\(\int \frac {-b+a x^3}{x^3 \sqrt [4]{b x+a x^4}} \, dx\) [1350]

Optimal result

Integrand size = 26, antiderivative size = 97 \[ \int \frac {-b+a x^3}{x^3 \sqrt [4]{b x+a x^4}} \, dx=\frac {4 \left (b x+a x^4\right )^{3/4}}{9 x^3}+\frac {2}{3} a^{3/4} \arctan \left (\frac {\sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right )+\frac {2}{3} a^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \left (b x+a x^4\right )^{3/4}}{b+a x^3}\right ) \] Output:

4/9*(a*x^4+b*x)^(3/4)/x^3+2/3*a^(3/4)*arctan(a^(1/4)*(a*x^4+b*x)^(3/4)/(a* 
x^3+b))+2/3*a^(3/4)*arctanh(a^(1/4)*(a*x^4+b*x)^(3/4)/(a*x^3+b))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 9.99 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.26 \[ \int \frac {-b+a x^3}{x^3 \sqrt [4]{b x+a x^4}} \, dx=\frac {2 \left (2 \left (b+a x^3\right )+3 a^{3/4} x^{9/4} \sqrt [4]{b+a x^3} \arctan \left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )+3 a^{3/4} x^{9/4} \sqrt [4]{b+a x^3} \text {arctanh}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{b+a x^3}}\right )\right )}{9 x^2 \sqrt [4]{x \left (b+a x^3\right )}} \] Input:

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

Output:

(2*(2*(b + a*x^3) + 3*a^(3/4)*x^(9/4)*(b + a*x^3)^(1/4)*ArcTan[(a^(1/4)*x^ 
(3/4))/(b + a*x^3)^(1/4)] + 3*a^(3/4)*x^(9/4)*(b + a*x^3)^(1/4)*ArcTanh[(a 
^(1/4)*x^(3/4))/(b + a*x^3)^(1/4)]))/(9*x^2*(x*(b + a*x^3))^(1/4))
 

Rubi [A] (warning: unable to verify)

Time = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.20, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1944, 1917, 851, 807, 770, 756, 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.

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

\(\Big \downarrow \) 1944

\(\displaystyle a \int \frac {1}{\sqrt [4]{a x^4+b x}}dx+\frac {4 \left (a x^4+b x\right )^{3/4}}{9 x^3}\)

\(\Big \downarrow \) 1917

\(\displaystyle \frac {a \sqrt [4]{x} \sqrt [4]{a x^3+b} \int \frac {1}{\sqrt [4]{x} \sqrt [4]{a x^3+b}}dx}{\sqrt [4]{a x^4+b x}}+\frac {4 \left (a x^4+b x\right )^{3/4}}{9 x^3}\)

\(\Big \downarrow \) 851

\(\displaystyle \frac {4 a \sqrt [4]{x} \sqrt [4]{a x^3+b} \int \frac {\sqrt {x}}{\sqrt [4]{a x^3+b}}d\sqrt [4]{x}}{\sqrt [4]{a x^4+b x}}+\frac {4 \left (a x^4+b x\right )^{3/4}}{9 x^3}\)

\(\Big \downarrow \) 807

\(\displaystyle \frac {4 a \sqrt [4]{x} \sqrt [4]{a x^3+b} \int \frac {1}{\sqrt [4]{b+a x}}dx^{3/4}}{3 \sqrt [4]{a x^4+b x}}+\frac {4 \left (a x^4+b x\right )^{3/4}}{9 x^3}\)

\(\Big \downarrow \) 770

\(\displaystyle \frac {4 a \sqrt [4]{x} \sqrt [4]{a x^3+b} \int \frac {1}{1-a x}d\frac {x^{3/4}}{\sqrt [4]{b+a x}}}{3 \sqrt [4]{a x^4+b x}}+\frac {4 \left (a x^4+b x\right )^{3/4}}{9 x^3}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {4 a \sqrt [4]{x} \sqrt [4]{a x^3+b} \left (\frac {1}{2} \int \frac {1}{1-\sqrt {a} \sqrt {x}}d\frac {x^{3/4}}{\sqrt [4]{b+a x}}+\frac {1}{2} \int \frac {1}{\sqrt {a} \sqrt {x}+1}d\frac {x^{3/4}}{\sqrt [4]{b+a x}}\right )}{3 \sqrt [4]{a x^4+b x}}+\frac {4 \left (a x^4+b x\right )^{3/4}}{9 x^3}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {4 a \sqrt [4]{x} \sqrt [4]{a x^3+b} \left (\frac {1}{2} \int \frac {1}{1-\sqrt {a} \sqrt {x}}d\frac {x^{3/4}}{\sqrt [4]{b+a x}}+\frac {\arctan \left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x+b}}\right )}{2 \sqrt [4]{a}}\right )}{3 \sqrt [4]{a x^4+b x}}+\frac {4 \left (a x^4+b x\right )^{3/4}}{9 x^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {4 a \sqrt [4]{x} \sqrt [4]{a x^3+b} \left (\frac {\arctan \left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} x^{3/4}}{\sqrt [4]{a x+b}}\right )}{2 \sqrt [4]{a}}\right )}{3 \sqrt [4]{a x^4+b x}}+\frac {4 \left (a x^4+b x\right )^{3/4}}{9 x^3}\)

Input:

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

Output:

