Integrand size = 18, antiderivative size = 149 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^{3/2}} \, dx=-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e \sqrt {d+e x}}+\frac {4 b \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \sqrt {1+c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),\frac {2 e}{\sqrt {-c^2} d+e}\right )}{c e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
-2*(a+b*arccsch(c*x))/e/(e*x+d)^(1/2)+4*b*EllipticPi(1/2*(1-x*(-c^2)^(1/2) )^(1/2)*2^(1/2),2,2^(1/2)*(e/(d*(-c^2)^(1/2)+e))^(1/2))*(c^2*x^2+1)^(1/2)* ((e*x+d)*(-c^2)^(1/2)/(d*(-c^2)^(1/2)+e))^(1/2)/c/e/x/(1+1/c^2/x^2)^(1/2)/ (e*x+d)^(1/2)
Result contains complex when optimal does not.
Time = 0.70 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.11 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^{3/2}} \, dx=\frac {-2 e \left (1+c^2 x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )+2 b c (i c d+e) \sqrt {2+\frac {2}{c^2 x^2}} x \sqrt {1+i c x} \sqrt {\frac {c e (i+c x) (d+e x)}{(i c d+e)^2}} \operatorname {EllipticPi}\left (1+\frac {i c d}{e},\arcsin \left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right ),\frac {i c d+e}{2 e}\right )}{e^2 \sqrt {d+e x} \left (1+c^2 x^2\right )} \]
(-2*e*(1 + c^2*x^2)*(a + b*ArcCsch[c*x]) + 2*b*c*(I*c*d + e)*Sqrt[2 + 2/(c ^2*x^2)]*x*Sqrt[1 + I*c*x]*Sqrt[(c*e*(I + c*x)*(d + e*x))/(I*c*d + e)^2]*E llipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c* d + e)/(2*e)])/(e^2*Sqrt[d + e*x]*(1 + c^2*x^2))
Leaf count is larger than twice the leaf count of optimal. \(736\) vs. \(2(149)=298\).
Time = 1.09 (sec) , antiderivative size = 736, normalized size of antiderivative = 4.94, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6844, 1898, 631, 1540, 1416, 2222}
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.
| \(\displaystyle \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6844 |
| \(\displaystyle -\frac {2 b \int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} x^2 \sqrt {d+e x}}dx}{c e}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1898 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c^2}+x^2} \int \frac {1}{x \sqrt {d+e x} \sqrt {x^2+\frac {1}{c^2}}}dx}{c e x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 631 |
| \(\displaystyle \frac {4 b \sqrt {\frac {1}{c^2}+x^2} \int -\frac {1}{e x \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{c e x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1540 |
| \(\displaystyle \frac {4 b \sqrt {\frac {1}{c^2}+x^2} \left (\frac {\left (c^2 d^2+e^2\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{e^2}-\frac {c \left (c d-\sqrt {c^2 d^2+e^2}\right ) \int \frac {1}{\sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{e^2}\right )}{c e x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {4 b \sqrt {\frac {1}{c^2}+x^2} \left (\frac {\left (c^2 d^2+e^2\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{e^2}-\frac {\sqrt {c} \sqrt [4]{c^2 d^2+e^2} \left (c d-\sqrt {c^2 d^2+e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}{\left (\frac {1}{c^2}+\frac {d^2}{e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 e^2 \sqrt {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}}\right )}{c e x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 2222 |
| \(\displaystyle \frac {4 b \sqrt {\frac {1}{c^2}+x^2} \left (\frac {\left (c^2 d^2+e^2\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}{\left (\frac {1}{c^2}+\frac {d^2}{e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (c d+\sqrt {c^2 d^2+e^2}\right )^2}{4 c d \sqrt {c^2 d^2+e^2}},2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {\sqrt {c^2 d^2+e^2} d}{2 c \left (\frac {d^2}{e^2}+\frac {1}{c^2}\right ) e^2}+\frac {1}{2}\right )}{4 \sqrt {c} d \sqrt {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}}+\frac {c \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{c \sqrt {d} \sqrt {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}}\right )}{2 \sqrt {d}}\right )}{e^2}-\frac {\sqrt {c} \sqrt [4]{c^2 d^2+e^2} \left (c d-\sqrt {c^2 d^2+e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}{\left (\frac {1}{c^2}+\frac {d^2}{e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 e^2 \sqrt {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}}\right )}{c e x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e \sqrt {d+e x}}\) |
