Integrand size = 32, antiderivative size = 717 \[ \int \left (d+e x+f x^2+g x^3\right ) \left (a+b x^2+c x^4\right )^{3/2} \, dx=-\frac {\left (18 b^3 c d-144 a b c^2 d-8 b^4 f+57 a b^2 c f-84 a^2 c^2 f\right ) x \sqrt {a+b x^2+c x^4}}{315 c^{5/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {3 \left (b^2-4 a c\right ) (2 c e-b g) \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{256 c^3}+\frac {x \left (9 b^2 c d+90 a c^2 d-4 b^3 f+9 a b c f+3 c \left (9 b c d-4 b^2 f+14 a c f\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{315 c^2}+\frac {(2 c e-b g) \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 c^2}+\frac {x \left (3 (3 c d+b f)+7 c f x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c}+\frac {g \left (a+b x^2+c x^4\right )^{5/2}}{10 c}+\frac {3 \left (b^2-4 a c\right )^2 (2 c e-b g) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{512 c^{7/2}}+\frac {\sqrt [4]{a} \left (18 b^3 c d-144 a b c^2 d-8 b^4 f+57 a b^2 c f-84 a^2 c^2 f\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{315 c^{11/4} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{a} \left (18 b^3 c d-144 a b c^2 d-8 b^4 f+57 a b^2 c f-84 a^2 c^2 f+\sqrt {a} \sqrt {c} \left (9 b^2 c d-180 a c^2 d-4 b^3 f+24 a b c f\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{630 c^{11/4} \sqrt {a+b x^2+c x^4}} \]
1/32*(-b*g+2*c*e)*(2*c*x^2+b)*(c*x^4+b*x^2+a)^(3/2)/c^2+1/63*x*(7*c*f*x^2+ 3*b*f+9*c*d)*(c*x^4+b*x^2+a)^(3/2)/c+1/10*g*(c*x^4+b*x^2+a)^(5/2)/c+3/512* (-4*a*c+b^2)^2*(-b*g+2*c*e)*arctanh(1/2*(2*c*x^2+b)/c^(1/2)/(c*x^4+b*x^2+a )^(1/2))/c^(7/2)-3/256*(-4*a*c+b^2)*(-b*g+2*c*e)*(2*c*x^2+b)*(c*x^4+b*x^2+ a)^(1/2)/c^3+1/315*x*(9*b^2*c*d+90*a*c^2*d-4*b^3*f+9*a*b*c*f+3*c*(14*a*c*f -4*b^2*f+9*b*c*d)*x^2)*(c*x^4+b*x^2+a)^(1/2)/c^2-1/315*(-84*a^2*c^2*f+57*a *b^2*c*f-144*a*b*c^2*d-8*b^4*f+18*b^3*c*d)*x*(c*x^4+b*x^2+a)^(1/2)/c^(5/2) /(a^(1/2)+x^2*c^(1/2))+1/315*a^(1/4)*(-84*a^2*c^2*f+57*a*b^2*c*f-144*a*b*c ^2*d-8*b^4*f+18*b^3*c*d)*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2* arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2* (2-b/a^(1/2)/c^(1/2))^(1/2))*(a^(1/2)+x^2*c^(1/2))*((c*x^4+b*x^2+a)/(a^(1/ 2)+x^2*c^(1/2))^2)^(1/2)/c^(11/4)/(c*x^4+b*x^2+a)^(1/2)-1/630*a^(1/4)*(cos (2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*El lipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))* (a^(1/2)+x^2*c^(1/2))*(18*b^3*c*d-144*a*b*c^2*d-8*b^4*f+57*a*b^2*c*f-84*a^ 2*c^2*f+(24*a*b*c*f-180*a*c^2*d-4*b^3*f+9*b^2*c*d)*a^(1/2)*c^(1/2))*((c*x^ 4+b*x^2+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/c^(11/4)/(c*x^4+b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.69 (sec) , antiderivative size = 2588, normalized size of antiderivative = 3.61 \[ \int \left (d+e x+f x^2+g x^3\right ) \left (a+b x^2+c x^4\right )^{3/2} \, dx=\text {Result too large to show} \]
(-2*Sqrt[c]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*(a + b*x^2 + c*x^4)*(-945*b^4* g + 2*b^3*c*(945*e + x*(512*f + 315*g*x)) - 12*b^2*c*(-525*a*g + c*x*(192* d + 105*e*x + 64*f*x^2 + 42*g*x^3)) - 8*b*c^2*(3*a*(525*e + 256*f*x + 147* g*x^2) + 2*c*x^3*(1152*d + 945*e*x + 800*f*x^2 + 693*g*x^3)) - 16*c^2*(504 *a^2*g + 2*c^2*x^5*(360*d + 7*x*(45*e + 40*f*x + 36*g*x^2)) + a*c*x*(2160* d + 7*x*(225*e + 16*x*(11*f + 9*g*x))))) + (2304*I)*Sqrt[2]*b^3*c^(3/2)*(b - Sqrt[b^2 - 4*a*c])*d*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b ^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(EllipticE[I*Arc Sinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/( b - Sqrt[b^2 - 4*a*c])] - EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]) + (18432 *I)*Sqrt[2]*a*b*c^(5/2)*(-b + Sqrt[b^2 - 4*a*c])*d*Sqrt[(b + Sqrt[b^2 - 4* a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x] , (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - EllipticF[I*ArcSinh[S qrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sq rt[b^2 - 4*a*c])]) + (7296*I)*Sqrt[2]*a*b^2*c^(3/2)*(b - Sqrt[b^2 - 4*a*c] )*f*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/ (b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*...
Time = 1.25 (sec) , antiderivative size = 693, normalized size of antiderivative = 0.97, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {2202, 1490, 1490, 25, 1511, 27, 1416, 1509, 1576, 1160, 1087, 1087, 1092, 219}
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.
| \(\displaystyle \int \left (a+b x^2+c x^4\right )^{3/2} \left (d+e x+f x^2+g x^3\right ) \, dx\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle \int \left (f x^2+d\right ) \left (c x^4+b x^2+a\right )^{3/2}dx+\int x \left (g x^2+e\right ) \left (c x^4+b x^2+a\right )^{3/2}dx\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {\int \left (\left (-4 f b^2+9 c d b+14 a c f\right ) x^2+a (18 c d-b f)\right ) \sqrt {c x^4+b x^2+a}dx}{21 c}+\int x \left (g x^2+e\right ) \left (c x^4+b x^2+a\right )^{3/2}dx+\frac {x \left (a+b x^2+c x^4\right )^{3/2} \left (3 (b f+3 c d)+7 c f x^2\right )}{63 c}\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {\frac {\int -\frac {\left (-8 f b^4+18 c d b^3+57 a c f b^2-144 a c^2 d b-84 a^2 c^2 f\right ) x^2+a \left (-4 f b^3+9 c d b^2+24 a c f b-180 a c^2 d\right )}{\sqrt {c x^4+b x^2+a}}dx}{15 c}+\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c f-4 b^2 f+9 b c d\right )+9 a b c f+90 a c^2 d-4 b^3 f+9 b^2 c d\right )}{15 c}}{21 c}+\int x \left (g x^2+e\right ) \left (c x^4+b x^2+a\right )^{3/2}dx+\frac {x \left (a+b x^2+c x^4\right )^{3/2} \left (3 (b f+3 c d)+7 c f x^2\right )}{63 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c f-4 b^2 f+9 b c d\right )+9 a b c f+90 a c^2 d-4 b^3 f+9 b^2 c d\right )}{15 c}-\frac {\int \frac {\left (-8 f b^4+18 c d b^3+57 a c f b^2-144 a c^2 d b-84 a^2 c^2 f\right ) x^2+a \left (-4 f b^3+9 c d b^2+24 a c f b-180 a c^2 d\right )}{\sqrt {c x^4+b x^2+a}}dx}{15 c}}{21 c}+\int x \left (g x^2+e\right ) \left (c x^4+b x^2+a\right )^{3/2}dx+\frac {x \left (a+b x^2+c x^4\right )^{3/2} \left (3 (b f+3 c d)+7 c f x^2\right )}{63 c}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c f-4 b^2 f+9 b c d\right )+9 a b c f+90 a c^2 d-4 b^3 f+9 b^2 c d\right )}{15 c}-\frac {\frac {\sqrt {a} \left (-84 a^2 c^2 f+57 a b^2 c f+\sqrt {a} \sqrt {c} \left (24 a b c f-180 a c^2 d-4 b^3 f+9 b^2 c d\right )-144 a b c^2 d-8 b^4 f+18 b^3 c d\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\sqrt {a} \left (-84 a^2 c^2 f+57 a b^2 c f-144 a b c^2 d-8 b^4 f+18 b^3 c d\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{15 c}}{21 c}+\int x \left (g x^2+e\right ) \left (c x^4+b x^2+a\right )^{3/2}dx+\frac {x \left (a+b x^2+c x^4\right )^{3/2} \left (3 (b f+3 c d)+7 c f x^2\right )}{63 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c f-4 b^2 f+9 b c d\right )+9 