\(\int \frac {1}{x^2 \sqrt [4]{a x^2+b x^3}} \, dx\) [391]

Optimal result

Integrand size = 19, antiderivative size = 113 \[ \int \frac {1}{x^2 \sqrt [4]{a x^2+b x^3}} \, dx=\frac {b}{3 a \sqrt [4]{a x^2+b x^3}}-\frac {2}{3 x \sqrt [4]{a x^2+b x^3}}+\frac {b^{3/2} \sqrt {x} \sqrt [4]{\frac {a+b x}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right |2\right )}{a^{3/2} \sqrt [4]{a x^2+b x^3}} \] Output:

1/3*b/a/(b*x^3+a*x^2)^(1/4)-2/3/x/(b*x^3+a*x^2)^(1/4)+b^(3/2)*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))/ 
a^(3/2)/(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 = 49, normalized size of antiderivative = 0.43 \[ \int \frac {1}{x^2 \sqrt [4]{a x^2+b x^3}} \, dx=-\frac {2 \sqrt [4]{1+\frac {b x}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},-\frac {1}{2},-\frac {b x}{a}\right )}{3 x \sqrt [4]{x^2 (a+b x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.89, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {1931, 1931, 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.

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

\(\Big \downarrow \) 1931

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

\(\Big \downarrow \) 1931

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

\(\Big \downarrow \) 1917

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 836

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 1390

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

\(\Big \downarrow \) 1389

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

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

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]
 

rule 1931

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[c^(j - 1)*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(m + j*p 
+ 1))), x] - Simp[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*p + 1)))   I 
nt[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] 
 &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[ 
m + j*p + 1, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b*x^3 + a*x^2)^(3/4)/(b*x^5 + a*x^4), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 9.41 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.39 \[ \int \frac {1}{x^2 \sqrt [4]{a x^2+b x^3}} \, dx=-\frac {4\,{\left (\frac {a}{b\,x}+1\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {7}{4};\ \frac {11}{4};\ -\frac {a}{b\,x}\right )}{7\,x\,{\left (b\,x^3+a\,x^2\right )}^{1/4}} \] Input:

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

Output:

-(4*(a/(b*x) + 1)^(1/4)*hypergeom([1/4, 7/4], 11/4, -a/(b*x)))/(7*x*(a*x^2 
 + b*x^3)^(1/4))
 

Reduce [F]

\[ \int \frac {1}{x^2 \sqrt [4]{a x^2+b x^3}} \, dx=\frac {-4 \sqrt {x}\, \left (b x +a \right )^{\frac {1}{4}}-\sqrt {b x +a}\, \left (\int \frac {\sqrt {x}\, \left (b x +a \right )^{\frac {3}{4}}}{b^{2} x^{5}+2 a b \,x^{4}+a^{2} x^{3}}d x \right ) a \,x^{2}}{5 \sqrt {b x +a}\, x^{2}} \] Input:

int(1/x^2/(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**3 + 2*a*b*x**4 + b**2*x**5),x)*a*x**2)/(5*sqrt(a + b*x)*x**2)