3.60 \(\int \frac {a+b \text {csch}^{-1}(c x)}{\sqrt {d+e x}} \, dx\)

Optimal. Leaf size=284 \[ \frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {4 b c \sqrt {c^2 x^2+1} \sqrt {\frac {d+e x}{\frac {e}{\sqrt {-c^2}}+d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{\left (-c^2\right )^{3/2} x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}-\frac {4 b d \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{c e x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}} \]

[Out]

2*(a+b*arccsch(c*x))*(e*x+d)^(1/2)/e+4*b*c*EllipticF(1/2*(1-(-c^2)^(1/2)*x)^(1/2)*2^(1/2),(-2*e*(-c^2)^(1/2)/(
c^2*d-e*(-c^2)^(1/2)))^(1/2))*(c^2*x^2+1)^(1/2)*((e*x+d)/(d+e/(-c^2)^(1/2)))^(1/2)/(-c^2)^(3/2)/x/(1+1/c^2/x^2
)^(1/2)/(e*x+d)^(1/2)-4*b*d*EllipticPi(1/2*(1-(-c^2)^(1/2)*x)^(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
)

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Rubi [A]  time = 0.41, antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6290, 1574, 944, 719, 419, 933, 168, 538, 537} \[ \frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {4 b c \sqrt {c^2 x^2+1} \sqrt {\frac {d+e x}{\frac {e}{\sqrt {-c^2}}+d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{\left (-c^2\right )^{3/2} x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}-\frac {4 b d \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{c e x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsch[c*x])/Sqrt[d + e*x],x]

[Out]

(2*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/e + (4*b*c*Sqrt[(d + e*x)/(d + e/Sqrt[-c^2])]*Sqrt[1 + c^2*x^2]*Ellipti
cF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (-2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e)])/((-c^2)^(3/2)*Sqrt[1 + 1
/(c^2*x^2)]*x*Sqrt[d + e*x]) - (4*b*d*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*Ellipt
icPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(c*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d
+ e*x])

Rule 168

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-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]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rule 538

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d*x^2)/c]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rule 719

Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(2*a*Rt[-(c/a), 2]*(d + e*x)^m*Sqrt[
1 + (c*x^2)/a])/(c*Sqrt[a + c*x^2]*((c*(d + e*x))/(c*d - a*e*Rt[-(c/a), 2]))^m), Subst[Int[(1 + (2*a*e*Rt[-(c/
a), 2]*x^2)/(c*d - a*e*Rt[-(c/a), 2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(1 - Rt[-(c/a), 2]*x)/2]], x] /; FreeQ[{a,
c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m^2, 1/4]

Rule 933

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[-(c
/a), 2]}, Dist[Sqrt[1 + (c*x^2)/a]/Sqrt[a + c*x^2], Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]
), x], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] &&  !GtQ[a, 0]

Rule 944

Int[Sqrt[(f_.) + (g_.)*(x_)]/(((d_.) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[g/e, Int[1/(S
qrt[f + g*x]*Sqrt[a + c*x^2]), x], x] + Dist[(e*f - d*g)/e, Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[a + c*x^2]), x
], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0]

Rule 1574

Int[(x_)^(m_.)*((a_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Dist[(x^(2*n*Fr
acPart[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] &&  !IntegerQ[p] &&  !IntegerQ[q] &&
PosQ[n]

Rule 6290

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(a +
b*ArcCsch[c*x]))/(e*(m + 1)), x] + Dist[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]

