Integrand size = 45, antiderivative size = 529 \[ \int \frac {A+B x+C x^2+D x^3+F x^4}{x \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=-\frac {2 (5 d D-7 c F) \sqrt {c+d x} \sqrt {a-b x^2}}{15 b d^2}-\frac {2 F (c+d x)^{3/2} \sqrt {a-b x^2}}{5 b d^2}-\frac {2 \sqrt {a} \left (15 b C d^2-10 b c d D+8 b c^2 F+9 a d^2 F\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{15 b^{3/2} d^3 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \left (a d^2 (5 d D-7 c F)-b \left (15 c C d^2-15 B d^3-10 c^2 d D+8 c^3 F\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{15 b^{3/2} d^3 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 A \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-2/15*(5*D*d-7*F*c)*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/b/d^2-2/5*F*(d*x+c)^(3/ 2)*(-b*x^2+a)^(1/2)/b/d^2-2/15*a^(1/2)*(15*C*b*d^2-10*D*b*c*d+9*F*a*d^2+8* F*b*c^2)*(d*x+c)^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticE(1/2*(1-b^(1/2)*x/a^( 1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/b^(3/ 2)/d^3/((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)-2/15*a^(1/2) *(a*d^2*(5*D*d-7*F*c)-b*(-15*B*d^3+15*C*c*d^2-10*D*c^2*d+8*F*c^3))*((d*x+c )/(c+a^(1/2)*d/b^(1/2)))^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticF(1/2*(1-b^(1/ 2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2 ))/b^(3/2)/d^3/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)-2*A*((d*x+c)/(c+a^(1/2)*d/b^ (1/2)))^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticPi(1/2*(1-b^(1/2)*x/a^(1/2))^(1 /2)*2^(1/2),2,2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/(d*x+c)^(1/ 2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 25.43 (sec) , antiderivative size = 792, normalized size of antiderivative = 1.50 \[ \int \frac {A+B x+C x^2+D x^3+F x^4}{x \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=-\frac {2 \left (c d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (15 b C d^2+9 a d^2 F+2 b c (-5 d D+4 c F)\right ) \left (a-b x^2\right )+b c d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} (c+d x) (5 d D-4 c F+3 d F x) \left (a-b x^2\right )+i c \left (b c-\sqrt {a} \sqrt {b} d\right ) \left (15 b C d^2+9 a d^2 F+2 b c (-5 d D+4 c F)\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \left (E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )+i b \left (a c d^2 (-5 d D+7 c F)+b \left (15 c^2 C d^2-15 B c d^3+15 A d^4-10 c^3 d D+8 c^4 F\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )-15 i A b^2 d^4 \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticPi}\left (\frac {\sqrt {b} c}{\sqrt {b} c-\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{15 b^2 c d^4 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \sqrt {c+d x} \sqrt {a-b x^2}} \] Input:
Integrate[(A + B*x + C*x^2 + D*x^3 + F*x^4)/(x*Sqrt[c + d*x]*Sqrt[a - b*x^ 2]),x]
Output:
(-2*(c*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(15*b*C*d^2 + 9*a*d^2*F + 2*b*c* (-5*d*D + 4*c*F))*(a - b*x^2) + b*c*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)*(5*d*D - 4*c*F + 3*d*F*x)*(a - b*x^2) + I*c*(b*c - Sqrt[a]*Sqrt[b]* d)*(15*b*C*d^2 + 9*a*d^2*F + 2*b*c*(-5*d*D + 4*c*F))*Sqrt[(d*(Sqrt[a]/Sqrt [b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d *x)^(3/2)*(EllipticE[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x ]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] - EllipticF[I*ArcSinh [Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(S qrt[b]*c - Sqrt[a]*d)]) + I*b*(a*c*d^2*(-5*d*D + 7*c*F) + b*(15*c^2*C*d^2 - 15*B*c*d^3 + 15*A*d^4 - 10*c^3*d*D + 8*c^4*F))*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^ (3/2)*EllipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], ( Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] - (15*I)*A*b^2*d^4*Sqrt[(d *(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticPi[(Sqrt[b]*c)/(Sqrt[b]*c - Sqrt[a]*d), I* ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a ]*d)/(Sqrt[b]*c - Sqrt[a]*d)]))/(15*b^2*c*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b ]]*Sqrt[c + d*x]*Sqrt[a - b*x^2])
Time = 2.37 (sec) , antiderivative size = 591, normalized size of antiderivative = 1.12, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.378, Rules used = {2351, 633, 632, 186, 413, 412, 2185, 27, 2185, 27, 600, 509, 508, 327, 512, 511, 321}
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.
| \(\displaystyle \int \frac {A+B x+C x^2+D x^3+F x^4}{x \sqrt {a-b x^2} \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 2351 |
| \(\displaystyle A \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+\int \frac {F x^3+D x^2+C x+B}{\sqrt {c+d x} \sqrt {a-b x^2}}dx\) |
| \(\Big \downarrow \) 633 |
| \(\displaystyle \frac {A \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{x \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}+\int \frac {F x^3+D x^2+C x+B}{\sqrt {c+d x} \sqrt {a-b x^2}}dx\) |
| \(\Big \downarrow \) 632 |
| \(\displaystyle \frac {A \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{x \sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}} \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {c+d x}}dx}{\sqrt {a-b x^2}}+\int \frac {F x^3+D x^2+C x+B}{\sqrt {c+d x} \sqrt {a-b x^2}}dx\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle \int \frac {F x^3+D x^2+C x+B}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {2 A \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {a}}{\sqrt {b} x \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {c+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}}}d\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {a-b x^2}}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \int \frac {F x^3+D x^2+C x+B}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {2 A \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \int \frac {\sqrt {a}}{\sqrt {b} x \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} c+\sqrt {a} d}}}d\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \int \frac {F x^3+D x^2+C x+B}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {2 A \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle -\frac {2 \int -\frac {b (5 d D-7 c F) x^2 d^2+(5 b B d+3 a c F) d^2+\left (-2 b F c^2+5 b C d^2+3 a d^2 F\right ) x d}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{5 b d^3}-\frac {2 A \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 F \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {b (5 d D-7 c F) x^2 d^2+(5 b B d+3 a c F) d^2+\left (-2 b F c^2+5 b C d^2+3 a d^2 F\right ) x d}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{5 b d^3}-\frac {2 A \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 F \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {b d^3 \left (d (15 b B d+5 a D d+2 a c F)+\left (15 b C d^2+9 a F d^2-2 b c (5 d D-4 c F)\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b d^2}-\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (5 d D-7 c F)}{5 b d^3}-\frac {2 A \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 F \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} d \int \frac {d (15 b B d+5 a D d+2 a c F)+\left (15 b C d^2+9 a F d^2-2 b c (5 d D-4 c F)\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (5 d D-7 c F)}{5 b d^3}-\frac {2 A \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 F \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {1}{3} d \left (\frac {\left (a d^2 (5 d D-7 c F)-b \left (-15 B d^3+8 c^3 F-10 c^2 d D+15 c C d^2\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {\left (9 a d^2 F-2 b c (5 d D-4 c F)+15 b C d^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}\right )-\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (5 d D-7 c F)}{5 b d^3}-\frac {2 A \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 F \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {1}{3} d \left (\frac {\left (a d^2 (5 d D-7 c F)-b \left (-15 B d^3+8 c^3 F-10 c^2 d D+15 c C d^2\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {\sqrt {1-\frac {b x^2}{a}} \left (9 a d^2 F-2 b c (5 d D-4 c F)+15 b C d^2\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )-\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (5 d D-7 c F)}{5 b d^3}-\frac {2 A \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 F \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\frac {1}{3} d \left (\frac {\left (a d^2 (5 d D-7 c F)-b \left (-15 B d^3+8 c^3 F-10 c^2 d D+15 c C d^2\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (9 a d^2 F-2 b c (5 d D-4 c F)+15 b C d^2\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (5 d D-7 c F)}{5 b d^3}-\frac {2 A \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 F \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {1}{3} d \left (\frac {\left (a d^2 (5 d D-7 c F)-b \left (-15 B d^3+8 c^3 F-10 c^2 d D+15 c C d^2\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (9 a d^2 F-2 b c (5 d D-4 c F)+15 b C d^2\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (5 d D-7 c F)}{5 b d^3}-\frac {2 A \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 F \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\frac {1}{3} d \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (a d^2 (5 d D-7 c F)-b \left (-15 B d^3+8 c^3 F-10 c^2 d D+15 c C d^2\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (9 a d^2 