Integrand size = 19, antiderivative size = 404 \[ \int x \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right ) \, dx=-\frac {4 b \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {8 b d \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \left (3 c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 c^4 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {8 b d^3 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
-2/3*d*(e*x+d)^(3/2)*(a+b*arccsc(c*x))/e^2+2/5*(e*x+d)^(5/2)*(a+b*arccsc(c *x))/e^2-4/15*b*(-c^2*x^2+1)*(e*x+d)^(1/2)/c^3/x/(1-1/c^2/x^2)^(1/2)-8/15* b*d*EllipticE(1/2*(-c*x+1)^(1/2)*2^(1/2),2^(1/2)*(e/(c*d+e))^(1/2))*(e*x+d )^(1/2)*(-c^2*x^2+1)^(1/2)/c^2/e/x/(1-1/c^2/x^2)^(1/2)/(c*(e*x+d)/(c*d+e)) ^(1/2)+4/15*b*(3*c^2*d^2-e^2)*EllipticF(1/2*(-c*x+1)^(1/2)*2^(1/2),2^(1/2) *(e/(c*d+e))^(1/2))*(c*(e*x+d)/(c*d+e))^(1/2)*(-c^2*x^2+1)^(1/2)/c^4/e/x/( 1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)+8/15*b*d^3*EllipticPi(1/2*(-c*x+1)^(1/2)* 2^(1/2),2,2^(1/2)*(e/(c*d+e))^(1/2))*(c*(e*x+d)/(c*d+e))^(1/2)*(-c^2*x^2+1 )^(1/2)/c/e^2/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)
Result contains complex when optimal does not.
Time = 12.42 (sec) , antiderivative size = 368, normalized size of antiderivative = 0.91 \[ \int x \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {1}{15} \left (\frac {4 b \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}{c}+\frac {2 a \sqrt {d+e x} \left (-2 d^2+d e x+3 e^2 x^2\right )}{e^2}+\frac {2 b \sqrt {d+e x} \left (-2 d^2+d e x+3 e^2 x^2\right ) \csc ^{-1}(c x)}{e^2}-\frac {4 i b \sqrt {\frac {e (1+c x)}{-c d+e}} \sqrt {\frac {e-c e x}{c d+e}} \left (-2 c d (c d-e) E\left (i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right )|\frac {c d+e}{c d-e}\right )+\left (-c^2 d^2-2 c d e+e^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right ),\frac {c d+e}{c d-e}\right )+2 c^2 d^2 \operatorname {EllipticPi}\left (1+\frac {e}{c d},i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right ),\frac {c d+e}{c d-e}\right )\right )}{c^3 e^2 \sqrt {-\frac {c}{c d+e}} \sqrt {1-\frac {1}{c^2 x^2}} x}\right ) \]
((4*b*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])/c + (2*a*Sqrt[d + e*x]*(-2*d^ 2 + d*e*x + 3*e^2*x^2))/e^2 + (2*b*Sqrt[d + e*x]*(-2*d^2 + d*e*x + 3*e^2*x ^2)*ArcCsc[c*x])/e^2 - ((4*I)*b*Sqrt[(e*(1 + c*x))/(-(c*d) + e)]*Sqrt[(e - c*e*x)/(c*d + e)]*(-2*c*d*(c*d - e)*EllipticE[I*ArcSinh[Sqrt[-(c/(c*d + e ))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] + (-(c^2*d^2) - 2*c*d*e + e^2)*El lipticF[I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e) ] + 2*c^2*d^2*EllipticPi[1 + e/(c*d), I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[ d + e*x]], (c*d + e)/(c*d - e)]))/(c^3*e^2*Sqrt[-(c/(c*d + e))]*Sqrt[1 - 1 /(c^2*x^2)]*x))/15
Time = 2.18 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.17, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.158, Rules used = {5770, 27, 7272, 2351, 27, 497, 27, 600, 508, 327, 511, 321, 634, 600, 508, 327, 511, 321, 632, 186, 413, 412}
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.
