Integrand size = 18, antiderivative size = 540 \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\frac {4 b e \left (1-c^2 x^2\right )}{15 c d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x (d+e x)^{3/2}}+\frac {16 b c e \left (1-c^2 x^2\right )}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b e \left (1-c^2 x^2\right )}{5 c d^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {4 b \left (7 c^2 d^2-3 e^2\right ) \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 \left (c^2 d^3-d e^2\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{5 c d^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
-2/5*(a+b*arccsc(c*x))/e/(e*x+d)^(5/2)+4/15*b*e*(-c^2*x^2+1)/c/d/(c^2*d^2- e^2)/x/(e*x+d)^(3/2)/(1-1/c^2/x^2)^(1/2)+16/15*b*c*e*(-c^2*x^2+1)/(c^2*d^2 -e^2)^2/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)+4/5*b*e*(-c^2*x^2+1)/c/d^2/(c^ 2*d^2-e^2)/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)-4/15*b*(7*c^2*d^2-3*e^2)*El lipticE(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*d^3-d*e^2)^2/x/(1-1/c^2/x^2)^(1/2)/(c*(e*x+d)/(c *d+e))^(1/2)+4/15*b*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)/d/(c^2*d^2-e^2)/x/( 1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)+4/5*b*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/d^2/e/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)
Time = 34.05 (sec) , antiderivative size = 1002, normalized size of antiderivative = 1.86 \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{7/2}} \, dx=-\frac {2 a}{5 e (d+e x)^{5/2}}+\frac {b \left (-\frac {c^4 \left (e+\frac {d}{x}\right )^4 x^4 \left (\frac {4 \left (-7 c^2 d^2+3 e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}}}{15 c^2 d^2 \left (-c^2 d^2+e^2\right )^2}+\frac {2 \csc ^{-1}(c x)}{5 c^3 d^3 e}-\frac {2 e^2 \csc ^{-1}(c x)}{5 c^3 d^3 \left (e+\frac {d}{x}\right )^3}-\frac {2 \left (2 c d e^2 \sqrt {1-\frac {1}{c^2 x^2}}-9 c^2 d^2 e \csc ^{-1}(c x)+9 e^3 \csc ^{-1}(c x)\right )}{15 c^3 d^3 \left (c^2 d^2-e^2\right ) \left (e+\frac {d}{x}\right )^2}-\frac {2 \left (-16 c^3 d^3 e \sqrt {1-\frac {1}{c^2 x^2}}+8 c d e^3 \sqrt {1-\frac {1}{c^2 x^2}}+9 c^4 d^4 \csc ^{-1}(c x)-18 c^2 d^2 e^2 \csc ^{-1}(c x)+9 e^4 \csc ^{-1}(c x)\right )}{15 c^3 d^3 \left (c^2 d^2-e^2\right )^2 \left (e+\frac {d}{x}\right )}\right )}{(d+e x)^{7/2}}+\frac {2 \left (e+\frac {d}{x}\right )^{7/2} (c x)^{7/2} \left (\frac {2 \left (c^2 d^2 e-e^3\right ) \sqrt {\frac {c d+c 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 )}{\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} (c x)^{3/2}}+\frac {2 \left (3 c^3 d^3+c d e^2\right ) \sqrt {\frac {c d+c 