3.21.96 \(\int \frac {1}{(2 b+a x) \sqrt [4]{b x^2+a x^3}} \, dx\) [2096]

3.21.96.1 Optimal result

Integrand size = 25, antiderivative size = 152 \[ \int \frac {1}{(2 b+a x) \sqrt [4]{b x^2+a x^3}} \, dx=-\frac {\arctan \left (\frac {-\frac {\sqrt {a} x^2}{2 \sqrt [4]{b}}+\frac {\sqrt [4]{b} \sqrt {b x^2+a x^3}}{\sqrt {a}}}{x \sqrt [4]{b x^2+a x^3}}\right )}{2 \sqrt {a} b^{3/4}}+\frac {\text {arctanh}\left (\frac {2 \sqrt {a} \sqrt [4]{b} x \sqrt [4]{b x^2+a x^3}}{a x^2+2 \sqrt {b} \sqrt {b x^2+a x^3}}\right )}{2 \sqrt {a} b^{3/4}} \]

output

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

3.21.96.2 Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(2 b+a x) \sqrt [4]{b x^2+a x^3}} \, dx=\frac {\sqrt {x} \sqrt [4]{b+a x} \left (-\arctan \left (\frac {-a x+2 \sqrt {b} \sqrt {b+a x}}{2 \sqrt {a} \sqrt [4]{b} \sqrt {x} \sqrt [4]{b+a x}}\right )+\text {arctanh}\left (\frac {2 \sqrt {a} \sqrt [4]{b} \sqrt {x} \sqrt [4]{b+a x}}{a x+2 \sqrt {b} \sqrt {b+a x}}\right )\right )}{2 \sqrt {a} b^{3/4} \sqrt [4]{x^2 (b+a x)}} \]

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

3.21.96.3 Rubi [C] (verified)

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

Time = 0.63 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.51, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {2467, 117, 116, 25, 27, 993, 1535, 762, 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

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

\(\Big \downarrow \) 117

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

\(\Big \downarrow \) 116

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 993

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

\(\Big \downarrow \) 1535

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 2213

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 219

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

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*ArcTanh[(Sqrt[2]*(b + a*x)^(1/4 
))/((-b)^(1/4)*Sqrt[1 - (b + a*x)/b])]/(Sqrt[2]*(-b)^(1/4)) - (b^(1/4)*Ell 
ipticF[ArcSin[(b + a*x)^(1/4)/b^(1/4)], -1])/(2*Sqrt[-b]))/2 + (ArcTan[(Sq 
rt[2]*(b + a*x)^(1/4))/((-b)^(1/4)*Sqrt[1 - (b + a*x)/b])]/(2*Sqrt[2]*(-b) 
^(1/4)) + (b^(1/4)*EllipticF[ArcSin[(b + a*x)^(1/4)/b^(1/4)], -1])/(2*Sqrt 
[-b]))/2))/(a*(b*x^2 + a*x^3)^(1/4))
 

3.21.96.3.1 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 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 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]
 

3.21.96.4 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)
 

3.21.96.5 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
 

3.21.96.6 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)
 

3.21.96.7 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)
 

3.21.96.8 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)
 

3.21.96.9 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)