Integrand size = 21, antiderivative size = 551 \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=-\frac {4 b \sqrt {d+e x} \left (1-c^2 x^2\right )}{15 c^3 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac {32 b d \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {8 b d^2 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b \left (2 c^2 d^2+e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 c^4 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {64 b d^3 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{5 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
-2*d*(e*x+d)^(3/2)*(a+b*arccsc(c*x))/e^4+2/5*(e*x+d)^(5/2)*(a+b*arccsc(c*x ))/e^4+2*d^3*(a+b*arccsc(c*x))/e^4/(e*x+d)^(1/2)+6*d^2*(a+b*arccsc(c*x))*( e*x+d)^(1/2)/e^4-4/15*b*(-c^2*x^2+1)*(e*x+d)^(1/2)/c^3/e^2/x/(1-1/c^2/x^2) ^(1/2)+32/15*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^3/x/(1-1/c^2/x^2)^(1/2)/(c*( e*x+d)/(c*d+e))^(1/2)-8*b*d^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^2/e^3/x /(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)-4/15*b*(2*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^3/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)-64/5*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^4/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^ (1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 34.38 (sec) , antiderivative size = 814, normalized size of antiderivative = 1.48 \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\frac {a d^4 \left (1+\frac {e x}{d}\right )^{3/2} B_{-\frac {e x}{d}}\left (4,-\frac {1}{2}\right )}{e^4 (d+e x)^{3/2}}+\frac {b \left (-\frac {c^2 \left (e+\frac {d}{x}\right )^2 x^2 \left (\frac {32 c d \sqrt {1-\frac {1}{c^2 x^2}}}{15 e^3}-\frac {32 c^2 d^2 \csc ^{-1}(c x)}{5 e^4}+\frac {2 c^2 d^2 \csc ^{-1}(c x)}{e^3 \left (e+\frac {d}{x}\right )}-\frac {2 c^2 x^2 \csc ^{-1}(c x)}{5 e^2}-\frac {2 c x \left (2 e \sqrt {1-\frac {1}{c^2 x^2}}-9 c d \csc ^{-1}(c x)\right )}{15 e^3}\right )}{(d+e x)^{3/2}}-\frac {2 \left (e+\frac {d}{x}\right )^{3/2} (c x)^{3/2} \left (\frac {2 \left (32 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 (48 c^3 d^3+8 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 {16 c d e \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 )}{\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 e^4 (d+e x)^{3/2}}\right )}{c^4} \]
(a*d^4*(1 + (e*x)/d)^(3/2)*Beta[-((e*x)/d), 4, -1/2])/(e^4*(d + e*x)^(3/2) ) + (b*(-((c^2*(e + d/x)^2*x^2*((32*c*d*Sqrt[1 - 1/(c^2*x^2)])/(15*e^3) - (32*c^2*d^2*ArcCsc[c*x])/(5*e^4) + (2*c^2*d^2*ArcCsc[c*x])/(e^3*(e + d/x)) - (2*c^2*x^2*ArcCsc[c*x])/(5*e^2) - (2*c*x*(2*e*Sqrt[1 - 1/(c^2*x^2)] - 9 *c*d*ArcCsc[c*x]))/(15*e^3)))/(d + e*x)^(3/2)) - (2*(e + d/x)^(3/2)*(c*x)^ (3/2)*((2*(32*c^2*d^2*e + e^3)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2* x^2]*EllipticF[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*(48*c^3*d^3 + 8*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)) - (16*c*d*e*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[Sqr t[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]*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]*Sqrt[c*x]*(-2 + c^2*x^2))))/(15*e^4*(d + e*x)^(3/2))))/c^4
Time = 2.04 (sec) , antiderivative size = 393, normalized size of antiderivative = 0.71, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {5770, 27, 7272, 2351, 632, 186, 413, 412, 2185, 27, 600, 508, 327, 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.
| \(\displaystyle \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 5770 |
| \(\displaystyle \frac {b \int \frac {2 \left (16 d^3+8 e x d^2-2 e^2 x^2 d+e^3 x^3\right )}{5 e^4 \sqrt {1-\frac {1}{c^2 x^2}} x^2 \sqrt {d+e x}}dx}{c}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \int \frac {16 d^3+8 e x d^2-2 e^2 x^2 d+e^3 x^3}{\sqrt {1-\frac {1}{c^2 x^2}} x^2 \sqrt {d+e x}}dx}{5 c e^4}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 7272 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \int \frac {16 d^3+8 e x d^2-2 e^2 x^2 d+e^3 x^3}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{5 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 2351 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (16 d^3 \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+\int \frac {x^2 e^3-2 d x e^2+8 d^2 e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx\right )}{5 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 632 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (\int \frac {x^2 e^3-2 d x e^2+8 d^2 e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+16 d^3 \int \frac {1}{x \sqrt {1-c x} \sqrt {c x+1} \sqrt {d+e x}}dx\right )}{5 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (\int \frac {x^2 e^3-2 d x e^2+8 d^2 e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-32 d^3 \int \frac {1}{c x \sqrt {c x+1} \sqrt {d+\frac {e}{c}-\frac {e (1-c x)}{c}}}d\sqrt {1-c x}\right )}{5 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (\int \frac {x^2 e^3-2 d x e^2+8 d^2 e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {32 d^3 \sqrt {1-\frac {e (1-c x)}{c d+e}} \int \frac {1}{c x \sqrt {c x+1} \sqrt {1-\frac {e (1-c x)}{c d+e}}}d\sqrt {1-c x}}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )}{5 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (\int \frac {x^2 e^3-2 d x e^2+8 d^2 e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {32 d^3 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )}{5 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {2 \int -\frac {e^3 \left (24 d^2 c^2-8 d e x c^2+e^2\right )}{2 \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{3 c^2 e^2}-\frac {32 d^3 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 e^2 \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )}{5 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (\frac {e \int \frac {24 d^2 c^2-8 d e x c^2+e^2}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {32 d^3 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 e^2 \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )}{5 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (\frac {e \left (\left (32 c^2 d^2+e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-8 c^2 d \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}}dx\right )}{3 c^2}-\frac {32 d^3 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 e^2 \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )}{5 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (\frac {e \left (\left (32 c^2 d^2+e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+\frac {16 c d \sqrt {d+e x} \int \frac {\sqrt {1-\frac {e (1-c x)}{c d+e}}}{\sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{\sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c^2}-\frac {32 d^3 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 e^2 \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )}{5 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (\frac {e \left (\left (32 c^2 d^2+e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+\frac {16 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c^2}-\frac {32 d^3 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 e^2 \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )}{5 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (\frac {e \left (\frac {16 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {2 \left (32 c^2 d^2+e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \int \frac {1}{\sqrt {1-\frac {e (1-c x)}{c d+e}} \sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {d+e x}}\right )}{3 c^2}-\frac {32 d^3 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 e^2 \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )}{5 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}+\frac {6 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {2 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {e \left (\frac {16 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {2 \left (32 c^2 d^2+e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}\right )}{3 c^2}-\frac {32 d^3 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 e^2 \sqrt {1-c^2 x^2} \sqrt {d+e x}}{3 c^2}\right )}{5 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}\) |
(2*d^3*(a + b*ArcCsc[c*x]))/(e^4*Sqrt[d + e*x]) + (6*d^2*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/e^4 - (2*d*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]))/e^4 + (2 *(d + e*x)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e^4) + (2*b*Sqrt[1 - c^2*x^2]*((- 2*e^2*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2])/(3*c^2) + (e*((16*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*(32*c^2*d^2 + e^2)*Sqrt[(c*(d + e*x))/(c*d + e)]*Elli pticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*Sqrt[d + e*x]))) /(3*c^2) - (32*d^3*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]) )/(5*c*e^4*Sqrt[1 - 1/(c^2*x^2)]*x)
3.1.63.3.1 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[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[(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]
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]
Time = 10.08 (sec) , antiderivative size = 880, normalized size of antiderivative = 1.60
| method | result | size |
| derivativedivides | \(\frac {-2 a \left (-\frac {\left (e x +d \right )^{\frac {5}{2}}}{5}+\left (e x +d \right )^{\frac {3}{2}} d -3 d^{2} \sqrt {e x +d}-\frac {d^{3}}{\sqrt {e x +d}}\right )-2 b \left (-\frac {\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {3}{2}} d -3 \,\operatorname {arccsc}\left (c x \right ) d^{2} \sqrt {e x +d}-\frac {\operatorname {arccsc}\left (c x \right ) d^{3}}{\sqrt {e x +d}}-\frac {2 \left (\sqrt {\frac {c}{c d -e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}+24 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}+8 \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}-48 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}-2 \sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}-8 \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 +8 \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}{c d -e}}\, c^{2} d^{2} \sqrt {e x +d}+\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^{4}}\) | \(880\) |
| default | \(\frac {-2 a \left (-\frac {\left (e x +d \right )^{\frac {5}{2}}}{5}+\left (e x +d \right )^{\frac {3}{2}} d -3 d^{2} \sqrt {e x +d}-\frac {d^{3}}{\sqrt {e x +d}}\right )-2 b \left (-\frac {\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {3}{2}} d -3 \,\operatorname {arccsc}\left (c x \right ) d^{2} \sqrt {e x +d}-\frac {\operatorname {arccsc}\left (c x \right ) d^{3}}{\sqrt {e x +d}}-\frac {2 \left (\sqrt {\frac {c}{c d -e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}+24 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}+8 \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}-48 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}-2 \sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}-8 \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 +8 \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}{c d -e}}\, c^{2} d^{2} \sqrt {e x +d}+\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^{4}}\) | \(880\) |
| parts | \(\frac {2 a \left (\frac {\left (e x +d \right )^{\frac {5}{2}}}{5}-\left (e x +d \right )^{\frac {3}{2}} d +3 d^{2} \sqrt {e x +d}+\frac {d^{3}}{\sqrt {e x +d}}\right )}{e^{4}}+\frac {2 b \left (\frac {\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}-\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {3}{2}} d +3 \,\operatorname {arccsc}\left (c x \right ) d^{2} \sqrt {e x +d}+\frac {\operatorname {arccsc}\left (c x \right ) d^{3}}{\sqrt {e x +d}}+\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 {16 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}}{5}+\frac {16 \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 {32 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}}{5}+\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \sqrt {e x +d}}{15}-\frac {16 \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 {16 \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^{4}}\) | \(893\) |
2/e^4*(-a*(-1/5*(e*x+d)^(5/2)+(e*x+d)^(3/2)*d-3*d^2*(e*x+d)^(1/2)-d^3/(e*x +d)^(1/2))-b*(-1/5*arccsc(c*x)*(e*x+d)^(5/2)+arccsc(c*x)*(e*x+d)^(3/2)*d-3 *arccsc(c*x)*d^2*(e*x+d)^(1/2)-arccsc(c*x)*d^3/(e*x+d)^(1/2)-2/15/c^3*((c/ (c*d-e))^(1/2)*c^2*(e*x+d)^(5/2)+24*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+8*((-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^2-48*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^2-2*(c/(c*d-e))^( 1/2)*c^2*d*(e*x+d)^(3/2)-8*((-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+8*((-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/(c*d-e))^(1/2)*c^2*d^2*(e*x+d)^(1/2)+((-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))*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)))
\[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int \frac {x^{3} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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)
\[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \]