3.58 \(\int \frac {x^2 (a+b \text {csch}^{-1}(c x))}{\sqrt {d+e x}} \, dx\)

Optimal. Leaf size=707 \[ \frac {2 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {32 b d^3 \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{15 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}+\frac {32 b c d^2 \sqrt {c^2 x^2+1} \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{15 \left (-c^2\right )^{3/2} e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}+\frac {4 b c \sqrt {c^2 x^2+1} \left (c^2 d^2+e^2\right ) \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{15 \left (-c^2\right )^{5/2} e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}-\frac {4 b c d \sqrt {c^2 x^2+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{5 \left (-c^2\right )^{3/2} e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {4 b \left (c^2 x^2+1\right ) \sqrt {d+e x}}{15 c^3 e x \sqrt {\frac {1}{c^2 x^2}+1}} \]

[Out]

-4/3*d*(e*x+d)^(3/2)*(a+b*arccsch(c*x))/e^3+2/5*(e*x+d)^(5/2)*(a+b*arccsch(c*x))/e^3+2*d^2*(a+b*arccsch(c*x))*
(e*x+d)^(1/2)/e^3+4/15*b*(c^2*x^2+1)*(e*x+d)^(1/2)/c^3/e/x/(1+1/c^2/x^2)^(1/2)-32/15*b*d^3*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^3/x/(1+1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)-4/5*b*c*d*EllipticE(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))*(e*x+d)^(1/2)*(c^2*x^2+1)^(1/2)/(-c^2)^(3/2)/e
^2/x/(1+1/c^2/x^2)^(1/2)/(c^2*(e*x+d)/(c^2*d-e*(-c^2)^(1/2)))^(1/2)+32/15*b*c*d^2*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)*(c^2*(e*x+d)/(c^2*d-e*(
-c^2)^(1/2)))^(1/2)/(-c^2)^(3/2)/e^2/x/(1+1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)+4/15*b*c*(c^2*d^2+e^2)*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)*(c^2*(e*x
+d)/(c^2*d-e*(-c^2)^(1/2)))^(1/2)/(-c^2)^(5/2)/e^2/x/(1+1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 2.14, antiderivative size = 707, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 15, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {43, 6310, 12, 6721, 6742, 719, 424, 944, 419, 932, 168, 538, 537, 833, 844} \[ \frac {2 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}+\frac {32 b c d^2 \sqrt {c^2 x^2+1} \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{15 \left (-c^2\right )^{3/2} e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}+\frac {4 b c \sqrt {c^2 x^2+1} \left (c^2 d^2+e^2\right ) \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{15 \left (-c^2\right )^{5/2} e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}-\frac {32 b d^3 \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{15 c e^3 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}-\frac {4 b c d \sqrt {c^2 x^2+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{5 \left (-c^2\right )^{3/2} e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {4 b \left (c^2 x^2+1\right ) \sqrt {d+e x}}{15 c^3 e x \sqrt {\frac {1}{c^2 x^2}+1}} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x],x]

[Out]

(4*b*Sqrt[d + e*x]*(1 + c^2*x^2))/(15*c^3*e*Sqrt[1 + 1/(c^2*x^2)]*x) + (2*d^2*Sqrt[d + e*x]*(a + b*ArcCsch[c*x
]))/e^3 - (4*d*(d + e*x)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e^3) + (2*(d + e*x)^(5/2)*(a + b*ArcCsch[c*x]))/(5*e^3
) - (4*b*c*d*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2]*EllipticE[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (-2*Sqrt[-c^2]*
e)/(c^2*d - Sqrt[-c^2]*e)])/(5*(-c^2)^(3/2)*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^
2]*e)]) + (32*b*c*d^2*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 -
Sqrt[-c^2]*x]/Sqrt[2]], (-2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e)])/(15*(-c^2)^(3/2)*e^2*Sqrt[1 + 1/(c^2*x^2)]
*x*Sqrt[d + e*x]) + (4*b*c*(c^2*d^2 + e^2)*Sqrt[(c^2*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*Elli
pticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (-2*Sqrt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e)])/(15*(-c^2)^(5/2)*e^2*
Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (32*b*d^3*Sqrt[(Sqrt[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^
2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (2*e)/(Sqrt[-c^2]*d + e)])/(15*c*e^3*Sqrt[1 + 1/(
c^2*x^2)]*x*Sqrt[d + e*x])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 424

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

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 833

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(g*(d + e*x)
^m*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p]) &&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 844

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

Rule 932

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[-(c
/a), 2]}, Dist[1/Sqrt[a], 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 6310

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*(u_), x_Symbol] :> With[{v = IntHide[u, x]}, Dist[a + b*ArcCsch[c*x],
v, x] + Dist[b/c, Int[SimplifyIntegrand[v/(x^2*Sqrt[1 + 1/(c^2*x^2)]), x], x], x] /; InverseFunctionFreeQ[v, x
]] /; FreeQ[{a, b, c}, x]

Rule 6721

Int[(u_.)*((a_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(b^IntPart[p]*(a + b*x^n)^FracPart[p])/(x^(n*FracP
art[p])*(1 + a/(x^n*b))^FracPart[p]), Int[u*x^(n*p)*(1 + a/(x^n*b))^p, x], x] /; FreeQ[{a, b, p}, x] &&  !Inte
gerQ[p] && ILtQ[n, 0] &&  !RationalFunctionQ[u, x] && IntegerQ[p + 1/2]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx &=\frac {2 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}+\frac {b \int \frac {2 \sqrt {d+e x} \left (8 d^2-4 d e x+3 e^2 x^2\right )}{15 e^3 \sqrt {1+\frac {1}{c^2 x^2}} x^2} \, dx}{c}\\ &=\frac {2 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}+\frac {(2 b) \int \frac {\sqrt {d+e x} \left (8 d^2-4 d e x+3 e^2 x^2\right )}{\sqrt {1+\frac {1}{c^2 x^2}} x^2} \, dx}{15 c e^3}\\ &=\frac {2 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {\sqrt {d+e x} \left (8 d^2-4 d e x+3 e^2 x^2\right )}{x \sqrt {1+c^2 x^2}} \, dx}{15 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=\frac {2 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \left (-\frac {4 d e \sqrt {d+e x}}{\sqrt {1+c^2 x^2}}+\frac {8 d^2 \sqrt {d+e x}}{x \sqrt {1+c^2 x^2}}+\frac {3 e^2 x \sqrt {d+e x}}{\sqrt {1+c^2 x^2}}\right ) \, dx}{15 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=\frac {2 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}+\frac {\left (16 b d^2 \sqrt {1+c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{x \sqrt {1+c^2 x^2}} \, dx}{15 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (8 b d \sqrt {1+c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1+c^2 x^2}} \, dx}{15 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {x \sqrt {d+e x}}{\sqrt {1+c^2 x^2}} \, dx}{5 c e \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=\frac {4 b \sqrt {d+e x} \left (1+c^2 x^2\right )}{15 c^3 e \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}+\frac {\left (16 b d^3 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{15 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (16 b d^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{15 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (4 b \sqrt {1+c^2 x^2}\right ) \int \frac {-\frac {e}{2}+\frac {1}{2} c^2 d x}{\sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{15 c^3 e \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (16 b \sqrt {-c^2} d \sqrt {d+e x} \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {-c^2} e x^2}{c^2 d-\sqrt {-c^2} e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )}{15 c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}\\ &=\frac {4 b \sqrt {d+e x} \left (1+c^2 x^2\right )}{15 c^3 e \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {16 b \sqrt {-c^2} d \sqrt {d+e x} \sqrt {1+c^2 x^2} E\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 )}{15 c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {\left (16 b d^3 \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}{15 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (2 b d \sqrt {1+c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1+c^2 x^2}} \, dx}{15 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (2 b \left (c^2 d^2+e^2\right ) \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{15 c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (32 b \sqrt {-c^2} d^2 \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \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 d-\sqrt {-c^2} e}}} \, dx,x,\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )}{15 c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=\frac {4 b \sqrt {d+e x} \left (1+c^2 x^2\right )}{15 c^3 e \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {16 b \sqrt {-c^2} d \sqrt {d+e x} \sqrt {1+c^2 x^2} E\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 )}{15 c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {32 b \sqrt {-c^2} d^2 \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \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 )}{15 c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (32 b d^3 \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 )}{15 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (4 b \sqrt {-c^2} d \sqrt {d+e x} \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {-c^2} e x^2}{c^2 d-\sqrt {-c^2} e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )}{15 c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}-\frac {\left (4 b \sqrt {-c^2} \left (c^2 d^2+e^2\right ) \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \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 d-\sqrt {-c^2} e}}} \, dx,x,\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )}{15 c^5 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=\frac {4 b \sqrt {d+e x} \left (1+c^2 x^2\right )}{15 c^3 e \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {4 b \sqrt {-c^2} d \sqrt {d+e x} \sqrt {1+c^2 x^2} E\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 )}{5 c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {32 b \sqrt {-c^2} d^2 \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \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 )}{15 c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b \sqrt {-c^2} \left (c^2 d^2+e^2\right ) \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \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 )}{15 c^5 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (32 b d^3 \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 )}{15 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=\frac {4 b \sqrt {d+e x} \left (1+c^2 x^2\right )}{15 c^3 e \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^3}-\frac {4 b \sqrt {-c^2} d \sqrt {d+e x} \sqrt {1+c^2 x^2} E\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 )}{5 c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {32 b \sqrt {-c^2} d^2 \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \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 )}{15 c^3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b \sqrt {-c^2} \left (c^2 d^2+e^2\right ) \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \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 )}{15 c^5 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {32 b d^3 \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 )}{15 c e^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ \end {align*}

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Mathematica [C]  time = 14.43, size = 1012, normalized size = 1.43 \[ \frac {b \left (-\frac {c \left (\frac {d}{x}+e\right ) x \left (-\frac {16 c^2 \text {csch}^{-1}(c x) d^2}{15 e^3}+\frac {4 c \sqrt {1+\frac {1}{c^2 x^2}} d}{5 e^2}-\frac {2 c^2 x^2 \text {csch}^{-1}(c x)}{5 e}-\frac {4 c x \left (e \sqrt {1+\frac {1}{c^2 x^2}}-2 c d \text {csch}^{-1}(c x)\right )}{15 e^2}\right )}{\sqrt {d+e x}}-\frac {2 \sqrt {\frac {d}{x}+e} \sqrt {c x} \left (-\frac {\sqrt {2} \left (7 c^2 d^2 e-e^3\right ) \sqrt {i c x+1} (c x+i) \sqrt {\frac {c d+c 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 )}{\sqrt {1+\frac {1}{c^2 x^2}} \sqrt {\frac {d}{x}+e} (c x)^{3/2} \sqrt {\frac {e (1-i c x)}{i c d+e}}}+\frac {i \sqrt {2} (c d-i e) \left (8 c^3 d^3-3 c d e^2\right ) \sqrt {i c x+1} \sqrt {\frac {e (c x+i) (c d+c e x)}{(i c d+e)^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 \sqrt {1+\frac {1}{c^2 x^2}} \sqrt {\frac {d}{x}+e} (c x)^{3/2}}+\frac {6 c d e \cosh \left (2 \text {csch}^{-1}(c x)\right ) \left (\frac {c x \left (c d \sqrt {2 i c x+2} (c x+i) \sqrt {\frac {c d+c 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 )+2 \sqrt {-\frac {e (c x-i)}{c d+i e}} (c x+i) \sqrt {\frac {c d+c e x}{c d-i e}} \left ((c d+i e) E\left (\sin ^{-1}\left (\sqrt {\frac {c d+c e x}{c d-i e}}\right )|\frac {c d-i e}{c d+i e}\right )-i e F\left (\sin ^{-1}\left (\sqrt {\frac {c d+c e x}{c d-i e}}\right )|\frac {c d-i e}{c d+i e}\right )\right )+(i c d+e) \sqrt {2 i c x+2} \sqrt {-\frac {e (c x+i)}{c d-i e}} \sqrt {\frac {e (c x+i) (c d+c e x)}{(i c d+e)^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 )\right )}{2 \sqrt {-\frac {e (c x+i)}{c d-i e}}}-(c d+c e x) \left (c^2 x^2+1\right )\right )}{\sqrt {1+\frac {1}{c^2 x^2}} \sqrt {\frac {d}{x}+e} \sqrt {c x} \left (c^2 x^2+2\right )}\right )}{15 e^3 \sqrt {d+e x}}\right )}{c^3}-\frac {a d^3 \sqrt {\frac {e x}{d}+1} B_{-\frac {e x}{d}}\left (3,\frac {1}{2}\right )}{e^3 \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x],x]

[Out]

-((a*d^3*Sqrt[1 + (e*x)/d]*Beta[-((e*x)/d), 3, 1/2])/(e^3*Sqrt[d + e*x])) + (b*(-((c*(e + d/x)*x*((4*c*d*Sqrt[
1 + 1/(c^2*x^2)])/(5*e^2) - (16*c^2*d^2*ArcCsch[c*x])/(15*e^3) - (2*c^2*x^2*ArcCsch[c*x])/(5*e) - (4*c*x*(e*Sq
rt[1 + 1/(c^2*x^2)] - 2*c*d*ArcCsch[c*x]))/(15*e^2)))/Sqrt[d + e*x]) - (2*Sqrt[e + d/x]*Sqrt[c*x]*(-((Sqrt[2]*
(7*c^2*d^2*e - e^3)*Sqrt[1 + I*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*EllipticF[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 + d/x]*(c*x)^(3/2)*Sqrt[(e*(1 - I*c*x)
)/(I*c*d + e)])) + (I*Sqrt[2]*(c*d - I*e)*(8*c^3*d^3 - 3*c*d*e^2)*Sqrt[1 + I*c*x]*Sqrt[(e*(I + c*x)*(c*d + c*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)])/
(e*Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) + (6*c*d*e*Cosh[2*ArcCsch[c*x]]*(-((c*d + c*e*x)*(1 + c^2*
x^2)) + (c*x*(c*d*Sqrt[2 + (2*I)*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[-((e*(I
+ c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)] + 2*Sqrt[-((e*(-I + c*x))/(c*d + I*e))]*(I + c*x)*Sqrt[(c*d + c*e*x
)/(c*d - I*e)]*((c*d + I*e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)] - I*e*
EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)]) + (I*c*d + e)*Sqrt[2 + (2*I)*c*x]
*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*Sqrt[(e*(I + c*x)*(c*d + c*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)]))/(2*Sqrt[-((e*(I + c*x))/(c*d - I*e))])))/(Sqr
t[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(2 + c^2*x^2))))/(15*e^3*Sqrt[d + e*x])))/c^3

