Integrand size = 28, antiderivative size = 417 \[ \int \frac {x^2 \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=-\frac {\left (b \left (80 c^2 d-3 b^2 f+12 a c f\right )+2 c \left (16 c^2 d-3 b^2 f+12 a c f\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c^2 f^2}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c f}-\frac {\left (128 c^4 d^2+192 a c^3 d f+3 b^4 f^2-24 a b^2 c f^2+48 c^2 f \left (b^2 d+a^2 f\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{5/2} f^3}+\frac {\sqrt {d} \left (c d-b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}-2 a \sqrt {f}+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x}{2 \sqrt {c d-b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f^3}+\frac {\sqrt {d} \left (c d+b \sqrt {d} \sqrt {f}+a f\right )^{3/2} \text {arctanh}\left (\frac {b \sqrt {d}+2 a \sqrt {f}+\left (2 c \sqrt {d}+b \sqrt {f}\right ) x}{2 \sqrt {c d+b \sqrt {d} \sqrt {f}+a f} \sqrt {a+b x+c x^2}}\right )}{2 f^3} \]
-1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c/f-1/128*(128*c^4*d^2+192*a*c^3*d*f+3* b^4*f^2-24*a*b^2*c*f^2+48*c^2*f*(a^2*f+b^2*d))*arctanh(1/2*(2*c*x+b)/c^(1/ 2)/(c*x^2+b*x+a)^(1/2))/c^(5/2)/f^3+1/2*arctanh(1/2*(b*d^(1/2)-2*a*f^(1/2) +x*(2*c*d^(1/2)-b*f^(1/2)))/(c*x^2+b*x+a)^(1/2)/(c*d+a*f-b*d^(1/2)*f^(1/2) )^(1/2))*d^(1/2)*(c*d+a*f-b*d^(1/2)*f^(1/2))^(3/2)/f^3+1/2*arctanh(1/2*(b* d^(1/2)+2*a*f^(1/2)+x*(2*c*d^(1/2)+b*f^(1/2)))/(c*x^2+b*x+a)^(1/2)/(c*d+a* f+b*d^(1/2)*f^(1/2))^(1/2))*d^(1/2)*(c*d+a*f+b*d^(1/2)*f^(1/2))^(3/2)/f^3- 1/64*(b*(12*a*c*f-3*b^2*f+80*c^2*d)+2*c*(12*a*c*f-3*b^2*f+16*c^2*d)*x)*(c* x^2+b*x+a)^(1/2)/c^2/f^2
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.77 (sec) , antiderivative size = 608, normalized size of antiderivative = 1.46 \[ \int \frac {x^2 \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\frac {-2 \sqrt {c} f \sqrt {a+x (b+c x)} \left (-3 b^3 f+2 b^2 c f x+8 c^2 x \left (4 c d+5 a f+2 c f x^2\right )+4 b c \left (20 c d+5 a f+6 c f x^2\right )\right )+\left (128 c^4 d^2+192 a c^3 d f+3 b^4 f^2-24 a b^2 c f^2+48 c^2 f \left (b^2 d+a^2 f\right )\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )-64 c^{5/2} d \text {RootSum}\left [b^2 d-a^2 f-4 b \sqrt {c} d \text {$\#$1}+4 c d \text {$\#$1}^2+2 a f \text {$\#$1}^2-f \text {$\#$1}^4\&,\frac {b c^2 d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )+b^3 d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-a^2 b f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right )-2 c^{5/2} d^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 b^2 \sqrt {c} d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-4 a c^{3/2} d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 a^2 \sqrt {c} f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 b c d f \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+2 a b f^2 \log \left (-\sqrt {c} x+\sqrt {a+b x+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{b \sqrt {c} d-2 c d \text {$\#$1}-a f \text {$\#$1}+f \text {$\#$1}^3}\&\right ]}{128 c^{5/2} f^3} \]
(-2*Sqrt[c]*f*Sqrt[a + x*(b + c*x)]*(-3*b^3*f + 2*b^2*c*f*x + 8*c^2*x*(4*c *d + 5*a*f + 2*c*f*x^2) + 4*b*c*(20*c*d + 5*a*f + 6*c*f*x^2)) + (128*c^4*d ^2 + 192*a*c^3*d*f + 3*b^4*f^2 - 24*a*b^2*c*f^2 + 48*c^2*f*(b^2*d + a^2*f) )*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]] - 64*c^(5/2)*d*RootSum[ b^2*d - a^2*f - 4*b*Sqrt[c]*d*#1 + 4*c*d*#1^2 + 2*a*f*#1^2 - f*#1^4 & , (b *c^2*d^2*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] + b^3*d*f*Log[-(Sq rt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] - a^2*b*f^2*Log[-(Sqrt[c]*x) + Sqrt [a + b*x + c*x^2] - #1] - 2*c^(5/2)*d^2*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 - 2*b^2*Sqrt[c]*d*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^ 2] - #1]*#1 - 4*a*c^(3/2)*d*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - # 1]*#1 - 2*a^2*Sqrt[c]*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*# 1 + 2*b*c*d*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1^2 + 2*a*b* f^2*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1^2)/(b*Sqrt[c]*d - 2* c*d*#1 - a*f*#1 + f*#1^3) & ])/(128*c^(5/2)*f^3)
Time = 1.42 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {2140, 27, 2140, 27, 2144, 27, 1092, 219, 1366, 25, 27, 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.
| \(\displaystyle \int \frac {x^2 \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx\) |
| \(\Big \downarrow \) 2140 |
| \(\displaystyle -\frac {\int -\frac {3 \sqrt {c x^2+b x+a} \left (f \left (-3 f b^2+16 c^2 d+12 a c f\right ) x^2+16 b c d f x+\left (3 b^2+4 a c\right ) d f\right )}{4 \left (d-f x^2\right )}dx}{12 c f^2}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {c x^2+b x+a} \left (f \left (-3 f b^2+16 c^2 d+12 a c f\right ) x^2+16 b c d f x+\left (3 b^2+4 a c\right ) d f\right )}{d-f x^2}dx}{16 c f^2}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c f}\) |
| \(\Big \downarrow \) 2140 |
| \(\displaystyle \frac {-\frac {\int \frac {-\left (\left (3 f^2 b^4-24 a c f^2 b^2+128 c^4 d^2+192 a c^3 d f+48 c^2 f \left (f a^2+b^2 d\right )\right ) x^2 f^2\right )+d \left (3 f b^4-8 c (10 c d+3 a f) b^2-16 a c^2 (4 c d+5 a f)\right ) f^2-256 b c^2 d (c d+a f) x f^2}{4 \sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{2 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c x \left (12 a c f-3 b^2 f+16 c^2 d\right )+b \left (12 a c f-3 b^2 f+80 c^2 d\right )\right )}{4 c}}{16 c f^2}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {-\left (\left (3 f^2 b^4-24 a c f^2 b^2+128 c^4 d^2+192 a c^3 d f+48 c^2 f \left (f a^2+b^2 d\right )\right ) x^2 f^2\right )+d \left (3 f b^4-8 c (10 c d+3 a f) b^2-16 a c^2 (4 c d+5 a f)\right ) f^2-256 b c^2 d (c d+a f) x f^2}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c x \left (12 a c f-3 b^2 f+16 c^2 d\right )+b \left (12 a c f-3 b^2 f+80 c^2 d\right )\right )}{4 c}}{16 c f^2}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c f}\) |
| \(\Big \downarrow \) 2144 |