(-2*(a + b*ArcCsch[c*x]))/(e*Sqrt[d + e*x]) + (4*b*Sqrt[c^(-2) + x^2]*(-1/ 2*(Sqrt[c]*(c^2*d^2 + e^2)^(1/4)*(c*d - Sqrt[c^2*d^2 + e^2])*(1 + (c*(d + e*x))/Sqrt[c^2*d^2 + e^2])*Sqrt[(c^(-2) + d^2/e^2 - (2*d*(d + e*x))/e^2 + (d + e*x)^2/e^2)/((c^(-2) + d^2/e^2)*(1 + (c*(d + e*x))/Sqrt[c^2*d^2 + e^2 ])^2)]*EllipticF[2*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/(c^2*d^2 + e^2)^(1/4)], (1 + (c*d)/Sqrt[c^2*d^2 + e^2])/2])/(e^2*Sqrt[c^(-2) + d^2/e^2 - (2*d*(d + e*x))/e^2 + (d + e*x)^2/e^2]) + ((c^2*d^2 + e^2)*(1 - (c*d)/Sqrt[c^2*d^2 + e^2])*((c*(1 + (c*d)/Sqrt[c^2*d^2 + e^2])*ArcTanh[Sqrt[d + e*x]/(c*Sqrt[ d]*Sqrt[c^(-2) + d^2/e^2 - (2*d*(d + e*x))/e^2 + (d + e*x)^2/e^2])])/(2*Sq rt[d]) + ((c^2*d^2 + e^2)^(1/4)*(1 - (c*d)/Sqrt[c^2*d^2 + e^2])*(1 + (c*(d + e*x))/Sqrt[c^2*d^2 + e^2])*Sqrt[(c^(-2) + d^2/e^2 - (2*d*(d + e*x))/e^2 + (d + e*x)^2/e^2)/((c^(-2) + d^2/e^2)*(1 + (c*(d + e*x))/Sqrt[c^2*d^2 + e^2])^2)]*EllipticPi[(c*d + Sqrt[c^2*d^2 + e^2])^2/(4*c*d*Sqrt[c^2*d^2 + e ^2]), 2*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/(c^2*d^2 + e^2)^(1/4)], 1/2 + (d*Sq rt[c^2*d^2 + e^2])/(2*c*(c^(-2) + d^2/e^2)*e^2)])/(4*Sqrt[c]*d*Sqrt[c^(-2) + d^2/e^2 - (2*d*(d + e*x))/e^2 + (d + e*x)^2/e^2])))/e^2))/(c*e*Sqrt[1 + 1/(c^2*x^2)]*x)
3.1.66.3.1 Defintions of rubi rules used
Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > Simp[-2 Subst[Int[1/((c - x^2)*Sqrt[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^2/d^ 2) + b*(x^4/d^2)]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && PosQ[b/a]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_S ymbol] :> With[{q = Rt[c/a, 2]}, Simp[(c*d + a*e*q)/(c*d^2 - a*e^2) Int[1 /Sqrt[a + b*x^2 + c*x^4], x], x] - Simp[(a*e*(e + d*q))/(c*d^2 - a*e^2) I nt[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
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_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(A rcTanh[Rt[b - c*(d/e) - a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*e*Rt[ b - c*(d/e) - a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*Ell ipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[-b + c*(d/e) + a*(e/d)]
Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbo l] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcCsch[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]
Result contains complex when optimal does not.
Time = 3.32 (sec) , antiderivative size = 328, normalized size of antiderivative = 2.20
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 a}{\sqrt {e x +d}}+2 b \left (-\frac {\operatorname {arccsch}\left (c x \right )}{\sqrt {e x +d}}+\frac {2 \sqrt {-\frac {i c e \left (e x +d \right )+c^{2} d \left (e x +d \right )-c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i c e \left (e x +d \right )-c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} e^{2} x^{2}}}\, x d \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )}{e}\) | \(328\) |
| default | \(\frac {-\frac {2 a}{\sqrt {e x +d}}+2 b \left (-\frac {\operatorname {arccsch}\left (c x \right )}{\sqrt {e x +d}}+\frac {2 \sqrt {-\frac {i c e \left (e x +d \right )+c^{2} d \left (e x +d \right )-c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i c e \left (e x +d \right )-c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} e^{2} x^{2}}}\, x d \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )}{e}\) | \(328\) |
| parts | \(-\frac {2 a}{\sqrt {e x +d}\, e}+\frac {2 b \left (-\frac {\operatorname {arccsch}\left (c x \right )}{\sqrt {e x +d}}+\frac {2 \sqrt {-\frac {i c e \left (e x +d \right )+c^{2} d \left (e x +d \right )-c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i c e \left (e x +d \right )-c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} e^{2} x^{2}}}\, x d \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )}{e}\) | \(330\) |
2/e*(-a/(e*x+d)^(1/2)+b*(-1/(e*x+d)^(1/2)*arccsch(c*x)+2/c/((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)/x/d/((c*d+I*e)*c/(c^2*d^2 +e^2))^(1/2)*(-(I*c*e*(e*x+d)+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1 /2)*((I*c*e*(e*x+d)-c^2*d*(e*x+d)+c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2)*Ellipt icPi((e*x+d)^(1/2)*((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2),1/(c*d+I*e)/c*(c^2*d^ 2+e^2)/d,(-(I*e-c*d)*c/(c^2*d^2+e^2))^(1/2)/((c*d+I*e)*c/(c^2*d^2+e^2))^(1 /2))))
\[ \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^{3/2}} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^{3/2}} \, dx=\int \frac {a + b \operatorname {acsch}{\left (c x \right )}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^{3/2}} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
-(2*c^2*integrate(x/((c^2*e*x^2 + e)*sqrt(c^2*x^2 + 1)*sqrt(e*x + d) + (c^ 2*e*x^2 + e)*sqrt(e*x + d)), x) + 2*log(sqrt(c^2*x^2 + 1) + 1)/(sqrt(e*x + d)*e) + integrate(((e*log(c) - 2*e)*c^2*x^2 - 2*c^2*d*x + e*log(c) + (c^2 *e*x^2 + e)*log(x))/((c^2*e^2*x^3 + c^2*d*e*x^2 + e^2*x + d*e)*sqrt(e*x + d)), x))*b - 2*a/(sqrt(e*x + d)*e)
\[ \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^{3/2}} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^{3/2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \]