Integrand size = 18, antiderivative size = 298 \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\frac {4 b e \left (1-c^2 x^2\right )}{3 c d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}-\frac {4 b \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 c d e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \] Output:
4/3*b*e*(-c^2*x^2+1)/c/d/(c^2*d^2-e^2)/(1-1/c^2/x^2)^(1/2)/x/(e*x+d)^(1/2) -2/3*(a+b*arccsc(c*x))/e/(e*x+d)^(3/2)-4/3*b*(e*x+d)^(1/2)*(-c^2*x^2+1)^(1 /2)*EllipticE(1/2*(-c*x+1)^(1/2)*2^(1/2),2^(1/2)*(e/(c*d+e))^(1/2))/d/(c^2 *d^2-e^2)/(1-1/c^2/x^2)^(1/2)/x/(c*(e*x+d)/(c*d+e))^(1/2)+4/3*b*(c*(e*x+d) /(c*d+e))^(1/2)*(-c^2*x^2+1)^(1/2)*EllipticPi(1/2*(-c*x+1)^(1/2)*2^(1/2),2 ,2^(1/2)*(e/(c*d+e))^(1/2))/c/d/e/(1-1/c^2/x^2)^(1/2)/x/(e*x+d)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(608\) vs. \(2(298)=596\).
Time = 20.22 (sec) , antiderivative size = 608, normalized size of antiderivative = 2.04 \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\frac {2 \left (-\frac {a}{e}+\frac {2 b c \sqrt {1-\frac {1}{c^2 x^2}} (d+e x)^2}{c^2 d^3-d e^2}-\frac {b e x^2 \csc ^{-1}(c x)}{d^2}-\frac {b (d+e x)^2 \csc ^{-1}(c x)}{d^2 e}+\frac {2 b x (d+e x) \left (-c d e \sqrt {1-\frac {1}{c^2 x^2}}+\left (c^2 d^2-e^2\right ) \csc ^{-1}(c x)\right )}{c^2 d^4-d^2 e^2}+\frac {2 b d \left (\frac {c (d+e x)}{c d+e}\right )^{3/2} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{(c d-e) e \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {2 b c (d+e x) \cos \left (2 \csc ^{-1}(c x)\right ) \left ((d+e x) \left (-1+c^2 x^2\right )+c d x \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )-\frac {x (1+c x) \sqrt {\frac {c (d+e x)}{c d-e}} \sqrt {\frac {e-c e x}{c d+e}} \left ((c d+e) E\left (\arcsin \left (\sqrt {\frac {c (d+e x)}{c d-e}}\right )|\frac {c d-e}{c d+e}\right )-e \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {c (d+e x)}{c d-e}}\right ),\frac {c d-e}{c d+e}\right )\right )}{\sqrt {\frac {e (1+c x)}{-c d+e}}}+e x \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )\right )}{d (c d-e) (c d+e) \sqrt {1-\frac {1}{c^2 x^2}} \left (-2+c^2 x^2\right )}\right )}{3 (d+e x)^{3/2}} \] Input:
Integrate[(a + b*ArcCsc[c*x])/(d + e*x)^(5/2),x]
Output:
(2*(-(a/e) + (2*b*c*Sqrt[1 - 1/(c^2*x^2)]*(d + e*x)^2)/(c^2*d^3 - d*e^2) - (b*e*x^2*ArcCsc[c*x])/d^2 - (b*(d + e*x)^2*ArcCsc[c*x])/(d^2*e) + (2*b*x* (d + e*x)*(-(c*d*e*Sqrt[1 - 1/(c^2*x^2)]) + (c^2*d^2 - e^2)*ArcCsc[c*x]))/ (c^2*d^4 - d^2*e^2) + (2*b*d*((c*(d + e*x))/(c*d + e))^(3/2)*Sqrt[1 - c^2* x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/((c*d - e)*e*Sqrt[1 - 1/(c^2*x^2)]*x) - (2*b*c*(d + e*x)*Cos[2*ArcCsc[c*x]]*((d + e*x)*(-1 + c^2*x^2) + c*d*x*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x ^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)] - (x*(1 + c* x)*Sqrt[(c*(d + e*x))/(c*d - e)]*Sqrt[(e - c*e*x)/(c*d + e)]*((c*d + e)*El lipticE[ArcSin[Sqrt[(c*(d + e*x))/(c*d - e)]], (c*d - e)/(c*d + e)] - e*El lipticF[ArcSin[Sqrt[(c*(d + e*x))/(c*d - e)]], (c*d - e)/(c*d + e)]))/Sqrt [(e*(1 + c*x))/(-(c*d) + e)] + e*x*Sqrt[(c*(d + 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)]))/( d*(c*d - e)*(c*d + e)*Sqrt[1 - 1/(c^2*x^2)]*(-2 + c^2*x^2))))/(3*(d + e*x) ^(3/2))
Time = 0.75 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.07, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5750, 1898, 635, 25, 27, 498, 27, 509, 508, 327, 633, 632, 186, 413, 412}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 5750 |
\(\displaystyle -\frac {2 b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}}dx}{3 c e}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 1898 |
\(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \int \frac {1}{x (d+e x)^{3/2} \sqrt {x^2-\frac {1}{c^2}}}dx}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 635 |
