\(\int \frac {1}{\sqrt [4]{a x^2+b x^3}} \, dx\) [122]

Optimal result

Integrand size = 15, antiderivative size = 87 \[ \int \frac {1}{\sqrt [4]{a x^2+b x^3}} \, dx=\frac {4 x}{\sqrt [4]{a x^2+b x^3}}-\frac {4 \sqrt {a} \sqrt {x} \sqrt [4]{\frac {a+b x}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b} \sqrt [4]{a x^2+b x^3}} \] Output:

4*x/(b*x^3+a*x^2)^(1/4)-4*a^(1/2)*x^(1/2)*((b*x+a)/a)^(1/4)*EllipticE(sin( 
1/2*arctan(b^(1/2)*x^(1/2)/a^(1/2))),2^(1/2))/b^(1/2)/(b*x^3+a*x^2)^(1/4)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

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

Time = 10.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.52 \[ \int \frac {1}{\sqrt [4]{a x^2+b x^3}} \, dx=\frac {2 x \sqrt [4]{1+\frac {b x}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},-\frac {b x}{a}\right )}{\sqrt [4]{x^2 (a+b x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.79, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {1917, 73, 836, 27, 765, 762, 1390, 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.

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

\(\Big \downarrow \) 1917

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 836

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 1390

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

\(\Big \downarrow \) 1389

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

(4*Sqrt[x]*(a + b*x)^(1/4)*((a^(3/4)*Sqrt[1 - (a + b*x)/a]*EllipticE[ArcSi 
n[(a + b*x)^(1/4)/a^(1/4)], -1])/Sqrt[-(a/b) + (a + b*x)/b] - (a^(3/4)*Sqr 
t[1 - (a + b*x)/a]*EllipticF[ArcSin[(a + b*x)^(1/4)/a^(1/4)], -1])/Sqrt[-( 
a/b) + (a + b*x)/b]))/(b*(a*x^2 + b*x^3)^(1/4))
 

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, 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 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], 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 1390

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

rule 1917

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + 
b*x^n)^FracPart[p]/(x^(j*FracPart[p])*(a + b*x^(n - j))^FracPart[p])   Int[ 
x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] &&  !Integ 
erQ[p] && NeQ[n, j] && PosQ[n - j]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int \frac {1}{\sqrt [4]{a x^2+b x^3}} \, dx=\int \frac {1}{\sqrt [4]{a x^{2} + b x^{3}}}\, dx \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 9.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.44 \[ \int \frac {1}{\sqrt [4]{a x^2+b x^3}} \, dx=\frac {2\,x\,{\left (\frac {b\,x}{a}+1\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{2};\ \frac {3}{2};\ -\frac {b\,x}{a}\right )}{{\left (b\,x^3+a\,x^2\right )}^{1/4}} \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

(4*sqrt(x)*(a + b*x)**(1/4) + sqrt(a + b*x)*int((sqrt(x)*(a + b*x)**(3/4)) 
/(a**2*x + 2*a*b*x**2 + b**2*x**3),x)*a)/(3*sqrt(a + b*x))