Integrand size = 29, antiderivative size = 758 \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^3 \sqrt {d+e x} \, dx=\frac {2 \left (19 a^3 d^3-6 a^2 c d e^2+8 b^3 e^3+3 a b e^2 (b d-9 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x}}{315 a^3 e^3}+\frac {2}{9} \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^4 \sqrt {d+e x}-\frac {4 \left (8 a^2 d^2+3 b^2 e^2+a e (4 b d-7 c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{3/2}}{315 a^2 e^3}+\frac {2 (a d+b e) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x (d+e x)^{5/2}}{63 a e^3}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (8 a^4 d^4+8 b^4 e^4-a^3 d^2 e (4 b d-9 c e)-4 a b^2 e^3 (b d+9 c e)-3 a^2 e^2 \left (b^2 d^2-5 b c d e-7 c^2 e^2\right )\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {d+e x} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{315 a^4 e^4 \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \left (c+b x+a x^2\right )}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (16 a^3 d^3+6 a^2 c d e^2-8 b^3 e^3-3 a b e^2 (b d-9 c e)\right ) \left (a d^2-e (b d-c e)\right ) \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (c+b x+a x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 a x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{315 a^4 e^4 \sqrt {d+e x} \left (c+b x+a x^2\right )} \] Output:
2/315*(19*a^3*d^3-6*a^2*c*d*e^2+8*b^3*e^3+3*a*b*e^2*(b*d-9*c*e))*(a+c/x^2+ b/x)^(1/2)*x*(e*x+d)^(1/2)/a^3/e^3+2/9*(a+c/x^2+b/x)^(1/2)*x^4*(e*x+d)^(1/ 2)-4/315*(8*a^2*d^2+3*b^2*e^2+a*e*(4*b*d-7*c*e))*(a+c/x^2+b/x)^(1/2)*x*(e* x+d)^(3/2)/a^2/e^3+2/63*(a*d+b*e)*(a+c/x^2+b/x)^(1/2)*x*(e*x+d)^(5/2)/a/e^ 3-2/315*2^(1/2)*(-4*a*c+b^2)^(1/2)*(8*a^4*d^4+8*b^4*e^4-a^3*d^2*e*(4*b*d-9 *c*e)-4*a*b^2*e^3*(b*d+9*c*e)-3*a^2*e^2*(b^2*d^2-5*b*c*d*e-7*c^2*e^2))*(a+ c/x^2+b/x)^(1/2)*x*(e*x+d)^(1/2)*(-a*(a*x^2+b*x+c)/(-4*a*c+b^2))^(1/2)*Ell ipticE(1/2*(1+(2*a*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*(-4*a*c+b^2) ^(1/2)*e/(2*a*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2))/a^4/e^4/(a*(e*x+d)/(2*a* d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2)/(a*x^2+b*x+c)+2/315*2^(1/2)*(-4*a*c+b^2 )^(1/2)*(16*a^3*d^3+6*a^2*c*d*e^2-8*b^3*e^3-3*a*b*e^2*(b*d-9*c*e))*(a*d^2- e*(b*d-c*e))*(a+c/x^2+b/x)^(1/2)*x*(a*(e*x+d)/(2*a*d-(b+(-4*a*c+b^2)^(1/2) )*e))^(1/2)*(-a*(a*x^2+b*x+c)/(-4*a*c+b^2))^(1/2)*EllipticF(1/2*(1+(2*a*x+ b)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*(-4*a*c+b^2)^(1/2)*e/(2*a*d-(b+(- 4*a*c+b^2)^(1/2))*e))^(1/2))/a^4/e^4/(e*x+d)^(1/2)/(a*x^2+b*x+c)
Result contains complex when optimal does not.
Time = 35.41 (sec) , antiderivative size = 7531, normalized size of antiderivative = 9.94 \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^3 \sqrt {d+e x} \, dx=\text {Result too large to show} \] Input:
Integrate[Sqrt[a + c/x^2 + b/x]*x^3*Sqrt[d + e*x],x]
Output:
Result too large to show
Time = 2.22 (sec) , antiderivative size = 799, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {1897, 1272, 25, 2184, 27, 2184, 27, 2184, 27, 1269, 1172, 321, 327}
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.
| \(\displaystyle \int x^3 \sqrt {d+e x} \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \, dx\) |
| \(\Big \downarrow \) 1897 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \int x^2 \sqrt {d+e x} \sqrt {a x^2+b x+c}dx}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 1272 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {2}{9} x^3 \sqrt {d+e x} \sqrt {a x^2+b x+c}-\frac {1}{9} \int -\frac {x^2 \left ((a d+b e) x^2+2 (b d+c e) x+3 c d\right )}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}dx\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{9} \int \frac {x^2 \left ((a d+b e) x^2+2 (b d+c e) x+3 c d\right )}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}dx+\frac {2}{9} x^3 \sqrt {d+e x} \sqrt {a x^2+b x+c}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{9} \left (\frac {2 \int -\frac {2 e^3 \left (8 a^2 d^2+3 b^2 e^2+a e (4 b d-7 c e)\right ) x^3+e^2 \left (11 a^2 d^3+8 a e (3 b d-2 c e) d+b e^2 (13 b d+5 c e)\right ) x^2+2 d e (a d+b e) \left (a d^2+e (4 b d+5 c e)\right ) x+d^2 e (a d+b e) (b d+5 c e)}{2 \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx}{7 a e^4}+\frac {2 (d+e x)^{5/2} \sqrt {a x^2+b x+c} (a d+b e)}{7 a e^3}\right )+\frac {2}{9} x^3 \sqrt {d+e x} \sqrt {a x^2+b x+c}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{9} \left (\frac {2 (d+e x)^{5/2} \sqrt {a x^2+b x+c} (a d+b e)}{7 a e^3}-\frac {\int \frac {2 e^3 \left (8 a^2 d^2+3 b^2 e^2+a e (4 b d-7 c e)\right ) x^3+e^2 \left (11 a^2 d^3+8 a e (3 b d-2 c e) d+b e^2 (13 b d+5 c e)\right ) x^2+2 d e (a d+b e) \left (a d^2+e (4 b d+5 c e)\right ) x+d^2 e (a d+b e) (b d+5 c e)}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}dx}{7 a e^4}\right )+\frac {2}{9} x^3 \sqrt {d+e x} \sqrt {a x^2+b x+c}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{9} \left (\frac {2 (d+e x)^{5/2} \sqrt {a x^2+b x+c} (a d+b e)}{7 a e^3}-\frac {\frac {2 \int -\frac {3 \left (19 a^3 d^3-6 a^2 c e^2 d+8 b^3 e^3+3 a b e^2 (b d-9 c e)\right ) x^2 e^5+d \left (a^2 (11 b d+23 c e) d^2+6 b^2 e^2 (b d+3 c e)+3 a e \left (b^2 d^2-5 b c e d-14 c^2 e^2\right )\right ) e^4+2 \left (11 a^3 d^4+a^2 e (23 b d-15 c e) d^2+3 b^2 e^3 (5 b d+3 c e)+3 a e^2 \left (2 b^2 d^2-16 b c e d-7 c^2 e^2\right )\right ) x e^4}{2 \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx}{5 a e^3}+\frac {4 e (d+e x)^{3/2} \sqrt {a x^2+b x+c} \left (8 a^2 d^2+a e (4 b d-7 c e)+3 b^2 e^2\right )}{5 a}}{7 a e^4}\right )+\frac {2}{9} x^3 \sqrt {d+e x} \sqrt {a x^2+b x+c}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{9} \left (\frac {2 (d+e x)^{5/2} \sqrt {a x^2+b x+c} (a d+b e)}{7 a e^3}-\frac {\frac {4 e (d+e x)^{3/2} \sqrt {a x^2+b x+c} \left (8 a^2 d^2+a e (4 b d-7 c e)+3 b^2 e^2\right )}{5 a}-\frac {\int \frac {3 \left (19 a^3 d^3-6 a^2 c e^2 d+8 b^3 e^3+3 a b e^2 (b d-9 c e)\right ) x^2 e^5+d \left (a^2 (11 b d+23 c e) d^2+6 b^2 e^2 (b d+3 c e)+3 a e \left (b^2 d^2-5 b c e d-14 c^2 e^2\right )\right ) e^4+2 \left (11 a^3 d^4+a^2 e (23 b d-15 c e) d^2+3 b^2 e^3 (5 b d+3 c e)+3 a e^2 \left (2 b^2 d^2-16 b c e d-7 c^2 e^2\right )\right ) x e^4}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}dx}{5 a e^3}}{7 a e^4}\right )+\frac {2}{9} x^3 \sqrt {d+e x} \sqrt {a x^2+b x+c}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{9} \left (\frac {2 (d+e x)^{5/2} \sqrt {a x^2+b x+c} (a d+b e)}{7 a e^3}-\frac {\frac {4 e (d+e x)^{3/2} \sqrt {a x^2+b x+c} \left (8 a^2 d^2+a e (4 b d-7 c e)+3 b^2 e^2\right )}{5 a}-\frac {\frac {2 \int -\frac {3 e^6 \left (4 a^3 (2 b d-c e) d^3-3 a^2 e \left (b^2 d^2-3 b c e d-12 c^2 e^2\right ) d+8 b^3 e^3 (b d+c e)-3 a b e^2 \left (b^2 d^2+14 b c e d+9 c^2 e^2\right )+2 \left (8 a^4 d^4-a^3 e (4 b d-9 c e) d^2+8 b^4 e^4-4 a b^2 e^3 (b d+9 c e)-3 a^2 e^2 \left (b^2 d^2-5 b c e d-7 c^2 e^2\right )\right ) x\right )}{2 \sqrt {d+e x} \sqrt {a x^2+b x+c}}dx}{3 a e^2}+\frac {2 e^4 \sqrt {d+e x} \sqrt {a x^2+b x+c} \left (19 a^3 d^3-6 a^2 c d e^2+3 a b e^2 (b d-9 c e)+8 b^3 e^3\right )}{a}}{5 a e^3}}{7 a e^4}\right )+\frac {2}{9} x^3 \sqrt {d+e x} \sqrt {a x^2+b x+c}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{9} \left (\frac {2 (d+e x)^{5/2} \sqrt {a x^2+b x+c} (a d+b e)}{7 a e^3}-\frac {\frac {4 e (d+e x)^{3/2} \sqrt {a x^2+b x+c} \left (8 a^2 d^2+a e (4 b d-7 c e)+3 b^2 e^2\right )}{5 a}-\frac {\frac {2 e^4 \sqrt {d+e x} \sqrt {a x^2+b x+c} \left (19 a^3 d^3-6 a^2 c d e^2+3 a b e^2 (b d-9 c e)+8 b^3 e^3\right )}{a}-\frac {e^4 \int \frac {4 a^3 (2 b d-c e) d^3-3 a^2 e \left (b^2 d^2-3 b c e d-12 c^2 e^2\right ) d+8 b^3 e^3 (b d+c e)-3 a b e^2 \left (b^2 d^2+14 b c e d+9 c^2 e^2\right )+2 \left (8 a^4 d^4-a^3 e (4 b d-9 c e) d^2+8 b^4 e^4-4 a b^2 e^3 (b d+9 c e)-3 a^2 e^2 \left (b^2 d^2-5 b c e d-7 c^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}dx}{a}}{5 a e^3}}{7 a e^4}\right )+\frac {2}{9} x^3 \sqrt {d+e x} \sqrt {a x^2+b x+c}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {x \sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \left (\frac {1}{9} \left (\frac {2 (d+e x)^{5/2} \sqrt {a x^2+b x+c} (a d+b e)}{7 a e^3}-\frac {\frac {4 e (d+e x)^{3/2} \sqrt {a x^2+b x+c} \left (8 a^2 d^2+a e (4 b d-7 c e)+3 b^2 e^2\right )}{5 a}-\frac {\frac {2 e^4 \sqrt {d+e x} \sqrt {a x^2+b x+c} \left (19 a^3 d^3-6 a^2 c d e^2+3 a b e^2 (b d-9 c e)+8 b^3 e^3\right )}{a}-\frac {e^4 \left (\frac {2 \left (8 a^4 d^4-a^3 d^2 e (4 b d-9 c e)-3 a^2 e^2 \left (b^2 d^2-5 b c d e-7 c^2 e^2\right )-4 a b^2 e^3 (b d+9 c e)+8 b^4 e^4\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a x^2+b x+c}}dx}{e}-\frac {\left (a d^2-b d e+c e^2\right ) \left (16 a^3 d^3+6 a^2 c d e^2-3 a b^2 d e^2+27 a b c e^3-8 b^3 e^3\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a x^2+b x+c}}dx}{e}\right )}{a}}{5 a e^3}}{7 a e^4}\right )+\frac {2}{9} x^3 \sqrt {d+e x} \sqrt {a x^2+b x+c}\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} x \left (\frac {2}{9} \sqrt {d+e x} \sqrt {a x^2+b x+c} x^3+\frac {1}{9} \left (\frac {2 (a d+b e) (d+e x)^{5/2} \sqrt {a x^2+b x+c}}{7 a e^3}-\frac {\frac {4 e \left (8 a^2 d^2+3 b^2 e^2+a e (4 b d-7 c e)\right ) (d+e x)^{3/2} \sqrt {a x^2+b x+c}}{5 a}-\frac {\frac {2 e^4 \left (19 a^3 d^3-6 a^2 c e^2 d+8 b^3 e^3+3 a b e^2 (b d-9 c e)\right ) \sqrt {d+e x} \sqrt {a x^2+b x+c}}{a}-\frac {e^4 \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (8 a^4 d^4-a^3 e (4 b d-9 c e) d^2+8 b^4 e^4-4 a b^2 e^3 (b d+9 c e)-3 a^2 e^2 \left (b^2 d^2-5 b c e d-7 c^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 a x+\sqrt {b^2-4 a c}\right )}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 a x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{a e \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a x^2+b x+c}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (a d^2-b e d+c e^2\right ) \left (16 a^3 d^3-3 a b^2 e^2 d+6 a^2 c e^2 d-8 b^3 e^3+27 a b c e^3\right ) \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {b+2 a x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 a x+\sqrt {b^2-4 a c}\right )}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{a e \sqrt {d+e x} \sqrt {a x^2+b x+c}}\right )}{a}}{5 a e^3}}{7 a e^4}\right )\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} x \left (\frac {2}{9} \sqrt {d+e x} \sqrt {a x^2+b x+c} x^3+\frac {1}{9} \left (\frac {2 (a d+b e) (d+e x)^{5/2} \sqrt {a x^2+b x+c}}{7 a e^3}-\frac {\frac {4 e \left (8 a^2 d^2+3 b^2 e^2+a e (4 b d-7 c e)\right ) (d+e x)^{3/2} \sqrt {a x^2+b x+c}}{5 a}-\frac {\frac {2 e^4 \left (19 a^3 d^3-6 a^2 c e^2 d+8 b^3 e^3+3 a b e^2 (b d-9 c e)\right ) \sqrt {d+e x} \sqrt {a x^2+b x+c}}{a}-\frac {e^4 \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (8 a^4 d^4-a^3 e (4 b d-9 c e) d^2+8 b^4 e^4-4 a b^2 e^3 (b d+9 c e)-3 a^2 e^2 \left (b^2 d^2-5 b c e d-7 c^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 a x+\sqrt {b^2-4 a c}\right )}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 a x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{a e \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a x^2+b x+c}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (a d^2-b e d+c e^2\right ) \left (16 a^3 d^3-3 a b^2 e^2 d+6 a^2 c e^2 d-8 b^3 e^3+27 a b c e^3\right ) \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a e \sqrt {d+e x} \sqrt {a x^2+b x+c}}\right )}{a}}{5 a e^3}}{7 a e^4}\right )\right )}{\sqrt {a x^2+b x+c}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} x \left (\frac {2}{9} \sqrt {d+e x} \sqrt {a x^2+b x+c} x^3+\frac {1}{9} \left (\frac {2 (a d+b e) (d+e x)^{5/2} \sqrt {a x^2+b x+c}}{7 a e^3}-\frac {\frac {4 e \left (8 a^2 d^2+3 b^2 e^2+a e (4 b d-7 c e)\right ) (d+e x)^{3/2} \sqrt {a x^2+b x+c}}{5 a}-\frac {\frac {2 e^4 \left (19 a^3 d^3-6 a^2 c e^2 d+8 b^3 e^3+3 a b e^2 (b d-9 c e)\right ) \sqrt {d+e x} \sqrt {a x^2+b x+c}}{a}-\frac {e^4 \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (8 a^4 d^4-a^3 e (4 b d-9 c e) d^2+8 b^4 e^4-4 a b^2 e^3 (b d+9 c e)-3 a^2 e^2 \left (b^2 d^2-5 b c e d-7 c^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a e \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a x^2+b x+c}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (a d^2-b e d+c e^2\right ) \left (16 a^3 d^3-3 a b^2 e^2 d+6 a^2 c e^2 d-8 b^3 e^3+27 a b c e^3\right ) \sqrt {\frac {a (d+e x)}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {a \left (a x^2+b x+c\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 a x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 a d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a e \sqrt {d+e x} \sqrt {a x^2+b x+c}}\right )}{a}}{5 a e^3}}{7 a e^4}\right )\right )}{\sqrt {a x^2+b x+c}}\) |
