\(\int \frac {1}{(c x)^{8/3} \sqrt [3]{a x+b x^2}} \, dx\) [132]

Optimal result

Integrand size = 21, antiderivative size = 245 \[ \int \frac {1}{(c x)^{8/3} \sqrt [3]{a x+b x^2}} \, dx=-\frac {\left (a x+b x^2\right )^{2/3}}{2 a (c x)^{8/3}}+\frac {2 b \left (a x+b x^2\right )^{2/3}}{3 a^2 c (c x)^{5/3}}+\frac {2 b^2 \sqrt [3]{c x} \sqrt [3]{a+b x} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3} c^3 \sqrt [3]{a x+b x^2}}-\frac {b^2 \sqrt [3]{c x} \sqrt [3]{a+b x} \log (x)}{9 a^{7/3} c^3 \sqrt [3]{a x+b x^2}}+\frac {b^2 \sqrt [3]{c x} \sqrt [3]{a+b x} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{3 a^{7/3} c^3 \sqrt [3]{a x+b x^2}} \] Output:

-1/2*(b*x^2+a*x)^(2/3)/a/(c*x)^(8/3)+2/3*b*(b*x^2+a*x)^(2/3)/a^2/c/(c*x)^( 
5/3)+2/9*b^2*(c*x)^(1/3)*(b*x+a)^(1/3)*arctan(1/3*(a^(1/3)+2*(b*x+a)^(1/3) 
)*3^(1/2)/a^(1/3))*3^(1/2)/a^(7/3)/c^3/(b*x^2+a*x)^(1/3)-1/9*b^2*(c*x)^(1/ 
3)*(b*x+a)^(1/3)*ln(x)/a^(7/3)/c^3/(b*x^2+a*x)^(1/3)+1/3*b^2*(c*x)^(1/3)*( 
b*x+a)^(1/3)*ln(a^(1/3)-(b*x+a)^(1/3))/a^(7/3)/c^3/(b*x^2+a*x)^(1/3)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.77 \[ \int \frac {1}{(c x)^{8/3} \sqrt [3]{a x+b x^2}} \, dx=\frac {x \left (-9 a^{7/3}+3 a^{4/3} b x+12 \sqrt [3]{a} b^2 x^2+4 \sqrt {3} b^2 x^2 \sqrt [3]{a+b x} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+4 b^2 x^2 \sqrt [3]{a+b x} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )-2 b^2 x^2 \sqrt [3]{a+b x} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )\right )}{18 a^{7/3} (c x)^{8/3} \sqrt [3]{x (a+b x)}} \] Input:

Integrate[1/((c*x)^(8/3)*(a*x + b*x^2)^(1/3)),x]
 

Output:

(x*(-9*a^(7/3) + 3*a^(4/3)*b*x + 12*a^(1/3)*b^2*x^2 + 4*Sqrt[3]*b^2*x^2*(a 
 + b*x)^(1/3)*ArcTan[(1 + (2*(a + b*x)^(1/3))/a^(1/3))/Sqrt[3]] + 4*b^2*x^ 
2*(a + b*x)^(1/3)*Log[a^(1/3) - (a + b*x)^(1/3)] - 2*b^2*x^2*(a + b*x)^(1/ 
3)*Log[a^(2/3) + a^(1/3)*(a + b*x)^(1/3) + (a + b*x)^(2/3)]))/(18*a^(7/3)* 
(c*x)^(8/3)*(x*(a + b*x))^(1/3))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.66, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1137, 52, 52, 67, 16, 1082, 217}

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.

\(\displaystyle \int \frac {1}{(c x)^{8/3} \sqrt [3]{a x+b x^2}} \, dx\)

\(\Big \downarrow \) 1137

\(\displaystyle \frac {x^3 \sqrt [3]{a+b x} \int \frac {1}{x^3 \sqrt [3]{a+b x}}dx}{(c x)^{8/3} \sqrt [3]{a x+b x^2}}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {x^3 \sqrt [3]{a+b x} \left (-\frac {2 b \int \frac {1}{x^2 \sqrt [3]{a+b x}}dx}{3 a}-\frac {(a+b x)^{2/3}}{2 a x^2}\right )}{(c x)^{8/3} \sqrt [3]{a x+b x^2}}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {x^3 \sqrt [3]{a+b x} \left (-\frac {2 b \left (-\frac {b \int \frac {1}{x \sqrt [3]{a+b x}}dx}{3 a}-\frac {(a+b x)^{2/3}}{a x}\right )}{3 a}-\frac {(a+b x)^{2/3}}{2 a x^2}\right )}{(c x)^{8/3} \sqrt [3]{a x+b x^2}}\)

