\(\int \frac {a+b \arccos (c x)}{\sqrt {d x}} \, dx\) [206]

Optimal result

Integrand size = 16, antiderivative size = 89 \[ \int \frac {a+b \arccos (c x)}{\sqrt {d x}} \, dx=\frac {2 \sqrt {d x} (a+b \arccos (c x))}{d}+\frac {4 b E\left (\left .\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {c} \sqrt {d}}-\frac {4 b \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{\sqrt {c} \sqrt {d}} \] Output:

2*(d*x)^(1/2)*(a+b*arccos(c*x))/d+4*b*EllipticE(c^(1/2)*(d*x)^(1/2)/d^(1/2 
),I)/c^(1/2)/d^(1/2)-4*b*EllipticF(c^(1/2)*(d*x)^(1/2)/d^(1/2),I)/c^(1/2)/ 
d^(1/2)
 

Mathematica [C] (verified)

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

Time = 0.03 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.51 \[ \int \frac {a+b \arccos (c x)}{\sqrt {d x}} \, dx=\frac {2 x \left (3 (a+b \arccos (c x))+2 b c x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},c^2 x^2\right )\right )}{3 \sqrt {d x}} \] Input:

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

Output:

(2*x*(3*(a + b*ArcCos[c*x]) + 2*b*c*x*Hypergeometric2F1[1/2, 3/4, 7/4, c^2 
*x^2]))/(3*Sqrt[d*x])
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5139, 266, 836, 27, 762, 1389, 327}

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.

\(\displaystyle \int \frac {a+b \arccos (c x)}{\sqrt {d x}} \, dx\)

\(\Big \downarrow \) 5139

\(\displaystyle \frac {2 b c \int \frac {\sqrt {d x}}{\sqrt {1-c^2 x^2}}dx}{d}+\frac {2 \sqrt {d x} (a+b \arccos (c x))}{d}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {4 b c \int \frac {d x}{\sqrt {1-c^2 x^2}}d\sqrt {d x}}{d^2}+\frac {2 \sqrt {d x} (a+b \arccos (c x))}{d}\)

\(\Big \downarrow \) 836

\(\displaystyle \frac {4 b c \left (\frac {d \int \frac {c x d+d}{d \sqrt {1-c^2 x^2}}d\sqrt {d x}}{c}-\frac {d \int \frac {1}{\sqrt {1-c^2 x^2}}d\sqrt {d x}}{c}\right )}{d^2}+\frac {2 \sqrt {d x} (a+b \arccos (c x))}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4 b c \left (\frac {\int \frac {c x d+d}{\sqrt {1-c^2 x^2}}d\sqrt {d x}}{c}-\frac {d \int \frac {1}{\sqrt {1-c^2 x^2}}d\sqrt {d x}}{c}\right )}{d^2}+\frac {2 \sqrt {d x} (a+b \arccos (c x))}{d}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {4 b c \left (\frac {\int \frac {c x d+d}{\sqrt {1-c^2 x^2}}d\sqrt {d x}}{c}-\frac {d^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{c^{3/2}}\right )}{d^2}+\frac {2 \sqrt {d x} (a+b \arccos (c x))}{d}\)

\(\Big \downarrow \) 1389

\(\displaystyle \frac {4 b c \left (\frac {d \int \frac {\sqrt {c x+1}}{\sqrt {1-c x}}d\sqrt {d x}}{c}-\frac {d^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{c^{3/2}}\right )}{d^2}+\frac {2 \sqrt {d x} (a+b \arccos (c x))}{d}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2 \sqrt {d x} (a+b \arccos (c x))}{d}+\frac {4 b c \left (\frac {d^{3/2} E\left (\left .\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{c^{3/2}}-\frac {d^{3/2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right ),-1\right )}{c^{3/2}}\right )}{d^2}\)

Input:

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

Output:

(2*Sqrt[d*x]*(a + b*ArcCos[c*x]))/d + (4*b*c*((d^(3/2)*EllipticE[ArcSin[(S 
qrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/c^(3/2) - (d^(3/2)*EllipticF[ArcSin[(Sqrt 
[c]*Sqrt[d*x])/Sqrt[d]], -1])/c^(3/2)))/d^2
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 327

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

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 836

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

rule 1389

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq 
rt[a]   Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, 
 d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
 

rule 5139

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

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.10

method	result	size
derivativedivides	\(\frac {2 \sqrt {d x}\, a +2 b \left (\sqrt {d x}\, \arccos \left (c x \right )-\frac {2 \sqrt {-c x +1}\, \sqrt {c x +1}\, \left (\operatorname {EllipticF}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )\right )}{\sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{d}\)	\(98\)
default	\(\frac {2 \sqrt {d x}\, a +2 b \left (\sqrt {d x}\, \arccos \left (c x \right )-\frac {2 \sqrt {-c x +1}\, \sqrt {c x +1}\, \left (\operatorname {EllipticF}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )\right )}{\sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{d}\)	\(98\)
parts	\(\frac {2 a \sqrt {d x}}{d}+\frac {2 b \left (\sqrt {d x}\, \arccos \left (c x \right )-\frac {2 \sqrt {-c x +1}\, \sqrt {c x +1}\, \left (\operatorname {EllipticF}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )\right )}{\sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{d}\)	\(101\)

Input:

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

Output:

2/d*((d*x)^(1/2)*a+b*((d*x)^(1/2)*arccos(c*x)-2/(c/d)^(1/2)*(-c*x+1)^(1/2) 
*(c*x+1)^(1/2)/(-c^2*x^2+1)^(1/2)*(EllipticF((d*x)^(1/2)*(c/d)^(1/2),I)-El 
lipticE((d*x)^(1/2)*(c/d)^(1/2),I))))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.58 \[ \int \frac {a+b \arccos (c x)}{\sqrt {d x}} \, dx=\frac {2 \, {\left (2 \, \sqrt {-c^{2} d} b {\rm weierstrassZeta}\left (\frac {4}{c^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{c^{2}}, 0, x\right )\right ) + {\left (b c \arccos \left (c x\right ) + a c\right )} \sqrt {d x}\right )}}{c d} \] Input:

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

Output:

2*(2*sqrt(-c^2*d)*b*weierstrassZeta(4/c^2, 0, weierstrassPInverse(4/c^2, 0 
, x)) + (b*c*arccos(c*x) + a*c)*sqrt(d*x))/(c*d)
 

Sympy [F(-2)]

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

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

Output:

Exception raised: TypeError >> Invalid comparison of non-real zoo
 

Maxima [F]

\[ \int \frac {a+b \arccos (c x)}{\sqrt {d x}} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{\sqrt {d x}} \,d x } \] Input:

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

Output:

(2*b*c*sqrt(d)*sqrt(x)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) - (2*b*c 
^2*d*integrate(sqrt(c*x + 1)*sqrt(-c*x + 1)*sqrt(x)/(c^2*d*x^2 - d), x) + 
4*b*c*sqrt(x) - (2*b*arctan(sqrt(c)*sqrt(x)) - b*log((c*x - 1)/(c*x + 2*sq 
rt(c)*sqrt(x) + 1)))*sqrt(c))*sqrt(d))/(c*d)
 

Giac [F]

\[ \int \frac {a+b \arccos (c x)}{\sqrt {d x}} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{\sqrt {d x}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \arccos (c x)}{\sqrt {d x}} \, dx=\int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{\sqrt {d\,x}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \arccos (c x)}{\sqrt {d x}} \, dx=\frac {2 \sqrt {x}\, a +\left (\int \frac {\mathit {acos} \left (c x \right )}{\sqrt {x}}d x \right ) b}{\sqrt {d}} \] Input:

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

Output:

(2*sqrt(x)*a + int(acos(c*x)/sqrt(x),x)*b)/sqrt(d)