Integrand size = 21, antiderivative size = 440 \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\frac {4 b d \left (1-c^2 x^2\right )}{3 c e \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {4 b d \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {32 b d \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
-2/3*d^2*(a+b*arccsc(c*x))/e^3/(e*x+d)^(3/2)+4*d*(a+b*arccsc(c*x))/e^3/(e* x+d)^(1/2)+4/3*b*d*(-c^2*x^2+1)/c/e/(c^2*d^2-e^2)/x/(1-1/c^2/x^2)^(1/2)/(e *x+d)^(1/2)+2*(a+b*arccsc(c*x))*(e*x+d)^(1/2)/e^3-4/3*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)/e^2/(c^2*d^2-e^2)/x/(1-1/c^2/x^2)^(1/2)/(c*(e*x+d)/(c*d+e))^(1/2)-4 *b*EllipticF(1/2*(-c*x+1)^(1/2)*2^(1/2),2^(1/2)*(e/(c*d+e))^(1/2))*(c*(e*x +d)/(c*d+e))^(1/2)*(-c^2*x^2+1)^(1/2)/c^2/e^2/x/(1-1/c^2/x^2)^(1/2)/(e*x+d )^(1/2)-32/3*b*d*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 = 34.07 (sec) , antiderivative size = 856, normalized size of antiderivative = 1.95 \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=-\frac {a d^3 \left (1+\frac {e x}{d}\right )^{5/2} B_{-\frac {e x}{d}}\left (3,-\frac {3}{2}\right )}{e^3 (d+e x)^{5/2}}+\frac {b \left (-\frac {c^3 \left (e+\frac {d}{x}\right )^3 x^3 \left (\frac {4 c d \sqrt {1-\frac {1}{c^2 x^2}}}{3 e^2 \left (-c^2 d^2+e^2\right )}-\frac {16 \csc ^{-1}(c x)}{3 e^3}+\frac {2 \csc ^{-1}(c x)}{3 e \left (e+\frac {d}{x}\right )^2}+\frac {4 \left (-c d e \sqrt {1-\frac {1}{c^2 x^2}}-2 c^2 d^2 \csc ^{-1}(c x)+2 e^2 \csc ^{-1}(c x)\right )}{3 e^2 \left (-c^2 d^2+e^2\right ) \left (e+\frac {d}{x}\right )}\right )}{(d+e x)^{5/2}}-\frac {2 \left (e+\frac {d}{x}\right )^{5/2} (c x)^{5/2} \left (\frac {2 \left (3 c^2 d^2 e-3 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^3 d^3-9 c d e^2\right ) \sqrt {\frac {c d+c e x}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} (c x)^{3/2}}+\frac {2 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 )}{3 (c d-e) e^3 (c d+e) (d+e x)^{5/2}}\right )}{c^3} \]
-((a*d^3*(1 + (e*x)/d)^(5/2)*Beta[-((e*x)/d), 3, -3/2])/(e^3*(d + e*x)^(5/ 2))) + (b*(-((c^3*(e + d/x)^3*x^3*((4*c*d*Sqrt[1 - 1/(c^2*x^2)])/(3*e^2*(- (c^2*d^2) + e^2)) - (16*ArcCsc[c*x])/(3*e^3) + (2*ArcCsc[c*x])/(3*e*(e + d /x)^2) + (4*(-(c*d*e*Sqrt[1 - 1/(c^2*x^2)]) - 2*c^2*d^2*ArcCsc[c*x] + 2*e^ 2*ArcCsc[c*x]))/(3*e^2*(-(c^2*d^2) + e^2)*(e + d/x))))/(d + e*x)^(5/2)) - (2*(e + d/x)^(5/2)*(c*x)^(5/2)*((2*(3*c^2*d^2*e - 3*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^3*d^3 - 9*c*d*e^2)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*Ellip ticPi[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*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[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)]))/Sq rt[(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))))/(3*(c* d - e)*e^3*(c*d + e)*(d + e*x)^(5/2))))/c^3
Time = 2.20 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.18, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.048, Rules used = {5770, 27, 7272, 2351, 635, 25, 27, 498, 27, 508, 327, 632, 186, 413, 412, 688, 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.
