Integrand size = 18, antiderivative size = 278 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{3 d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}-\frac {4 b c \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 d \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 d e \sqrt {d+e x}} \]
-2/3*(a+b*arcsech(c*x))/e/(e*x+d)^(3/2)-4/3*b*c*EllipticE(1/2*(-c*x+1)^(1/ 2)*2^(1/2),2^(1/2)*(e/(c*d+e))^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(e*x +d)^(1/2)/d/(c^2*d^2-e^2)/(c*(e*x+d)/(c*d+e))^(1/2)+4/3*b*EllipticPi(1/2*( -c*x+1)^(1/2)*2^(1/2),2,2^(1/2)*(e/(c*d+e))^(1/2))*(1/(c*x+1))^(1/2)*(c*x+ 1)^(1/2)*(c*(e*x+d)/(c*d+e))^(1/2)/d/e/(e*x+d)^(1/2)+4/3*b*e*(1/(c*x+1))^( 1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/d/(c^2*d^2-e^2)/(e*x+d)^(1/2)
Result contains complex when optimal does not.
Time = 25.13 (sec) , antiderivative size = 4527, normalized size of antiderivative = 16.28 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\text {Result too large to show} \]
(-2*a)/(3*e*(d + e*x)^(3/2)) + Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[d + e*x]*((4 *b*c)/(3*d*(c^2*d^2 - e^2)) - (4*b)/(3*d*(c*d + e)*(d + e*x))) - (2*b*ArcS ech[c*x])/(3*e*(d + e*x)^(3/2)) - (4*b*((e*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[ c*(1 + (1 - c*x)/(1 + c*x))]*(c*d + e + (c*d*(1 - c*x))/(1 + c*x) - (e*(1 - c*x))/(1 + c*x)))/((1 + (1 - c*x)/(1 + c*x))*Sqrt[c + (c*(1 - c*x))/(1 + c*x)]*Sqrt[(c*d + e + (c*d*(1 - c*x))/(1 + c*x) - (e*(1 - c*x))/(1 + c*x) )/(c + (c*(1 - c*x))/(1 + c*x))]) - ((c*d - e)*Sqrt[c*(1 + (1 - c*x)/(1 + c*x))]*Sqrt[c*(1 + (1 - c*x)/(1 + c*x))*(c*d + e + (c*d*(1 - c*x))/(1 + c* x) - (e*(1 - c*x))/(1 + c*x))]*((I*(-(c*d) - e)*e*Sqrt[1 + (1 - c*x)/(1 + c*x)]*Sqrt[1 - ((c*d - e)*(1 - c*x))/((-(c*d) - e)*(1 + c*x))]*(EllipticE[ I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], -((c*d - e)/(-(c*d) - e))] - Ellipti cF[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], -((c*d - e)/(-(c*d) - e))]))/((c* d - e)*Sqrt[c*(1 + (1 - c*x)/(1 + c*x))*(c*d + e + ((c*d - e)*(1 - c*x))/( 1 + c*x))]) + (I*c*d*Sqrt[1 + (1 - c*x)/(1 + c*x)]*Sqrt[1 - ((c*d - e)*(1 - c*x))/((-(c*d) - e)*(1 + c*x))]*EllipticF[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], -((c*d - e)/(-(c*d) - e))])/Sqrt[c*(1 + (1 - c*x)/(1 + c*x))*(c*d + e + ((c*d - e)*(1 - c*x))/(1 + c*x))] + (I*e*Sqrt[1 + (1 - c*x)/(1 + c*x )]*Sqrt[1 - ((c*d - e)*(1 - c*x))/((-(c*d) - e)*(1 + c*x))]*EllipticF[I*Ar cSinh[Sqrt[(1 - c*x)/(1 + c*x)]], -((c*d - e)/(-(c*d) - e))])/Sqrt[c*(1 + (1 - c*x)/(1 + c*x))*(c*d + e + ((c*d - e)*(1 - c*x))/(1 + c*x))] - (I*...
Time = 0.56 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.91, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6842, 635, 25, 27, 498, 27, 508, 327, 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.
| \(\displaystyle \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6842 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {1}{x (d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx}{3 e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 635 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\int -\frac {e}{d (d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx+\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}\right )}{3 e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}-\int \frac {e}{d (d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )}{3 e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}-\frac {e \int \frac {1}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx}{d}\right )}{3 e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 498 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c^2 \int -\frac {\sqrt {d+e x}}{2 \sqrt {1-c^2 x^2}}dx}{c^2 d^2-e^2}\right )}{d}\right )}{3 e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}-\frac {e \left (\frac {c^2 \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}}dx}{c^2 d^2-e^2}+\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{d}\right )}{3 e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} \int \frac {\sqrt {1-\frac {e (1-c x)}{c d+e}}}{\sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}\right )}{3 e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}\right )}{3 e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 632 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {\int \frac {1}{x \sqrt {1-c x} \sqrt {c x+1} \sqrt {d+e x}}dx}{d}-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}\right )}{3 e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {2 \int \frac {1}{c x \sqrt {c x+1} \sqrt {d+\frac {e}{c}-\frac {e (1-c x)}{c}}}d\sqrt {1-c x}}{d}-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}\right )}{3 e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \int \frac {1}{c x \sqrt {c x+1} \sqrt {1-\frac {e (1-c x)}{c d+e}}}d\sqrt {1-c x}}{d \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}\right )}{3 e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle -\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 e (d+e x)^{3/2}}-\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )}{3 e}\) |
(-2*(a + b*ArcSech[c*x]))/(3*e*(d + e*x)^(3/2)) - (2*b*Sqrt[(1 + c*x)^(-1) ]*Sqrt[1 + c*x]*(-((e*((2*e*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*Sqrt[d + e *x]) - (2*c*Sqrt[d + e*x]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/( c*d + e)])/((c^2*d^2 - e^2)*Sqrt[(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*e)
3.1.85.3.1 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[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/((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[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbo l] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcSech[c*x])/(e*(m + 1))), x] + Simp[ b*(Sqrt[1 + c*x]/(e*(m + 1)))*Sqrt[1/(1 + c*x)] Int[(d + e*x)^(m + 1)/(x* Sqrt[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. \(889\) vs. \(2(246)=492\).
Time = 13.39 (sec) , antiderivative size = 890, normalized size of antiderivative = 3.20
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 a}{3 \left (e x +d \right )^{\frac {3}{2}}}+2 b \left (-\frac {\operatorname {arcsech}\left (c x \right )}{3 \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 c \,e^{2} \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c e x}}\, x \sqrt {-\frac {-c \left (e x +d \right )+c d -e}{c e x}}\, \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}+c^{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 ) 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 \left (c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}\right ) d^{2} \sqrt {\frac {c}{c d +e}}\, \left (c d +e \right ) \left (c d -e \right ) \sqrt {e x +d}}\right )}{e}\) | \(890\) |
| default | \(\frac {-\frac {2 a}{3 \left (e x +d \right )^{\frac {3}{2}}}+2 b \left (-\frac {\operatorname {arcsech}\left (c x \right )}{3 \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 c \,e^{2} \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c e x}}\, x \sqrt {-\frac {-c \left (e x +d \right )+c d -e}{c e x}}\, \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}+c^{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 ) 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 \left (c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}\right ) d^{2} \sqrt {\frac {c}{c d +e}}\, \left (c d +e \right ) \left (c d -e \right ) \sqrt {e x +d}}\right )}{e}\) | \(890\) |
| parts | \(-\frac {2 a}{3 \left (e x +d \right )^{\frac {3}{2}} e}+\frac {2 b \left (-\frac {\operatorname {arcsech}\left (c x \right )}{3 \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 c \,e^{2} \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c e x}}\, x \sqrt {\frac {c \left (e x +d \right )-c d +e}{c e x}}\, \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 ) \sqrt {e x +d}\, c^{2} d^{2}+c^{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 ) 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 ) \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 \sqrt {e x +d}\, \left (c d -e \right ) \left (c d +e \right ) \sqrt {\frac {c}{c d +e}}\, d^{2} \left (c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}\right )}\right )}{e}\) | \(904\) |
2/e*(-1/3*a/(e*x+d)^(3/2)+b*(-1/3/(e*x+d)^(3/2)*arcsech(c*x)+2/3*c*e^2*((- c*(e*x+d)+c*d+e)/c/e/x)^(1/2)*x*(-(-c*(e*x+d)+c*d-e)/c/e/x)^(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^2*d^2*(e*x+d)^(1/2)+c^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))*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^2*(e*x+d)^2-2*c^2*d*(e*x+d)+c^2*d^2-e^2)/d^2/(c/(c*d+e))^(1/2)/(c*d +e)/(c*d-e)/(e*x+d)^(1/2)))
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\int \frac {a + b \operatorname {asech}{\left (c x \right )}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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)
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{5/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^{5/2}} \,d x \]