3.16.31 \(\int \frac {1}{(-2 b+a x) \sqrt [4]{-b x^2+a x^3}} \, dx\) [1531]

3.16.31.1 Optimal result

Integrand size = 26, antiderivative size = 106 \[ \int \frac {1}{(-2 b+a x) \sqrt [4]{-b x^2+a x^3}} \, dx=\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b x^2+a x^3}}{\sqrt {a} x}\right )}{\sqrt {2} \sqrt {a} b^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b x^2+a x^3}}{\sqrt {a} x}\right )}{\sqrt {2} \sqrt {a} b^{3/4}} \]

output

1/2*arctan(2^(1/2)*b^(1/4)*(a*x^3-b*x^2)^(1/4)/a^(1/2)/x)*2^(1/2)/a^(1/2)/ 
b^(3/4)-1/2*arctanh(2^(1/2)*b^(1/4)*(a*x^3-b*x^2)^(1/4)/a^(1/2)/x)*2^(1/2) 
/a^(1/2)/b^(3/4)
 

3.16.31.2 Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(-2 b+a x) \sqrt [4]{-b x^2+a x^3}} \, dx=\frac {\sqrt {x} \sqrt [4]{-b+a x} \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x}}{\sqrt {a} \sqrt {x}}\right )-\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x}}{\sqrt {a} \sqrt {x}}\right )\right )}{\sqrt {2} \sqrt {a} b^{3/4} \sqrt [4]{x^2 (-b+a x)}} \]

input

Integrate[1/((-2*b + a*x)*(-(b*x^2) + a*x^3)^(1/4)),x]
 

output

(Sqrt[x]*(-b + a*x)^(1/4)*(ArcTan[(Sqrt[2]*b^(1/4)*(-b + a*x)^(1/4))/(Sqrt 
[a]*Sqrt[x])] - ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x)^(1/4))/(Sqrt[a]*Sqrt[x 
])]))/(Sqrt[2]*Sqrt[a]*b^(3/4)*(x^2*(-b + a*x))^(1/4))
 

3.16.31.3 Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.65 (sec) , antiderivative size = 347, normalized size of antiderivative = 3.27, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2467, 25, 117, 116, 25, 27, 993, 1535, 761, 2213, 216, 219}

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.

\(\displaystyle \int \frac {1}{(a x-2 b) \sqrt [4]{a x^3-b x^2}} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt {x} \sqrt [4]{a x-b} \int -\frac {1}{\sqrt {x} (2 b-a x) \sqrt [4]{a x-b}}dx}{\sqrt [4]{a x^3-b x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt {x} \sqrt [4]{a x-b} \int \frac {1}{\sqrt {x} (2 b-a x) \sqrt [4]{a x-b}}dx}{\sqrt [4]{a x^3-b x^2}}\)

\(\Big \downarrow \) 117

\(\displaystyle -\frac {\sqrt {\frac {a x}{b}} \sqrt [4]{a x-b} \int \frac {1}{\sqrt {\frac {a x}{b}} (2 b-a x) \sqrt [4]{a x-b}}dx}{\sqrt [4]{a x^3-b x^2}}\)

\(\Big \downarrow \) 116

\(\displaystyle \frac {4 \sqrt {\frac {a x}{b}} \sqrt [4]{a x-b} \int -\frac {\sqrt {a x-b}}{a (2 b-a x) \sqrt {\frac {a x-b}{b}+1}}d\sqrt [4]{a x-b}}{\sqrt [4]{a x^3-b x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {4 \sqrt {\frac {a x}{b}} \sqrt [4]{a x-b} \int \frac {\sqrt {a x-b}}{a (2 b-a x) \sqrt {\frac {a x-b}{b}+1}}d\sqrt [4]{a x-b}}{\sqrt [4]{a x^3-b x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {4 \sqrt {\frac {a x}{b}} \sqrt [4]{a x-b} \int \frac {\sqrt {a x-b}}{(2 b-a x) \sqrt {\frac {a x-b}{b}+1}}d\sqrt [4]{a x-b}}{a \sqrt [4]{a x^3-b x^2}}\)

\(\Big \downarrow \) 993

\(\displaystyle -\frac {4 \sqrt {\frac {a x}{b}} \sqrt [4]{a x-b} \left (\frac {1}{2} \int \frac {1}{\left (\sqrt {b}-\sqrt {a x-b}\right ) \sqrt {\frac {a x-b}{b}+1}}d\sqrt [4]{a x-b}-\frac {1}{2} \int \frac {1}{\left (\sqrt {b}+\sqrt {a x-b}\right ) \sqrt {\frac {a x-b}{b}+1}}d\sqrt [4]{a x-b}\right )}{a \sqrt [4]{a x^3-b x^2}}\)

\(\Big \downarrow \) 1535

\(\displaystyle -\frac {4 \sqrt {\frac {a x}{b}} \sqrt [4]{a x-b} \left (\frac {1}{2} \left (-\frac {\int \frac {1}{\sqrt {\frac {a x-b}{b}+1}}d\sqrt [4]{a x-b}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {b}-\sqrt {a x-b}}{\left (\sqrt {b}+\sqrt {a x-b}\right ) \sqrt {\frac {a x-b}{b}+1}}d\sqrt [4]{a x-b}}{2 \sqrt {b}}\right )+\frac {1}{2} \left (\frac {\int \frac {1}{\sqrt {\frac {a x-b}{b}+1}}d\sqrt [4]{a x-b}}{2 \sqrt {b}}+\frac {\int \frac {\sqrt {b}+\sqrt {a x-b}}{\left (\sqrt {b}-\sqrt {a x-b}\right ) \sqrt {\frac {a x-b}{b}+1}}d\sqrt [4]{a x-b}}{2 \sqrt {b}}\right )\right )}{a \sqrt [4]{a x^3-b x^2}}\)

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 2213

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 219

\(\displaystyle -\frac {4 \sqrt {\frac {a x}{b}} \sqrt [4]{a x-b} \left (\frac {1}{2} \left (\frac {\sqrt {\frac {a x}{\left (\sqrt {a x-b}+\sqrt {b}\right )^2}} \left (\sqrt {a x-b}+\sqrt {b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 b^{3/4} \sqrt {\frac {a x-b}{b}+1}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a x-b}}{\sqrt [4]{b} \sqrt {\frac {a x-b}{b}+1}}\right )}{2 \sqrt {2} \sqrt [4]{b}}\right )+\frac {1}{2} \left (-\frac {\sqrt {\frac {a x}{\left (\sqrt {a x-b}+\sqrt {b}\right )^2}} \left (\sqrt {a x-b}+\sqrt {b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a x-b}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{4 b^{3/4} \sqrt {\frac {a x-b}{b}+1}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x-b}}{\sqrt [4]{b} \sqrt {\frac {a x-b}{b}+1}}\right )}{2 \sqrt {2} \sqrt [4]{b}}\right )\right )}{a \sqrt [4]{a x^3-b x^2}}\)

input

Int[1/((-2*b + a*x)*(-(b*x^2) + a*x^3)^(1/4)),x]
 

output

(-4*Sqrt[(a*x)/b]*(-b + a*x)^(1/4)*((-1/2*ArcTan[(Sqrt[2]*(-b + a*x)^(1/4) 
)/(b^(1/4)*Sqrt[1 + (-b + a*x)/b])]/(Sqrt[2]*b^(1/4)) - (Sqrt[(a*x)/(Sqrt[ 
b] + Sqrt[-b + a*x])^2]*(Sqrt[b] + Sqrt[-b + a*x])*EllipticF[2*ArcTan[(-b 
+ a*x)^(1/4)/b^(1/4)], 1/2])/(4*b^(3/4)*Sqrt[1 + (-b + a*x)/b]))/2 + (ArcT 
anh[(Sqrt[2]*(-b + a*x)^(1/4))/(b^(1/4)*Sqrt[1 + (-b + a*x)/b])]/(2*Sqrt[2 
]*b^(1/4)) + (Sqrt[(a*x)/(Sqrt[b] + Sqrt[-b + a*x])^2]*(Sqrt[b] + Sqrt[-b 
+ a*x])*EllipticF[2*ArcTan[(-b + a*x)^(1/4)/b^(1/4)], 1/2])/(4*b^(3/4)*Sqr 
t[1 + (-b + a*x)/b]))/2))/(a*(-(b*x^2) + a*x^3)^(1/4))
 

3.16.31.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 116

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^( 
1/4)), x_] :> Simp[-4   Subst[Int[x^2/((b*e - a*f - b*x^4)*Sqrt[c - d*(e/f) 
 + d*(x^4/f)]), x], x, (e + f*x)^(1/4)], x] /; FreeQ[{a, b, c, d, e, f}, x] 
 && GtQ[-f/(d*e - c*f), 0]
 

rule 117

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^( 
1/4)), x_] :> Simp[Sqrt[(-f)*((c + d*x)/(d*e - c*f))]/Sqrt[c + d*x]   Int[1 
/((a + b*x)*Sqrt[(-c)*(f/(d*e - c*f)) - d*f*(x/(d*e - c*f))]*(e + f*x)^(1/4 
)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !GtQ[-f/(d*e - c*f), 0]
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

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 993

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> 
With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2* 
b)   Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Simp[s/(2*b)   Int[1/((r 
 - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
a*d, 0]
 

rule 1535

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

rule 2213

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

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

3.16.31.4 Maple [F]

\[\int \frac {1}{\left (a x -2 b \right ) \left (a \,x^{3}-b \,x^{2}\right )^{\frac {1}{4}}}d x\]

input

int(1/(a*x-2*b)/(a*x^3-b*x^2)^(1/4),x)
 

output

int(1/(a*x-2*b)/(a*x^3-b*x^2)^(1/4),x)
 

3.16.31.5 Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(-2 b+a x) \sqrt [4]{-b x^2+a x^3}} \, dx=\text {Timed out} \]

input

integrate(1/(a*x-2*b)/(a*x^3-b*x^2)^(1/4),x, algorithm="fricas")
 

output

Timed out
 

3.16.31.6 Sympy [F]

\[ \int \frac {1}{(-2 b+a x) \sqrt [4]{-b x^2+a x^3}} \, dx=\int \frac {1}{\sqrt [4]{x^{2} \left (a x - b\right )} \left (a x - 2 b\right )}\, dx \]

input

integrate(1/(a*x-2*b)/(a*x**3-b*x**2)**(1/4),x)
 

output

Integral(1/((x**2*(a*x - b))**(1/4)*(a*x - 2*b)), x)
 

3.16.31.7 Maxima [F]

\[ \int \frac {1}{(-2 b+a x) \sqrt [4]{-b x^2+a x^3}} \, dx=\int { \frac {1}{{\left (a x^{3} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x - 2 \, b\right )}} \,d x } \]

input

integrate(1/(a*x-2*b)/(a*x^3-b*x^2)^(1/4),x, algorithm="maxima")
 

output

integrate(1/((a*x^3 - b*x^2)^(1/4)*(a*x - 2*b)), x)
 

3.16.31.8 Giac [F]

\[ \int \frac {1}{(-2 b+a x) \sqrt [4]{-b x^2+a x^3}} \, dx=\int { \frac {1}{{\left (a x^{3} - b x^{2}\right )}^{\frac {1}{4}} {\left (a x - 2 \, b\right )}} \,d x } \]

input

integrate(1/(a*x-2*b)/(a*x^3-b*x^2)^(1/4),x, algorithm="giac")
 

output

integrate(1/((a*x^3 - b*x^2)^(1/4)*(a*x - 2*b)), x)
 

3.16.31.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(-2 b+a x) \sqrt [4]{-b x^2+a x^3}} \, dx=-\int \frac {1}{\left (2\,b-a\,x\right )\,{\left (a\,x^3-b\,x^2\right )}^{1/4}} \,d x \]

input

int(-1/((2*b - a*x)*(a*x^3 - b*x^2)^(1/4)),x)
 

output

-int(1/((2*b - a*x)*(a*x^3 - b*x^2)^(1/4)), x)