\(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\int -\frac {e}{d (d+e x)^{3/2} \sqrt {x^2-\frac {1}{c^2}}}dx+\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx}{d}\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx}{d}-\int \frac {e}{d (d+e x)^{3/2} \sqrt {x^2-\frac {1}{c^2}}}dx\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx}{d}-\frac {e \int \frac {1}{(d+e x)^{3/2} \sqrt {x^2-\frac {1}{c^2}}}dx}{d}\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 498 |
\(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx}{d}-\frac {e \left (-\frac {2 \int -\frac {\sqrt {d+e x}}{2 \sqrt {x^2-\frac {1}{c^2}}}dx}{d^2-\frac {e^2}{c^2}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}}}{\left (d^2-\frac {e^2}{c^2}\right ) \sqrt {d+e x}}\right )}{d}\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx}{d}-\frac {e \left (\frac {\int \frac {\sqrt {d+e x}}{\sqrt {x^2-\frac {1}{c^2}}}dx}{d^2-\frac {e^2}{c^2}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}}}{\left (d^2-\frac {e^2}{c^2}\right ) \sqrt {d+e x}}\right )}{d}\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 509 |
\(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx}{d}-\frac {e \left (\frac {\sqrt {1-c^2 x^2} \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}}dx}{\sqrt {x^2-\frac {1}{c^2}} \left (d^2-\frac {e^2}{c^2}\right )}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}}}{\left (d^2-\frac {e^2}{c^2}\right ) \sqrt {d+e x}}\right )}{d}\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 508 |
\(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx}{d}-\frac {e \left (-\frac {2 \sqrt {1-c^2 x^2} \sqrt {d+e x} \int \frac {\sqrt {1-\frac {e (1-c x)}{c d+e}}}{\sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {x^2-\frac {1}{c^2}} \left (d^2-\frac {e^2}{c^2}\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}}}{\left (d^2-\frac {e^2}{c^2}\right ) \sqrt {d+e x}}\right )}{d}\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx}{d}-\frac {e \left (-\frac {2 \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \left (d^2-\frac {e^2}{c^2}\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}}}{\left (d^2-\frac {e^2}{c^2}\right ) \sqrt {d+e x}}\right )}{d}\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 633 |
\(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\frac {\sqrt {1-c^2 x^2} \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d \sqrt {x^2-\frac {1}{c^2}}}-\frac {e \left (-\frac {2 \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \left (d^2-\frac {e^2}{c^2}\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}}}{\left (d^2-\frac {e^2}{c^2}\right ) \sqrt {d+e x}}\right )}{d}\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 632 |
\(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\frac {\sqrt {1-c^2 x^2} \int \frac {1}{x \sqrt {1-c x} \sqrt {c x+1} \sqrt {d+e x}}dx}{d \sqrt {x^2-\frac {1}{c^2}}}-\frac {e \left (-\frac {2 \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \left (d^2-\frac {e^2}{c^2}\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}}}{\left (d^2-\frac {e^2}{c^2}\right ) \sqrt {d+e x}}\right )}{d}\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 186 |
\(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {2 \sqrt {1-c^2 x^2} \int \frac {1}{c x \sqrt {c x+1} \sqrt {d+\frac {e}{c}-\frac {e (1-c x)}{c}}}d\sqrt {1-c x}}{d \sqrt {x^2-\frac {1}{c^2}}}-\frac {e \left (-\frac {2 \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \left (d^2-\frac {e^2}{c^2}\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}}}{\left (d^2-\frac {e^2}{c^2}\right ) \sqrt {d+e x}}\right )}{d}\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 413 |
\(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \int \frac {1}{c x \sqrt {c x+1} \sqrt {1-\frac {e (1-c x)}{c d+e}}}d\sqrt {1-c x}}{d \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {e \left (-\frac {2 \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \left (d^2-\frac {e^2}{c^2}\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}}}{\left (d^2-\frac {e^2}{c^2}\right ) \sqrt {d+e x}}\right )}{d}\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 412 |
\(\displaystyle -\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}-\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {e \left (-\frac {2 \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \left (d^2-\frac {e^2}{c^2}\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}}}{\left (d^2-\frac {e^2}{c^2}\right ) \sqrt {d+e x}}\right )}{d}-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )}{3 c e x \sqrt {1-\frac {1}{c^2 x^2}}}\) |
Input:
Int[(a + b*ArcCsc[c*x])/(d + e*x)^(5/2),x]
Output:
(-2*(a + b*ArcCsc[c*x]))/(3*e*(d + e*x)^(3/2)) - (2*b*Sqrt[-c^(-2) + x^2]* (-((e*((-2*e*Sqrt[-c^(-2) + x^2])/((d^2 - e^2/c^2)*Sqrt[d + e*x]) - (2*Sqr t[d + e*x]*Sqrt[1 - c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e )/(c*d + e)])/(c*(d^2 - e^2/c^2)*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[-c^(-2 ) + x^2])))/d) - (2*Sqrt[1 - c^2*x^2]*Sqrt[1 - (e*(1 - c*x))/(c*d + e)]*El lipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(d*Sqrt[-c^(- 2) + x^2]*Sqrt[d + e/c - (e*(1 - c*x))/c])))/(3*c*e*Sqrt[1 - 1/(c^2*x^2)]* x)
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[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[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > With[{q = Rt[-b/a, 2]}, Simp[1/Sqrt[a] Int[1/(x*Sqrt[c + d*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > Simp[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(x*Sqrt[c + d*x]*Sqrt[1 + b*(x^2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[((c_) + (d_.)*(x_))^(n_)/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp[c^(n + 1/2) Int[1/(x*Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] + Int[( (c + d*x)^n/Sqrt[a + b*x^2])*ExpandToSum[(1 - c^(n + 1/2)*(c + d*x)^(-n - 1 /2))/x, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[n + 1/2, 0]
Int[(x_)^(m_.)*((a_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^ (q_.), x_Symbol] :> Simp[x^(2*n*FracPart[p])*((a + c/x^(2*n))^FracPart[p]/( c + a*x^(2*n))^FracPart[p]) Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + a*x^(2*n ))^p, x], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[mn2, -2*n] && !I ntegerQ[p] && !IntegerQ[q] && PosQ[n]
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol ] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcCsc[c*x])/(e*(m + 1))), x] + Simp[b/ (c*e*(m + 1)) Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x] / ; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(873\) vs. \(2(270)=540\).
Time = 10.74 (sec) , antiderivative size = 874, normalized size of antiderivative = 2.93
method | result | size |
derivativedivides | \(\frac {-\frac {2 a}{3 \left (e x +d \right )^{\frac {3}{2}}}+2 b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (\sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{2}-\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2} d^{2} \sqrt {e x +d}+\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2} d^{2} \sqrt {e x +d}-\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c^{2} d^{2} \sqrt {e x +d}-2 \sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \left (e x +d \right )-\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e \sqrt {e x +d}+\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e \sqrt {e x +d}+\sqrt {\frac {c}{c d -e}}\, c^{2} d^{3}+\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) e^{2} \sqrt {e x +d}-\sqrt {\frac {c}{c d -e}}\, d \,e^{2}\right )}{3 c \left (c d -e \right ) \sqrt {\frac {c}{c d -e}}\, \sqrt {e x +d}\, \left (c d +e \right ) d^{2} x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e}\) | \(874\) |
default | \(\frac {-\frac {2 a}{3 \left (e x +d \right )^{\frac {3}{2}}}+2 b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (\sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{2}-\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2} d^{2} \sqrt {e x +d}+\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2} d^{2} \sqrt {e x +d}-\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c^{2} d^{2} \sqrt {e x +d}-2 \sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \left (e x +d \right )-\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e \sqrt {e x +d}+\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e \sqrt {e x +d}+\sqrt {\frac {c}{c d -e}}\, c^{2} d^{3}+\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) e^{2} \sqrt {e x +d}-\sqrt {\frac {c}{c d -e}}\, d \,e^{2}\right )}{3 c \left (c d -e \right ) \sqrt {\frac {c}{c d -e}}\, \sqrt {e x +d}\, \left (c d +e \right ) d^{2} x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e}\) | \(874\) |