a b c f+90 a c^2 d-4 b^3 f+9 b^2 c d\right )}{15 c}-\frac {\frac {\sqrt {a} \left (-84 a^2 c^2 f+57 a b^2 c f+\sqrt {a} \sqrt {c} \left (24 a b c f-180 a c^2 d-4 b^3 f+9 b^2 c d\right )-144 a b c^2 d-8 b^4 f+18 b^3 c d\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\left (-84 a^2 c^2 f+57 a b^2 c f-144 a b c^2 d-8 b^4 f+18 b^3 c d\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{15 c}}{21 c}+\int x \left (g x^2+e\right ) \left (c x^4+b x^2+a\right )^{3/2}dx+\frac {x \left (a+b x^2+c x^4\right )^{3/2} \left (3 (b f+3 c d)+7 c f x^2\right )}{63 c}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c f-4 b^2 f+9 b c d\right )+9 a b c f+90 a c^2 d-4 b^3 f+9 b^2 c d\right )}{15 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (-84 a^2 c^2 f+57 a b^2 c f+\sqrt {a} \sqrt {c} \left (24 a b c f-180 a c^2 d-4 b^3 f+9 b^2 c d\right )-144 a b c^2 d-8 b^4 f+18 b^3 c d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (-84 a^2 c^2 f+57 a b^2 c f-144 a b c^2 d-8 b^4 f+18 b^3 c d\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{15 c}}{21 c}+\int x \left (g x^2+e\right ) \left (c x^4+b x^2+a\right )^{3/2}dx+\frac {x \left (a+b x^2+c x^4\right )^{3/2} \left (3 (b f+3 c d)+7 c f x^2\right )}{63 c}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \int x \left (g x^2+e\right ) \left (c x^4+b x^2+a\right )^{3/2}dx+\frac {\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c f-4 b^2 f+9 b c d\right )+9 a b c f+90 a c^2 d-4 b^3 f+9 b^2 c d\right )}{15 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (-84 a^2 c^2 f+57 a b^2 c f+\sqrt {a} \sqrt {c} \left (24 a b c f-180 a c^2 d-4 b^3 f+9 b^2 c d\right )-144 a b c^2 d-8 b^4 f+18 b^3 c d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (-84 a^2 c^2 f+57 a b^2 c f-144 a b c^2 d-8 b^4 f+18 b^3 c d\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}}{21 c}+\frac {x \left (a+b x^2+c x^4\right )^{3/2} \left (3 (b f+3 c d)+7 c f x^2\right )}{63 c}\) |
| \(\Big \downarrow \) 1576 |
| \(\displaystyle \frac {1}{2} \int \left (g x^2+e\right ) \left (c x^4+b x^2+a\right )^{3/2}dx^2+\frac {\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c f-4 b^2 f+9 b c d\right )+9 a b c f+90 a c^2 d-4 b^3 f+9 b^2 c d\right )}{15 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (-84 a^2 c^2 f+57 a b^2 c f+\sqrt {a} \sqrt {c} \left (24 a b c f-180 a c^2 d-4 b^3 f+9 b^2 c d\right )-144 a b c^2 d-8 b^4 f+18 b^3 c d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (-84 a^2 c^2 f+57 a b^2 c f-144 a b c^2 d-8 b^4 f+18 b^3 c d\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}}{21 c}+\frac {x \left (a+b x^2+c x^4\right )^{3/2} \left (3 (b f+3 c d)+7 c f x^2\right )}{63 c}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {1}{2} \left (\frac {(2 c e-b g) \int \left (c x^4+b x^2+a\right )^{3/2}dx^2}{2 c}+\frac {g \left (a+b x^2+c x^4\right )^{5/2}}{5 c}\right )+\frac {\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c f-4 b^2 f+9 b c d\right )+9 a b c f+90 a c^2 d-4 b^3 f+9 b^2 c d\right )}{15 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (-84 a^2 c^2 f+57 a b^2 c f+\sqrt {a} \sqrt {c} \left (24 a b c f-180 a c^2 d-4 b^3 f+9 b^2 c d\right )-144 a b c^2 d-8 b^4 f+18 b^3 c d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (-84 a^2 c^2 f+57 a b^2 c f-144 a b c^2 d-8 b^4 f+18 b^3 c d\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}}{21 