| \(\displaystyle \frac {-\frac {f \left (48 c^2 f \left (a^2 f+b^2 d\right )-24 a b^2 c f^2+192 a c^3 d f+3 b^4 f^2+128 c^4 d^2\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx-\frac {\int \frac {128 c^2 d f^2 \left (c^2 d^2+2 a c f d+f \left (f a^2+b^2 d\right )+2 b f (c d+a f) x\right )}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{f}}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c x \left (12 a c f-3 b^2 f+16 c^2 d\right )+b \left (12 a c f-3 b^2 f+80 c^2 d\right )\right )}{4 c}}{16 c f^2}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {f \left (48 c^2 f \left (a^2 f+b^2 d\right )-24 a b^2 c f^2+192 a c^3 d f+3 b^4 f^2+128 c^4 d^2\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx-128 c^2 d f \int \frac {c^2 d^2+2 a c f d+f \left (f a^2+b^2 d\right )+2 b f (c d+a f) x}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c x \left (12 a c f-3 b^2 f+16 c^2 d\right )+b \left (12 a c f-3 b^2 f+80 c^2 d\right )\right )}{4 c}}{16 c f^2}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c f}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {-\frac {2 f \left (48 c^2 f \left (a^2 f+b^2 d\right )-24 a b^2 c f^2+192 a c^3 d f+3 b^4 f^2+128 c^4 d^2\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}-128 c^2 d f \int \frac {c^2 d^2+2 a c f d+f \left (f a^2+b^2 d\right )+2 b f (c d+a f) x}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c x \left (12 a c f-3 b^2 f+16 c^2 d\right )+b \left (12 a c f-3 b^2 f+80 c^2 d\right )\right )}{4 c}}{16 c f^2}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {\frac {f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (48 c^2 f \left (a^2 f+b^2 d\right )-24 a b^2 c f^2+192 a c^3 d f+3 b^4 f^2+128 c^4 d^2\right )}{\sqrt {c}}-128 c^2 d f \int \frac {c^2 d^2+2 a c f d+f \left (f a^2+b^2 d\right )+2 b f (c d+a f) x}{\sqrt {c x^2+b x+a} \left (d-f x^2\right )}dx}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c x \left (12 a c f-3 b^2 f+16 c^2 d\right )+b \left (12 a c f-3 b^2 f+80 c^2 d\right )\right )}{4 c}}{16 c f^2}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c f}\) |
| \(\Big \downarrow \) 1366 |
| \(\displaystyle \frac {-\frac {\frac {f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (48 c^2 f \left (a^2 f+b^2 d\right )-24 a b^2 c f^2+192 a c^3 d f+3 b^4 f^2+128 c^4 d^2\right )}{\sqrt {c}}-128 c^2 d f \left (\frac {\sqrt {f} \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^2 \int \frac {1}{\sqrt {f} \left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d}}-\frac {\sqrt {f} \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^2 \int -\frac {1}{\sqrt {f} \left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d}}\right )}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c x \left (12 a c f-3 b^2 f+16 c^2 d\right )+b \left (12 a c f-3 b^2 f+80 c^2 d\right )\right )}{4 c}}{16 c f^2}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (48 c^2 f \left (a^2 f+b^2 d\right )-24 a b^2 c f^2+192 a c^3 d f+3 b^4 f^2+128 c^4 d^2\right )}{\sqrt {c}}-128 c^2 d f \left (\frac {\sqrt {f} \left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^2 \int \frac {1}{\sqrt {f} \left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d}}+\frac {\sqrt {f} \left (a f+b \sqrt {d} \sqrt {f}+c d\right )^2 \int \frac {1}{\sqrt {f} \left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d}}\right )}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c x \left (12 a c f-3 b^2 f+16 c^2 d\right )+b \left (12 a c f-3 b^2 f+80 c^2 d\right )\right )}{4 c}}{16 c f^2}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (48 c^2 f \left (a^2 f+b^2 d\right )-24 a b^2 c f^2+192 a c^3 d f+3 b^4 f^2+128 c^4 d^2\right )}{\sqrt {c}}-128 c^2 d f \left (\frac {\left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^2 \int \frac {1}{\left (\sqrt {f} x+\sqrt {d}\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d}}+\frac {\left (a f+b \sqrt {d} \sqrt {f}+c d\right )^2 \int \frac {1}{\left (\sqrt {d}-\sqrt {f} x\right ) \sqrt {c x^2+b x+a}}dx}{2 \sqrt {d}}\right )}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c x \left (12 a c f-3 b^2 f+16 c^2 d\right )+b \left (12 a c f-3 b^2 f+80 c^2 d\right )\right )}{4 c}}{16 c f^2}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c f}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {-\frac {\frac {f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (48 c^2 f \left (a^2 f+b^2 d\right )-24 a b^2 c f^2+192 a c^3 d f+3 b^4 f^2+128 c^4 d^2\right )}{\sqrt {c}}-128 c^2 d f \left (-\frac {\left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^2 \int \frac {1}{4 \left (-\sqrt {d} \sqrt {f} b+c d+a f\right )-\frac {\left (-2 \sqrt {f} a+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x+b \sqrt {d}\right )^2}{c x^2+b x+a}}d\left (-\frac {-2 \sqrt {f} a+\left (2 c \sqrt {d}-b \sqrt {f}\right ) x+b \sqrt {d}}{\sqrt {c x^2+b x+a}}\right )}{\sqrt {d}}-\frac {\left (a f+b \sqrt {d} \sqrt {f}+c d\right )^2 \int \frac {1}{4 \left (\sqrt {d} \sqrt {f} b+c d+a f\right )-\frac {\left (2 \sqrt {f} a+\left (\sqrt {f} b+2 c \sqrt {d}\right ) x+b \sqrt {d}\right )^2}{c x^2+b x+a}}d\left (-\frac {2 \sqrt {f} a+\left (\sqrt {f} b+2 c \sqrt {d}\right ) x+b \sqrt {d}}{\sqrt {c x^2+b x+a}}\right )}{\sqrt {d}}\right )}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c x \left (12 a c f-3 b^2 f+16 c^2 d\right )+b \left (12 a c f-3 b^2 f+80 c^2 d\right )\right )}{4 c}}{16 c f^2}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {\frac {f \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (48 c^2 f \left (a^2 f+b^2 d\right )-24 a b^2 c f^2+192 a c^3 d f+3 b^4 f^2+128 c^4 d^2\right )}{\sqrt {c}}-128 c^2 d f \left (\frac {\left (a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d\right )^{3/2} \text {arctanh}\left (\frac {-2 a \sqrt {f}+x \left (2 c \sqrt {d}-b \sqrt {f}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \left (-\sqrt {d}\right ) \sqrt {f}+c d}}\right )}{2 \sqrt {d}}+\frac {\left (a f+b \sqrt {d} \sqrt {f}+c d\right )^{3/2} \text {arctanh}\left (\frac {2 a \sqrt {f}+x \left (b \sqrt {f}+2 c \sqrt {d}\right )+b \sqrt {d}}{2 \sqrt {a+b x+c x^2} \sqrt {a f+b \sqrt {d} \sqrt {f}+c d}}\right )}{2 \sqrt {d}}\right )}{8 c f^2}-\frac {\sqrt {a+b x+c x^2} \left (2 c x \left (12 a c f-3 b^2 f+16 c^2 d\right )+b \left (12 a c f-3 b^2 f+80 c^2 d\right )\right )}{4 c}}{16 c f^2}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c f}\) |
-1/8*((b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(c*f) + (-1/4*((b*(80*c^2*d - 3 *b^2*f + 12*a*c*f) + 2*c*(16*c^2*d - 3*b^2*f + 12*a*c*f)*x)*Sqrt[a + b*x + c*x^2])/c - ((f*(128*c^4*d^2 + 192*a*c^3*d*f + 3*b^4*f^2 - 24*a*b^2*c*f^2 + 48*c^2*f*(b^2*d + a^2*f))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/Sqrt[c] - 128*c^2*d*f*(((c*d - b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*A rcTanh[(b*Sqrt[d] - 2*a*Sqrt[f] + (2*c*Sqrt[d] - b*Sqrt[f])*x)/(2*Sqrt[c*d - b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[d]) + ((c*d + b*Sqrt[d]*Sqrt[f] + a*f)^(3/2)*ArcTanh[(b*Sqrt[d] + 2*a*Sqrt[f] + (2*c*Sq rt[d] + b*Sqrt[f])*x)/(2*Sqrt[c*d + b*Sqrt[d]*Sqrt[f] + a*f]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[d])))/(8*c*f^2))/(16*c*f^2)
3.1.85.3.1 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[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + ( f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Simp[(h/2 + c*(g/(2*q ))) Int[1/((-q + c*x)*Sqrt[d + e*x + f*x^2]), x], x] + Simp[(h/2 - c*(g/( 2*q))) Int[1/((q + c*x)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d , e, f, g, h}, x] && NeQ[e^2 - 4*d*f, 0] && PosQ[(-a)*c]
Int[(Px_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_ ), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[P x, x, 2]}, Simp[(B*c*f*(2*p + 2*q + 3) + C*(b*f*p) + 2*c*C*f*(p + q + 1)*x) *(a + b*x + c*x^2)^p*((d + f*x^2)^(q + 1)/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3))), x] - Simp[1/(2*c*f^2*(p + q + 1)*(2*p + 2*q + 3)) Int[(a + b*x + c *x^2)^(p - 1)*(d + f*x^2)^q*Simp[p*(b*d)*(C*((-b)*f)*(q + 1) - c*((-B)*f)*( 2*p + 2*q + 3)) + (p + q + 1)*(b^2*C*d*f*p + a*c*(C*(2*d*f) + f*(-2*A*f)*(2 *p + 2*q + 3))) + (2*p*(c*d - a*f)*(C*((-b)*f)*(q + 1) - c*((-B)*f)*(2*p + 2*q + 3)) + (p + q + 1)*((-b)*c*(C*(-4*d*f)*(2*p + q + 2) + f*(2*C*d + 2*A* f)*(2*p + 2*q + 3))))*x + (p*((-b)*f)*(C*((-b)*f)*(q + 1) - c*((-B)*f)*(2*p + 2*q + 3)) + (p + q + 1)*(C*f^2*p*(b^2 - 4*a*c) - c^2*(C*(-4*d*f)*(2*p + q + 2) + f*(2*C*d + 2*A*f)*(2*p + 2*q + 3))))*x^2, x], x], x]] /; FreeQ[{a, b, c, d, f, q}, x] && PolyQ[Px, x, 2] && GtQ[p, 0] && NeQ[p + q + 1, 0] && NeQ[2*p + 2*q + 3, 0] && !IGtQ[p, 0] && !IGtQ[q, 0]
Int[(Px_)/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2]}, Simp[C/c Int[1/Sqrt[d + e*x + f*x^2], x], x] + Simp[1/c Int[(A* c - a*C + B*c*x)/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f}, x] && PolyQ[Px, x, 2]
Time = 0.76 (sec) , antiderivative size = 648, normalized size of antiderivative = 1.55
| method | result | size |
| risch | \(-\frac {\left (16 f \,c^{3} x^{3}+24 b \,c^{2} f \,x^{2}+40 a \,c^{2} f x +2 b^{2} c f x +32 c^{3} d x +20 a b c f -3 b^{3} f +80 b d \,c^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{64 c^{2} f^{2}}-\frac {\frac {\left (48 a^{2} c^{2} f^{2}-24 a \,b^{2} c \,f^{2}+192 a \,c^{3} d f +3 b^{4} f^{2}+48 b^{2} c^{2} d f +128 c^{4} d^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{f \sqrt {c}}-\frac {64 c^{2} d \left (2 \sqrt {d f}\, a b f +2 \sqrt {d f}\, b c d -a^{2} f^{2}-2 a c d f -b^{2} d f -c^{2} d^{2}\right ) \ln \left (\frac {\frac {-2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x +\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (-2 c \sqrt {d f}+b f \right ) \left (x +\frac {\sqrt {d f}}{f}\right )}{f}+\frac {-b \sqrt {d f}+f a +c d}{f}}}{x +\frac {\sqrt {d f}}{f}}\right )}{\sqrt {d f}\, f \sqrt {\frac {-b \sqrt {d f}+f a +c d}{f}}}-\frac {64 c^{2} d \left (2 \sqrt {d f}\, a b f +2 \sqrt {d f}\, b c d +a^{2} f^{2}+2 a c d f +b^{2} d f +c^{2} d^{2}\right ) \ln \left (\frac {\frac {2 b \sqrt {d f}+2 f a +2 c d}{f}+\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+2 \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}\, \sqrt {\left (x -\frac {\sqrt {d f}}{f}\right )^{2} c +\frac {\left (2 c \sqrt {d f}+b f \right ) \left (x -\frac {\sqrt {d f}}{f}\right )}{f}+\frac {b \sqrt {d f}+f a +c d}{f}}}{x -\frac {\sqrt {d f}}{f}}\right )}{\sqrt {d f}\, f \sqrt {\frac {b \sqrt {d f}+f a +c d}{f}}}}{128 c^{2} f^{2}}\) | \(648\) |
| default | \(\text {Expression too large to display}\) | \(1593\) |
-1/64*(16*c^3*f*x^3+24*b*c^2*f*x^2+40*a*c^2*f*x+2*b^2*c*f*x+32*c^3*d*x+20* a*b*c*f-3*b^3*f+80*b*c^2*d)*(c*x^2+b*x+a)^(1/2)/c^2/f^2-1/128/c^2/f^2*(1/f *(48*a^2*c^2*f^2-24*a*b^2*c*f^2+192*a*c^3*d*f+3*b^4*f^2+48*b^2*c^2*d*f+128 *c^4*d^2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-64*c^2*d*(2* (d*f)^(1/2)*a*b*f+2*(d*f)^(1/2)*b*c*d-a^2*f^2-2*a*c*d*f-b^2*d*f-c^2*d^2)/( d*f)^(1/2)/f/(1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2)*ln((2/f*(-b*(d*f)^(1/2)+ f*a+c*d)+1/f*(-2*c*(d*f)^(1/2)+b*f)*(x+(d*f)^(1/2)/f)+2*(1/f*(-b*(d*f)^(1/ 2)+f*a+c*d))^(1/2)*((x+(d*f)^(1/2)/f)^2*c+1/f*(-2*c*(d*f)^(1/2)+b*f)*(x+(d *f)^(1/2)/f)+1/f*(-b*(d*f)^(1/2)+f*a+c*d))^(1/2))/(x+(d*f)^(1/2)/f))-64*c^ 2*d*(2*(d*f)^(1/2)*a*b*f+2*(d*f)^(1/2)*b*c*d+a^2*f^2+2*a*c*d*f+b^2*d*f+c^2 *d^2)/(d*f)^(1/2)/f/((b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2)*ln((2*(b*(d*f)^(1/2) +f*a+c*d)/f+(2*c*(d*f)^(1/2)+b*f)/f*(x-(d*f)^(1/2)/f)+2*((b*(d*f)^(1/2)+f* a+c*d)/f)^(1/2)*((x-(d*f)^(1/2)/f)^2*c+(2*c*(d*f)^(1/2)+b*f)/f*(x-(d*f)^(1 /2)/f)+(b*(d*f)^(1/2)+f*a+c*d)/f)^(1/2))/(x-(d*f)^(1/2)/f)))
Timed out. \[ \int \frac {x^2 \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\text {Timed out} \]
\[ \int \frac {x^2 \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=- \int \frac {a x^{2} \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx - \int \frac {b x^{3} \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx - \int \frac {c x^{4} \sqrt {a + b x + c x^{2}}}{- d + f x^{2}}\, dx \]
-Integral(a*x**2*sqrt(a + b*x + c*x**2)/(-d + f*x**2), x) - Integral(b*x** 3*sqrt(a + b*x + c*x**2)/(-d + f*x**2), x) - Integral(c*x**4*sqrt(a + b*x + c*x**2)/(-d + f*x**2), x)
Exception generated. \[ \int \frac {x^2 \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(((c*sqrt(4*d*f))/(2*f^2)>0)', se e `assume?
Exception generated. \[ \int \frac {x^2 \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {x^2 \left (a+b x+c x^2\right )^{3/2}}{d-f x^2} \, dx=\int \frac {x^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{d-f\,x^2} \,d x \]