| \(\displaystyle \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5770 |
| \(\displaystyle \frac {b \int \frac {2 \left (8 d^2+12 e x d+3 e^2 x^2\right )}{3 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}}dx}{c}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \int \frac {8 d^2+12 e x d+3 e^2 x^2}{\sqrt {1-\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}}dx}{3 c e^3}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}\) |
| \(\Big \downarrow \) 7272 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \int \frac {8 d^2+12 e x d+3 e^2 x^2}{x (d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}\) |
| \(\Big \downarrow \) 2351 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \int \frac {1}{x (d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx+\int \frac {3 x e^2+12 d e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}\) |
| \(\Big \downarrow \) 635 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \left (\int -\frac {e}{d (d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx+\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}\right )+\int \frac {3 x e^2+12 d e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}-\int \frac {e}{d (d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )+\int \frac {3 x e^2+12 d e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}-\frac {e \int \frac {1}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx}{d}\right )+\int \frac {3 x e^2+12 d e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}\) |
| \(\Big \downarrow \) 498 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c^2 \int -\frac {\sqrt {d+e x}}{2 \sqrt {1-c^2 x^2}}dx}{c^2 d^2-e^2}\right )}{d}\right )+\int \frac {3 x e^2+12 d e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}-\frac {e \left (\frac {c^2 \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}}dx}{c^2 d^2-e^2}+\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{d}\right )+\int \frac {3 x e^2+12 d e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} \int \frac {\sqrt {1-\frac {e (1-c x)}{c d+e}}}{\sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}\right )+\int \frac {3 x e^2+12 d e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}\right )+\int \frac {3 x e^2+12 d e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}\) |
| \(\Big \downarrow \) 632 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \left (\frac {\int \frac {1}{x \sqrt {1-c x} \sqrt {c x+1} \sqrt {d+e x}}dx}{d}-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}\right )+\int \frac {3 x e^2+12 d e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \left (-\frac {2 \int \frac {1}{c x \sqrt {c x+1} \sqrt {d+\frac {e}{c}-\frac {e (1-c x)}{c}}}d\sqrt {1-c x}}{d}-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}\right )+\int \frac {3 x e^2+12 d e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \left (-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \int \frac {1}{c x \sqrt {c x+1} \sqrt {1-\frac {e (1-c x)}{c d+e}}}d\sqrt {1-c x}}{d \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}\right )+\int \frac {3 x e^2+12 d e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (\int \frac {3 x e^2+12 d e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx+8 d^2 \left (-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (\frac {2 \int \frac {3 e \left (4 d^2 c^2+3 d e x c^2-e^2\right )}{2 \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{c^2 d^2-e^2}+8 d^2 \left (-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )+\frac {18 d e^2 \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (\frac {3 e \int \frac {4 d^2 c^2+3 d e x c^2-e^2}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{c^2 d^2-e^2}+8 d^2 \left (-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )+\frac {18 d e^2 \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (\frac {3 e \left (\left (c^2 d^2-e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+3 c^2 d \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}}dx\right )}{c^2 d^2-e^2}+8 d^2 \left (-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )+\frac {18 d e^2 \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (\frac {3 e \left (\left (c^2 d^2-e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {6 c d \sqrt {d+e x} \int \frac {\sqrt {1-\frac {e (1-c x)}{c d+e}}}{\sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{\sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{c^2 d^2-e^2}+8 d^2 \left (-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )+\frac {18 d e^2 \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (\frac {3 e \left (\left (c^2 d^2-e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {6 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{c^2 d^2-e^2}+8 d^2 \left (-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )+\frac {18 d e^2 \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (\frac {3 e \left (-\frac {2 \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \int \frac {1}{\sqrt {1-\frac {e (1-c x)}{c d+e}} \sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {d+e x}}-\frac {6 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{c^2 d^2-e^2}+8 d^2 \left (-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )+\frac {18 d e^2 \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{3 e^3 (d+e x)^{3/2}}+\frac {4 d \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}+\frac {2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \left (-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )+\frac {3 e \left (-\frac {2 \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {6 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{c^2 d^2-e^2}+\frac {18 d e^2 \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}\) |
(-2*d^2*(a + b*ArcCsc[c*x]))/(3*e^3*(d + e*x)^(3/2)) + (4*d*(a + b*ArcCsc[ c*x]))/(e^3*Sqrt[d + e*x]) + (2*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/e^3 + ( 2*b*Sqrt[1 - c^2*x^2]*((18*d*e^2*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*Sqrt[ d + e*x]) + (3*e*((-6*c*d*Sqrt[d + e*x]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqr t[2]], (2*e)/(c*d + e)])/Sqrt[(c*(d + e*x))/(c*d + e)] - (2*(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])))/(c^2*d^2 - e^2) + 8*d^2*(-((e*((2*e*S qrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*Sqrt[d + e*x]) - (2*c*Sqrt[d + e*x]*Ell ipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/((c^2*d^2 - e^2)*S qrt[(c*(d + e*x))/(c*d + e)])))/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)])/(d*Sqrt[d + e/c - (e*(1 - c*x))/c]))))/(3*c*e^3*Sqrt[1 - 1/(c^2*x^2)]*x)
3.1.70.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ )]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2 Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c*f)/d, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + S imp[b/((n + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p*(c*(n + 1) - d*(n + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[n , -1] && ((LtQ[n, -1] && IntQuadraticQ[a, 0, b, c, d, n, p, x]) || (SumSimp lerQ[n, 1] && IntegerQ[p]) || ILtQ[Simplify[n + 2*p + 3], 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[( (c + d*x)^n/Sqrt[a + b*x^2])*ExpandToSum[(1 - c^(n + 1/2)*(c + d*x)^(-n - 1 /2))/x, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[n + 1/2, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( (m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((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. \(1025\) vs. \(2(401)=802\).
Time = 9.03 (sec) , antiderivative size = 1026, normalized size of antiderivative = 2.33
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1026\) |
| default | \(\text {Expression too large to display}\) | \(1026\) |
| parts | \(\text {Expression too large to display}\) | \(1041\) |
2/e^3*(a*((e*x+d)^(1/2)-1/3*d^2/(e*x+d)^(3/2)+2*d/(e*x+d)^(1/2))+b*((e*x+d )^(1/2)*arccsc(c*x)-1/3*arccsc(c*x)*d^2/(e*x+d)^(3/2)+2*arccsc(c*x)*d/(e*x +d)^(1/2)+2/3/c*(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^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)*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*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)*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*d^2*(e*x+d) ^(1/2)-(c/(c*d-e))^(1/2)*c^2*d*(e*x+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*d*e*(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)*EllipticE((e*x+d)^(1/2)*(c/( c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*c*d*e*(e*x+d)^(1/2)+2*(c/(c*d-e))^( 1/2)*c^2*d^2*(e*x+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)*EllipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d +e))^(1/2))*e^2*(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)*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))*e^2*(e*x+d)^(1/2)-(c/(c*d- e))^(1/2)*c^2*d^3+(c/(c*d-e))^(1/2)*d*e^2)/(c*d-e)/(c/(c*d-e))^(1/2)/(e...
\[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{2}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
integral((b*x^2*arccsc(c*x) + a*x^2)*sqrt(e*x + d)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3), x)
Timed out. \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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)
\[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{2}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (d+e\,x\right )}^{5/2}} \,d x \]