Integrand size = 32, antiderivative size = 680 \[ \int \frac {d+e x+f x^2+g x^3}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx=\frac {x \left (b^2 d-2 a c d-a b f+c (b d-2 a f) x^2\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}-\frac {b e-2 a g+(2 c e-b g) x^2}{3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}+\frac {4 (2 c e-b g) \left (b+2 c x^2\right )}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x^2+c x^4}}+\frac {x \left (2 b^4 d-17 a b^2 c d+20 a^2 c^2 d+a b^3 f+4 a^2 b c f+c \left (2 b^3 d-16 a b c d+a b^2 f+12 a^2 c f\right ) x^2\right )}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x^2+c x^4}}-\frac {\sqrt {c} \left (2 b^3 d-16 a b c d+a b^2 f+12 a^2 c f\right ) x \sqrt {a+b x^2+c x^4}}{3 a^2 \left (b^2-4 a c\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {\sqrt [4]{c} \left (2 b^3 d-16 a b c d+a b^2 f+12 a^2 c f\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 a^{7/4} \left (b^2-4 a c\right )^2 \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{c} \left (2 b^2 d-3 \sqrt {a} b \sqrt {c} d-10 a c d+a b f+6 a^{3/2} \sqrt {c} f\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{6 a^{7/4} \left (b-2 \sqrt {a} \sqrt {c}\right ) \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \] Output:
1/3*x*(b^2*d-2*a*c*d-a*b*f+c*(-2*a*f+b*d)*x^2)/a/(-4*a*c+b^2)/(c*x^4+b*x^2 +a)^(3/2)-1/3*(b*e-2*a*g+(-b*g+2*c*e)*x^2)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^(3 /2)+4/3*(-b*g+2*c*e)*(2*c*x^2+b)/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)^(1/2)+1/3* x*(2*b^4*d-17*a*b^2*c*d+20*a^2*c^2*d+a*b^3*f+4*a^2*b*c*f+c*(12*a^2*c*f+a*b ^2*f-16*a*b*c*d+2*b^3*d)*x^2)/a^2/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)^(1/2)-1/3 *c^(1/2)*(12*a^2*c*f+a*b^2*f-16*a*b*c*d+2*b^3*d)*x*(c*x^4+b*x^2+a)^(1/2)/a ^2/(-4*a*c+b^2)^2/(a^(1/2)+c^(1/2)*x^2)+1/3*c^(1/4)*(12*a^2*c*f+a*b^2*f-16 *a*b*c*d+2*b^3*d)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)* x^2)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/ c^(1/2))^(1/2))/a^(7/4)/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)^(1/2)-1/6*c^(1/4)*( 2*b^2*d-3*a^(1/2)*b*c^(1/2)*d-10*a*c*d+a*b*f+6*a^(3/2)*c^(1/2)*f)*(a^(1/2) +c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*InverseJacob iAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/a^(7/4)/( b-2*a^(1/2)*c^(1/2))/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 12.97 (sec) , antiderivative size = 598, normalized size of antiderivative = 0.88 \[ \int \frac {d+e x+f x^2+g x^3}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx=\frac {-4 a \left (b^2-4 a c\right ) \left (-2 a^2 g-b d x \left (b+c x^2\right )+2 a c x (d+x (e+f x))+a b (e+x (f-g x))\right )+4 \left (a+b x^2+c x^4\right ) \left (2 b^3 d x \left (b+c x^2\right )+a b x \left (-17 b c d+b^2 f-16 c^2 d x^2+b c f x^2\right )+4 a^2 \left (-b^2 g+c^2 x (5 d+x (4 e+3 f x))+b c (2 e+x (f-2 g x))\right )\right )+\frac {i \sqrt {2} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \left (a+b x^2+c x^4\right ) \left (-\left (\left (-b+\sqrt {b^2-4 a c}\right ) \left (2 b^3 d-16 a b c d+a b^2 f+12 a^2 c f\right ) E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )+\left (-2 b^4 d+b^3 \left (2 \sqrt {b^2-4 a c} d-a f\right )+4 a b c \left (-4 \sqrt {b^2-4 a c} d+a f\right )+a b^2 \left (18 c d+\sqrt {b^2-4 a c} f\right )+4 a^2 c \left (-10 c d+3 \sqrt {b^2-4 a c} f\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{\sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}}}}{12 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )^{3/2}} \] Input:
Integrate[(d + e*x + f*x^2 + g*x^3)/(a + b*x^2 + c*x^4)^(5/2),x]
Output:
(-4*a*(b^2 - 4*a*c)*(-2*a^2*g - b*d*x*(b + c*x^2) + 2*a*c*x*(d + x*(e + f* x)) + a*b*(e + x*(f - g*x))) + 4*(a + b*x^2 + c*x^4)*(2*b^3*d*x*(b + c*x^2 ) + a*b*x*(-17*b*c*d + b^2*f - 16*c^2*d*x^2 + b*c*f*x^2) + 4*a^2*(-(b^2*g) + c^2*x*(5*d + x*(4*e + 3*f*x)) + b*c*(2*e + x*(f - 2*g*x)))) + (I*Sqrt[2 ]*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])]*(a + b*x^2 + c*x^4)*(-((-b + Sqrt[b^2 - 4*a*c])*(2*b^3*d - 16*a*b*c*d + a*b^2*f + 12*a^2*c*f)*EllipticE[I*ArcSin h[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])]) + (-2*b^4*d + b^3*(2*Sqrt[b^2 - 4*a*c]*d - a*f) + 4* a*b*c*(-4*Sqrt[b^2 - 4*a*c]*d + a*f) + a*b^2*(18*c*d + Sqrt[b^2 - 4*a*c]*f ) + 4*a^2*c*(-10*c*d + 3*Sqrt[b^2 - 4*a*c]*f))*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])]))/Sqrt[c/(b + Sqrt[b^2 - 4*a*c])])/(12*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)^(3/2))
Time = 1.04 (sec) , antiderivative size = 650, normalized size of antiderivative = 0.96, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2202, 1492, 25, 1492, 27, 1511, 27, 1416, 1509, 1576, 1159, 1088}
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.
| \(\displaystyle \int \frac {d+e x+f x^2+g x^3}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle \int \frac {f x^2+d}{\left (c x^4+b x^2+a\right )^{5/2}}dx+\int \frac {x \left (g x^2+e\right )}{\left (c x^4+b x^2+a\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle -\frac {\int -\frac {2 d b^2+a f b+3 c (b d-2 a f) x^2-10 a c d}{\left (c x^4+b x^2+a\right )^{3/2}}dx}{3 a \left (b^2-4 a c\right )}+\int \frac {x \left (g x^2+e\right )}{\left (c x^4+b x^2+a\right )^{5/2}}dx+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {2 d b^2+a f b+3 c (b d-2 a f) x^2-10 a c d}{\left (c x^4+b x^2+a\right )^{3/2}}dx}{3 a \left (b^2-4 a c\right )}+\int \frac {x \left (g x^2+e\right )}{\left (c x^4+b x^2+a\right )^{5/2}}dx+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle \frac {\frac {x \left (c x^2 \left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right )+4 a^2 b c f+20 a^2 c^2 d+a b^3 f-17 a b^2 c d+2 b^4 d\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\int \frac {c \left (\left (2 d b^3+a f b^2-16 a c d b+12 a^2 c f\right ) x^2+a \left (d b^2+8 a f b-20 a c d\right )\right )}{\sqrt {c x^4+b x^2+a}}dx}{a \left (b^2-4 a c\right )}}{3 a \left (b^2-4 a c\right )}+\int \frac {x \left (g x^2+e\right )}{\left (c x^4+b x^2+a\right )^{5/2}}dx+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {x \left (c x^2 \left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right )+4 a^2 b c f+20 a^2 c^2 d+a b^3 f-17 a b^2 c d+2 b^4 d\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {c \int \frac {\left (2 d b^3+a f b^2-16 a c d b+12 a^2 c f\right ) x^2+a \left (d b^2+8 a f b-20 a c d\right )}{\sqrt {c x^4+b x^2+a}}dx}{a \left (b^2-4 a c\right )}}{3 a \left (b^2-4 a c\right )}+\int \frac {x \left (g x^2+e\right )}{\left (c x^4+b x^2+a\right )^{5/2}}dx+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {\frac {x \left (c x^2 \left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right )+4 a^2 b c f+20 a^2 c^2 d+a b^3 f-17 a b^2 c d+2 b^4 d\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {c \left (\frac {\sqrt {a} \left (12 a^2 c f+\sqrt {a} \sqrt {c} \left (8 a b f-20 a c d+b^2 d\right )+a b^2 f-16 a b c d+2 b^3 d\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\sqrt {a} \left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}\right )}{a \left (b^2-4 a c\right )}}{3 a \left (b^2-4 a c\right )}+\int \frac {x \left (g x^2+e\right )}{\left (c x^4+b x^2+a\right )^{5/2}}dx+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {x \left (c x^2 \left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right )+4 a^2 b c f+20 a^2 c^2 d+a b^3 f-17 a b^2 c d+2 b^4 d\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {c \left (\frac {\sqrt {a} \left (12 a^2 c f+\sqrt {a} \sqrt {c} \left (8 a b f-20 a c d+b^2 d\right )+a b^2 f-16 a b c d+2 b^3 d\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}\right )}{a \left (b^2-4 a c\right )}}{3 a \left (b^2-4 a c\right )}+\int \frac {x \left (g x^2+e\right )}{\left (c x^4+b x^2+a\right )^{5/2}}dx+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\frac {x \left (c x^2 \left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right )+4 a^2 b c f+20 a^2 c^2 d+a b^3 f-17 a b^2 c d+2 b^4 d\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {c \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (12 a^2 c f+\sqrt {a} \sqrt {c} \left (8 a b f-20 a c d+b^2 d\right )+a b^2 f-16 a b c d+2 b^3 d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}\right )}{a \left (b^2-4 a c\right )}}{3 a \left (b^2-4 a c\right )}+\int \frac {x \left (g x^2+e\right )}{\left (c x^4+b x^2+a\right )^{5/2}}dx+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \int \frac {x \left (g x^2+e\right )}{\left (c x^4+b x^2+a\right )^{5/2}}dx+\frac {\frac {x \left (c x^2 \left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right )+4 a^2 b c f+20 a^2 c^2 d+a b^3 f-17 a b^2 c d+2 b^4 d\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {c \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (12 a^2 c f+\sqrt {a} \sqrt {c} \left (8 a b f-20 a c d+b^2 d\right )+a b^2 f-16 a b c d+2 b^3 d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}\right )}{a \left (b^2-4 a c\right )}}{3 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 1576 |
| \(\displaystyle \frac {1}{2} \int \frac {g x^2+e}{\left (c x^4+b x^2+a\right )^{5/2}}dx^2+\frac {\frac {x \left (c x^2 \left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right )+4 a^2 b c f+20 a^2 c^2 d+a b^3 f-17 a b^2 c d+2 b^4 d\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {c \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (12 a^2 c f+\sqrt {a} \sqrt {c} \left (8 a b f-20 a c d+b^2 d\right )+a b^2 f-16 a b c d+2 b^3 d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}\right )}{a \left (b^2-4 a c\right )}}{3 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 1159 |