\(\Big \downarrow \) 67

\(\displaystyle \frac {x^3 \sqrt [3]{a+b x} \left (-\frac {2 b \left (-\frac {b \left (\frac {3}{2} \int \frac {1}{a^{2/3}+\sqrt [3]{a+b x} \sqrt [3]{a}+(a+b x)^{2/3}}d\sqrt [3]{a+b x}-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{a+b x}}d\sqrt [3]{a+b x}}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}\right )}{3 a}-\frac {(a+b x)^{2/3}}{a x}\right )}{3 a}-\frac {(a+b x)^{2/3}}{2 a x^2}\right )}{(c x)^{8/3} \sqrt [3]{a x+b x^2}}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {x^3 \sqrt [3]{a+b x} \left (-\frac {2 b \left (-\frac {b \left (\frac {3}{2} \int \frac {1}{a^{2/3}+\sqrt [3]{a+b x} \sqrt [3]{a}+(a+b x)^{2/3}}d\sqrt [3]{a+b x}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}\right )}{3 a}-\frac {(a+b x)^{2/3}}{a x}\right )}{3 a}-\frac {(a+b x)^{2/3}}{2 a x^2}\right )}{(c x)^{8/3} \sqrt [3]{a x+b x^2}}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {x^3 \sqrt [3]{a+b x} \left (-\frac {2 b \left (-\frac {b \left (-\frac {3 \int \frac {1}{-(a+b x)^{2/3}-3}d\left (\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}\right )}{3 a}-\frac {(a+b x)^{2/3}}{a x}\right )}{3 a}-\frac {(a+b x)^{2/3}}{2 a x^2}\right )}{(c x)^{8/3} \sqrt [3]{a x+b x^2}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {x^3 \sqrt [3]{a+b x} \left (-\frac {2 b \left (-\frac {b \left (\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}\right )}{3 a}-\frac {(a+b x)^{2/3}}{a x}\right )}{3 a}-\frac {(a+b x)^{2/3}}{2 a x^2}\right )}{(c x)^{8/3} \sqrt [3]{a x+b x^2}}\)

Input:

Int[1/((c*x)^(8/3)*(a*x + b*x^2)^(1/3)),x]
 

Output:

(x^3*(a + b*x)^(1/3)*(-1/2*(a + b*x)^(2/3)/(a*x^2) - (2*b*(-((a + b*x)^(2/ 
3)/(a*x)) - (b*((Sqrt[3]*ArcTan[(1 + (2*(a + b*x)^(1/3))/a^(1/3))/Sqrt[3]] 
)/a^(1/3) - Log[x]/(2*a^(1/3)) + (3*Log[a^(1/3) - (a + b*x)^(1/3)])/(2*a^( 
1/3))))/(3*a)))/(3*a)))/((c*x)^(8/3)*(a*x + b*x^2)^(1/3))
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
 

rule 67

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ 
{q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x 
] + (Simp[3/(2*b)   Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], 
 x] - Simp[3/(2*b*q)   Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] / 
; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
 

rule 217

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

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1137

Int[((e_.)*(x_))^(m_)*((b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[( 
e*x)^m*((b*x + c*x^2)^p/(x^(m + p)*(b + c*x)^p))   Int[x^(m + p)*(b + c*x)^ 
p, x], x] /; FreeQ[{b, c, e, m}, x]
 