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 )}{\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} (c x)^{3/2}}+\frac {2 \left (-7 c^2 d^2 e+3 e^3\right ) \cos \left (2 \csc ^{-1}(c x)\right ) \left ((c d+c e x) \left (-1+c^2 x^2\right )+c^2 d x \sqrt {\frac {c d+c 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 )-\frac {c x (1+c x) \sqrt {\frac {e-c e x}{c d+e}} \sqrt {\frac {c d+c e x}{c d-e}} \left ((c d+e) E\left (\arcsin \left (\sqrt {\frac {c d+c e x}{c d-e}}\right )|\frac {c d-e}{c d+e}\right )-e \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {c d+c e x}{c d-e}}\right ),\frac {c d-e}{c d+e}\right )\right )}{\sqrt {\frac {e (1+c x)}{-c d+e}}}+c e x \sqrt {\frac {c d+c 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 )\right )}{c d \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} \sqrt {c x} \left (-2+c^2 x^2\right )}\right )}{15 c d (c d-e)^2 e (c d+e)^2 (d+e x)^{7/2}}\right )}{c} \]
(-2*a)/(5*e*(d + e*x)^(5/2)) + (b*(-((c^4*(e + d/x)^4*x^4*((4*(-7*c^2*d^2 + 3*e^2)*Sqrt[1 - 1/(c^2*x^2)])/(15*c^2*d^2*(-(c^2*d^2) + e^2)^2) + (2*Arc Csc[c*x])/(5*c^3*d^3*e) - (2*e^2*ArcCsc[c*x])/(5*c^3*d^3*(e + d/x)^3) - (2 *(2*c*d*e^2*Sqrt[1 - 1/(c^2*x^2)] - 9*c^2*d^2*e*ArcCsc[c*x] + 9*e^3*ArcCsc [c*x]))/(15*c^3*d^3*(c^2*d^2 - e^2)*(e + d/x)^2) - (2*(-16*c^3*d^3*e*Sqrt[ 1 - 1/(c^2*x^2)] + 8*c*d*e^3*Sqrt[1 - 1/(c^2*x^2)] + 9*c^4*d^4*ArcCsc[c*x] - 18*c^2*d^2*e^2*ArcCsc[c*x] + 9*e^4*ArcCsc[c*x]))/(15*c^3*d^3*(c^2*d^2 - e^2)^2*(e + d/x))))/(d + e*x)^(7/2)) + (2*(e + d/x)^(7/2)*(c*x)^(7/2)*((2 *(c^2*d^2*e - e^3)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*Ellipti cF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c^2*x^2)] *Sqrt[e + d/x]*(c*x)^(3/2)) + (2*(3*c^3*d^3 + c*d*e^2)*Sqrt[(c*d + c*e*x)/ (c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) + (2*( -7*c^2*d^2*e + 3*e^3)*Cos[2*ArcCsc[c*x]]*((c*d + c*e*x)*(-1 + c^2*x^2) + c ^2*d*x*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sq rt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)] - (c*x*(1 + c*x)*Sqrt[(e - c*e*x)/( c*d + e)]*Sqrt[(c*d + c*e*x)/(c*d - e)]*((c*d + e)*EllipticE[ArcSin[Sqrt[( c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)] - e*EllipticF[ArcSin[Sqrt[( c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)]))/Sqrt[(e*(1 + c*x))/(-(c*d ) + e)] + c*e*x*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*Ellipti...
Time = 1.11 (sec) , antiderivative size = 516, normalized size of antiderivative = 0.96, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.056, Rules used = {5750, 1898, 635, 633, 632, 186, 413, 412, 688, 27, 688, 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.