| \(\displaystyle \frac {1}{2} \left (-\frac {4 (2 c e-b g) \int \frac {1}{\left (c x^4+b x^2+a\right )^{3/2}}dx^2}{3 \left (b^2-4 a c\right )}-\frac {2 \left (-2 a g+x^2 (2 c e-b g)+b e\right )}{3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}\right )+\frac {\frac {x \left (c x^2 \left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right )+4 a^2 b c f+20 a^2 c^2 d+a b^3 f-17 a b^2 c d+2 b^4 d\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {c \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (12 a^2 c f+\sqrt {a} \sqrt {c} \left (8 a b f-20 a c d+b^2 d\right )+a b^2 f-16 a b c d+2 b^3 d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}\right )}{a \left (b^2-4 a c\right )}}{3 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}\) |
| \(\Big \downarrow \) 1088 |
| \(\displaystyle \frac {\frac {x \left (c x^2 \left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right )+4 a^2 b c f+20 a^2 c^2 d+a b^3 f-17 a b^2 c d+2 b^4 d\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {c \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (12 a^2 c f+\sqrt {a} \sqrt {c} \left (8 a b f-20 a c d+b^2 d\right )+a b^2 f-16 a b c d+2 b^3 d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (12 a^2 c f+a b^2 f-16 a b c d+2 b^3 d\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}\right )}{a \left (b^2-4 a c\right )}}{3 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}+\frac {1}{2} \left (\frac {8 \left (b+2 c x^2\right ) (2 c e-b g)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x^2+c x^4}}-\frac {2 \left (-2 a g+x^2 (2 c e-b g)+b e\right )}{3 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^{3/2}}\right )\) |
Input:
Int[(d + e*x + f*x^2 + g*x^3)/(a + b*x^2 + c*x^4)^(5/2),x]
Output:
(x*(b^2*d - 2*a*c*d - a*b*f + c*(b*d - 2*a*f)*x^2))/(3*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^(3/2)) + ((-2*(b*e - 2*a*g + (2*c*e - b*g)*x^2))/(3*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^(3/2)) + (8*(2*c*e - b*g)*(b + 2*c*x^2))/(3*( b^2 - 4*a*c)^2*Sqrt[a + b*x^2 + c*x^4]))/2 + ((x*(2*b^4*d - 17*a*b^2*c*d + 20*a^2*c^2*d + a*b^3*f + 4*a^2*b*c*f + c*(2*b^3*d - 16*a*b*c*d + a*b^2*f + 12*a^2*c*f)*x^2))/(a*(b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4]) - (c*(-(((2* b^3*d - 16*a*b*c*d + a*b^2*f + 12*a^2*c*f)*(-((x*Sqrt[a + b*x^2 + c*x^4])/ (Sqrt[a] + Sqrt[c]*x^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^ 2 + c*x^4)/(Sqrt[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])))/Sqr t[c]) + (a^(1/4)*(2*b^3*d - 16*a*b*c*d + a*b^2*f + 12*a^2*c*f + Sqrt[a]*Sq rt[c]*(b^2*d - 20*a*c*d + 8*a*b*f))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^ 2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[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])))/( a*(b^2 - 4*a*c)))/(3*a*(b^2 - 4*a*c))
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_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[((b*d - 2*a*e + (2*c*d - b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b* x + c*x^2)^(p + 1), x] - Simp[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 - 4*a* c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] & & LtQ[p, -1] && NeQ[p, -3/2]
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*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*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] && LtQ[p, -1] && 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 = 1.77 (sec) , antiderivative size = 829, normalized size of antiderivative = 1.22
| method | result | size |
| elliptic | \(\frac {\left (\frac {\left (2 a f -b d \right ) x^{3}}{3 c a \left (4 a c -b^{2}\right )}-\frac {\left (b g -2 c e \right ) x^{2}}{3 \left (4 a c -b^{2}\right ) c^{2}}+\frac {\left (a b f +2 d a c -b^{2} d \right ) x}{3 a \left (4 a c -b^{2}\right ) c^{2}}-\frac {2 a g -b e}{3 \left (4 a c -b^{2}\right ) c^{2}}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{\left (x^{4}+\frac {b \,x^{2}}{c}+\frac {a}{c}\right )^{2}}-\frac {2 c \left (-\frac {\left (12 a^{2} c f +a \,b^{2} f -16 a b c d +2 b^{3} d \right ) x^{3}}{6 a^{2} \left (4 a c -b^{2}\right )^{2}}+\frac {4 \left (b g -2 c e \right ) x^{2}}{3 \left (4 a c -b^{2}\right )^{2}}-\frac {\left (4 a^{2} b c f +20 a^{2} c^{2} d +a \,b^{3} f -17 a \,b^{2} c d +2 b^{4} d \right ) x}{6 a^{2} \left (4 a c -b^{2}\right )^{2} c}+\frac {2 b \left (b g -2 c e \right )}{3 \left (4 a c -b^{2}\right )^{2} c}\right )}{\sqrt {\left (x^{4}+\frac {b \,x^{2}}{c}+\frac {a}{c}\right ) c}}+\frac {\left (-\frac {a b f -10 d a c +2 b^{2} d}{3 \left (4 a c -b^{2}\right ) a^{2}}-\frac {4 a^{2} b c f +20 a^{2} c^{2} d +a \,b^{3} f -17 a \,b^{2} c d +2 b^{4} d}{3 a^{2} \left (4 a c -b^{2}\right )^{2}}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {c \left (12 a^{2} c f +a \,b^{2} f -16 a b c d +2 b^{3} d \right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{6 \left (4 a c -b^{2}\right )^{2} a \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\) | \(829\) |
| default | \(\text {Expression too large to display}\) | \(1395\) |
Input:
int((g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
(1/3/c*(2*a*f-b*d)/a/(4*a*c-b^2)*x^3-1/3*(b*g-2*c*e)/(4*a*c-b^2)/c^2*x^2+1 /3*(a*b*f+2*a*c*d-b^2*d)/a/(4*a*c-b^2)/c^2*x-1/3*(2*a*g-b*e)/(4*a*c-b^2)/c ^2)*(c*x^4+b*x^2+a)^(1/2)/(x^4+1/c*b*x^2+1/c*a)^2-2*c*(-1/6*(12*a^2*c*f+a* b^2*f-16*a*b*c*d+2*b^3*d)/a^2/(4*a*c-b^2)^2*x^3+4/3*(b*g-2*c*e)/(4*a*c-b^2 )^2*x^2-1/6*(4*a^2*b*c*f+20*a^2*c^2*d+a*b^3*f-17*a*b^2*c*d+2*b^4*d)/a^2/(4 *a*c-b^2)^2/c*x+2/3*b*(b*g-2*c*e)/(4*a*c-b^2)^2/c)/((x^4+1/c*b*x^2+1/c*a)* c)^(1/2)+1/4*(-1/3/(4*a*c-b^2)*(a*b*f-10*a*c*d+2*b^2*d)/a^2-1/3*(4*a^2*b*c *f+20*a^2*c^2*d+a*b^3*f-17*a*b^2*c*d+2*b^4*d)/a^2/(4*a*c-b^2)^2)*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))+1/6*c*(12*a^2*c*f+a*b^2*f-16*a*b*c*d+2*b^3*d)/(4*a*c- b^2)^2/a*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)/(b+(-4*a*c+b^2)^(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))-EllipticE(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)))
Leaf count of result is larger than twice the leaf count of optimal. 1948 vs. \(2 (591) = 1182\).
Time = 0.13 (sec) , antiderivative size = 1948, normalized size of antiderivative = 2.86 \[ \int \frac {d+e x+f x^2+g x^3}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate((g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^(5/2),x, algorithm="fricas")
Output:
-1/6*(sqrt(1/2)*((2*(b^4*c^2 - 8*a*b^2*c^3)*d + (a*b^3*c^2 + 12*a^2*b*c^3) *f)*x^8 + 2*(2*(b^5*c - 8*a*b^3*c^2)*d + (a*b^4*c + 12*a^2*b^2*c^2)*f)*x^6 + (2*(b^6 - 6*a*b^4*c - 16*a^2*b^2*c^2)*d + (a*b^5 + 14*a^2*b^3*c + 24*a^ 3*b*c^2)*f)*x^4 + 2*(2*(a*b^5 - 8*a^2*b^3*c)*d + (a^2*b^4 + 12*a^3*b^2*c)* f)*x^2 + 2*(a^2*b^4 - 8*a^3*b^2*c)*d + (a^3*b^3 + 12*a^4*b*c)*f - ((2*(a*b ^3*c^2 - 8*a^2*b*c^3)*d + (a^2*b^2*c^2 + 12*a^3*c^3)*f)*x^8 + 2*(2*(a*b^4* c - 8*a^2*b^2*c^2)*d + (a^2*b^3*c + 12*a^3*b*c^2)*f)*x^6 + (2*(a*b^5 - 6*a ^2*b^3*c - 16*a^3*b*c^2)*d + (a^2*b^4 + 14*a^3*b^2*c + 24*a^4*c^2)*f)*x^4 + 2*(2*(a^2*b^4 - 8*a^3*b^2*c)*d + (a^3*b^3 + 12*a^4*b*c)*f)*x^2 + 2*(a^3* b^3 - 8*a^4*b*c)*d + (a^4*b^2 + 12*a^5*c)*f)*sqrt((b^2 - 4*a*c)/a^2))*sqrt (a)*sqrt((a*sqrt((b^2 - 4*a*c)/a^2) - b)/a)*elliptic_e(arcsin(sqrt(1/2)*x* sqrt((a*sqrt((b^2 - 4*a*c)/a^2) - b)/a)), 1/2*(a*b*sqrt((b^2 - 4*a*c)/a^2) + b^2 - 2*a*c)/(a*c)) + sqrt(1/2)*(((4*(5*a^2*b + 4*a*b^2)*c^3 - (a*b^3 + 2*b^4)*c^2)*d - (12*a^2*b*c^3 + (8*a^2*b^2 + a*b^3)*c^2)*f)*x^8 + 2*((4*( 5*a^2*b^2 + 4*a*b^3)*c^2 - (a*b^4 + 2*b^5)*c)*d - (12*a^2*b^2*c^2 + (8*a^2 *b^3 + a*b^4)*c)*f)*x^6 - ((a*b^5 + 2*b^6 - 8*(5*a^3*b + 4*a^2*b^2)*c^2 - 6*(3*a^2*b^3 + 2*a*b^4)*c)*d + (8*a^2*b^4 + a*b^5 + 24*a^3*b*c^2 + 2*(8*a^ 3*b^2 + 7*a^2*b^3)*c)*f)*x^4 - 2*((a^2*b^4 + 2*a*b^5 - 4*(5*a^3*b^2 + 4*a^ 2*b^3)*c)*d + (8*a^3*b^3 + a^2*b^4 + 12*a^3*b^2*c)*f)*x^2 - (a^3*b^3 + 2*a ^2*b^4 - 4*(5*a^4*b + 4*a^3*b^2)*c)*d - (8*a^4*b^2 + a^3*b^3 + 12*a^4*b...
Timed out. \[ \int \frac {d+e x+f x^2+g x^3}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((g*x**3+f*x**2+e*x+d)/(c*x**4+b*x**2+a)**(5/2),x)
Output:
Timed out
\[ \int \frac {d+e x+f x^2+g x^3}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx=\int { \frac {g x^{3} + f x^{2} + e x + d}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^(5/2),x, algorithm="maxima")
Output:
integrate((g*x^3 + f*x^2 + e*x + d)/(c*x^4 + b*x^2 + a)^(5/2), x)
\[ \int \frac {d+e x+f x^2+g x^3}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx=\int { \frac {g x^{3} + f x^{2} + e x + d}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^(5/2),x, algorithm="giac")
Output:
integrate((g*x^3 + f*x^2 + e*x + d)/(c*x^4 + b*x^2 + a)^(5/2), x)
Timed out. \[ \int \frac {d+e x+f x^2+g x^3}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx=\int \frac {g\,x^3+f\,x^2+e\,x+d}{{\left (c\,x^4+b\,x^2+a\right )}^{5/2}} \,d x \] Input:
int((d + e*x + f*x^2 + g*x^3)/(a + b*x^2 + c*x^4)^(5/2),x)
Output:
int((d + e*x + f*x^2 + g*x^3)/(a + b*x^2 + c*x^4)^(5/2), x)
\[ \int \frac {d+e x+f x^2+g x^3}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx=\text {too large to display} \] Input:
int((g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^(5/2),x)
Output:
( - 8*sqrt(a + b*x**2 + c*x**4)*a**3*c*g - 2*sqrt(a + b*x**2 + c*x**4)*a** 2*b**2*g + 12*sqrt(a + b*x**2 + c*x**4)*a**2*b*c*e - 12*sqrt(a + b*x**2 + c*x**4)*a**2*b*c*g*x**2 + 24*sqrt(a + b*x**2 + c*x**4)*a**2*c**2*e*x**2 + 16*sqrt(a + b*x**2 + c*x**4)*a**2*c**2*f*x**3 - sqrt(a + b*x**2 + c*x**4)* a*b**3*e - 3*sqrt(a + b*x**2 + c*x**4)*a*b**3*g*x**2 + 6*sqrt(a + b*x**2 + c*x**4)*a*b**2*c*e*x**2 - 8*sqrt(a + b*x**2 + c*x**4)*a*b**2*c*f*x**3 - 1 2*sqrt(a + b*x**2 + c*x**4)*a*b**2*c*g*x**4 + 24*sqrt(a + b*x**2 + c*x**4) *a*b*c**2*e*x**4 - 8*sqrt(a + b*x**2 + c*x**4)*a*b*c**2*g*x**6 + 16*sqrt(a + b*x**2 + c*x**4)*a*c**3*e*x**6 + sqrt(a + b*x**2 + c*x**4)*b**4*f*x**3 + 48*int(sqrt(a + b*x**2 + c*x**4)/(a**3 + 3*a**2*b*x**2 + 3*a**2*c*x**4 + 3*a*b**2*x**4 + 6*a*b*c*x**6 + 3*a*c**2*x**8 + b**3*x**6 + 3*b**2*c*x**8 + 3*b*c**2*x**10 + c**3*x**12),x)*a**5*c**2*d - 24*int(sqrt(a + b*x**2 + c *x**4)/(a**3 + 3*a**2*b*x**2 + 3*a**2*c*x**4 + 3*a*b**2*x**4 + 6*a*b*c*x** 6 + 3*a*c**2*x**8 + b**3*x**6 + 3*b**2*c*x**8 + 3*b*c**2*x**10 + c**3*x**1 2),x)*a**4*b**2*c*d + 96*int(sqrt(a + b*x**2 + c*x**4)/(a**3 + 3*a**2*b*x* *2 + 3*a**2*c*x**4 + 3*a*b**2*x**4 + 6*a*b*c*x**6 + 3*a*c**2*x**8 + b**3*x **6 + 3*b**2*c*x**8 + 3*b*c**2*x**10 + c**3*x**12),x)*a**4*b*c**2*d*x**2 + 96*int(sqrt(a + b*x**2 + c*x**4)/(a**3 + 3*a**2*b*x**2 + 3*a**2*c*x**4 + 3*a*b**2*x**4 + 6*a*b*c*x**6 + 3*a*c**2*x**8 + b**3*x**6 + 3*b**2*c*x**8 + 3*b*c**2*x**10 + c**3*x**12),x)*a**4*c**3*d*x**4 + 3*int(sqrt(a + b*x*...