Input:
Int[Sqrt[a + c/x^2 + b/x]*x^3*Sqrt[d + e*x],x]
Output:
(Sqrt[a + c/x^2 + b/x]*x*((2*x^3*Sqrt[d + e*x]*Sqrt[c + b*x + a*x^2])/9 + ((2*(a*d + b*e)*(d + e*x)^(5/2)*Sqrt[c + b*x + a*x^2])/(7*a*e^3) - ((4*e*( 8*a^2*d^2 + 3*b^2*e^2 + a*e*(4*b*d - 7*c*e))*(d + e*x)^(3/2)*Sqrt[c + b*x + a*x^2])/(5*a) - ((2*e^4*(19*a^3*d^3 - 6*a^2*c*d*e^2 + 8*b^3*e^3 + 3*a*b* e^2*(b*d - 9*c*e))*Sqrt[d + e*x]*Sqrt[c + b*x + a*x^2])/a - (e^4*((2*Sqrt[ 2]*Sqrt[b^2 - 4*a*c]*(8*a^4*d^4 + 8*b^4*e^4 - a^3*d^2*e*(4*b*d - 9*c*e) - 4*a*b^2*e^3*(b*d + 9*c*e) - 3*a^2*e^2*(b^2*d^2 - 5*b*c*d*e - 7*c^2*e^2))*S qrt[d + e*x]*Sqrt[-((a*(c + b*x + a*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin [Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*a*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqr t[b^2 - 4*a*c]*e)/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(a*e*Sqrt[(a*(d + e*x))/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[c + b*x + a*x^2]) - (2*Sqr t[2]*Sqrt[b^2 - 4*a*c]*(a*d^2 - b*d*e + c*e^2)*(16*a^3*d^3 - 3*a*b^2*d*e^2 + 6*a^2*c*d*e^2 - 8*b^3*e^3 + 27*a*b*c*e^3)*Sqrt[(a*(d + e*x))/(2*a*d - ( b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((a*(c + b*x + a*x^2))/(b^2 - 4*a*c))]*El lipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*a*x)/Sqrt[b^2 - 4*a*c]]/Sqr t[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(a*e *Sqrt[d + e*x]*Sqrt[c + b*x + a*x^2])))/a)/(5*a*e^3))/(7*a*e^4))/9))/Sqrt[ c + b*x + a*x^2]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 )/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e *Rt[b^2 - 4*a*c, 2])))^m)) Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e }, x] && EqQ[m^2, 1/4]
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[((d_.) + (e_.)*(x_))^(m_.)*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)* (x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2*(d + e*x)^(m + 1)*Sqrt[f + g*x]*( Sqrt[a + b*x + c*x^2]/(e*(2*m + 5))), x] - Simp[1/(e*(2*m + 5)) Int[((d + e*x)^m/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]))*Simp[b*d*f - 3*a*e*f + a*d*g + 2*(c*d*f - b*e*f + b*d*g - a*e*g)*x - (c*e*f - 3*c*d*g + b*e*g)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && IntegerQ[2*m] && !LtQ[m, -1]
Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^(mn_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Simp[x^(2*n*FracPart[p])*((a + b/x^ n + c/x^(2*n))^FracPart[p]/(c + b*x^n + a*x^(2*n))^FracPart[p]) Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + b*x^n + a*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && EqQ[mn, -n] && EqQ[mn2, 2*mn] && !IntegerQ[p] && !IntegerQ[q] && PosQ[n]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c *d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !(IGt Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(3660\) vs. \(2(686)=1372\).
