\(\int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x)) \, dx\) [169]

Optimal result

Integrand size = 23, antiderivative size = 142 \[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x)) \, dx=-\frac {4 b \sqrt {e (c+d x)} \sqrt {1+(c+d x)^2}}{9 d}+\frac {2 (e (c+d x))^{3/2} (a+b \text {arcsinh}(c+d x))}{3 d e}+\frac {2 b \sqrt {e} (1+c+d x) \sqrt {\frac {1+(c+d x)^2}{(1+c+d x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),\frac {1}{2}\right )}{9 d \sqrt {1+(c+d x)^2}} \] Output:

-4/9*b*(e*(d*x+c))^(1/2)*(1+(d*x+c)^2)^(1/2)/d+2/3*(e*(d*x+c))^(3/2)*(a+b* 
arcsinh(d*x+c))/d/e+2/9*b*e^(1/2)*(d*x+c+1)*((1+(d*x+c)^2)/(d*x+c+1)^2)^(1 
/2)*InverseJacobiAM(2*arctan((e*(d*x+c))^(1/2)/e^(1/2)),1/2*2^(1/2))/d/(1+ 
(d*x+c)^2)^(1/2)
 

Mathematica [C] (verified)

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

Time = 0.02 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.61 \[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x)) \, dx=\frac {2 \sqrt {e (c+d x)} \left (3 a c+3 a d x-2 b \sqrt {1+(c+d x)^2}+3 b c \text {arcsinh}(c+d x)+3 b d x \text {arcsinh}(c+d x)+2 b \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-(c+d x)^2\right )\right )}{9 d} \] Input:

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

Output:

(2*Sqrt[e*(c + d*x)]*(3*a*c + 3*a*d*x - 2*b*Sqrt[1 + (c + d*x)^2] + 3*b*c* 
ArcSinh[c + d*x] + 3*b*d*x*ArcSinh[c + d*x] + 2*b*Hypergeometric2F1[1/4, 1 
/2, 5/4, -(c + d*x)^2]))/(9*d)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6274, 6191, 262, 266, 761}

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.

\(\displaystyle \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x)) \, dx\)

\(\Big \downarrow \) 6274

\(\displaystyle \frac {\int \sqrt {e (c+d x)} (a+b \text {arcsinh}(c+d x))d(c+d x)}{d}\)

\(\Big \downarrow \) 6191

\(\displaystyle \frac {\frac {2 (e (c+d x))^{3/2} (a+b \text {arcsinh}(c+d x))}{3 e}-\frac {2 b \int \frac {(e (c+d x))^{3/2}}{\sqrt {(c+d x)^2+1}}d(c+d x)}{3 e}}{d}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {\frac {2 (e (c+d x))^{3/2} (a+b \text {arcsinh}(c+d x))}{3 e}-\frac {2 b \left (\frac {2}{3} e \sqrt {(c+d x)^2+1} \sqrt {e (c+d x)}-\frac {1}{3} e^2 \int \frac {1}{\sqrt {e (c+d x)} \sqrt {(c+d x)^2+1}}d(c+d x)\right )}{3 e}}{d}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {\frac {2 (e (c+d x))^{3/2} (a+b \text {arcsinh}(c+d x))}{3 e}-\frac {2 b \left (\frac {2}{3} e \sqrt {(c+d x)^2+1} \sqrt {e (c+d x)}-\frac {2}{3} e \int \frac {1}{\sqrt {(c+d x)^2+1}}d\sqrt {e (c+d x)}\right )}{3 e}}{d}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {\frac {2 (e (c+d x))^{3/2} (a+b \text {arcsinh}(c+d x))}{3 e}-\frac {2 b \left (\frac {2}{3} e \sqrt {(c+d x)^2+1} \sqrt {e (c+d x)}-\frac {\sqrt {e} (e (c+d x)+e) \sqrt {\frac {e^2 (c+d x)^2+e^2}{(e (c+d x)+e)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),\frac {1}{2}\right )}{3 \sqrt {(c+d x)^2+1}}\right )}{3 e}}{d}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 761

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 6191

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
 :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[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 6274

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[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]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.47 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.26

method	result	size
derivativedivides	\(\frac {\frac {2 \left (d e x +c e \right )^{\frac {3}{2}} a}{3}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {3}{2}} \operatorname {arcsinh}\left (\frac {d e x +c e}{e}\right )}{3}-\frac {2 \left (\frac {e^{2} \sqrt {d e x +c e}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{3}-\frac {e^{2} \sqrt {1-\frac {i \left (d e x +c e \right )}{e}}\, \sqrt {1+\frac {i \left (d e x +c e \right )}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )}{3 \sqrt {\frac {i}{e}}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{3 e}\right )}{d e}\)	\(179\)
default	\(\frac {\frac {2 \left (d e x +c e \right )^{\frac {3}{2}} a}{3}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {3}{2}} \operatorname {arcsinh}\left (\frac {d e x +c e}{e}\right )}{3}-\frac {2 \left (\frac {e^{2} \sqrt {d e x +c e}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{3}-\frac {e^{2} \sqrt {1-\frac {i \left (d e x +c e \right )}{e}}\, \sqrt {1+\frac {i \left (d e x +c e \right )}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )}{3 \sqrt {\frac {i}{e}}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{3 e}\right )}{d e}\)	\(179\)
parts	\(\frac {2 a \left (d e x +c e \right )^{\frac {3}{2}}}{3 d e}+\frac {2 b \left (\frac {\left (d e x +c e \right )^{\frac {3}{2}} \operatorname {arcsinh}\left (\frac {d e x +c e}{e}\right )}{3}-\frac {2 \left (\frac {e^{2} \sqrt {d e x +c e}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{3}-\frac {e^{2} \sqrt {1-\frac {i \left (d e x +c e \right )}{e}}\, \sqrt {1+\frac {i \left (d e x +c e \right )}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )}{3 \sqrt {\frac {i}{e}}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{3 e}\right )}{d e}\)	\(184\)

Input:

int((d*e*x+c*e)^(1/2)*(a+b*arcsinh(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00 \[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x)) \, dx=-\frac {2 \, {\left (2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} \sqrt {d e x + c e} b d^{2} - 3 \, {\left (b d^{3} x + b c d^{2}\right )} \sqrt {d e x + c e} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 2 \, \sqrt {d^{3} e} b {\rm weierstrassPInverse}\left (-\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right ) - 3 \, {\left (a d^{3} x + a c d^{2}\right )} \sqrt {d e x + c e}\right )}}{9 \, d^{3}} \] Input:

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

Output:

-2/9*(2*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*sqrt(d*e*x + c*e)*b*d^2 - 3*(b*d 
^3*x + b*c*d^2)*sqrt(d*e*x + c*e)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c 
^2 + 1)) - 2*sqrt(d^3*e)*b*weierstrassPInverse(-4/d^2, 0, (d*x + c)/d) - 3 
*(a*d^3*x + a*c*d^2)*sqrt(d*e*x + c*e))/d^3
 

Sympy [F]

\[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x)) \, dx=\int \sqrt {e \left (c + d x\right )} \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )\, dx \] Input:

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x)) \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

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>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

\[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x)) \, dx=\int { \sqrt {d e x + c e} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x)) \, dx=\int \sqrt {c\,e+d\,e\,x}\,\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right ) \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x)) \, dx=\frac {\sqrt {e}\, \left (2 \sqrt {d x +c}\, a c +2 \sqrt {d x +c}\, a d x +3 \left (\int \sqrt {d x +c}\, \mathit {asinh} \left (d x +c \right )d x \right ) b d \right )}{3 d} \] Input:

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

Output:

(sqrt(e)*(2*sqrt(c + d*x)*a*c + 2*sqrt(c + d*x)*a*d*x + 3*int(sqrt(c + d*x 
)*asinh(c + d*x),x)*b*d))/(3*d)