Integrand size = 29, antiderivative size = 482 \[ \int \frac {x^5 \left (a+b x^2+c x^4\right )^{3/2}}{d+e x^2} \, dx=\frac {\left (128 c^4 d^4+3 b^4 e^4-32 c^3 d^2 e (5 b d-4 a e)+8 b c^2 d e^2 (2 b d-3 a e)+6 b^2 c e^3 (b d-2 a e)-2 c e \left (32 c^3 d^3-3 b^3 e^3-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)\right ) x^2\right ) \sqrt {a+b x^2+c x^4}}{256 c^3 e^5}+\frac {\left (16 c^2 d^2-6 b c d e-3 b^2 e^2-6 c e (2 c d+b e) x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{96 c^2 e^3}+\frac {\left (a+b x^2+c x^4\right )^{5/2}}{10 c e}-\frac {\left (256 c^5 d^5+3 b^5 e^5+6 b^3 c e^4 (b d-4 a e)-384 c^4 d^3 e (b d-a e)+96 c^3 d e^2 (b d-a e)^2+16 b c^2 e^3 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{512 c^{7/2} e^6}+\frac {d^2 \left (c d^2-b d e+a e^2\right )^{3/2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x^2}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x^2+c x^4}}\right )}{2 e^6} \]
1/96*(16*c^2*d^2-6*b*c*d*e-3*b^2*e^2-6*c*e*(b*e+2*c*d)*x^2)*(c*x^4+b*x^2+a )^(3/2)/c^2/e^3+1/10*(c*x^4+b*x^2+a)^(5/2)/c/e-1/512*(256*c^5*d^5+3*b^5*e^ 5+6*b^3*c*e^4*(-4*a*e+b*d)-384*c^4*d^3*e*(-a*e+b*d)+96*c^3*d*e^2*(-a*e+b*d )^2+16*b*c^2*e^3*(3*a^2*e^2-3*a*b*d*e+b^2*d^2))*arctanh(1/2*(2*c*x^2+b)/c^ (1/2)/(c*x^4+b*x^2+a)^(1/2))/c^(7/2)/e^6+1/2*d^2*(a*e^2-b*d*e+c*d^2)^(3/2) *arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x^2)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^4 +b*x^2+a)^(1/2))/e^6+1/256*(128*c^4*d^4+3*b^4*e^4-32*c^3*d^2*e*(-4*a*e+5*b *d)+8*b*c^2*d*e^2*(-3*a*e+2*b*d)+6*b^2*c*e^3*(-2*a*e+b*d)-2*c*e*(32*c^3*d^ 3-3*b^3*e^3-8*c^2*d*e*(-3*a*e+2*b*d)-6*b*c*e^2*(-2*a*e+b*d))*x^2)*(c*x^4+b *x^2+a)^(1/2)/c^3/e^5
Time = 10.67 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.13 \[ \int \frac {x^5 \left (a+b x^2+c x^4\right )^{3/2}}{d+e x^2} \, dx=\frac {1280 d^2 \left (a+b x^2+c x^4\right )^{3/2}-\frac {480 d e \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{c}+\frac {768 e^2 \left (a+b x^2+c x^4\right )^{5/2}}{c}-\frac {90 \left (b^2-4 a c\right ) d e \left (-2 \sqrt {c} \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}+\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )\right )}{c^{5/2}}+\frac {15 b e^2 \left (-16 \left (b+2 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}+3 \left (b^2-4 a c\right ) \left (\frac {2 \left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{c}+\frac {\left (-b^2+4 a c\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{c^{3/2}}\right )\right )}{c^2}-\frac {240 d^2 \left ((2 c d-b e) \left (8 c^2 d^2-b^2 e^2+4 c e (-2 b d+3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )+2 \sqrt {c} \left (e \sqrt {a+b x^2+c x^4} \left (-b^2 e^2+4 c^2 d \left (-2 d+e x^2\right )-2 c e \left (-5 b d+4 a e+b e x^2\right )\right )+8 c \left (c d^2+e (-b d+a e)\right )^{3/2} \text {arctanh}\left (\frac {-b d+2 a e-2 c d x^2+b e x^2}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+b x^2+c x^4}}\right )\right )\right )}{c^{3/2} e^3}}{7680 e^3} \]
(1280*d^2*(a + b*x^2 + c*x^4)^(3/2) - (480*d*e*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(3/2))/c + (768*e^2*(a + b*x^2 + c*x^4)^(5/2))/c - (90*(b^2 - 4*a*c )*d*e*(-2*Sqrt[c]*(b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4] + (b^2 - 4*a*c)*Ar cTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])]))/c^(5/2) + (15*b *e^2*(-16*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(3/2) + 3*(b^2 - 4*a*c)*((2*(b + 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/c + ((-b^2 + 4*a*c)*ArcTanh[(b + 2*c* x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/c^(3/2))))/c^2 - (240*d^2*((2*c *d - b*e)*(8*c^2*d^2 - b^2*e^2 + 4*c*e*(-2*b*d + 3*a*e))*ArcTanh[(b + 2*c* x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])] + 2*Sqrt[c]*(e*Sqrt[a + b*x^2 + c*x^4]*(-(b^2*e^2) + 4*c^2*d*(-2*d + e*x^2) - 2*c*e*(-5*b*d + 4*a*e + b*e* x^2)) + 8*c*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*ArcTanh[(-(b*d) + 2*a*e - 2*c *d*x^2 + b*e*x^2)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + b*x^2 + c*x^4 ])])))/(c^(3/2)*e^3))/(7680*e^3)
Time = 1.05 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {1578, 1267, 27, 1231, 27, 1231, 27, 1269, 1092, 219, 1154, 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.
| \(\displaystyle \int \frac {x^5 \left (a+b x^2+c x^4\right )^{3/2}}{d+e x^2} \, dx\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {1}{2} \int \frac {x^4 \left (c x^4+b x^2+a\right )^{3/2}}{e x^2+d}dx^2\) |
| \(\Big \downarrow \) 1267 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {5 e \left ((2 c d+b e) x^2+b d\right ) \left (c x^4+b x^2+a\right )^{3/2}}{2 \left (e x^2+d\right )}dx^2}{5 c e^2}+\frac {\left (a+b x^2+c x^4\right )^{5/2}}{5 c e}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {\left (a+b x^2+c x^4\right )^{5/2}}{5 c e}-\frac {\int \frac {\left ((2 c d+b e) x^2+b d\right ) \left (c x^4+b x^2+a\right )^{3/2}}{e x^2+d}dx^2}{2 c e}\right )\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {1}{2} \left (\frac {\left (a+b x^2+c x^4\right )^{5/2}}{5 c e}-\frac {-\frac {\int \frac {\left (d \left (3 e^2 b^3+6 c d e b^2-4 c \left (4 c d^2+3 a e^2\right ) b+8 a c^2 d e\right )-\left (32 c^3 d^3-8 c^2 e (2 b d-3 a e) d-3 b^3 e^3-6 b c e^2 (b d-2 a e)\right ) x^2\right ) \sqrt {c x^4+b x^2+a}}{2 \left (e x^2+d\right )}dx^2}{8 c e^2}-\frac {\left (a+b x^2+c x^4\right )^{3/2} \left (-3 b^2 e^2-6 c e x^2 (b e+2 c d)-6 b c d e+16 c^2 d^2\right )}{24 c e^2}}{2 c e}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {\left (a+b x^2+c x^4\right )^{5/2}}{5 c e}-\frac {-\frac {\int \frac {\left (d \left (3 e^2 b^3+6 c d e b^2-4 c \left (4 c d^2+3 a e^2\right ) b+8 a c^2 d e\right )-\left (32 c^3 d^3-8 c^2 e (2 b d-3 a e) d-3 b^3 e^3-6 b c e^2 (b d-2 a e)\right ) x^2\right ) \sqrt {c x^4+b x^2+a}}{e x^2+d}dx^2}{16 c e^2}-\frac {\left (a+b x^2+c x^4\right )^{3/2} \left (-3 b^2 e^2-6 c e x^2 (b e+2 c d)-6 b c d e+16 c^2 d^2\right )}{24 c e^2}}{2 c e}\right )\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {1}{2} \left (\frac {\left (a+b x^2+c x^4\right )^{5/2}}{5 c e}-\frac {-\frac {\frac {\sqrt {a+b x^2+c x^4} \left (-2 c e x^2 \left (-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)-3 b^3 e^3+32 c^3 d^3\right )+6 b^2 c e^3 (b d-2 a e)-32 c^3 d^2 e (5 b d-4 a e)+8 b c^2 d e^2 (2 b d-3 a e)+3 b^4 e^4+128 c^4 d^4\right )}{4 c e^2}-\frac {\int \frac {\left (256 c^5 d^5-384 c^4 e (b d-a e) d^3+96 c^3 e^2 (b d-a e)^2 d+3 b^5 e^5+6 b^3 c e^4 (b d-4 a e)+16 b c^2 e^3 \left (b^2 d^2-3 a b e d+3 a^2 e^2\right )\right ) x^2+d \left (3 e^4 b^5+6 c d e^3 b^4+8 c e^2 \left (2 c d^2-3 a e^2\right ) b^3-16 c^2 d e \left (10 c d^2+3 a e^2\right ) b^2+16 c^2 \left (8 c^2 d^4+20 a c e^2 d^2+3 a^2 e^4\right ) b-32 a c^3 d e \left (4 c d^2+5 a e^2\right )\right )}{2 \left (e x^2+d\right ) \sqrt {c x^4+b x^2+a}}dx^2}{4 c e^2}}{16 c e^2}-\frac {\left (a+b x^2+c x^4\right )^{3/2} \left (-3 b^2 e^2-6 c e x^2 (b e+2 c d)-6 b c d e+16 c^2 d^2\right )}{24 c e^2}}{2 c e}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {\left (a+b x^2+c x^4\right )^{5/2}}{5 c e}-\frac {-\frac {\frac {\sqrt {a+b x^2+c x^4} \left (-2 c e x^2 \left (-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)-3 b^3 e^3+32 c^3 d^3\right )+6 b^2 c e^3 (b d-2 a e)-32 c^3 d^2 e (5 b d-4 a e)+8 b c^2 d e^2 (2 b d-3 a e)+3 b^4 e^4+128 c^4 d^4\right )}{4 c e^2}-\frac {\int \frac {\left (256 c^5 d^5-384 c^4 e (b d-a e) d^3+96 c^3 e^2 (b d-a e)^2 d+3 b^5 e^5+6 b^3 c e^4 (b d-4 a e)+16 b c^2 e^3 \left (b^2 d^2-3 a b e d+3 a^2 e^2\right )\right ) x^2+d \left (3 e^4 b^5+6 c d e^3 b^4+8 c e^2 \left (2 c d^2-3 a e^2\right ) b^3-16 c^2 d e \left (10 c d^2+3 a e^2\right ) b^2+16 c^2 \left (8 c^2 d^4+20 a c e^2 d^2+3 a^2 e^4\right ) b-32 a c^3 d e \left (4 c d^2+5 a e^2\right )\right )}{\left (e x^2+d\right ) \sqrt {c x^4+b x^2+a}}dx^2}{8 c e^2}}{16 c e^2}-\frac {\left (a+b x^2+c x^4\right )^{3/2} \left (-3 b^2 e^2-6 c e x^2 (b e+2 c d)-6 b c d e+16 c^2 d^2\right )}{24 c e^2}}{2 c e}\right )\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{2} \left (\frac {\left (a+b x^2+c x^4\right )^{5/2}}{5 c e}-\frac {-\frac {\frac {\sqrt {a+b x^2+c x^4} \left (-2 c e x^2 \left (-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)-3 b^3 e^3+32 c^3 d^3\right )+6 b^2 c e^3 (b d-2 a e)-32 c^3 d^2 e (5 b d-4 a e)+8 b c^2 d e^2 (2 b d-3 a e)+3 b^4 e^4+128 c^4 d^4\right )}{4 c e^2}-\frac {\frac {\left (16 b c^2 e^3 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )+6 b^3 c e^4 (b d-4 a e)-384 c^4 d^3 e (b d-a e)+96 c^3 d e^2 (b d-a e)^2+3 b^5 e^5+256 c^5 d^5\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx^2}{e}-\frac {256 c^3 d^2 \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{\left (e x^2+d\right ) \sqrt {c x^4+b x^2+a}}dx^2}{e}}{8 c e^2}}{16 c e^2}-\frac {\left (a+b x^2+c x^4\right )^{3/2} \left (-3 b^2 e^2-6 c e x^2 (b e+2 c d)-6 b c d e+16 c^2 d^2\right )}{24 c e^2}}{2 c e}\right )\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1}{2} \left (\frac {\left (a+b x^2+c x^4\right )^{5/2}}{5 c e}-\frac {-\frac {\frac {\sqrt {a+b x^2+c x^4} \left (-2 c e x^2 \left (-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)-3 b^3 e^3+32 c^3 d^3\right )+6 b^2 c e^3 (b d-2 a e)-32 c^3 d^2 e (5 b d-4 a e)+8 b c^2 d e^2 (2 b d-3 a e)+3 b^4 e^4+128 c^4 d^4\right )}{4 c e^2}-\frac {\frac {2 \left (16 b c^2 e^3 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )+6 b^3 c e^4 (b d-4 a e)-384 c^4 d^3 e (b d-a e)+96 c^3 d e^2 (b d-a e)^2+3 b^5 e^5+256 c^5 d^5\right ) \int \frac {1}{4 c-x^4}d\frac {2 c x^2+b}{\sqrt {c x^4+b x^2+a}}}{e}-\frac {256 c^3 d^2 \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{\left (e x^2+d\right ) \sqrt {c x^4+b x^2+a}}dx^2}{e}}{8 c e^2}}{16 c e^2}-\frac {\left (a+b x^2+c x^4\right )^{3/2} \left (-3 b^2 e^2-6 c e x^2 (b e+2 c d)-6 b c d e+16 c^2 d^2\right )}{24 c e^2}}{2 c e}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {\left (a+b x^2+c x^4\right )^{5/2}}{5 c e}-\frac {-\frac {\frac {\sqrt {a+b x^2+c x^4} \left (-2 c e x^2 \left (-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)-3 b^3 e^3+32 c^3 d^3\right )+6 b^2 c e^3 (b d-2 a e)-32 c^3 d^2 e (5 b d-4 a e)+8 b c^2 d e^2 (2 b d-3 a e)+3 b^4 e^4+128 c^4 d^4\right )}{4 c e^2}-\frac {\frac {\text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right ) \left (16 b c^2 e^3 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )+6 b^3 c e^4 (b d-4 a e)-384 c^4 d^3 e (b d-a e)+96 c^3 d e^2 (b d-a e)^2+3 b^5 e^5+256 c^5 d^5\right )}{\sqrt {c} e}-\frac {256 c^3 d^2 \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{\left (e x^2+d\right ) \sqrt {c x^4+b x^2+a}}dx^2}{e}}{8 c e^2}}{16 c e^2}-\frac {\left (a+b x^2+c x^4\right )^{3/2} \left (-3 b^2 e^2-6 c e x^2 (b e+2 c d)-6 b c d e+16 c^2 d^2\right )}{24 c e^2}}{2 c e}\right )\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{2} \left (\frac {\left (a+b x^2+c x^4\right )^{5/2}}{5 c e}-\frac {-\frac {\frac {\sqrt {a+b x^2+c x^4} \left (-2 c e x^2 \left (-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)-3 b^3 e^3+32 c^3 d^3\right )+6 b^2 c e^3 (b d-2 a e)-32 c^3 d^2 e (5 b d-4 a e)+8 b c^2 d e^2 (2 b d-3 a e)+3 b^4 e^4+128 c^4 d^4\right )}{4 c e^2}-\frac {\frac {512 c^3 d^2 \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-x^4}d\left (-\frac {(2 c d-b e) x^2+b d-2 a e}{\sqrt {c x^4+b x^2+a}}\right )}{e}+\frac {\text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right ) \left (16 b c^2 e^3 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )+6 b^3 c e^4 (b d-4 a e)-384 c^4 d^3 e (b d-a e)+96 c^3 d e^2 (b d-a e)^2+3 b^5 e^5+256 c^5 d^5\right )}{\sqrt {c} e}}{8 c e^2}}{16 c e^2}-\frac {\left (a+b x^2+c x^4\right )^{3/2} \left (-3 b^2 e^2-6 c e x^2 (b e+2 c d)-6 b c d e+16 c^2 d^2\right )}{24 c e^2}}{2 c e}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {\left (a+b x^2+c x^4\right )^{5/2}}{5 c e}-\frac {-\frac {\frac {\sqrt {a+b x^2+c x^4} \left (-2 c e x^2 \left (-8 c^2 d e (2 b d-3 a e)-6 b c e^2 (b d-2 a e)-3 b^3 e^3+32 c^3 d^3\right )+6 b^2 c e^3 (b d-2 a e)-32 c^3 d^2 e (5 b d-4 a e)+8 b c^2 d e^2 (2 b d-3 a e)+3 b^4 e^4+128 c^4 d^4\right )}{4 c e^2}-\frac {\frac {\text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right ) \left (16 b c^2 e^3 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )+6 b^3 c e^4 (b d-4 a e)-384 c^4 d^3 e (b d-a e)+96 c^3 d e^2 (b d-a e)^2+3 b^5 e^5+256 c^5 d^5\right )}{\sqrt {c} e}-\frac {256 c^3 d^2 \left (a e^2-b d e+c d^2\right )^{3/2} \text {arctanh}\left (\frac {-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt {a+b x^2+c x^4} \sqrt {a e^2-b d e+c d^2}}\right )}{e}}{8 c e^2}}{16 c e^2}-\frac {\left (a+b x^2+c x^4\right )^{3/2} \left (-3 b^2 e^2-6 c e x^2 (b e+2 c d)-6 b c d e+16 c^2 d^2\right )}{24 c e^2}}{2 c e}\right )\) |
((a + b*x^2 + c*x^4)^(5/2)/(5*c*e) - (-1/24*((16*c^2*d^2 - 6*b*c*d*e - 3*b ^2*e^2 - 6*c*e*(2*c*d + b*e)*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(c*e^2) - ((( 128*c^4*d^4 + 3*b^4*e^4 - 32*c^3*d^2*e*(5*b*d - 4*a*e) + 8*b*c^2*d*e^2*(2* b*d - 3*a*e) + 6*b^2*c*e^3*(b*d - 2*a*e) - 2*c*e*(32*c^3*d^3 - 3*b^3*e^3 - 8*c^2*d*e*(2*b*d - 3*a*e) - 6*b*c*e^2*(b*d - 2*a*e))*x^2)*Sqrt[a + b*x^2 + c*x^4])/(4*c*e^2) - (((256*c^5*d^5 + 3*b^5*e^5 + 6*b^3*c*e^4*(b*d - 4*a* e) - 384*c^4*d^3*e*(b*d - a*e) + 96*c^3*d*e^2*(b*d - a*e)^2 + 16*b*c^2*e^3 *(b^2*d^2 - 3*a*b*d*e + 3*a^2*e^2))*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[ a + b*x^2 + c*x^4])])/(Sqrt[c]*e) - (256*c^3*d^2*(c*d^2 - b*d*e + a*e^2)^( 3/2)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x^2)/(2*Sqrt[c*d^2 - b*d*e + a*e ^2]*Sqrt[a + b*x^2 + c*x^4])])/e)/(8*c*e^2))/(16*c*e^2))/(2*c*e))/2
3.4.22.3.1 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[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[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g^n*(d + e*x)^(m + n - 1)*((a + b *x + c*x^2)^(p + 1)/(c*e^(n - 1)*(m + n + 2*p + 1))), x] + Simp[1/(c*e^n*(m + n + 2*p + 1)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^n*(m + n + 2*p + 1)*(f + g*x)^n - c*g^n*(m + n + 2*p + 1)*(d + e*x)^n - g^n*(d + e*x)^(n - 2)*(b*d*e*(p + 1) + a*e^2*(m + n - 1) - c*d^2*(m + n + 2*p + 1) - e*(2*c*d - b*e)*(m + n + p)*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g , m, p}, x] && IGtQ[n, 1] && IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(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 egerQ[(m - 1)/2]
