Integrand size = 21, antiderivative size = 496 \[ \int x^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\frac {4 b d \sqrt {d+e x} \left (1-c^2 x^2\right )}{105 c^3 e \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {4 b (d+e x)^{3/2} \left (1-c^2 x^2\right )}{35 c^3 e \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}+\frac {4 b \left (5 c^2 d^2-9 e^2\right ) \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{105 c^4 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b d \left (9 c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{105 c^4 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {32 b d^4 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{105 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
2/3*d^2*(e*x+d)^(3/2)*(a+b*arccsc(c*x))/e^3-4/5*d*(e*x+d)^(5/2)*(a+b*arccs c(c*x))/e^3+2/7*(e*x+d)^(7/2)*(a+b*arccsc(c*x))/e^3-4/35*b*(e*x+d)^(3/2)*( -c^2*x^2+1)/c^3/e/x/(1-1/c^2/x^2)^(1/2)+4/105*b*d*(-c^2*x^2+1)*(e*x+d)^(1/ 2)/c^3/e/x/(1-1/c^2/x^2)^(1/2)+4/105*b*(5*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^2/x/(1-1/c^2/x^2)^(1/2)/(c*(e*x+d)/(c*d+e))^(1/2)-4/105*b*d*(9 *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^2/x/(1-1/c^2/x^2)^( 1/2)/(e*x+d)^(1/2)-32/105*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^ 3/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 = 44.49 (sec) , antiderivative size = 870, normalized size of antiderivative = 1.75 \[ \int x^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right ) \, dx=-\frac {a d^3 \sqrt {d+e x} B_{-\frac {e x}{d}}\left (3,\frac {3}{2}\right )}{e^3 \sqrt {1+\frac {e x}{d}}}+\frac {b \left (-\frac {c \left (e+\frac {d}{x}\right ) x \left (-\frac {4 \left (-5 c^2 d^2+9 e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}}}{105 e^2}-\frac {16 c^3 d^3 \csc ^{-1}(c x)}{105 e^3}-\frac {2}{7} c^3 x^3 \csc ^{-1}(c x)-\frac {2 c^2 x^2 \left (2 e \sqrt {1-\frac {1}{c^2 x^2}}+c d \csc ^{-1}(c x)\right )}{35 e}-\frac {8 c x \left (c d e \sqrt {1-\frac {1}{c^2 x^2}}-c^2 d^2 \csc ^{-1}(c x)\right )}{105 e^2}\right )}{\sqrt {d+e x}}-\frac {2 \sqrt {e+\frac {d}{x}} \sqrt {c x} \left (\frac {2 \left (9 c^3 d^3 e-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 (8 c^4 d^4+5 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 (-5 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^3 \sqrt {d+e x}}\right )}{c^4} \]
-((a*d^3*Sqrt[d + e*x]*Beta[-((e*x)/d), 3, 3/2])/(e^3*Sqrt[1 + (e*x)/d])) + (b*(-((c*(e + d/x)*x*((-4*(-5*c^2*d^2 + 9*e^2)*Sqrt[1 - 1/(c^2*x^2)])/(1 05*e^2) - (16*c^3*d^3*ArcCsc[c*x])/(105*e^3) - (2*c^3*x^3*ArcCsc[c*x])/7 - (2*c^2*x^2*(2*e*Sqrt[1 - 1/(c^2*x^2)] + c*d*ArcCsc[c*x]))/(35*e) - (8*c*x *(c*d*e*Sqrt[1 - 1/(c^2*x^2)] - c^2*d^2*ArcCsc[c*x]))/(105*e^2)))/Sqrt[d + e*x]) - (2*Sqrt[e + d/x]*Sqrt[c*x]*((2*(9*c^3*d^3*e - 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*(8*c^4*d^4 + 5*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*(-5*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)]*S qrt[(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)]))/(c*d*Sqrt[1 - 1/(c^2*x^2)]*Sqrt [e + d/x]*Sqrt[c*x]*(-2 + c^2*x^2))))/(105*e^3*Sqrt[d + e*x])))/c^4
Time = 2.51 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.10, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.095, Rules used = {5770, 27, 7272, 2351, 634, 600, 508, 327, 511, 321, 632, 186, 413, 412, 687, 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.
| \(\displaystyle \int x^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right ) \, dx\) |
| \(\Big \downarrow \) 5770 |
| \(\displaystyle \frac {b \int \frac {2 (d+e x)^{3/2} \left (8 d^2-12 e x d+15 e^2 x^2\right )}{105 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^2}dx}{c}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \int \frac {(d+e x)^{3/2} \left (8 d^2-12 e x d+15 e^2 x^2\right )}{\sqrt {1-\frac {1}{c^2 x^2}} x^2}dx}{105 c e^3}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 7272 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \int \frac {(d+e x)^{3/2} \left (8 d^2-12 e x d+15 e^2 x^2\right )}{x \sqrt {1-c^2 x^2}}dx}{105 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 2351 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \int \frac {(d+e x)^{3/2}}{x \sqrt {1-c^2 x^2}}dx+\int \frac {(d+e x)^{3/2} \left (15 e^2 x-12 d e\right )}{\sqrt {1-c^2 x^2}}dx\right )}{105 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 634 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \left (d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\int \frac {-x e^2-2 d e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx\right )+\int \frac {(d+e x)^{3/2} \left (15 e^2 x-12 d e\right )}{\sqrt {1-c^2 x^2}}dx\right )}{105 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \left (d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+d e \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+e \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}}dx\right )+\int \frac {(d+e x)^{3/2} \left (15 e^2 x-12 d e\right )}{\sqrt {1-c^2 x^2}}dx\right )}{105 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \left (d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+d e \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {2 e \sqrt {d+e x} \int \frac {\sqrt {1-\frac {e (1-c x)}{c d+e}}}{\sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )+\int \frac {(d+e x)^{3/2} \left (15 e^2 x-12 d e\right )}{\sqrt {1-c^2 x^2}}dx\right )}{105 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \left (d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+d e \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )+\int \frac {(d+e x)^{3/2} \left (15 e^2 x-12 d e\right )}{\sqrt {1-c^2 x^2}}dx\right )}{105 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \left (d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {2 d e \sqrt {\frac {c (d+e x)}{c d+e}} \int \frac {1}{\sqrt {1-\frac {e (1-c x)}{c d+e}} \sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {d+e x}}-\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )+\int \frac {(d+e x)^{3/2} \left (15 e^2 x-12 d e\right )}{\sqrt {1-c^2 x^2}}dx\right )}{105 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \left (d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {2 d e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )+\int \frac {(d+e x)^{3/2} \left (15 e^2 x-12 d e\right )}{\sqrt {1-c^2 x^2}}dx\right )}{105 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 632 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \left (d^2 \int \frac {1}{x \sqrt {1-c x} \sqrt {c x+1} \sqrt {d+e x}}dx-\frac {2 d e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )+\int \frac {(d+e x)^{3/2} \left (15 e^2 x-12 d e\right )}{\sqrt {1-c^2 x^2}}dx\right )}{105 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \left (-2 d^2 \int \frac {1}{c x \sqrt {c x+1} \sqrt {d+\frac {e}{c}-\frac {e (1-c x)}{c}}}d\sqrt {1-c x}-\frac {2 d e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )+\int \frac {(d+e x)^{3/2} \left (15 e^2 x-12 d e\right )}{\sqrt {1-c^2 x^2}}dx\right )}{105 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \left (-\frac {2 d^2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \int \frac {1}{c x \sqrt {c x+1} \sqrt {1-\frac {e (1-c x)}{c d+e}}}d\sqrt {1-c x}}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 d e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )+\int \frac {(d+e x)^{3/2} \left (15 e^2 x-12 d e\right )}{\sqrt {1-c^2 x^2}}dx\right )}{105 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (\int \frac {(d+e x)^{3/2} \left (15 e^2 x-12 d e\right )}{\sqrt {1-c^2 x^2}}dx+8 d^2 \left (-\frac {2 d^2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 d e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )\right )}{105 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {2 \int \frac {15 e \sqrt {d+e x} \left (4 d^2 c^2+d e x c^2-3 e^2\right )}{2 \sqrt {1-c^2 x^2}}dx}{5 c^2}+8 d^2 \left (-\frac {2 d^2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 d e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )-\frac {6 e^2 \sqrt {1-c^2 x^2} (d+e x)^{3/2}}{c^2}\right )}{105 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 e \int \frac {\sqrt {d+e x} \left (4 d^2 c^2+d e x c^2-3 e^2\right )}{\sqrt {1-c^2 x^2}}dx}{c^2}+8 d^2 \left (-\frac {2 d^2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 d e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )-\frac {6 e^2 \sqrt {1-c^2 x^2} (d+e x)^{3/2}}{c^2}\right )}{105 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 e \left (-\frac {2 \int -\frac {c^2 \left (4 d \left (3 c^2 d^2-2 e^2\right )+e \left (13 c^2 d^2-9 e^2\right ) x\right )}{2 \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {2}{3} d e \sqrt {1-c^2 x^2} \sqrt {d+e x}\right )}{c^2}+8 d^2 \left (-\frac {2 d^2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 d e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )-\frac {6 e^2 \sqrt {1-c^2 x^2} (d+e x)^{3/2}}{c^2}\right )}{105 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 e \left (\frac {1}{3} \int \frac {4 d \left (3 c^2 d^2-2 e^2\right )+e \left (13 c^2 d^2-9 e^2\right ) x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {2}{3} d e \sqrt {1-c^2 x^2} \sqrt {d+e x}\right )}{c^2}+8 d^2 \left (-\frac {2 d^2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 d e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )-\frac {6 e^2 \sqrt {1-c^2 x^2} (d+e x)^{3/2}}{c^2}\right )}{105 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 e \left (\frac {1}{3} \left (\left (13 c^2 d^2-9 e^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}}dx-d (c d-e) (c d+e) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx\right )-\frac {2}{3} d e \sqrt {1-c^2 x^2} \sqrt {d+e x}\right )}{c^2}+8 d^2 \left (-\frac {2 d^2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 d e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )-\frac {6 e^2 \sqrt {1-c^2 x^2} (d+e x)^{3/2}}{c^2}\right )}{105 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 e \left (\frac {1}{3} \left (-\frac {2 \left (13 c^2 d^2-9 e^2\right ) \sqrt {d+e x} \int \frac {\sqrt {1-\frac {e (1-c x)}{c d+e}}}{\sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {\frac {c (d+e x)}{c d+e}}}-d (c d-e) (c d+e) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx\right )-\frac {2}{3} d e \sqrt {1-c^2 x^2} \sqrt {d+e x}\right )}{c^2}+8 d^2 \left (-\frac {2 d^2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 d e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )-\frac {6 e^2 \sqrt {1-c^2 x^2} (d+e x)^{3/2}}{c^2}\right )}{105 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 e \left (\frac {1}{3} \left (-d (c d-e) (c d+e) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {2 \left (13 c^2 d^2-9 e^2\right ) \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )-\frac {2}{3} d e \sqrt {1-c^2 x^2} \sqrt {d+e x}\right )}{c^2}+8 d^2 \left (-\frac {2 d^2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 d e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )-\frac {6 e^2 \sqrt {1-c^2 x^2} (d+e x)^{3/2}}{c^2}\right )}{105 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 e \left (\frac {1}{3} \left (\frac {2 d (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \int \frac {1}{\sqrt {1-\frac {e (1-c x)}{c d+e}} \sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {d+e x}}-\frac {2 \left (13 c^2 d^2-9 e^2\right ) \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )-\frac {2}{3} d e \sqrt {1-c^2 x^2} \sqrt {d+e x}\right )}{c^2}+8 d^2 \left (-\frac {2 d^2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 d e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )-\frac {6 e^2 \sqrt {1-c^2 x^2} (d+e x)^{3/2}}{c^2}\right )}{105 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2 d^2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^3}-\frac {4 d (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {3 e \left (\frac {1}{3} \left (\frac {2 d (c d-e) (c d+e) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 \left (13 c^2 d^2-9 e^2\right ) \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )-\frac {2}{3} d e \sqrt {1-c^2 x^2} \sqrt {d+e x}\right )}{c^2}+8 d^2 \left (-\frac {2 d^2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 d e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )-\frac {6 e^2 \sqrt {1-c^2 x^2} (d+e x)^{3/2}}{c^2}\right )}{105 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}\) |
(2*d^2*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e^3) - (4*d*(d + e*x)^(5/2) *(a + b*ArcCsc[c*x]))/(5*e^3) + (2*(d + e*x)^(7/2)*(a + b*ArcCsc[c*x]))/(7 *e^3) + (2*b*Sqrt[1 - c^2*x^2]*((-6*e^2*(d + e*x)^(3/2)*Sqrt[1 - c^2*x^2]) /c^2 - (3*e*((-2*d*e*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2])/3 + ((-2*(13*c^2*d^2 - 9*e^2)*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*(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 + 8*d^2*((-2*e*Sqrt[d + e*x]*Ellipti cE[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])))/(105*c*e^3*Sqrt[1 - 1/(c^2* x^2)]*x)
3.1.51.3.1 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[((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[((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[((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. \(1203\) vs. \(2(445)=890\).
Time = 12.33 (sec) , antiderivative size = 1204, normalized size of antiderivative = 2.43
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1204\) |
| default | \(\text {Expression too large to display}\) | \(1204\) |
| parts | \(\text {Expression too large to display}\) | \(1225\) |
2/e^3*(a*(1/7*(e*x+d)^(7/2)-2/5*d*(e*x+d)^(5/2)+1/3*d^2*(e*x+d)^(3/2))+b*( 1/7*arccsc(c*x)*(e*x+d)^(7/2)-2/5*arccsc(c*x)*d*(e*x+d)^(5/2)+1/3*arccsc(c *x)*d^2*(e*x+d)^(3/2)+2/105/c^4*(3*(c/(c*d-e))^(1/2)*c^3*(e*x+d)^(7/2)-7*( c/(c*d-e))^(1/2)*c^3*d*(e*x+d)^(5/2)+4*((-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+5*((-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^3-8*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+5*(c/(c*d-e)) ^(1/2)*c^3*d^2*(e*x+d)^(3/2)-5*((-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+5*((-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-(c/(c*d-e))^(1/2)*c^3*d^3*(e*x+d)^(1/2)+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)*Ellip ticF((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)*Ellipt icE((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*c*d*e^2-3*(c/ (c*d-e))^(1/2)*c*e^2*(e*x+d)^(3/2)+9*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)...
\[ \int x^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int { \sqrt {e x + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{2} \,d x } \]
\[ \int x^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int x^{2} \left (a + b \operatorname {acsc}{\left (c x \right )}\right ) \sqrt {d + e x}\, dx \]
Exception generated. \[ \int x^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\text {Exception raised: ValueError} \]
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)
\[ \int x^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int { \sqrt {e x + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{2} \,d x } \]
Timed out. \[ \int x^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right ) \, dx=\int x^2\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )\,\sqrt {d+e\,x} \,d x \]