F-2 b c (5 d D-4 c F)+15 b C d^2\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (5 d D-7 c F)}{5 b d^3}-\frac {2 A \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 F \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\frac {1}{3} d \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (a d^2 (5 d D-7 c F)-b \left (-15 B d^3+8 c^3 F-10 c^2 d D+15 c C d^2\right )\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (9 a d^2 F-2 b c (5 d D-4 c F)+15 b C d^2\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (5 d D-7 c F)}{5 b d^3}-\frac {2 A \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 F \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {2 A \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}+\frac {\frac {1}{3} d \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (a d^2 (5 d D-7 c F)-b \left (-15 B d^3+8 c^3 F-10 c^2 d D+15 c C d^2\right )\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (9 a d^2 F-2 b c (5 d D-4 c F)+15 b C d^2\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} d \sqrt {a-b x^2} \sqrt {c+d x} (5 d D-7 c F)}{5 b d^3}-\frac {2 F \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b d^2}\) |
Input:
Int[(A + B*x + C*x^2 + D*x^3 + F*x^4)/(x*Sqrt[c + d*x]*Sqrt[a - b*x^2]),x]
Output:
(-2*F*(c + d*x)^(3/2)*Sqrt[a - b*x^2])/(5*b*d^2) + ((-2*d*(5*d*D - 7*c*F)* Sqrt[c + d*x]*Sqrt[a - b*x^2])/3 + (d*((-2*Sqrt[a]*(15*b*C*d^2 + 9*a*d^2*F - 2*b*c*(5*d*D - 4*c*F))*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcS in[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d) ])/(Sqrt[b]*d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b *x^2]) - (2*Sqrt[a]*(a*d^2*(5*d*D - 7*c*F) - b*(15*c*C*d^2 - 15*B*d^3 - 10 *c^2*d*D + 8*c^3*F))*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqr t[1 - (b*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2] )))/3)/(5*b*d^3) - (2*A*Sqrt[1 - (b*x^2)/a]*Sqrt[1 - (Sqrt[a]*d*(1 - (Sqrt [b]*x)/Sqrt[a]))/(Sqrt[b]*c + Sqrt[a]*d)]*EllipticPi[2, ArcSin[Sqrt[1 - (S qrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*Sqrt[a]*d)/(Sqrt[b]*c + Sqrt[a]*d)])/(Sqrt [a - b*x^2]*Sqrt[c + (Sqrt[a]*d)/Sqrt[b] - (Sqrt[a]*d*(1 - (Sqrt[b]*x)/Sqr t[a]))/Sqrt[b]])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ )]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2 Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c*f)/d, 0]
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[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[b/a]
Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > With[{q = Rt[-b/a, 2]}, Simp[1/Sqrt[a] Int[1/(x*Sqrt[c + d*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > Simp[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(x*Sqrt[c + d*x]*Sqrt[1 + b*(x^2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) ^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x )^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p )*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d , e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && !(EqQ[d, 0] && True) && !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Int[((Px_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.))/(x_), x_S ymbol] :> Int[PolynomialQuotient[Px, x, x]*(c + d*x)^n*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, x, x] Int[(c + d*x)^n*((a + b*x^2)^p/x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && PolynomialQ[Px, x]
Leaf count of result is larger than twice the leaf count of optimal. \(893\) vs. \(2(442)=884\).
Time = 3.14 (sec) , antiderivative size = 894, normalized size of antiderivative = 1.69
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (d x +c \right )}\, \left (-\frac {2 F x \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{5 b d}-\frac {2 \left (D-\frac {4 F c}{5 d}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 b d}+\frac {2 \left (B +\frac {2 F a c}{5 b d}+\frac {\left (D-\frac {4 F c}{5 d}\right ) a}{3 b}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (C +\frac {3 F a}{5 b}-\frac {2 \left (D-\frac {4 F c}{5 d}\right ) c}{3 d}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {2 A \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, d \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, -\frac {\left (-\frac {c}{d}+\frac {\sqrt {a b}}{b}\right ) d}{c}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}\, c}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}}\) | \(894\) |
| default | \(\text {Expression too large to display}\) | \(3088\) |
Input:
int((F*x^4+D*x^3+C*x^2+B*x+A)/x/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x,method=_R ETURNVERBOSE)
Output:
((-b*x^2+a)*(d*x+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x+c)^(1/2)*(-2/5*F/b/d*x*(- b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/3*(D-4/5*F/d*c)/b/d*(-b*d*x^3-b*c*x^2+a *d*x+a*c)^(1/2)+2*(B+2/5*F/b/d*a*c+1/3*(D-4/5*F/d*c)/b*a)*(c/d-1/b*(a*b)^( 1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b *(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(- b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2))) ^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+2*(C+3/5*F/b *a-2/3*(D-4/5*F/d*c)/d*c)*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1 /2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a* b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2) *((-c/d-1/b*(a*b)^(1/2))*EllipticE(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),( (-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+1/b*(a*b)^(1/2)*Elli pticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d- 1/b*(a*b)^(1/2)))^(1/2)))-2*A*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b )^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b *(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^( 1/2)/c*d*EllipticPi(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),-(-c/d+1/b*(a*b) ^(1/2))/c*d,((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)))
Timed out. \[ \int \frac {A+B x+C x^2+D x^3+F x^4}{x \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\text {Timed out} \] Input:
integrate((F*x^4+D*x^3+C*x^2+B*x+A)/x/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, al gorithm="fricas")
Output:
Timed out
\[ \int \frac {A+B x+C x^2+D x^3+F x^4}{x \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int \frac {A + B x + C x^{2} + D x^{3} + F x^{4}}{x \sqrt {a - b x^{2}} \sqrt {c + d x}}\, dx \] Input:
integrate((F*x**4+D*x**3+C*x**2+B*x+A)/x/(d*x+c)**(1/2)/(-b*x**2+a)**(1/2) ,x)
Output:
Integral((A + B*x + C*x**2 + D*x**3 + F*x**4)/(x*sqrt(a - b*x**2)*sqrt(c + d*x)), x)
\[ \int \frac {A+B x+C x^2+D x^3+F x^4}{x \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int { \frac {F x^{4} + D x^{3} + C x^{2} + B x + A}{\sqrt {-b x^{2} + a} \sqrt {d x + c} x} \,d x } \] Input:
integrate((F*x^4+D*x^3+C*x^2+B*x+A)/x/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, al gorithm="maxima")
Output:
integrate((F*x^4 + D*x^3 + C*x^2 + B*x + A)/(sqrt(-b*x^2 + a)*sqrt(d*x + c )*x), x)
\[ \int \frac {A+B x+C x^2+D x^3+F x^4}{x \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int { \frac {F x^{4} + D x^{3} + C x^{2} + B x + A}{\sqrt {-b x^{2} + a} \sqrt {d x + c} x} \,d x } \] Input:
integrate((F*x^4+D*x^3+C*x^2+B*x+A)/x/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, al gorithm="giac")
Output:
integrate((F*x^4 + D*x^3 + C*x^2 + B*x + A)/(sqrt(-b*x^2 + a)*sqrt(d*x + c )*x), x)
Timed out. \[ \int \frac {A+B x+C x^2+D x^3+F x^4}{x \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int \frac {A+B\,x+C\,x^2+F\,x^4+x^3\,D}{x\,\sqrt {a-b\,x^2}\,\sqrt {c+d\,x}} \,d x \] Input:
int((A + B*x + C*x^2 + F*x^4 + x^3*D)/(x*(a - b*x^2)^(1/2)*(c + d*x)^(1/2) ),x)
Output:
int((A + B*x + C*x^2 + F*x^4 + x^3*D)/(x*(a - b*x^2)^(1/2)*(c + d*x)^(1/2) ), x)
\[ \int \frac {A+B x+C x^2+D x^3+F x^4}{x \sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\frac {-6 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, a d f -10 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b c d -4 \sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, b c f x -9 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a b \,d^{2} f -8 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b^{2} c^{2} f -5 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}\, x^{2}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b^{2} c \,d^{2}+10 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{4}-b c \,x^{3}+a d \,x^{2}+a c x}d x \right ) a \,b^{2} c d +3 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a^{2} d^{2} f +4 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a b \,c^{2} f +5 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) a b c \,d^{2}+10 \left (\int \frac {\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}{-b d \,x^{3}-b c \,x^{2}+a d x +a c}d x \right ) b^{3} c d}{10 b^{2} c d} \] Input:
int((F*x^4+D*x^3+C*x^2+B*x+A)/x/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x)
Output:
( - 6*sqrt(c + d*x)*sqrt(a - b*x**2)*a*d*f - 10*sqrt(c + d*x)*sqrt(a - b*x **2)*b*c*d - 4*sqrt(c + d*x)*sqrt(a - b*x**2)*b*c*f*x - 9*int((sqrt(c + d* x)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a*b*d**2* f - 8*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*b**2*c**2*f - 5*int((sqrt(c + d*x)*sqrt(a - b*x**2)*x**2)/(a* c + a*d*x - b*c*x**2 - b*d*x**3),x)*b**2*c*d**2 + 10*int((sqrt(c + d*x)*sq rt(a - b*x**2))/(a*c*x + a*d*x**2 - b*c*x**3 - b*d*x**4),x)*a*b**2*c*d + 3 *int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c + a*d*x - b*c*x**2 - b*d*x**3), x)*a**2*d**2*f + 4*int((sqrt(c + d*x)*sqrt(a - b*x**2))/(a*c + a*d*x - b*c *x**2 - b*d*x**3),x)*a*b*c**2*f + 5*int((sqrt(c + d*x)*sqrt(a - b*x**2))/( a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*a*b*c*d**2 + 10*int((sqrt(c + d*x)*s qrt(a - b*x**2))/(a*c + a*d*x - b*c*x**2 - b*d*x**3),x)*b**3*c*d)/(10*b**2 *c*d)