(4*(b*x + a*x^4)^(3/4))/(9*x^3) + (4*a*x^(1/4)*(b + a*x^3)^(1/4)*(ArcTan[( 
a^(1/4)*x^(3/4))/(b + a*x)^(1/4)]/(2*a^(1/4)) + ArcTanh[(a^(1/4)*x^(3/4))/ 
(b + a*x)^(1/4)]/(2*a^(1/4))))/(3*(b*x + a*x^4)^(1/4))
 

Defintions of rubi rules used

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 756

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

rule 770

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n)   Subst[In 
t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, 
 b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 
/n]
 

rule 807

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m 
+ 1, n]}, Simp[1/k   Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, 
x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
 

rule 851

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = 
 Denominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ 
n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && 
FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
 

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 1944

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + 
(d_.)*(x_)^(n_.)), x_Symbol] :> Simp[c*e^(j - 1)*(e*x)^(m - j + 1)*((a*x^j 
+ b*x^(j + n))^(p + 1)/(a*(m + j*p + 1))), x] + Simp[(a*d*(m + j*p + 1) - b 
*c*(m + n + p*(j + n) + 1))/(a*e^n*(m + j*p + 1))   Int[(e*x)^(m + n)*(a*x^ 
j + b*x^(j + n))^p, x], x] /; FreeQ[{a, b, c, d, e, j, p}, x] && EqQ[jn, j 
+ n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && (LtQ[m + j*p, -1 
] || (IntegersQ[m - 1/2, p - 1/2] && LtQ[p, 0] && LtQ[m, (-n)*p - 1])) && ( 
GtQ[e, 0] || IntegersQ[j, n]) && NeQ[m + j*p + 1, 0] && NeQ[m - n + j*p + 1 
, 0]
 

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.01

method	result	size
pseudoelliptic	\(\frac {-6 \arctan \left (\frac {{\left (\left (a \,x^{3}+b \right ) x \right )}^{\frac {1}{4}}}{x \,a^{\frac {1}{4}}}\right ) a^{\frac {3}{4}} x^{3}+3 \ln \left (\frac {-a^{\frac {1}{4}} x -{\left (\left (a \,x^{3}+b \right ) x \right )}^{\frac {1}{4}}}{a^{\frac {1}{4}} x -{\left (\left (a \,x^{3}+b \right ) x \right )}^{\frac {1}{4}}}\right ) a^{\frac {3}{4}} x^{3}+4 {\left (\left (a \,x^{3}+b \right ) x \right )}^{\frac {3}{4}}}{9 x^{3}}\)	\(98\)

Input:

int((a*x^3-b)/x^3/(a*x^4+b*x)^(1/4),x,method=_RETURNVERBOSE)
 

Output:

1/9*(-6*arctan(((a*x^3+b)*x)^(1/4)/x/a^(1/4))*a^(3/4)*x^3+3*ln((-a^(1/4)*x 
-((a*x^3+b)*x)^(1/4))/(a^(1/4)*x-((a*x^3+b)*x)^(1/4)))*a^(3/4)*x^3+4*((a*x 
^3+b)*x)^(3/4))/x^3
 

Fricas [F(-1)]

Timed out. \[ \int \frac {-b+a x^3}{x^3 \sqrt [4]{b x+a x^4}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (77) = 154\).

Time = 0.24 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.91 \[ \int \frac {-b+a x^3}{x^3 \sqrt [4]{b x+a x^4}} \, dx=\frac {1}{3} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) + \frac {1}{3} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) - \frac {1}{6} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right ) + \frac {1}{6} \, \sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{3}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{3}}}\right ) + \frac {4}{9} \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{4}} \] Input:

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

Output:

1/3*sqrt(2)*(-a)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a + b/x 
^3)^(1/4))/(-a)^(1/4)) + 1/3*sqrt(2)*(-a)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt( 
2)*(-a)^(1/4) - 2*(a + b/x^3)^(1/4))/(-a)^(1/4)) - 1/6*sqrt(2)*(-a)^(3/4)* 
log(sqrt(2)*(-a)^(1/4)*(a + b/x^3)^(1/4) + sqrt(-a) + sqrt(a + b/x^3)) + 1 
/6*sqrt(2)*(-a)^(3/4)*log(-sqrt(2)*(-a)^(1/4)*(a + b/x^3)^(1/4) + sqrt(-a) 
 + sqrt(a + b/x^3)) + 4/9*(a + b/x^3)^(3/4)
 

Mupad [B] (verification not implemented)

Time = 9.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.60 \[ \int \frac {-b+a x^3}{x^3 \sqrt [4]{b x+a x^4}} \, dx=\frac {4\,{\left (a\,x^4+b\,x\right )}^{3/4}}{9\,x^3}+\frac {4\,a\,x\,{\left (\frac {a\,x^3}{b}+1\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ -\frac {a\,x^3}{b}\right )}{3\,{\left (a\,x^4+b\,x\right )}^{1/4}} \] Input:

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

Output:

(4*(b*x + a*x^4)^(3/4))/(9*x^3) + (4*a*x*((a*x^3)/b + 1)^(1/4)*hypergeom([ 
1/4, 1/4], 5/4, -(a*x^3)/b))/(3*(b*x + a*x^4)^(1/4))
 

Reduce [F]

\[ \int \frac {-b+a x^3}{x^3 \sqrt [4]{b x+a x^4}} \, dx=-\left (\int \frac {1}{x^{\frac {13}{4}} \left (a \,x^{3}+b \right )^{\frac {1}{4}}}d x \right ) b +\left (\int \frac {1}{x^{\frac {1}{4}} \left (a \,x^{3}+b \right )^{\frac {1}{4}}}d x \right ) a \] Input:

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

Output:

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