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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(x^2*(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 \[ \int \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{2}}{\sqrt {e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.07, size = 1991, normalized size = 2.82 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e^3*(a*(1/5*(e*x+d)^(5/2)-2/3*d*(e*x+d)^(3/2)+d^2*(e*x+d)^(1/2))+b*(1/5*arccsch(c*x)*(e*x+d)^(5/2)-2/3*arccs
ch(c*x)*d*(e*x+d)^(3/2)+arccsch(c*x)*d^2*(e*x+d)^(1/2)+2/15/c^3*(I*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*(e*x+d)^(
1/2)*e^3-((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*(e*x+d)^(5/2)*c^3*d+I*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*(e*x+d)^(1/
2)*c^2*d^2*e-2*I*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*(e*x+d)^(3/2)*c^2*d*e-I*(-(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))*e^3+2*((I*e+c*d)*c/(
c^2*d^2+e^2))^(1/2)*(e*x+d)^(3/2)*c^3*d^2-4*(-(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))*c^3*d^3-3*(-(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)*EllipticE((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))*c^3*d^3+8*(-(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)*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))*c^3*d^3+7*I*(-(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))*c^2*d^2*e-((I*e+c*d)*c/(c^2*d^2
+e^2))^(1/2)*(e*x+d)^(1/2)*c^3*d^3-8*I*(-(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)*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))*c^2
*d^2*e+4*(-(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))*c*d*e^2-3*(-(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)*EllipticE((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))*c*d*e^2+I*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*(e*x+d)^(5/2)*c^2
*e-((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)*(e*x+d)^(1/2)*c*d*e^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)/(I*e-c*d)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2}{15} \, a {\left (\frac {3 \, {\left (e x + d\right )}^{\frac {5}{2}}}{e^{3}} - \frac {10 \, {\left (e x + d\right )}^{\frac {3}{2}} d}{e^{3}} + \frac {15 \, \sqrt {e x + d} d^{2}}{e^{3}}\right )} + \frac {1}{15} \, b {\left (\frac {2 \, {\left (3 \, e^{3} x^{3} - d e^{2} x^{2} + 4 \, d^{2} e x + 8 \, d^{3}\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{\sqrt {e x + d} e^{3}} + 15 \, \int \frac {2 \, {\left (3 \, c^{2} e^{3} x^{4} - c^{2} d e^{2} x^{3} + 4 \, c^{2} d^{2} e x^{2} + 8 \, c^{2} d^{3} x\right )}}{15 \, {\left ({\left (c^{2} e^{3} x^{2} + e^{3}\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {e x + d} + {\left (c^{2} e^{3} x^{2} + e^{3}\right )} \sqrt {e x + d}\right )}}\,{d x} - 15 \, \int -\frac {2 \, c^{2} d e^{2} x^{3} - 3 \, {\left (5 \, e^{3} \log \relax (c) + 2 \, e^{3}\right )} c^{2} x^{4} - 16 \, c^{2} d^{3} x - {\left (8 \, c^{2} d^{2} e + 15 \, e^{3} \log \relax (c)\right )} x^{2} - 15 \, {\left (c^{2} e^{3} x^{4} + e^{3} x^{2}\right )} \log \relax (x)}{15 \, {\left (c^{2} e^{3} x^{2} + e^{3}\right )} \sqrt {e x + d}}\,{d x}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*a*(3*(e*x + d)^(5/2)/e^3 - 10*(e*x + d)^(3/2)*d/e^3 + 15*sqrt(e*x + d)*d^2/e^3) + 1/15*b*(2*(3*e^3*x^3 -
d*e^2*x^2 + 4*d^2*e*x + 8*d^3)*log(sqrt(c^2*x^2 + 1) + 1)/(sqrt(e*x + d)*e^3) + 15*integrate(2/15*(3*c^2*e^3*x
^4 - c^2*d*e^2*x^3 + 4*c^2*d^2*e*x^2 + 8*c^2*d^3*x)/((c^2*e^3*x^2 + e^3)*sqrt(c^2*x^2 + 1)*sqrt(e*x + d) + (c^
2*e^3*x^2 + e^3)*sqrt(e*x + d)), x) - 15*integrate(-1/15*(2*c^2*d*e^2*x^3 - 3*(5*e^3*log(c) + 2*e^3)*c^2*x^4 -
16*c^2*d^3*x - (8*c^2*d^2*e + 15*e^3*log(c))*x^2 - 15*(c^2*e^3*x^4 + e^3*x^2)*log(x))/((c^2*e^3*x^2 + e^3)*sq
rt(e*x + d)), x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((x^2*(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 \[ \int \frac {x^{2} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\sqrt {d + e x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*(a + b*acsch(c*x))/sqrt(d + e*x), x)

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