Rubi steps

\begin {align*} \int \frac {a+b \text {csch}^{-1}(c x)}{\sqrt {d+e x}} \, dx &=\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {(2 b) \int \frac {\sqrt {d+e x}}{\sqrt {1+\frac {1}{c^2 x^2}} x^2} \, dx}{c e}\\ &=\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {\left (2 b \sqrt {\frac {1}{c^2}+x^2}\right ) \int \frac {\sqrt {d+e x}}{x \sqrt {\frac {1}{c^2}+x^2}} \, dx}{c e \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {\left (2 b \sqrt {\frac {1}{c^2}+x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {\frac {1}{c^2}+x^2}} \, dx}{c \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (2 b d \sqrt {\frac {1}{c^2}+x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {\frac {1}{c^2}+x^2}} \, dx}{c e \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {\left (2 b d \sqrt {1+c^2 x^2}\right ) \int \frac {1}{x \sqrt {1-\sqrt {-c^2} x} \sqrt {1+\sqrt {-c^2} x} \sqrt {d+e x}} \, dx}{c e \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (4 b \sqrt {-c^2} \sqrt {\frac {d+e x}{d-\frac {\sqrt {-c^2} e}{c^2}}} \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {-c^2} e x^2}{c^2 \left (d-\frac {\sqrt {-c^2} e}{c^2}\right )}}} \, dx,x,\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )}{c^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {4 b \sqrt {-c^2} \sqrt {\frac {d+e x}{d+\frac {e}{\sqrt {-c^2}}}} \sqrt {1+c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{c^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (4 b d \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {d+\frac {e}{\sqrt {-c^2}}-\frac {e x^2}{\sqrt {-c^2}}}} \, dx,x,\sqrt {1-\sqrt {-c^2} x}\right )}{c e \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {4 b \sqrt {-c^2} \sqrt {\frac {d+e x}{d+\frac {e}{\sqrt {-c^2}}}} \sqrt {1+c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{c^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (4 b d \sqrt {1+c^2 x^2} \sqrt {1+\frac {e \left (-1+\sqrt {-c^2} x\right )}{\sqrt {-c^2} d+e}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {1-\frac {e x^2}{\sqrt {-c^2} \left (d+\frac {e}{\sqrt {-c^2}}\right )}}} \, dx,x,\sqrt {1-\sqrt {-c^2} x}\right )}{c e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {4 b \sqrt {-c^2} \sqrt {\frac {d+e x}{d+\frac {e}{\sqrt {-c^2}}}} \sqrt {1+c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{c^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b d \sqrt {1+c^2 x^2} \sqrt {1-\frac {e \left (1-\sqrt {-c^2} x\right )}{\sqrt {-c^2} d+e}} \Pi \left (2;\sin ^{-1}\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}}\\ \end {align*}

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Mathematica [C]  time = 5.22, size = 307, normalized size = 1.08 \[ \frac {2 \left (a e (d+e x)-\frac {b \left (\frac {d}{x}+e\right ) \left (-c e x \text {csch}^{-1}(c x)+\frac {\sqrt {2} \sqrt {1+i c x} \left (c d (e+i c d) \sqrt {-\frac {e (c x+i)}{c d-i e}} \sqrt {\frac {c e (c x+i) (d+e x)}{(e+i c d)^2}} \Pi \left (\frac {i c d}{e}+1;\sin ^{-1}\left (\sqrt {-\frac {e (c x+i)}{c d-i e}}\right )|\frac {i c d+e}{2 e}\right )-e^2 (c x+i) \sqrt {\frac {c (d+e x)}{c d-i e}} F\left (\sin ^{-1}\left (\sqrt {-\frac {e (c x+i)}{c d-i e}}\right )|\frac {i c d+e}{2 e}\right )\right )}{\sqrt {\frac {1}{c^2 x^2}+1} \sqrt {-\frac {e (c x+i)}{c d-i e}} (c d+c e x)}\right )}{c}\right )}{e^2 \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCsch[c*x])/Sqrt[d + e*x],x]

[Out]