| \(\displaystyle \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 5750 |
| \(\displaystyle -\frac {2 b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^2 (d+e x)^{5/2}}dx}{5 c e}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 1898 |
| \(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \int \frac {1}{x (d+e x)^{5/2} \sqrt {x^2-\frac {1}{c^2}}}dx}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 635 |
| \(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\int \frac {-\frac {x e^2}{d^2}-\frac {2 e}{d}}{(d+e x)^{5/2} \sqrt {x^2-\frac {1}{c^2}}}dx+\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx}{d^2}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 633 |
| \(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\int \frac {-\frac {x e^2}{d^2}-\frac {2 e}{d}}{(d+e x)^{5/2} \sqrt {x^2-\frac {1}{c^2}}}dx+\frac {\sqrt {1-c^2 x^2} \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d^2 \sqrt {x^2-\frac {1}{c^2}}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 632 |
| \(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\int \frac {-\frac {x e^2}{d^2}-\frac {2 e}{d}}{(d+e x)^{5/2} \sqrt {x^2-\frac {1}{c^2}}}dx+\frac {\sqrt {1-c^2 x^2} \int \frac {1}{x \sqrt {1-c x} \sqrt {c x+1} \sqrt {d+e x}}dx}{d^2 \sqrt {x^2-\frac {1}{c^2}}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\int \frac {-\frac {x e^2}{d^2}-\frac {2 e}{d}}{(d+e x)^{5/2} \sqrt {x^2-\frac {1}{c^2}}}dx-\frac {2 \sqrt {1-c^2 x^2} \int \frac {1}{c x \sqrt {c x+1} \sqrt {d+\frac {e}{c}-\frac {e (1-c x)}{c}}}d\sqrt {1-c x}}{d^2 \sqrt {x^2-\frac {1}{c^2}}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\int \frac {-\frac {x e^2}{d^2}-\frac {2 e}{d}}{(d+e x)^{5/2} \sqrt {x^2-\frac {1}{c^2}}}dx-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \int \frac {1}{c x \sqrt {c x+1} \sqrt {1-\frac {e (1-c x)}{c d+e}}}d\sqrt {1-c x}}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\int \frac {-\frac {x e^2}{d^2}-\frac {2 e}{d}}{(d+e x)^{5/2} \sqrt {x^2-\frac {1}{c^2}}}dx-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {2 \int \frac {e \left (3 d \left (2-\frac {e^2}{c^2 d^2}\right )-e x\right )}{2 d (d+e x)^{3/2} \sqrt {x^2-\frac {1}{c^2}}}dx}{3 \left (d^2-\frac {e^2}{c^2}\right )}-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2 e^2 \sqrt {x^2-\frac {1}{c^2}}}{3 d \left (d^2-\frac {e^2}{c^2}\right ) (d+e x)^{3/2}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {e \int \frac {3 \left (2 d-\frac {e^2}{c^2 d}\right )-e x}{(d+e x)^{3/2} \sqrt {x^2-\frac {1}{c^2}}}dx}{3 d \left (d^2-\frac {e^2}{c^2}\right )}-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2 e^2 \sqrt {x^2-\frac {1}{c^2}}}{3 d \left (d^2-\frac {e^2}{c^2}\right ) (d+e x)^{3/2}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {e \left (-\frac {2 \int -\frac {6 d^2-\frac {2 e^2}{c^2}+e \left (7 d-\frac {3 e^2}{c^2 d}\right ) x}{2 \sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx}{d^2-\frac {e^2}{c^2}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}} \left (7 c^2 d^2-3 e^2\right )}{d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 d \left (d^2-\frac {e^2}{c^2}\right )}-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2 e^2 \sqrt {x^2-\frac {1}{c^2}}}{3 d \left (d^2-\frac {e^2}{c^2}\right ) (d+e x)^{3/2}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {e \left (\frac {\int \frac {2 \left (3 d^2-\frac {e^2}{c^2}\right )+e \left (7 d-\frac {3 e^2}{c^2 d}\right ) x}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx}{d^2-\frac {e^2}{c^2}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}} \left (7 c^2 d^2-3 e^2\right )}{d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 d \left (d^2-\frac {e^2}{c^2}\right )}-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2 e^2 \sqrt {x^2-\frac {1}{c^2}}}{3 d \left (d^2-\frac {e^2}{c^2}\right ) (d+e x)^{3/2}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {e \left (\frac {\left (7 d-\frac {3 e^2}{c^2 d}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x^2-\frac {1}{c^2}}}dx-\left (d^2-\frac {e^2}{c^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx}{d^2-\frac {e^2}{c^2}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}} \left (7 c^2 d^2-3 e^2\right )}{d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 d \left (d^2-\frac {e^2}{c^2}\right )}-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2 e^2 \sqrt {x^2-\frac {1}{c^2}}}{3 d \left (d^2-\frac {e^2}{c^2}\right ) (d+e x)^{3/2}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {e \left (\frac {\frac {\sqrt {1-c^2 x^2} \left (7 d-\frac {3 e^2}{c^2 d}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}}dx}{\sqrt {x^2-\frac {1}{c^2}}}-\left (d^2-\frac {e^2}{c^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx}{d^2-\frac {e^2}{c^2}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}} \left (7 c^2 d^2-3 e^2\right )}{d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 d \left (d^2-\frac {e^2}{c^2}\right )}-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2 e^2 \sqrt {x^2-\frac {1}{c^2}}}{3 d \left (d^2-\frac {e^2}{c^2}\right ) (d+e x)^{3/2}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {e \left (\frac {-\left (\left (d^2-\frac {e^2}{c^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx\right )-\frac {2 \sqrt {1-c^2 x^2} \left (7 d-\frac {3 e^2}{c^2 d}\right ) \sqrt {d+e x} \int \frac {\sqrt {1-\frac {e (1-c x)}{c d+e}}}{\sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}}{d^2-\frac {e^2}{c^2}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}} \left (7 c^2 d^2-3 e^2\right )}{d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 d \left (d^2-\frac {e^2}{c^2}\right )}-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2 e^2 \sqrt {x^2-\frac {1}{c^2}}}{3 d \left (d^2-\frac {e^2}{c^2}\right ) (d+e x)^{3/2}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {e \left (\frac {-\left (d^2-\frac {e^2}{c^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx-\frac {2 \sqrt {1-c^2 x^2} \left (7 d-\frac {3 e^2}{c^2 d}\right ) \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}}{d^2-\frac {e^2}{c^2}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}} \left (7 c^2 d^2-3 e^2\right )}{d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 d \left (d^2-\frac {e^2}{c^2}\right )}-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2 e^2 \sqrt {x^2-\frac {1}{c^2}}}{3 d \left (d^2-\frac {e^2}{c^2}\right ) (d+e x)^{3/2}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {e \left (\frac {-\frac {\sqrt {1-c^2 x^2} \left (d^2-\frac {e^2}{c^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{\sqrt {x^2-\frac {1}{c^2}}}-\frac {2 \sqrt {1-c^2 x^2} \left (7 d-\frac {3 e^2}{c^2 d}\right ) \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}}{d^2-\frac {e^2}{c^2}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}} \left (7 c^2 d^2-3 e^2\right )}{d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 d \left (d^2-\frac {e^2}{c^2}\right )}-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2 e^2 \sqrt {x^2-\frac {1}{c^2}}}{3 d \left (d^2-\frac {e^2}{c^2}\right ) (d+e x)^{3/2}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {e \left (\frac {\frac {2 \sqrt {1-c^2 x^2} \left (d^2-\frac {e^2}{c^2}\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \int \frac {1}{\sqrt {1-\frac {e (1-c x)}{c d+e}} \sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {d+e x}}-\frac {2 \sqrt {1-c^2 x^2} \left (7 d-\frac {3 e^2}{c^2 d}\right ) \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}}{d^2-\frac {e^2}{c^2}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}} \left (7 c^2 d^2-3 e^2\right )}{d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 d \left (d^2-\frac {e^2}{c^2}\right )}-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2 e^2 \sqrt {x^2-\frac {1}{c^2}}}{3 d \left (d^2-\frac {e^2}{c^2}\right ) (d+e x)^{3/2}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {e \left (\frac {\frac {2 \sqrt {1-c^2 x^2} \left (d^2-\frac {e^2}{c^2}\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {d+e x}}-\frac {2 \sqrt {1-c^2 x^2} \left (7 d-\frac {3 e^2}{c^2 d}\right ) \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}}{d^2-\frac {e^2}{c^2}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}} \left (7 c^2 d^2-3 e^2\right )}{d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 d \left (d^2-\frac {e^2}{c^2}\right )}-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2 e^2 \sqrt {x^2-\frac {1}{c^2}}}{3 d \left (d^2-\frac {e^2}{c^2}\right ) (d+e x)^{3/2}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\) |
(-2*(a + b*ArcCsc[c*x]))/(5*e*(d + e*x)^(5/2)) - (2*b*Sqrt[-c^(-2) + x^2]* ((2*e^2*Sqrt[-c^(-2) + x^2])/(3*d*(d^2 - e^2/c^2)*(d + e*x)^(3/2)) - (e*(( -2*e*(7*c^2*d^2 - 3*e^2)*Sqrt[-c^(-2) + x^2])/(d*(c^2*d^2 - e^2)*Sqrt[d + e*x]) + ((-2*(7*d - (3*e^2)/(c^2*d))*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*Ellip ticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*Sqrt[(c*(d + e*x) )/(c*d + e)]*Sqrt[-c^(-2) + x^2]) + (2*(d^2 - e^2/c^2)*Sqrt[(c*(d + e*x))/ (c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e )/(c*d + e)])/(c*Sqrt[d + e*x]*Sqrt[-c^(-2) + x^2]))/(d^2 - e^2/c^2)))/(3* d*(d^2 - e^2/c^2)) - (2*Sqrt[1 - c^2*x^2]*Sqrt[1 - (e*(1 - c*x))/(c*d + e) ]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(d^2*Sqrt [-c^(-2) + x^2]*Sqrt[d + e/c - (e*(1 - c*x))/c])))/(5*c*e*Sqrt[1 - 1/(c^2* x^2)]*x)
3.1.75.3.1 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[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[((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[( (c + d*x)^n/Sqrt[a + b*x^2])*ExpandToSum[(1 - c^(n + 1/2)*(c + d*x)^(-n - 1 /2))/x, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[n + 1/2, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( (m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[(x_)^(m_.)*((a_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^ (q_.), x_Symbol] :> Simp[x^(2*n*FracPart[p])*((a + c/x^(2*n))^FracPart[p]/( c + a*x^(2*n))^FracPart[p]) Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + a*x^(2*n ))^p, x], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[mn2, -2*n] && !I ntegerQ[p] && !IntegerQ[q] && PosQ[n]
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol ] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcCsc[c*x])/(e*(m + 1))), x] + Simp[b/ (c*e*(m + 1)) Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x] / ; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1617\) vs. \(2(491)=982\).
Time = 9.34 (sec) , antiderivative size = 1618, normalized size of antiderivative = 3.00
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1618\) |
| default | \(\text {Expression too large to display}\) | \(1618\) |
| parts | \(\text {Expression too large to display}\) | \(1642\) |
2/e*(-1/5*a/(e*x+d)^(5/2)+b*(-1/5/(e*x+d)^(5/2)*arccsc(c*x)-2/15/c*(7*(c/( c*d-e))^(1/2)*c^4*d^3*(e*x+d)^3-6*((-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^4*d^4*(e*x+d)^(3/2)+7*((-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^4*d^4*(e*x+d)^(3/2)-3*((-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^4*d^4*(e*x+d)^(3/2)-13*(c/(c*d-e))^(1/2)*c^4*d^4*(e*x+d)^2-7*((-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^3*d^3*e*(e*x+d)^(3 /2)+7*((-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^3*d ^3*e*(e*x+d)^(3/2)+5*(c/(c*d-e))^(1/2)*c^4*d^5*(e*x+d)-3*(c/(c*d-e))^(1/2) *c^2*d*e^2*(e*x+d)^3+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*d^2*e^2*(e*x+d)^(3/2)-3*((-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*e^2*(e*x+d)^(3/2)+6*((-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)...
Timed out. \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e+c*d>0)', see `assume?` for mor e details)
\[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^{7/2}} \,d x \]