Time = 0.64 (sec) , antiderivative size = 620, normalized size of antiderivative = 1.29
| method | result | size |
| pseudoelliptic | \(\frac {-5 \left (e^{2} \left (a e -b d \right )^{2} c^{\frac {7}{2}}+\left (2 e^{2} d^{2} a -2 d^{3} e b \right ) c^{\frac {9}{2}}+c^{\frac {11}{2}} d^{4}\right ) d^{2} \ln \left (\frac {2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, e +\left (b \,x^{2}+2 a \right ) e -d \left (2 c \,x^{2}+b \right )}{e \,x^{2}+d}\right )+\left (\left (-\frac {15 b \left (a c -\frac {b^{2}}{4}\right )^{2} e^{5}}{16}-\frac {15 c d \left (a c -\frac {b^{2}}{4}\right )^{2} e^{4}}{8}+\frac {15 c^{2} d^{2} \left (a c -\frac {b^{2}}{12}\right ) b \,e^{3}}{4}-\frac {15 \left (a c +\frac {b^{2}}{4}\right ) c^{3} d^{3} e^{2}}{2}+\frac {15 b \,c^{4} d^{4} e}{2}-5 c^{5} d^{5}\right ) \ln \left (\frac {2 c \,x^{2}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}+b}{\sqrt {c}}\right )+e \left (\frac {20 e \left (\frac {3 x^{4} \left (\frac {11 b \,x^{2}}{16}+a \right ) e^{3}}{10}-\frac {15 d \left (\frac {3 b \,x^{2}}{5}+a \right ) x^{2} e^{2}}{32}+d^{2} \left (\frac {7 b \,x^{2}}{16}+a \right ) e -\frac {15 b \,d^{3}}{16}\right ) c^{\frac {7}{2}}}{3}+\left (\frac {5}{3} d^{2} e^{2} x^{4}-\frac {5}{2} d^{3} e \,x^{2}-\frac {5}{4} d \,e^{3} x^{6}+e^{4} x^{8}+5 d^{4}\right ) c^{\frac {9}{2}}+\left (\left (\left (\frac {1}{16} b^{2} x^{4}+a^{2}+\frac {7}{16} a b \,x^{2}\right ) e^{2}-\frac {25 d b \left (\frac {b \,x^{2}}{10}+a \right ) e}{16}+\frac {5 b^{2} d^{2}}{8}\right ) c^{\frac {5}{2}}-\frac {25 \left (\left (\left (\frac {b \,x^{2}}{10}+a \right ) e -\frac {3 b d}{10}\right ) c^{\frac {3}{2}}-\frac {3 b^{2} e \sqrt {c}}{20}\right ) e \,b^{2}}{32}\right ) e^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}+\frac {15 \left (b \left (a c -\frac {b^{2}}{4}\right )^{2} e^{5}+2 c d \left (a c -\frac {b^{2}}{4}\right )^{2} e^{4}-4 c^{2} d^{2} \left (a c -\frac {b^{2}}{12}\right ) b \,e^{3}+\left (8 a \,c^{4}+2 b^{2} c^{3}\right ) d^{3} e^{2}-8 b \,c^{4} d^{4} e +\frac {16 c^{5} d^{5}}{3}\right ) \ln \left (2\right )}{16}\right ) e \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{10 c^{\frac {7}{2}} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, e^{7}}\) | \(620\) |
| risch | \(\frac {\left (384 c^{4} e^{4} x^{8}+528 b \,c^{3} e^{4} x^{6}-480 c^{4} d \,e^{3} x^{6}+768 a \,c^{3} e^{4} x^{4}+24 b^{2} c^{2} e^{4} x^{4}-720 b \,c^{3} d \,e^{3} x^{4}+640 c^{4} d^{2} e^{2} x^{4}+168 a b \,c^{2} e^{4} x^{2}-1200 a \,c^{3} d \,e^{3} x^{2}-30 b^{3} c \,e^{4} x^{2}-60 b^{2} c^{2} d \,e^{3} x^{2}+1120 b \,c^{3} d^{2} e^{2} x^{2}-960 c^{4} d^{3} e \,x^{2}+384 a^{2} c^{2} e^{4}-300 a \,b^{2} c \,e^{4}-600 a b \,c^{2} d \,e^{3}+2560 a \,c^{3} d^{2} e^{2}+45 b^{4} e^{4}+90 b^{3} c d \,e^{3}+240 b^{2} c^{2} d^{2} e^{2}-2400 b \,c^{3} e \,d^{3}+1920 c^{4} d^{4}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{3840 c^{3} e^{5}}-\frac {\frac {128 d^{2} \left (a^{2} e^{4}-2 a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right ) c^{3} \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x^{2}+\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x^{2}+\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x^{2}+\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}+\frac {\left (48 a^{2} b \,c^{2} e^{5}+96 a^{2} c^{3} d \,e^{4}-24 a \,b^{3} c \,e^{5}-48 a \,b^{2} c^{2} d \,e^{4}-192 a b \,c^{3} d^{2} e^{3}+384 a \,c^{4} d^{3} e^{2}+3 b^{5} e^{5}+6 b^{4} c d \,e^{4}+16 b^{3} c^{2} d^{2} e^{3}+96 b^{2} c^{3} d^{3} e^{2}-384 b \,c^{4} d^{4} e +256 c^{5} d^{5}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 e \sqrt {c}}}{256 e^{5} c^{3}}\) | \(691\) |
| default | \(\text {Expression too large to display}\) | \(1428\) |
| elliptic | \(\text {Expression too large to display}\) | \(1808\) |
1/10/c^(7/2)/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(-5*(e^2*(a*e-b*d)^2*c^(7/2)+ (2*a*d^2*e^2-2*b*d^3*e)*c^(9/2)+c^(11/2)*d^4)*d^2*ln((2*(c*x^4+b*x^2+a)^(1 /2)*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*e+(b*x^2+2*a)*e-d*(2*c*x^2+b))/(e*x^2+ d))+((-15/16*b*(a*c-1/4*b^2)^2*e^5-15/8*c*d*(a*c-1/4*b^2)^2*e^4+15/4*c^2*d ^2*(a*c-1/12*b^2)*b*e^3-15/2*(a*c+1/4*b^2)*c^3*d^3*e^2+15/2*b*c^4*d^4*e-5* c^5*d^5)*ln((2*c*x^2+2*(c*x^4+b*x^2+a)^(1/2)*c^(1/2)+b)/c^(1/2))+e*(20/3*e *(3/10*x^4*(11/16*b*x^2+a)*e^3-15/32*d*(3/5*b*x^2+a)*x^2*e^2+d^2*(7/16*b*x ^2+a)*e-15/16*b*d^3)*c^(7/2)+(5/3*d^2*e^2*x^4-5/2*d^3*e*x^2-5/4*d*e^3*x^6+ e^4*x^8+5*d^4)*c^(9/2)+(((1/16*b^2*x^4+a^2+7/16*a*b*x^2)*e^2-25/16*d*b*(1/ 10*b*x^2+a)*e+5/8*b^2*d^2)*c^(5/2)-25/32*(((1/10*b*x^2+a)*e-3/10*b*d)*c^(3 /2)-3/20*b^2*e*c^(1/2))*e*b^2)*e^2)*(c*x^4+b*x^2+a)^(1/2)+15/16*(b*(a*c-1/ 4*b^2)^2*e^5+2*c*d*(a*c-1/4*b^2)^2*e^4-4*c^2*d^2*(a*c-1/12*b^2)*b*e^3+(8*a *c^4+2*b^2*c^3)*d^3*e^2-8*b*c^4*d^4*e+16/3*c^5*d^5)*ln(2))*e*((a*e^2-b*d*e +c*d^2)/e^2)^(1/2))/e^7
Timed out. \[ \int \frac {x^5 \left (a+b x^2+c x^4\right )^{3/2}}{d+e x^2} \, dx=\text {Timed out} \]
\[ \int \frac {x^5 \left (a+b x^2+c x^4\right )^{3/2}}{d+e x^2} \, dx=\int \frac {x^{5} \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}{d + e x^{2}}\, dx \]
Exception generated. \[ \int \frac {x^5 \left (a+b x^2+c x^4\right )^{3/2}}{d+e x^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Exception generated. \[ \int \frac {x^5 \left (a+b x^2+c x^4\right )^{3/2}}{d+e x^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {x^5 \left (a+b x^2+c x^4\right )^{3/2}}{d+e x^2} \, dx=\int \frac {x^5\,{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{e\,x^2+d} \,d x \]