(2*(a*e*(d + e*x) - (b*(e + d/x)*(-(c*e*x*ArcCsch[c*x]) + (Sqrt[2]*Sqrt[1 + I*c*x]*(-(e^2*(I + c*x)*Sqrt[(c*(d
+ e*x))/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)]) + c*d*(I*c*d +
e)*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*Sqrt[(c*e*(I + c*x)*(d + e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e,
ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)]))/(Sqrt[1 + 1/(c^2*x^2)]*Sqrt[-((e*(I + c*x))/
(c*d - I*e))]*(c*d + c*e*x))))/c))/(e^2*Sqrt[d + e*x])

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arcsch}\left (c x\right ) + a}{\sqrt {e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arccsch(c*x) + a)/sqrt(e*x + d), x)

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maple [C]  time = 0.07, size = 395, normalized size = 1.39 \[ \frac {2 a \sqrt {e x +d}+2 b \left (\sqrt {e x +d}\, \mathrm {arccsch}\left (c x \right )+\frac {2 \sqrt {-\frac {i \left (e x +d \right ) c e +\left (e x +d \right ) c^{2} d -c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i \left (e x +d \right ) c e -\left (e x +d \right ) c^{2} d +c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \left (\EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right )-\EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )\right )}{c \sqrt {\frac {\left (e x +d \right )^{2} c^{2}-2 \left (e x +d \right ) c^{2} d +c^{2} d^{2}+e^{2}}{c^{2} x^{2} e^{2}}}\, x \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccsch(c*x))/(e*x+d)^(1/2),x)

[Out]

2/e*(a*(e*x+d)^(1/2)+b*((e*x+d)^(1/2)*arccsch(c*x)+2/c*(-(I*(e*x+d)*c*e+(e*x+d)*c^2*d-c^2*d^2-e^2)/(c^2*d^2+e^
2))^(1/2)*((I*(e*x+d)*c*e-(e*x+d)*c^2*d+c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2)*(EllipticF((e*x+d)^(1/2)*((I*e+c*d)*
c/(c^2*d^2+e^2))^(1/2),(-(2*I*c*d*e-c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2))-EllipticPi((e*x+d)^(1/2)*((I*e+c*d)*c/(
c^2*d^2+e^2))^(1/2),1/(I*e+c*d)/c*(c^2*d^2+e^2)/d,(-(I*e-c*d)*c/(c^2*d^2+e^2))^(1/2)/((I*e+c*d)*c/(c^2*d^2+e^2
))^(1/2)))/(((e*x+d)^2*c^2-2*(e*x+d)*c^2*d+c^2*d^2+e^2)/c^2/x^2/e^2)^(1/2)/x/((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)
))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ b {\left (\frac {2 \, \sqrt {e x + d} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{e} + \int \frac {2 \, {\left (c^{2} e x^{2} + c^{2} d x\right )}}{{\left (c^{2} e x^{2} + e\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {e x + d} + {\left (c^{2} e x^{2} + e\right )} \sqrt {e x + d}}\,{d x} - \int \frac {{\left (e \log \relax (c) + 2 \, e\right )} c^{2} x^{2} + 2 \, c^{2} d x + e \log \relax (c) + {\left (c^{2} e x^{2} + e\right )} \log \relax (x)}{{\left (c^{2} e x^{2} + e\right )} \sqrt {e x + d}}\,{d x}\right )} + \frac {2 \, \sqrt {e x + d} a}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

b*(2*sqrt(e*x + d)*log(sqrt(c^2*x^2 + 1) + 1)/e + integrate(2*(c^2*e*x^2 + c^2*d*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) - 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*x^2 + e)*sqrt(e*x + d)), x)) + 2*sqrt(e*x + d)*a/e

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{\sqrt {d+e\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asinh(1/(c*x)))/(d + e*x)^(1/2),x)

[Out]

int((a + b*asinh(1/(c*x)))/(d + e*x)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {acsch}{\left (c x \right )}}{\sqrt {d + e x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acsch(c*x))/(e*x+d)**(1/2),x)

[Out]

Integral((a + b*acsch(c*x))/sqrt(d + e*x), x)

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