parts | \(-\frac {2 a}{3 \left (e x +d \right )^{\frac {3}{2}} e}+\frac {2 b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (\sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{2}-\sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) \sqrt {e x +d}\, c^{2} d^{2}-\sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) \sqrt {e x +d}\, c^{2} d^{2}+\sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) \sqrt {e x +d}\, c^{2} d^{2}-2 \sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \left (e x +d \right )-\sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) \sqrt {e x +d}\, c d e +\sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) \sqrt {e x +d}\, c d e +\sqrt {\frac {c}{c d -e}}\, c^{2} d^{3}+\sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) \sqrt {e x +d}\, e^{2}-\sqrt {\frac {c}{c d -e}}\, d \,e^{2}\right )}{3 c \left (c d -e \right ) \sqrt {e x +d}\, \left (c d +e \right ) \sqrt {\frac {c}{c d -e}}\, d^{2} x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e}\) | \(888\) |
Input:
int((a+b*arccsc(c*x))/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/e*(-1/3*a/(e*x+d)^(3/2)+b*(-1/3/(e*x+d)^(3/2)*arccsc(c*x)-2/3/c*((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^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)-((-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)-2*(c/(c*d-e))^(1/2)*c^2*d^2*(e*x+d)-((-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)+(c/(c*d -e))^(1/2)*c^2*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))*e^2*(e*x+d)^(1/2)-(c/(c*d-e))^(1/2)*d*e ^2)/(c*d-e)/(c/(c*d-e))^(1/2)/(e*x+d)^(1/2)/(c*d+e)/d^2/x/((c^2*(e*x+d)^2- 2*c^2*d*(e*x+d)+c^2*d^2-e^2)/c^2/e^2/x^2)^(1/2)))
\[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((a+b*arccsc(c*x))/(e*x+d)^(5/2),x, algorithm="fricas")
Output:
integral(sqrt(e*x + d)*(b*arccsc(c*x) + a)/(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2* e*x + d^3), x)
\[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\int \frac {a + b \operatorname {acsc}{\left (c x \right )}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((a+b*acsc(c*x))/(e*x+d)**(5/2),x)
Output:
Integral((a + b*acsc(c*x))/(d + e*x)**(5/2), x)
Exception generated. \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a+b*arccsc(c*x))/(e*x+d)^(5/2),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e+c*d>0)', see `assume?` for mor e details)
\[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((a+b*arccsc(c*x))/(e*x+d)^(5/2),x, algorithm="giac")
Output:
integrate((b*arccsc(c*x) + a)/(e*x + d)^(5/2), x)
Timed out. \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^{5/2}} \,d x \] Input:
int((a + b*asin(1/(c*x)))/(d + e*x)^(5/2),x)
Output:
int((a + b*asin(1/(c*x)))/(d + e*x)^(5/2), x)
\[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\frac {3 \sqrt {e x +d}\, \left (\int \frac {\mathit {acsc} \left (c x \right )}{\sqrt {e x +d}\, d^{2}+2 \sqrt {e x +d}\, d e x +\sqrt {e x +d}\, e^{2} x^{2}}d x \right ) b d e +3 \sqrt {e x +d}\, \left (\int \frac {\mathit {acsc} \left (c x \right )}{\sqrt {e x +d}\, d^{2}+2 \sqrt {e x +d}\, d e x +\sqrt {e x +d}\, e^{2} x^{2}}d x \right ) b \,e^{2} x -2 a}{3 \sqrt {e x +d}\, e \left (e x +d \right )} \] Input:
int((a+b*acsc(c*x))/(e*x+d)^(5/2),x)
Output:
(3*sqrt(d + e*x)*int(acsc(c*x)/(sqrt(d + e*x)*d**2 + 2*sqrt(d + e*x)*d*e*x + sqrt(d + e*x)*e**2*x**2),x)*b*d*e + 3*sqrt(d + e*x)*int(acsc(c*x)/(sqrt (d + e*x)*d**2 + 2*sqrt(d + e*x)*d*e*x + sqrt(d + e*x)*e**2*x**2),x)*b*e** 2*x - 2*a)/(3*sqrt(d + e*x)*e*(d + e*x))