c}+\frac {x \left (a+b x^2+c x^4\right )^{3/2} \left (3 (b f+3 c d)+7 c f x^2\right )}{63 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{2} \left (\frac {(2 c e-b g) \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^4+b x^2+a}dx^2}{16 c}\right )}{2 c}+\frac {g \left (a+b x^2+c x^4\right )^{5/2}}{5 c}\right )+\frac {\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c f-4 b^2 f+9 b c d\right )+9 a b c f+90 a c^2 d-4 b^3 f+9 b^2 c d\right )}{15 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (-84 a^2 c^2 f+57 a b^2 c f+\sqrt {a} \sqrt {c} \left (24 a b c f-180 a c^2 d-4 b^3 f+9 b^2 c d\right )-144 a b c^2 d-8 b^4 f+18 b^3 c d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (-84 a^2 c^2 f+57 a b^2 c f-144 a b c^2 d-8 b^4 f+18 b^3 c d\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}}{21 c}+\frac {x \left (a+b x^2+c x^4\right )^{3/2} \left (3 (b f+3 c d)+7 c f x^2\right )}{63 c}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{2} \left (\frac {(2 c e-b g) \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx^2}{8 c}\right )}{16 c}\right )}{2 c}+\frac {g \left (a+b x^2+c x^4\right )^{5/2}}{5 c}\right )+\frac {\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c f-4 b^2 f+9 b c d\right )+9 a b c f+90 a c^2 d-4 b^3 f+9 b^2 c d\right )}{15 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (-84 a^2 c^2 f+57 a b^2 c f+\sqrt {a} \sqrt {c} \left (24 a b c f-180 a c^2 d-4 b^3 f+9 b^2 c d\right )-144 a b c^2 d-8 b^4 f+18 b^3 c d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (-84 a^2 c^2 f+57 a b^2 c f-144 a b c^2 d-8 b^4 f+18 b^3 c d\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}}{21 c}+\frac {x \left (a+b x^2+c x^4\right )^{3/2} \left (3 (b f+3 c d)+7 c f x^2\right )}{63 c}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1}{2} \left (\frac {(2 c e-b g) \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-x^4}d\frac {2 c x^2+b}{\sqrt {c x^4+b x^2+a}}}{4 c}\right )}{16 c}\right )}{2 c}+\frac {g \left (a+b x^2+c x^4\right )^{5/2}}{5 c}\right )+\frac {\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c f-4 b^2 f+9 b c d\right )+9 a b c f+90 a c^2 d-4 b^3 f+9 b^2 c d\right )}{15 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (-84 a^2 c^2 f+57 a b^2 c f+\sqrt {a} \sqrt {c} \left (24 a b c f-180 a c^2 d-4 b^3 f+9 b^2 c d\right )-144 a b c^2 d-8 b^4 f+18 b^3 c d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (-84 a^2 c^2 f+57 a b^2 c f-144 a b c^2 d-8 b^4 f+18 b^3 c d\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}}{21 c}+\frac {x \left (a+b x^2+c x^4\right )^{3/2} \left (3 (b f+3 c d)+7 c f x^2\right )}{63 c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {x \sqrt {a+b x^2+c x^4} \left (3 c x^2 \left (14 a c f-4 b^2 f+9 b c d\right )+9 a b c f+90 a c^2 d-4 b^3 f+9 b^2 c d\right )}{15 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (-84 a^2 c^2 f+57 a b^2 c f+\sqrt {a} \sqrt {c} \left (24 a b c f-180 a c^2 d-4 b^3 f+9 b^2 c d\right )-144 a b c^2 d-8 b^4 f+18 b^3 c d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (-84 a^2 c^2 f+57 a b^2 c f-144 a b c^2 d-8 b^4 f+18 b^3 c d\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}}{21 c}+\frac {1}{2} \left (\frac {(2 c e-b g) \left (\frac {\left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2}}\right )}{16 c}\right )}{2 c}+\frac {g \left (a+b x^2+c x^4\right )^{5/2}}{5 c}\right )+\frac {x \left (a+b x^2+c x^4\right )^{3/2} \left (3 (b f+3 c d)+7 c f x^2\right )}{63 c}\) |
(x*(3*(3*c*d + b*f) + 7*c*f*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(63*c) + ((g*( a + b*x^2 + c*x^4)^(5/2))/(5*c) + ((2*c*e - b*g)*(((b + 2*c*x^2)*(a + b*x^ 2 + c*x^4)^(3/2))/(8*c) - (3*(b^2 - 4*a*c)*(((b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(4*c) - ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(8*c^(3/2))))/(16*c)))/(2*c))/2 + ((x*(9*b^2*c*d + 90*a *c^2*d - 4*b^3*f + 9*a*b*c*f + 3*c*(9*b*c*d - 4*b^2*f + 14*a*c*f)*x^2)*Sqr t[a + b*x^2 + c*x^4])/(15*c) - (-(((18*b^3*c*d - 144*a*b*c^2*d - 8*b^4*f + 57*a*b^2*c*f - 84*a^2*c^2*f)*(-((x*Sqrt[a + b*x^2 + c*x^4])/(Sqrt[a] + Sq rt[c]*x^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(S qrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/( Sqrt[a]*Sqrt[c]))/4])/(c^(1/4)*Sqrt[a + b*x^2 + c*x^4])))/Sqrt[c]) + (a^(1 /4)*(18*b^3*c*d - 144*a*b*c^2*d - 8*b^4*f + 57*a*b^2*c*f - 84*a^2*c^2*f + Sqrt[a]*Sqrt[c]*(9*b^2*c*d - 180*a*c^2*d - 4*b^3*f + 24*a*b*c*f))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ellipt icF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*c^(3/4 )*Sqrt[a + b*x^2 + c*x^4]))/(15*c))/(21*c)
3.2.3.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(2*b*e*p + c*d*(4*p + 3) + c*e*(4*p + 1)*x^2)*((a + b*x^2 + c *x^4)^p/(c*(4*p + 1)*(4*p + 3))), x] + Simp[2*(p/(c*(4*p + 1)*(4*p + 3))) Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) - b^2*e*(2*p + 1))*x^2, x]*(a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] , x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b *x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] && !PolyQ[Pn, x^2]
Time = 5.79 (sec) , antiderivative size = 1209, normalized size of antiderivative = 1.69
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1209\) |
| elliptic | \(\text {Expression too large to display}\) | \(1376\) |
| default | \(\text {Expression too large to display}\) | \(1580\) |
1/80640/c^3*(8064*c^4*g*x^8+8960*c^4*f*x^7+11088*b*c^3*g*x^6+10080*c^4*e*x ^6+12800*b*c^3*f*x^5+11520*c^4*d*x^5+16128*a*c^3*g*x^4+504*b^2*c^2*g*x^4+1 5120*b*c^3*e*x^4+19712*a*c^3*f*x^3+768*b^2*c^2*f*x^3+18432*b*c^3*d*x^3+352 8*a*b*c^2*g*x^2+25200*a*c^3*e*x^2-630*b^3*c*g*x^2+1260*b^2*c^2*e*x^2+6144* a*b*c^2*f*x+34560*a*c^3*d*x-1024*b^3*c*f*x+2304*b^2*c^2*d*x+8064*a^2*c^2*g -6300*a*b^2*c*g+12600*a*b*c^2*e+945*b^4*g-1890*b^3*c*e)*(c*x^4+b*x^2+a)^(1 /2)-1/80640/c^3*(-11520*a^2*c^3*d*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2 )*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a* x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2) ^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))+1536*a^2*b *c^2*f*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/ 2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^ (1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2* b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))-256*a*b^3*c*f*2^(1/2)/((-b+(-4*a*c+b^ 2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a *c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)* ((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^ (1/2))+576*a*b^2*c^2*d*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+ (-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/ (c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/...
Time = 0.31 (sec) , antiderivative size = 911, normalized size of antiderivative = 1.27 \[ \int \left (d+e x+f x^2+g x^3\right ) \left (a+b x^2+c x^4\right )^{3/2} \, dx=-\frac {512 \, \sqrt {\frac {1}{2}} {\left ({\left (18 \, {\left (b^{3} c^{2} - 8 \, a b c^{3}\right )} d - {\left (8 \, b^{4} c - 57 \, a b^{2} c^{2} + 84 \, a^{2} c^{3}\right )} f\right )} x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (18 \, {\left (b^{4} c - 8 \, a b^{2} c^{2}\right )} d - {\left (8 \, b^{5} - 57 \, a b^{3} c + 84 \, a^{2} b c^{2}\right )} f\right )} x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - 512 \, \sqrt {\frac {1}{2}} {\left ({\left (9 \, {\left (2 \, b^{3} c^{2} + 20 \, a c^{4} - {\left (16 \, a b + b^{2}\right )} c^{3}\right )} d - {\left (8 \, b^{4} c + 12 \, {\left (7 \, a^{2} + 2 \, a b\right )} c^{3} - {\left (57 \, a b^{2} + 4 \, b^{3}\right )} c^{2}\right )} f\right )} x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (9 \, {\left (2 \, b^{4} c - 20 \, a b c^{3} - {\left (16 \, a b^{2} - b^{3}\right )} c^{2}\right )} d - {\left (8 \, b^{5} + 12 \, {\left (7 \, a^{2} b - 2 \, a b^{2}\right )} c^{2} - {\left (57 \, a b^{3} - 4 \, b^{4}\right )} c\right )} f\right )} x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) + 945 \, {\left (2 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} e - {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} g\right )} \sqrt {c} x \log \left (8 \, c^{2} x^{4} + 8 \, b c x^{2} + b^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} + 4 \, a c\right ) - 4 \, {\left (8064 \, c^{5} g x^{9} + 8960 \, c^{5} f x^{8} + 1008 \, {\left (10 \, c^{5} e + 11 \, b c^{4} g\right )} x^{7} + 1280 \, {\left (9 \, c^{5} d + 10 \, b c^{4} f\right )} x^{6} + 504 \, {\left (30 \, b c^{4} e + {\left (b^{2} c^{3} + 32 \, a c^{4}\right )} g\right )} x^{5} + 256 \, {\left (72 \, b c^{4} d + {\left (3 \, b^{2} c^{3} + 77 \, a c^{4}\right )} f\right )} x^{4} + 126 \, {\left (10 \, {\left (b^{2} c^{3} + 20 \, a c^{4}\right )} e - {\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} g\right )} x^{3} + 256 \, {\left (9 \, {\left (b^{2} c^{3} + 15 \, a c^{4}\right )} d - 4 \, {\left (b^{3} c^{2} - 6 \, a b c^{3}\right )} f\right )} x^{2} - 4608 \, {\left (b^{3} c^{2} - 8 \, a b c^{3}\right )} d + 256 \, {\left (8 \, b^{4} c - 57 \, a b^{2} c^{2} + 84 \, a^{2} c^{3}\right )} f - 63 \, {\left (10 \, {\left (3 \, b^{3} c^{2} - 20 \, a b c^{3}\right )} e - {\left (15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3}\right )} g\right )} x\right )} \sqrt {c x^{4} + b x^{2} + a}}{322560 \, c^{4} x} \]
-1/322560*(512*sqrt(1/2)*((18*(b^3*c^2 - 8*a*b*c^3)*d - (8*b^4*c - 57*a*b^ 2*c^2 + 84*a^2*c^3)*f)*x*sqrt((b^2 - 4*a*c)/c^2) - (18*(b^4*c - 8*a*b^2*c^ 2)*d - (8*b^5 - 57*a*b^3*c + 84*a^2*b*c^2)*f)*x)*sqrt(c)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)*elliptic_e(arcsin(sqrt(1/2)*sqrt((c*sqrt((b^2 - 4*a *c)/c^2) - b)/c)/x), 1/2*(b*c*sqrt((b^2 - 4*a*c)/c^2) + b^2 - 2*a*c)/(a*c) ) - 512*sqrt(1/2)*((9*(2*b^3*c^2 + 20*a*c^4 - (16*a*b + b^2)*c^3)*d - (8*b ^4*c + 12*(7*a^2 + 2*a*b)*c^3 - (57*a*b^2 + 4*b^3)*c^2)*f)*x*sqrt((b^2 - 4 *a*c)/c^2) - (9*(2*b^4*c - 20*a*b*c^3 - (16*a*b^2 - b^3)*c^2)*d - (8*b^5 + 12*(7*a^2*b - 2*a*b^2)*c^2 - (57*a*b^3 - 4*b^4)*c)*f)*x)*sqrt(c)*sqrt((c* sqrt((b^2 - 4*a*c)/c^2) - b)/c)*elliptic_f(arcsin(sqrt(1/2)*sqrt((c*sqrt(( b^2 - 4*a*c)/c^2) - b)/c)/x), 1/2*(b*c*sqrt((b^2 - 4*a*c)/c^2) + b^2 - 2*a *c)/(a*c)) + 945*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*e - (b^5 - 8*a*b^3* c + 16*a^2*b*c^2)*g)*sqrt(c)*x*log(8*c^2*x^4 + 8*b*c*x^2 + b^2 - 4*sqrt(c* x^4 + b*x^2 + a)*(2*c*x^2 + b)*sqrt(c) + 4*a*c) - 4*(8064*c^5*g*x^9 + 8960 *c^5*f*x^8 + 1008*(10*c^5*e + 11*b*c^4*g)*x^7 + 1280*(9*c^5*d + 10*b*c^4*f )*x^6 + 504*(30*b*c^4*e + (b^2*c^3 + 32*a*c^4)*g)*x^5 + 256*(72*b*c^4*d + (3*b^2*c^3 + 77*a*c^4)*f)*x^4 + 126*(10*(b^2*c^3 + 20*a*c^4)*e - (5*b^3*c^ 2 - 28*a*b*c^3)*g)*x^3 + 256*(9*(b^2*c^3 + 15*a*c^4)*d - 4*(b^3*c^2 - 6*a* b*c^3)*f)*x^2 - 4608*(b^3*c^2 - 8*a*b*c^3)*d + 256*(8*b^4*c - 57*a*b^2*c^2 + 84*a^2*c^3)*f - 63*(10*(3*b^3*c^2 - 20*a*b*c^3)*e - (15*b^4*c - 100*...
\[ \int \left (d+e x+f x^2+g x^3\right ) \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}} \left (d + e x + f x^{2} + g x^{3}\right )\, dx \]
\[ \int \left (d+e x+f x^2+g x^3\right ) \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (g x^{3} + f x^{2} + e x + d\right )} \,d x } \]
\[ \int \left (d+e x+f x^2+g x^3\right ) \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} {\left (g x^{3} + f x^{2} + e x + d\right )} \,d x } \]
Timed out. \[ \int \left (d+e x+f x^2+g x^3\right ) \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int {\left (c\,x^4+b\,x^2+a\right )}^{3/2}\,\left (g\,x^3+f\,x^2+e\,x+d\right ) \,d x \]