Time = 2.21 (sec) , antiderivative size = 3661, normalized size of antiderivative = 4.83
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(3661\) |
| default | \(\text {Expression too large to display}\) | \(9182\) |
Input:
int((a+c/x^2+b/x)^(1/2)*x^3*(e*x+d)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/315*(35*a^3*e^3*x^3+5*a^3*d*e^2*x^2+5*a^2*b*e^3*x^2-6*a^3*d^2*e*x+2*a^2* b*d*e^2*x+14*a^2*c*e^3*x-6*a*b^2*e^3*x+8*a^3*d^3-3*a^2*b*d^2*e+8*a^2*c*d*e ^2-3*a*b^2*d*e^2-27*a*b*c*e^3+8*b^3*e^3)*(e*x+d)^(1/2)/a^3/e^3*((a*x^2+b*x +c)/x^2)^(1/2)*x-1/315/a^3/e^3*(2*(16*a^4*d^4-8*a^3*b*d^3*e+18*a^3*c*d^2*e ^2-6*a^2*b^2*d^2*e^2+30*a^2*b*c*d*e^3+42*a^2*c^2*e^4-8*a*b^3*d*e^3-72*a*b^ 2*c*e^4+16*b^4*e^4)*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/a)*((x+d/e)/(d/e-1/2*( b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x-1/2*(-b+(-4*a*c+b^2)^(1/2))/a)/(-d/e-1 /2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-d /e+1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2)/(a*e*x^3+a*d*x^2+b*e*x^2+b*d*x+c*e *x+c*d)^(1/2)*((-d/e-1/2*(-b+(-4*a*c+b^2)^(1/2))/a)*EllipticE(((x+d/e)/(d/ e-1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/a )/(-d/e-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2))+1/2*(-b+(-4*a*c+b^2)^(1/2)) /a*EllipticF(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2),((-d/e+1/2 *(b+(-4*a*c+b^2)^(1/2))/a)/(-d/e-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/2)))+1 6*a^3*b*d^4*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/a)*((x+d/e)/(d/e-1/2*(b+(-4*a* c+b^2)^(1/2))/a))^(1/2)*((x-1/2*(-b+(-4*a*c+b^2)^(1/2))/a)/(-d/e-1/2*(-b+( -4*a*c+b^2)^(1/2))/a))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-d/e+1/2*( b+(-4*a*c+b^2)^(1/2))/a))^(1/2)/(a*e*x^3+a*d*x^2+b*e*x^2+b*d*x+c*e*x+c*d)^ (1/2)*EllipticF(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/a))^(1/2),((-d/e+ 1/2*(b+(-4*a*c+b^2)^(1/2))/a)/(-d/e-1/2*(-b+(-4*a*c+b^2)^(1/2))/a))^(1/...
