\(\int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^{5/2}} \, dx\) [234]

Optimal result

Integrand size = 23, antiderivative size = 266 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^{5/2}} \, dx=-\frac {4 b \sqrt {1+(c+d x)^2}}{3 d e^2 \sqrt {e (c+d x)}}+\frac {4 b \sqrt {e (c+d x)} \sqrt {1+(c+d x)^2}}{3 d e^3 (1+c+d x)}-\frac {2 (a+b \text {arcsinh}(c+d x))}{3 d e (e (c+d x))^{3/2}}-\frac {4 b (1+c+d x) \sqrt {\frac {1+(c+d x)^2}{(1+c+d x)^2}} E\left (2 \arctan \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )|\frac {1}{2}\right )}{3 d e^{5/2} \sqrt {1+(c+d x)^2}}+\frac {2 b (1+c+d x) \sqrt {\frac {1+(c+d x)^2}{(1+c+d x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),\frac {1}{2}\right )}{3 d e^{5/2} \sqrt {1+(c+d x)^2}} \]

[Out]

-2/3*(a+b*arcsinh(d*x+c))/d/e/(e*(d*x+c))^(3/2)-4/3*b*(1+(d*x+c)^2)^(1/2)/d/e^2/(e*(d*x+c))^(1/2)+4/3*b*(e*(d*
x+c))^(1/2)*(1+(d*x+c)^2)^(1/2)/d/e^3/(d*x+c+1)-4/3*b*(d*x+c+1)*(cos(2*arctan((e*(d*x+c))^(1/2)/e^(1/2)))^2)^(
1/2)/cos(2*arctan((e*(d*x+c))^(1/2)/e^(1/2)))*EllipticE(sin(2*arctan((e*(d*x+c))^(1/2)/e^(1/2))),1/2*2^(1/2))*
((1+(d*x+c)^2)/(d*x+c+1)^2)^(1/2)/d/e^(5/2)/(1+(d*x+c)^2)^(1/2)+2/3*b*(d*x+c+1)*(cos(2*arctan((e*(d*x+c))^(1/2
)/e^(1/2)))^2)^(1/2)/cos(2*arctan((e*(d*x+c))^(1/2)/e^(1/2)))*EllipticF(sin(2*arctan((e*(d*x+c))^(1/2)/e^(1/2)
)),1/2*2^(1/2))*((1+(d*x+c)^2)/(d*x+c+1)^2)^(1/2)/d/e^(5/2)/(1+(d*x+c)^2)^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5859, 5776, 331, 335, 311, 226, 1210} \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^{5/2}} \, dx=-\frac {2 (a+b \text {arcsinh}(c+d x))}{3 d e (e (c+d x))^{3/2}}+\frac {2 b (c+d x+1) \sqrt {\frac {(c+d x)^2+1}{(c+d x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),\frac {1}{2}\right )}{3 d e^{5/2} \sqrt {(c+d x)^2+1}}-\frac {4 b (c+d x+1) \sqrt {\frac {(c+d x)^2+1}{(c+d x+1)^2}} E\left (2 \arctan \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )|\frac {1}{2}\right )}{3 d e^{5/2} \sqrt {(c+d x)^2+1}}+\frac {4 b \sqrt {(c+d x)^2+1} \sqrt {e (c+d x)}}{3 d e^3 (c+d x+1)}-\frac {4 b \sqrt {(c+d x)^2+1}}{3 d e^2 \sqrt {e (c+d x)}} \]

[In]

Int[(a + b*ArcSinh[c + d*x])/(c*e + d*e*x)^(5/2),x]

[Out]

(-4*b*Sqrt[1 + (c + d*x)^2])/(3*d*e^2*Sqrt[e*(c + d*x)]) + (4*b*Sqrt[e*(c + d*x)]*Sqrt[1 + (c + d*x)^2])/(3*d*
e^3*(1 + c + d*x)) - (2*(a + b*ArcSinh[c + d*x]))/(3*d*e*(e*(c + d*x))^(3/2)) - (4*b*(1 + c + d*x)*Sqrt[(1 + (
c + d*x)^2)/(1 + c + d*x)^2]*EllipticE[2*ArcTan[Sqrt[e*(c + d*x)]/Sqrt[e]], 1/2])/(3*d*e^(5/2)*Sqrt[1 + (c + d
*x)^2]) + (2*b*(1 + c + d*x)*Sqrt[(1 + (c + d*x)^2)/(1 + c + d*x)^2]*EllipticF[2*ArcTan[Sqrt[e*(c + d*x)]/Sqrt
[e]], 1/2])/(3*d*e^(5/2)*Sqrt[1 + (c + d*x)^2])

Rule 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 311

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^4], x],
x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 1210

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a +
c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rule 5776

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcS
inh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[
1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5859

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a+b \text {arcsinh}(x)}{(e x)^{5/2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {2 (a+b \text {arcsinh}(c+d x))}{3 d e (e (c+d x))^{3/2}}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{(e x)^{3/2} \sqrt {1+x^2}} \, dx,x,c+d x\right )}{3 d e} \\ & = -\frac {4 b \sqrt {1+(c+d x)^2}}{3 d e^2 \sqrt {e (c+d x)}}-\frac {2 (a+b \text {arcsinh}(c+d x))}{3 d e (e (c+d x))^{3/2}}+\frac {(2 b) \text {Subst}\left (\int \frac {\sqrt {e x}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{3 d e^3} \\ & = -\frac {4 b \sqrt {1+(c+d x)^2}}{3 d e^2 \sqrt {e (c+d x)}}-\frac {2 (a+b \text {arcsinh}(c+d x))}{3 d e (e (c+d x))^{3/2}}+\frac {(4 b) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+\frac {x^4}{e^2}}} \, dx,x,\sqrt {e (c+d x)}\right )}{3 d e^4} \\ & = -\frac {4 b \sqrt {1+(c+d x)^2}}{3 d e^2 \sqrt {e (c+d x)}}-\frac {2 (a+b \text {arcsinh}(c+d x))}{3 d e (e (c+d x))^{3/2}}+\frac {(4 b) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{e^2}}} \, dx,x,\sqrt {e (c+d x)}\right )}{3 d e^3}-\frac {(4 b) \text {Subst}\left (\int \frac {1-\frac {x^2}{e}}{\sqrt {1+\frac {x^4}{e^2}}} \, dx,x,\sqrt {e (c+d x)}\right )}{3 d e^3} \\ & = -\frac {4 b \sqrt {1+(c+d x)^2}}{3 d e^2 \sqrt {e (c+d x)}}+\frac {4 b \sqrt {e (c+d x)} \sqrt {1+(c+d x)^2}}{3 d e^3 (1+c+d x)}-\frac {2 (a+b \text {arcsinh}(c+d x))}{3 d e (e (c+d x))^{3/2}}-\frac {4 b (1+c+d x) \sqrt {\frac {1+(c+d x)^2}{(1+c+d x)^2}} E\left (2 \arctan \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )|\frac {1}{2}\right )}{3 d e^{5/2} \sqrt {1+(c+d x)^2}}+\frac {2 b (1+c+d x) \sqrt {\frac {1+(c+d x)^2}{(1+c+d x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),\frac {1}{2}\right )}{3 d e^{5/2} \sqrt {1+(c+d x)^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.22 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^{5/2}} \, dx=-\frac {2 \left (a+b \text {arcsinh}(c+d x)+2 b (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-(c+d x)^2\right )\right )}{3 d e (e (c+d x))^{3/2}} \]

[In]

Integrate[(a + b*ArcSinh[c + d*x])/(c*e + d*e*x)^(5/2),x]

[Out]

(-2*(a + b*ArcSinh[c + d*x] + 2*b*(c + d*x)*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c + d*x)^2]))/(3*d*e*(e*(c + d
*x))^(3/2))

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.51 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.76

method	result	size
derivativedivides	\(\frac {-\frac {2 a}{3 \left (d e x +c e \right )^{\frac {3}{2}}}+2 b \left (-\frac {\operatorname {arcsinh}\left (\frac {d e x +c e}{e}\right )}{3 \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {-\frac {2 \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{3 \sqrt {d e x +c e}}+\frac {2 i \sqrt {1-\frac {i \left (d e x +c e \right )}{e}}\, \sqrt {1+\frac {i \left (d e x +c e \right )}{e}}\, \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )\right )}{3 e \sqrt {\frac {i}{e}}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{d e}\)	\(202\)
default	\(\frac {-\frac {2 a}{3 \left (d e x +c e \right )^{\frac {3}{2}}}+2 b \left (-\frac {\operatorname {arcsinh}\left (\frac {d e x +c e}{e}\right )}{3 \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {-\frac {2 \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{3 \sqrt {d e x +c e}}+\frac {2 i \sqrt {1-\frac {i \left (d e x +c e \right )}{e}}\, \sqrt {1+\frac {i \left (d e x +c e \right )}{e}}\, \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )\right )}{3 e \sqrt {\frac {i}{e}}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{d e}\)	\(202\)
parts	\(-\frac {2 a}{3 \left (d e x +c e \right )^{\frac {3}{2}} d e}+\frac {2 b \left (-\frac {\operatorname {arcsinh}\left (\frac {d e x +c e}{e}\right )}{3 \left (d e x +c e \right )^{\frac {3}{2}}}+\frac {-\frac {2 \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{3 \sqrt {d e x +c e}}+\frac {2 i \sqrt {1-\frac {i \left (d e x +c e \right )}{e}}\, \sqrt {1+\frac {i \left (d e x +c e \right )}{e}}\, \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )\right )}{3 e \sqrt {\frac {i}{e}}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}}{e}\right )}{d e}\)	\(207\)

[In]

int((a+b*arcsinh(d*x+c))/(d*e*x+c*e)^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/d/e*(-1/3*a/(d*e*x+c*e)^(3/2)+b*(-1/3/(d*e*x+c*e)^(3/2)*arcsinh(1/e*(d*e*x+c*e))+2/3/e*(-(1/e^2*(d*e*x+c*e)^
2+1)^(1/2)/(d*e*x+c*e)^(1/2)+I/e/(I/e)^(1/2)*(1-I/e*(d*e*x+c*e))^(1/2)*(1+I/e*(d*e*x+c*e))^(1/2)/(1/e^2*(d*e*x
+c*e)^2+1)^(1/2)*(EllipticF((d*e*x+c*e)^(1/2)*(I/e)^(1/2),I)-EllipticE((d*e*x+c*e)^(1/2)*(I/e)^(1/2),I)))))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.10 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.68 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^{5/2}} \, dx=-\frac {2 \, {\left (\sqrt {d e x + c e} b d \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) + \sqrt {d e x + c e} a d + 2 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \sqrt {d^{3} e} {\rm weierstrassZeta}\left (-\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (-\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right ) + 2 \, {\left (b d^{2} x + b c d\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} \sqrt {d e x + c e}\right )}}{3 \, {\left (d^{4} e^{3} x^{2} + 2 \, c d^{3} e^{3} x + c^{2} d^{2} e^{3}\right )}} \]

[In]

integrate((a+b*arcsinh(d*x+c))/(d*e*x+c*e)^(5/2),x, algorithm="fricas")

[Out]

-2/3*(sqrt(d*e*x + c*e)*b*d*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) + sqrt(d*e*x + c*e)*a*d + 2*(b*d^
2*x^2 + 2*b*c*d*x + b*c^2)*sqrt(d^3*e)*weierstrassZeta(-4/d^2, 0, weierstrassPInverse(-4/d^2, 0, (d*x + c)/d))
+ 2*(b*d^2*x + b*c*d)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*sqrt(d*e*x + c*e))/(d^4*e^3*x^2 + 2*c*d^3*e^3*x + c^2
*d^2*e^3)

Sympy [F]

\[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^{5/2}} \, dx=\int \frac {a + b \operatorname {asinh}{\left (c + d x \right )}}{\left (e \left (c + d x\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate((a+b*asinh(d*x+c))/(d*e*x+c*e)**(5/2),x)

[Out]

Integral((a + b*asinh(c + d*x))/(e*(c + d*x))**(5/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*arcsinh(d*x+c))/(d*e*x+c*e)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^{5/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((a+b*arcsinh(d*x+c))/(d*e*x+c*e)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arcsinh(d*x + c) + a)/(d*e*x + c*e)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^{5/2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^{5/2}} \,d x \]

[In]

int((a + b*asinh(c + d*x))/(c*e + d*e*x)^(5/2),x)

[Out]

int((a + b*asinh(c + d*x))/(c*e + d*e*x)^(5/2), x)