| \(\displaystyle \int x \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right ) \, dx\) |
| \(\Big \downarrow \) 5770 |
| \(\displaystyle \frac {b \int -\frac {2 (2 d-3 e x) (d+e x)^{3/2}}{15 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2}dx}{c}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \int \frac {(2 d-3 e x) (d+e x)^{3/2}}{\sqrt {1-\frac {1}{c^2 x^2}} x^2}dx}{15 c e^2}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 7272 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \int \frac {(2 d-3 e x) (d+e x)^{3/2}}{x \sqrt {1-c^2 x^2}}dx}{15 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 2351 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\int -\frac {3 e (d+e x)^{3/2}}{\sqrt {1-c^2 x^2}}dx+2 d \int \frac {(d+e x)^{3/2}}{x \sqrt {1-c^2 x^2}}dx\right )}{15 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \int \frac {(d+e x)^{3/2}}{x \sqrt {1-c^2 x^2}}dx-3 e \int \frac {(d+e x)^{3/2}}{\sqrt {1-c^2 x^2}}dx\right )}{15 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 497 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \int \frac {(d+e x)^{3/2}}{x \sqrt {1-c^2 x^2}}dx-3 e \left (-\frac {2 \int -\frac {3 d^2 c^2+4 d e x c^2+e^2}{2 \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )\right )}{15 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \int \frac {(d+e x)^{3/2}}{x \sqrt {1-c^2 x^2}}dx-3 e \left (\frac {\int \frac {3 d^2 c^2+4 d e x c^2+e^2}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )\right )}{15 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \int \frac {(d+e x)^{3/2}}{x \sqrt {1-c^2 x^2}}dx-3 e \left (\frac {4 c^2 d \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}}dx-(c d-e) (c d+e) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )\right )}{15 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \int \frac {(d+e x)^{3/2}}{x \sqrt {1-c^2 x^2}}dx-3 e \left (\frac {-\left ((c d-e) (c d+e) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx\right )-\frac {8 c d \sqrt {d+e x} \int \frac {\sqrt {1-\frac {e (1-c x)}{c d+e}}}{\sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{\sqrt {\frac {c (d+e x)}{c d+e}}}}{3 c^2}-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )\right )}{15 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \int \frac {(d+e x)^{3/2}}{x \sqrt {1-c^2 x^2}}dx-3 e \left (\frac {-(c d-e) (c d+e) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {8 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}}{3 c^2}-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )\right )}{15 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \int \frac {(d+e x)^{3/2}}{x \sqrt {1-c^2 x^2}}dx-3 e \left (\frac {\frac {2 (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \int \frac {1}{\sqrt {1-\frac {e (1-c x)}{c d+e}} \sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {d+e x}}-\frac {8 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}}{3 c^2}-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )\right )}{15 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \int \frac {(d+e x)^{3/2}}{x \sqrt {1-c^2 x^2}}dx-3 e \left (\frac {\frac {2 (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {8 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}}{3 c^2}-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )\right )}{15 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 634 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \left (d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\int \frac {-x e^2-2 d e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx\right )-3 e \left (\frac {\frac {2 (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {8 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}}{3 c^2}-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )\right )}{15 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \left (d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+d e \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+e \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}}dx\right )-3 e \left (\frac {\frac {2 (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {8 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}}{3 c^2}-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )\right )}{15 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \left (d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+d e \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {2 e \sqrt {d+e x} \int \frac {\sqrt {1-\frac {e (1-c x)}{c d+e}}}{\sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )-3 e \left (\frac {\frac {2 (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {8 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}}{3 c^2}-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )\right )}{15 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \left (d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+d e \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )-3 e \left (\frac {\frac {2 (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {8 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}}{3 c^2}-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )\right )}{15 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \left (d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {2 d e \sqrt {\frac {c (d+e x)}{c d+e}} \int \frac {1}{\sqrt {1-\frac {e (1-c x)}{c d+e}} \sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {d+e x}}-\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )-3 e \left (\frac {\frac {2 (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {8 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}}{3 c^2}-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )\right )}{15 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \left (d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {2 d e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )-3 e \left (\frac {\frac {2 (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {8 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}}{3 c^2}-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )\right )}{15 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 632 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \left (d^2 \int \frac {1}{x \sqrt {1-c x} \sqrt {c x+1} \sqrt {d+e x}}dx-\frac {2 d e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )-3 e \left (\frac {\frac {2 (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {8 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}}{3 c^2}-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )\right )}{15 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \left (-2 d^2 \int \frac {1}{c x \sqrt {c x+1} \sqrt {d+\frac {e}{c}-\frac {e (1-c x)}{c}}}d\sqrt {1-c x}-\frac {2 d e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )-3 e \left (\frac {\frac {2 (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {8 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}}{3 c^2}-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )\right )}{15 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \left (-\frac {2 d^2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \int \frac {1}{c x \sqrt {c x+1} \sqrt {1-\frac {e (1-c x)}{c d+e}}}d\sqrt {1-c x}}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 d e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )-3 e \left (\frac {\frac {2 (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {8 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}}{3 c^2}-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )\right )}{15 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^2}-\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \left (-\frac {2 d^2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 d e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )-3 e \left (\frac {\frac {2 (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {8 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}}{3 c^2}-\frac {2 e \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )\right )}{15 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}\) |
(-2*d*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e^2) + (2*(d + e*x)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e^2) - (2*b*Sqrt[1 - c^2*x^2]*(-3*e*((-2*e*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2])/(3*c^2) + ((-8*c*d*Sqrt[d + e*x]*EllipticE[ArcSin[ Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/Sqrt[(c*(d + e*x))/(c*d + e)] + (2*(c*d - e)*(c*d + e)*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticF[ArcSin[Sqrt [1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*Sqrt[d + e*x]))/(3*c^2)) + 2*d*(( -2*e*Sqrt[d + e*x]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e )])/(c*Sqrt[(c*(d + e*x))/(c*d + e)]) - (2*d*e*Sqrt[(c*(d + e*x))/(c*d + e )]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*Sqrt[d + e*x]) - (2*d^2*Sqrt[1 - (e*(1 - c*x))/(c*d + e)]*EllipticPi[2, ArcSin[Sqrt [1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/Sqrt[d + e/c - (e*(1 - c*x))/c])))/( 15*c*e^2*Sqrt[1 - 1/(c^2*x^2)]*x)
3.1.52.3.1 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[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[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[1/(b *(n + 2*p + 1)) Int[(c + d*x)^(n - 2)*(a + b*x^2)^p*Simp[b*c^2*(n + 2*p + 1) - a*d^2*(n - 1) + 2*b*c*d*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, n , p}, x] && If[RationalQ[n], GtQ[n, 1], SumSimplerQ[n, -2]] && NeQ[n + 2*p + 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
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[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[((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[((c_) + (d_.)*(x_))^(n_)/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp[c^(n + 1/2) Int[1/(x*Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] - Int[( 1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]))*ExpandToSum[(c^(n + 1/2) - (c + d*x)^(n + 1/2))/x, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n - 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]
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*(u_), x_Symbol] :> With[{v = IntHide [u, x]}, Simp[(a + b*ArcCsc[c*x]) v, x] + Simp[b/c Int[SimplifyIntegran d[v/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x], x] /; InverseFunctionFreeQ[v, x]] /; FreeQ[{a, b, c}, x]
Int[(u_.)*((a_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[b^IntPart[p]*(( a + b*x^n)^FracPart[p]/(x^(n*FracPart[p])*(1 + a*(1/(x^n*b)))^FracPart[p])) Int[u*x^(n*p)*(1 + a*(1/(x^n*b)))^p, x], x] /; FreeQ[{a, b, p}, x] && ! IntegerQ[p] && ILtQ[n, 0] && !RationalFunctionQ[u, x] && IntegerQ[p + 1/2]
Leaf count of result is larger than twice the leaf count of optimal. \(825\) vs. \(2(363)=726\).
Time = 9.99 (sec) , antiderivative size = 826, normalized size of antiderivative = 2.04
| method | result | size |
| derivativedivides | \(\frac {-2 a \left (-\frac {\left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {\left (e x +d \right )^{\frac {3}{2}} d}{3}\right )-2 b \left (-\frac {\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {3}{2}} d}{3}-\frac {2 \left (\sqrt {\frac {c}{c d -e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}-2 \sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}-d^{2} \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2}-2 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2} d^{2}+2 d^{2} \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c^{2}+\sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \sqrt {e x +d}+2 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e -2 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e +\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e^{2}-\sqrt {\frac {c}{c d -e}}\, e^{2} \sqrt {e x +d}\right )}{15 c^{3} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e^{2}}\) | \(826\) |
| default | \(\frac {-2 a \left (-\frac {\left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {\left (e x +d \right )^{\frac {3}{2}} d}{3}\right )-2 b \left (-\frac {\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {3}{2}} d}{3}-\frac {2 \left (\sqrt {\frac {c}{c d -e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}-2 \sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}-d^{2} \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2}-2 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2} d^{2}+2 d^{2} \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c^{2}+\sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \sqrt {e x +d}+2 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e -2 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e +\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e^{2}-\sqrt {\frac {c}{c d -e}}\, e^{2} \sqrt {e x +d}\right )}{15 c^{3} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e^{2}}\) | \(826\) |
| parts | \(\frac {2 a \left (\frac {\left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {\left (e x +d \right )^{\frac {3}{2}} d}{3}\right )}{e^{2}}+\frac {2 b \left (\frac {\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {3}{2}} d}{3}+\frac {\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}}{15}-\frac {4 \sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{15}-\frac {2 d^{2} \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2}}{15}-\frac {4 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2} d^{2}}{15}+\frac {4 d^{2} \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c^{2}}{15}+\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \sqrt {e x +d}}{15}+\frac {4 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e}{15}-\frac {4 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e}{15}+\frac {2 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e^{2}}{15}-\frac {2 \sqrt {\frac {c}{c d -e}}\, e^{2} \sqrt {e x +d}}{15}}{c^{3} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e^{2}}\) | \(839\) |
2/e^2*(-a*(-1/5*(e*x+d)^(5/2)+1/3*(e*x+d)^(3/2)*d)-b*(-1/5*arccsc(c*x)*(e* x+d)^(5/2)+1/3*arccsc(c*x)*(e*x+d)^(3/2)*d-2/15/c^3*((c/(c*d-e))^(1/2)*c^2 *(e*x+d)^(5/2)-2*(c/(c*d-e))^(1/2)*c^2*d*(e*x+d)^(3/2)-d^2*((-c*(e*x+d)+c* d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticF((e*x+d)^( 1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*c^2-2*((-c*(e*x+d)+c*d-e)/ (c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticE((e*x+d)^(1/2)* (c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*c^2*d^2+2*d^2*((-c*(e*x+d)+c*d- e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticPi((e*x+d)^(1 /2)*(c/(c*d-e))^(1/2),1/c*(c*d-e)/d,(c/(c*d+e))^(1/2)/(c/(c*d-e))^(1/2))*c ^2+(c/(c*d-e))^(1/2)*c^2*d^2*(e*x+d)^(1/2)+2*((-c*(e*x+d)+c*d-e)/(c*d-e))^ (1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticF((e*x+d)^(1/2)*(c/(c*d-e ))^(1/2),((c*d-e)/(c*d+e))^(1/2))*c*d*e-2*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/ 2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticE((e*x+d)^(1/2)*(c/(c*d-e))^ (1/2),((c*d-e)/(c*d+e))^(1/2))*c*d*e+((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*(( -c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2) ,((c*d-e)/(c*d+e))^(1/2))*e^2-(c/(c*d-e))^(1/2)*e^2*(e*x+d)^(1/2))/(c/(c*d -e))^(1/2)/x/((c^2*(e*x+d)^2-2*c^2*d*(e*x+d)+c^2*d^2-e^2)/c^2/e^2/x^2)^(1/ 2)))
\[ \int x \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int { \sqrt {e x + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x \,d x } \]
\[ \int x \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int x \left (a + b \operatorname {acsc}{\left (c x \right )}\right ) \sqrt {d + e x}\, dx \]
Exception generated. \[ \int x \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\text {Exception raised: ValueError} \]
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+c*d>0)', see `assume?` for mor e details)
\[ \int x \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int { \sqrt {e x + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x \,d x } \]
Timed out. \[ \int x \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int x\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )\,\sqrt {d+e\,x} \,d x \]