Time = 0.10 (sec) , antiderivative size = 734, normalized size of antiderivative = 0.97 \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^3 \sqrt {d+e x} \, dx =\text {Too large to display} \] Input:
integrate((a+c/x^2+b/x)^(1/2)*x^3*(e*x+d)^(1/2),x, algorithm="fricas")
Output:
2/945*((16*a^5*d^5 - 16*a^4*b*d^4*e - 5*(a^3*b^2 - 6*a^4*c)*d^3*e^2 - (5*a ^2*b^3 - 21*a^3*b*c)*d^2*e^3 - 2*(8*a*b^4 - 42*a^2*b^2*c + 33*a^3*c^2)*d*e ^4 + (16*b^5 - 96*a*b^3*c + 123*a^2*b*c^2)*e^5)*sqrt(a*e)*weierstrassPInve rse(4/3*(a^2*d^2 - a*b*d*e + (b^2 - 3*a*c)*e^2)/(a^2*e^2), -4/27*(2*a^3*d^ 3 - 3*a^2*b*d^2*e - 3*(a*b^2 - 6*a^2*c)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)/(a^ 3*e^3), 1/3*(3*a*e*x + a*d + b*e)/(a*e)) + 6*(8*a^5*d^4*e - 4*a^4*b*d^3*e^ 2 - 3*(a^3*b^2 - 3*a^4*c)*d^2*e^3 - (4*a^2*b^3 - 15*a^3*b*c)*d*e^4 + (8*a* b^4 - 36*a^2*b^2*c + 21*a^3*c^2)*e^5)*sqrt(a*e)*weierstrassZeta(4/3*(a^2*d ^2 - a*b*d*e + (b^2 - 3*a*c)*e^2)/(a^2*e^2), -4/27*(2*a^3*d^3 - 3*a^2*b*d^ 2*e - 3*(a*b^2 - 6*a^2*c)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)/(a^3*e^3), weiers trassPInverse(4/3*(a^2*d^2 - a*b*d*e + (b^2 - 3*a*c)*e^2)/(a^2*e^2), -4/27 *(2*a^3*d^3 - 3*a^2*b*d^2*e - 3*(a*b^2 - 6*a^2*c)*d*e^2 + (2*b^3 - 9*a*b*c )*e^3)/(a^3*e^3), 1/3*(3*a*e*x + a*d + b*e)/(a*e))) + 3*(35*a^5*e^5*x^4 + 5*(a^5*d*e^4 + a^4*b*e^5)*x^3 - 2*(3*a^5*d^2*e^3 - a^4*b*d*e^4 + (3*a^3*b^ 2 - 7*a^4*c)*e^5)*x^2 + (8*a^5*d^3*e^2 - 3*a^4*b*d^2*e^3 - (3*a^3*b^2 - 8* a^4*c)*d*e^4 + (8*a^2*b^3 - 27*a^3*b*c)*e^5)*x)*sqrt(e*x + d)*sqrt((a*x^2 + b*x + c)/x^2))/(a^5*e^5)
\[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^3 \sqrt {d+e x} \, dx=\int x^{3} \sqrt {d + e x} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}}\, dx \] Input:
integrate((a+c/x**2+b/x)**(1/2)*x**3*(e*x+d)**(1/2),x)
Output:
Integral(x**3*sqrt(d + e*x)*sqrt(a + b/x + c/x**2), x)
\[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^3 \sqrt {d+e x} \, dx=\int { \sqrt {e x + d} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}} x^{3} \,d x } \] Input:
integrate((a+c/x^2+b/x)^(1/2)*x^3*(e*x+d)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(e*x + d)*sqrt(a + b/x + c/x^2)*x^3, x)
\[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^3 \sqrt {d+e x} \, dx=\int { \sqrt {e x + d} \sqrt {a + \frac {b}{x} + \frac {c}{x^{2}}} x^{3} \,d x } \] Input:
integrate((a+c/x^2+b/x)^(1/2)*x^3*(e*x+d)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(e*x + d)*sqrt(a + b/x + c/x^2)*x^3, x)
Timed out. \[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^3 \sqrt {d+e x} \, dx=\int x^3\,\sqrt {d+e\,x}\,\sqrt {a+\frac {b}{x}+\frac {c}{x^2}} \,d x \] Input:
int(x^3*(d + e*x)^(1/2)*(a + b/x + c/x^2)^(1/2),x)
Output:
int(x^3*(d + e*x)^(1/2)*(a + b/x + c/x^2)^(1/2), x)
\[ \int \sqrt {a+\frac {c}{x^2}+\frac {b}{x}} x^3 \sqrt {d+e x} \, dx=\int \sqrt {a +\frac {c}{x^{2}}+\frac {b}{x}}\, x^{3} \sqrt {e x +d}d x \] Input:
int((a+c/x^2+b/x)^(1/2)*x^3*(e*x+d)^(1/2),x)
Output:
int((a+c/x^2+b/x)^(1/2)*x^3*(e*x+d)^(1/2),x)