Integrand size = 21, antiderivative size = 714 \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=-\frac {4 b \sqrt {d+e x} \left (1-c^2 x^2\right )}{35 c^3 e \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {4 b d \sqrt {d+e x} \left (1-c^2 x^2\right )}{21 c^3 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}-\frac {24 b d^2 \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{35 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \left (2 c^2 d^2-9 e^2\right ) \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{105 c^4 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {64 b d^3 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{35 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {32 b d (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{105 c^4 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {64 b d^4 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{35 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
2*d^2*(e*x+d)^(3/2)*(a+b*arccsc(c*x))/e^4-6/5*d*(e*x+d)^(5/2)*(a+b*arccsc( c*x))/e^4+2/7*(e*x+d)^(7/2)*(a+b*arccsc(c*x))/e^4-2*d^3*(a+b*arccsc(c*x))* (e*x+d)^(1/2)/e^4-4/35*b*(-c^2*x^2+1)*(e*x+d)^(1/2)/c^3/e/(1-1/c^2/x^2)^(1 /2)+4/21*b*d*(-c^2*x^2+1)*(e*x+d)^(1/2)/c^3/e^2/x/(1-1/c^2/x^2)^(1/2)-24/3 5*b*d^2*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)+4/105*b*(2*c^2*d^2-9*e^2)*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^4/e^3/x/(1 -1/c^2/x^2)^(1/2)/(c*(e*x+d)/(c*d+e))^(1/2)+64/35*b*d^3*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)-32/105*b*d*(c *d-e)*(c*d+e)*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)+64/35*b*d^4*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 = 33.90 (sec) , antiderivative size = 873, normalized size of antiderivative = 1.22 \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\frac {a d^4 \sqrt {1+\frac {e x}{d}} B_{-\frac {e x}{d}}\left (4,\frac {1}{2}\right )}{e^4 \sqrt {d+e x}}+\frac {b \left (-\frac {c \left (e+\frac {d}{x}\right ) x \left (-\frac {4 \left (16 c^2 d^2+9 e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}}}{105 e^3}+\frac {32 c^3 d^3 \csc ^{-1}(c x)}{35 e^4}-\frac {2 c^3 x^3 \csc ^{-1}(c x)}{7 e}-\frac {4 c^2 x^2 \left (e \sqrt {1-\frac {1}{c^2 x^2}}-3 c d \csc ^{-1}(c x)\right )}{35 e^2}+\frac {4 c x \left (5 c d e \sqrt {1-\frac {1}{c^2 x^2}}-12 c^2 d^2 \csc ^{-1}(c x)\right )}{105 e^3}\right )}{\sqrt {d+e x}}+\frac {2 \sqrt {e+\frac {d}{x}} \sqrt {c x} \left (\frac {2 \left (40 c^3 d^3 e+8 c d 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^4 d^4+16 c^2 d^2 e^2+9 e^4\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 (-16 c^3 d^3 e-9 c d 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 )}{105 e^4 \sqrt {d+e x}}\right )}{c^4} \]
(a*d^4*Sqrt[1 + (e*x)/d]*Beta[-((e*x)/d), 4, 1/2])/(e^4*Sqrt[d + e*x]) + ( b*(-((c*(e + d/x)*x*((-4*(16*c^2*d^2 + 9*e^2)*Sqrt[1 - 1/(c^2*x^2)])/(105* e^3) + (32*c^3*d^3*ArcCsc[c*x])/(35*e^4) - (2*c^3*x^3*ArcCsc[c*x])/(7*e) - (4*c^2*x^2*(e*Sqrt[1 - 1/(c^2*x^2)] - 3*c*d*ArcCsc[c*x]))/(35*e^2) + (4*c *x*(5*c*d*e*Sqrt[1 - 1/(c^2*x^2)] - 12*c^2*d^2*ArcCsc[c*x]))/(105*e^3)))/S qrt[d + e*x]) + (2*Sqrt[e + d/x]*Sqrt[c*x]*((2*(40*c^3*d^3*e + 8*c*d*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^4*d^4 + 16*c^2*d^2*e^2 + 9*e^4)*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*(-16 *c^3*d^3*e - 9*c*d*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[ Sqrt[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]*Elliptic Pi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)]))/(c*d*Sqrt[1 - 1/(c ^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(-2 + c^2*x^2))))/(105*e^4*Sqrt[d + e*...
Time = 2.35 (sec) , antiderivative size = 489, normalized size of antiderivative = 0.68, 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, 637, 2009, 2185, 27, 687, 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.
| \(\displaystyle \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 5770 |
| \(\displaystyle \frac {b \int -\frac {2 \sqrt {d+e x} \left (16 d^3-8 e x d^2+6 e^2 x^2 d-5 e^3 x^3\right )}{35 e^4 \sqrt {1-\frac {1}{c^2 x^2}} x^2}dx}{c}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \int \frac {\sqrt {d+e x} \left (16 d^3-8 e x d^2+6 e^2 x^2 d-5 e^3 x^3\right )}{\sqrt {1-\frac {1}{c^2 x^2}} x^2}dx}{35 c e^4}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 7272 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \int \frac {\sqrt {d+e x} \left (16 d^3-8 e x d^2+6 e^2 x^2 d-5 e^3 x^3\right )}{x \sqrt {1-c^2 x^2}}dx}{35 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 2351 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (16 d^3 \int \frac {\sqrt {d+e x}}{x \sqrt {1-c^2 x^2}}dx+\int \frac {\sqrt {d+e x} \left (-5 x^2 e^3+6 d x e^2-8 d^2 e\right )}{\sqrt {1-c^2 x^2}}dx\right )}{35 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (16 d^3 \int \left (\frac {d}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}+\frac {e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}\right )dx+\int \frac {\sqrt {d+e x} \left (-5 x^2 e^3+6 d x e^2-8 d^2 e\right )}{\sqrt {1-c^2 x^2}}dx\right )}{35 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\int \frac {\sqrt {d+e x} \left (-5 x^2 e^3+6 d x e^2-8 d^2 e\right )}{\sqrt {1-c^2 x^2}}dx+16 d^3 \left (-\frac {2 e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 d \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {d+e x}}\right )\right )}{35 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {2 \int \frac {5 e^3 \sqrt {d+e x} \left (8 d^2 c^2-8 d e x c^2+3 e^2\right )}{2 \sqrt {1-c^2 x^2}}dx}{5 c^2 e^2}+16 d^3 \left (-\frac {2 e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 d \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {d+e x}}\right )+\frac {2 e^2 \sqrt {1-c^2 x^2} (d+e x)^{3/2}}{c^2}\right )}{35 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {e \int \frac {\sqrt {d+e x} \left (8 d^2 c^2-8 d e x c^2+3 e^2\right )}{\sqrt {1-c^2 x^2}}dx}{c^2}+16 d^3 \left (-\frac {2 e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 d \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {d+e x}}\right )+\frac {2 e^2 \sqrt {1-c^2 x^2} (d+e x)^{3/2}}{c^2}\right )}{35 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {e \left (\frac {16}{3} d e \sqrt {1-c^2 x^2} \sqrt {d+e x}-\frac {2 \int -\frac {c^2 \left (d \left (24 c^2 d^2+e^2\right )+e \left (16 c^2 d^2+9 e^2\right ) x\right )}{2 \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{3 c^2}\right )}{c^2}+16 d^3 \left (-\frac {2 e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 d \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {d+e x}}\right )+\frac {2 e^2 \sqrt {1-c^2 x^2} (d+e x)^{3/2}}{c^2}\right )}{35 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {e \left (\frac {1}{3} \int \frac {d \left (24 c^2 d^2+e^2\right )+e \left (16 c^2 d^2+9 e^2\right ) x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+\frac {16}{3} d e \sqrt {1-c^2 x^2} \sqrt {d+e x}\right )}{c^2}+16 d^3 \left (-\frac {2 e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 d \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {d+e x}}\right )+\frac {2 e^2 \sqrt {1-c^2 x^2} (d+e x)^{3/2}}{c^2}\right )}{35 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {e \left (\frac {1}{3} \left (8 d \left (c^2 d^2-e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+\left (16 c^2 d^2+9 e^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}}dx\right )+\frac {16}{3} d e \sqrt {1-c^2 x^2} \sqrt {d+e x}\right )}{c^2}+16 d^3 \left (-\frac {2 e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 d \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {d+e x}}\right )+\frac {2 e^2 \sqrt {1-c^2 x^2} (d+e x)^{3/2}}{c^2}\right )}{35 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {e \left (\frac {1}{3} \left (8 d \left (c^2 d^2-e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {2 \left (16 c^2 d^2+9 e^2\right ) \sqrt {d+e x} \int \frac {\sqrt {1-\frac {e (1-c x)}{c d+e}}}{\sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )+\frac {16}{3} d e \sqrt {1-c^2 x^2} \sqrt {d+e x}\right )}{c^2}+16 d^3 \left (-\frac {2 e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 d \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {d+e x}}\right )+\frac {2 e^2 \sqrt {1-c^2 x^2} (d+e x)^{3/2}}{c^2}\right )}{35 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {e \left (\frac {1}{3} \left (8 d \left (c^2 d^2-e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {2 \left (16 c^2 d^2+9 e^2\right ) \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )+\frac {16}{3} d e \sqrt {1-c^2 x^2} \sqrt {d+e x}\right )}{c^2}+16 d^3 \left (-\frac {2 e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 d \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {d+e x}}\right )+\frac {2 e^2 \sqrt {1-c^2 x^2} (d+e x)^{3/2}}{c^2}\right )}{35 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {e \left (\frac {1}{3} \left (-\frac {16 d \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \int \frac {1}{\sqrt {1-\frac {e (1-c x)}{c d+e}} \sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {d+e x}}-\frac {2 \left (16 c^2 d^2+9 e^2\right ) \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )+\frac {16}{3} d e \sqrt {1-c^2 x^2} \sqrt {d+e x}\right )}{c^2}+16 d^3 \left (-\frac {2 e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 d \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {d+e x}}\right )+\frac {2 e^2 \sqrt {1-c^2 x^2} (d+e x)^{3/2}}{c^2}\right )}{35 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {2 d^3 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^4}-\frac {6 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^4}-\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {e \left (\frac {1}{3} \left (-\frac {16 d \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 \left (16 c^2 d^2+9 e^2\right ) \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )+\frac {16}{3} d e \sqrt {1-c^2 x^2} \sqrt {d+e x}\right )}{c^2}+16 d^3 \left (-\frac {2 e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 d \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {d+e x}}\right )+\frac {2 e^2 \sqrt {1-c^2 x^2} (d+e x)^{3/2}}{c^2}\right )}{35 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}}}\) |
(-2*d^3*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/e^4 + (2*d^2*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]))/e^4 - (6*d*(d + e*x)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e^4) + (2*(d + e*x)^(7/2)*(a + b*ArcCsc[c*x]))/(7*e^4) - (2*b*Sqrt[1 - c^2*x^2 ]*((2*e^2*(d + e*x)^(3/2)*Sqrt[1 - c^2*x^2])/c^2 - (e*((16*d*e*Sqrt[d + e* x]*Sqrt[1 - c^2*x^2])/3 + ((-2*(16*c^2*d^2 + 9*e^2)*Sqrt[d + e*x]*Elliptic E[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*Sqrt[(c*(d + e*x))/( c*d + e)]) - (16*d*(c^2*d^2 - e^2)*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 + 16*d^3*((-2*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*Sqrt[(c*(d + e* x))/(c*d + e)]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e )])/Sqrt[d + e*x])))/(35*c*e^4*Sqrt[1 - 1/(c^2*x^2)]*x)
3.1.57.3.1 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/(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[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[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Int[ExpandIntegrand[(a + b*x^2)^p/Sqrt[c + d*x], x^m*(c + d*x)^(n + 1 /2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[p + 1/2] && IntegerQ[n + 1/2] && IntegerQ[m]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) ), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp [c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x ] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && Eq Q[f, 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 = 10.84 (sec) , antiderivative size = 1233, normalized size of antiderivative = 1.73
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1233\) |
| default | \(\text {Expression too large to display}\) | \(1233\) |
| parts | \(\text {Expression too large to display}\) | \(1251\) |
2/e^4*(-a*(-1/7*(e*x+d)^(7/2)+3/5*d*(e*x+d)^(5/2)-d^2*(e*x+d)^(3/2)+d^3*(e *x+d)^(1/2))-b*(-1/7*arccsc(c*x)*(e*x+d)^(7/2)+3/5*arccsc(c*x)*d*(e*x+d)^( 5/2)-arccsc(c*x)*d^2*(e*x+d)^(3/2)+arccsc(c*x)*d^3*(e*x+d)^(1/2)+2/105/c^4 *(-3*(c/(c*d-e))^(1/2)*c^3*(e*x+d)^(7/2)+14*(c/(c*d-e))^(1/2)*c^3*d*(e*x+d )^(5/2)-19*(c/(c*d-e))^(1/2)*c^3*d^2*(e*x+d)^(3/2)+24*((-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+16*((-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^3-48*d^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^3+8*(c/(c*d-e))^(1/2)*c^3*d^3*(e*x+d)^(1/2)-16*((-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+16*((-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+3*(c/(c*d-e))^(1/2)*c*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*d*e^2+9*((-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))^...
\[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{3}}{\sqrt {e x + d}} \,d x } \]
\[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int \frac {x^{3} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )}{\sqrt {d + e x}}\, dx \]
Exception generated. \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\text {Exception raised: ValueError} \]
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)
\[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{3}}{\sqrt {e x + d}} \,d x } \]
Timed out. \[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {d+e\,x}} \,d x \]