large to display}\) | \(4115\) |
Input:
int(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2)/(f*x^2+e),x,method=_RETURNVERBOSE)
Output:
((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(-(b*d*x^2+a*d )/c*d^3/(a*d-b*c)^3*x/(c*f-d*e)/((x^2+c/d)*(b*d*x^2+a*d))^(1/2)-1/3*b/a/(a *d-b*c)*x/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^ (1/2)/(x^2+a/b)^2-1/3*(b*d*x^2+b*c)*b^2/a^2/(a*d-b*c)^2*x*(10*a^2*d*f-5*a* b*c*f-7*a*b*d*e+2*b^2*c*e)/(a*f-b*e)/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/((x ^2+a/b)*(b*d*x^2+b*c))^(1/2)+10/3/(-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))*b^2/(a*d-b*c)/(a*f-b*e)/(a^2*d*f-a*b*c*f-a*b*d*e+b^2 *c*e)*d*f-5/3/(-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) )*b^3/(a*d-b*c)/a/(a*f-b*e)/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)*c*f-7/3/(-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))*b^3/(a*d-b*c)/a/ (a*f-b*e)/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)*d*e+2/3/(-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))*b^4/(a*d-b*c)/a^2/(a*f-b*e)/(a^2*d *f-a*b*c*f-a*b*d*e+b^2*c*e)*c*e+10/3*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*b^3/(a*d-b*c)^2/(a*f-b *e)/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)*f*EllipticE(x*(-b/a)^(1/2),(-1+(a*d+ b*c)/c/b)^(1/2))+f^3/(a*f-b*e)^2/(c*f-d*e)/e/(-b/a)^(1/2)*(1+b*x^2/a)^(...
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2)/(f*x^2+e),x, algorithm="fricas ")
Output:
Timed out
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {5}{2}} \left (c + d x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )}\, dx \] Input:
integrate(1/(b*x**2+a)**(5/2)/(d*x**2+c)**(3/2)/(f*x**2+e),x)
Output:
Integral(1/((a + b*x**2)**(5/2)*(c + d*x**2)**(3/2)*(e + f*x**2)), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2)/(f*x^2+e),x, algorithm="maxima ")
Output:
integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)^(3/2)*(f*x^2 + e)), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}} \,d x } \] Input:
integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2)/(f*x^2+e),x, algorithm="giac")
Output:
integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)^(3/2)*(f*x^2 + e)), x)
Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{5/2}\,{\left (d\,x^2+c\right )}^{3/2}\,\left (f\,x^2+e\right )} \,d x \] Input:
int(1/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)*(e + f*x^2)),x)
Output:
int(1/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)*(e + f*x^2)), x)
\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{3} d^{2} f \,x^{12}+3 a \,b^{2} d^{2} f \,x^{10}+2 b^{3} c d f \,x^{10}+b^{3} d^{2} e \,x^{10}+3 a^{2} b \,d^{2} f \,x^{8}+6 a \,b^{2} c d f \,x^{8}+3 a \,b^{2} d^{2} e \,x^{8}+b^{3} c^{2} f \,x^{8}+2 b^{3} c d e \,x^{8}+a^{3} d^{2} f \,x^{6}+6 a^{2} b c d f \,x^{6}+3 a^{2} b \,d^{2} e \,x^{6}+3 a \,b^{2} c^{2} f \,x^{6}+6 a \,b^{2} c d e \,x^{6}+b^{3} c^{2} e \,x^{6}+2 a^{3} c d f \,x^{4}+a^{3} d^{2} e \,x^{4}+3 a^{2} b \,c^{2} f \,x^{4}+6 a^{2} b c d e \,x^{4}+3 a \,b^{2} c^{2} e \,x^{4}+a^{3} c^{2} f \,x^{2}+2 a^{3} c d e \,x^{2}+3 a^{2} b \,c^{2} e \,x^{2}+a^{3} c^{2} e}d x \] Input:
int(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2)/(f*x^2+e),x)
Output:
int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**3*c**2*e + a**3*c**2*f*x**2 + 2*a**3*c*d*e*x**2 + 2*a**3*c*d*f*x**4 + a**3*d**2*e*x**4 + a**3*d**2*f*x** 6 + 3*a**2*b*c**2*e*x**2 + 3*a**2*b*c**2*f*x**4 + 6*a**2*b*c*d*e*x**4 + 6* a**2*b*c*d*f*x**6 + 3*a**2*b*d**2*e*x**6 + 3*a**2*b*d**2*f*x**8 + 3*a*b**2 *c**2*e*x**4 + 3*a*b**2*c**2*f*x**6 + 6*a*b**2*c*d*e*x**6 + 6*a*b**2*c*d*f *x**8 + 3*a*b**2*d**2*e*x**8 + 3*a*b**2*d**2*f*x**10 + b**3*c**2*e*x**6 + b**3*c**2*f*x**8 + 2*b**3*c*d*e*x**8 + 2*b**3*c*d*f*x**10 + b**3*d**2*e*x* *10 + b**3*d**2*f*x**12),x)