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.84

method	result	size
risch	\(-\frac {\left (b x +a \right ) \left (-4 b x +3 a \right )}{6 a^{2} x \,c^{2} \left (c x \right )^{\frac {2}{3}} \left (x \left (b x +a \right )\right )^{\frac {1}{3}}}+\frac {2 b^{2} \left (\frac {\ln \left (\left (c^{2} b x +a \,c^{2}\right )^{\frac {1}{3}}-\left (a \,c^{2}\right )^{\frac {1}{3}}\right )}{\left (a \,c^{2}\right )^{\frac {1}{3}}}-\frac {\ln \left (\left (c^{2} b x +a \,c^{2}\right )^{\frac {2}{3}}+\left (a \,c^{2}\right )^{\frac {1}{3}} \left (c^{2} b x +a \,c^{2}\right )^{\frac {1}{3}}+\left (a \,c^{2}\right )^{\frac {2}{3}}\right )}{2 \left (a \,c^{2}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (c^{2} b x +a \,c^{2}\right )^{\frac {1}{3}}}{\left (a \,c^{2}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{\left (a \,c^{2}\right )^{\frac {1}{3}}}\right ) \left (c^{2} \left (b x +a \right )\right )^{\frac {1}{3}} x}{9 a^{2} c^{2} \left (c x \right )^{\frac {2}{3}} \left (x \left (b x +a \right )\right )^{\frac {1}{3}}}\)	\(207\)

Input:

int(1/(c*x)^(8/3)/(b*x^2+a*x)^(1/3),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 667, normalized size of antiderivative = 2.72 \[ \int \frac {1}{(c x)^{8/3} \sqrt [3]{a x+b x^2}} \, dx =\text {Too large to display} \] Input:

integrate(1/(c*x)^(8/3)/(b*x^2+a*x)^(1/3),x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

[1/18*(6*sqrt(1/3)*a*b^2*c*x^3*sqrt(-(a*c^2)^(1/3)/a)*log(-(b*c^2*x^2 + 3* 
a*c^2*x - 3*(a*c^2)^(1/3)*(b*x^2 + a*x)^(2/3)*(c*x)^(1/3)*c - 3*sqrt(1/3)* 
(2*(b*x^2 + a*x)^(1/3)*(c*x)^(2/3)*a*c - (a*c^2)^(2/3)*(b*x^2 + a*x)^(2/3) 
*(c*x)^(1/3) - (b*c*x^2 + a*c*x)*(a*c^2)^(1/3))*sqrt(-(a*c^2)^(1/3)/a))/x^ 
2) + 4*(a*c^2)^(2/3)*b^2*x^3*log(((b*x^2 + a*x)^(2/3)*(c*x)^(1/3)*a*c - (a 
*c^2)^(2/3)*(b*x^2 + a*x))/(b*x^2 + a*x)) - 2*(a*c^2)^(2/3)*b^2*x^3*log((( 
b*x^2 + a*x)^(1/3)*(c*x)^(2/3)*a*c + (a*c^2)^(2/3)*(b*x^2 + a*x)^(2/3)*(c* 
x)^(1/3) + (b*c*x^2 + a*c*x)*(a*c^2)^(1/3))/(b*x^2 + a*x)) + 3*(4*a*b*c*x 
- 3*a^2*c)*(b*x^2 + a*x)^(2/3)*(c*x)^(1/3))/(a^3*c^4*x^3), -1/18*(12*sqrt( 
1/3)*a*b^2*c*x^3*sqrt((a*c^2)^(1/3)/a)*arctan(sqrt(1/3)*(2*(a*c^2)^(2/3)*( 
b*x^2 + a*x)^(2/3)*(c*x)^(1/3) + (b*c*x^2 + a*c*x)*(a*c^2)^(1/3))*sqrt((a* 
c^2)^(1/3)/a)/(b*c^2*x^2 + a*c^2*x)) - 4*(a*c^2)^(2/3)*b^2*x^3*log(((b*x^2 
 + a*x)^(2/3)*(c*x)^(1/3)*a*c - (a*c^2)^(2/3)*(b*x^2 + a*x))/(b*x^2 + a*x) 
) + 2*(a*c^2)^(2/3)*b^2*x^3*log(((b*x^2 + a*x)^(1/3)*(c*x)^(2/3)*a*c + (a* 
c^2)^(2/3)*(b*x^2 + a*x)^(2/3)*(c*x)^(1/3) + (b*c*x^2 + a*c*x)*(a*c^2)^(1/ 
3))/(b*x^2 + a*x)) - 3*(4*a*b*c*x - 3*a^2*c)*(b*x^2 + a*x)^(2/3)*(c*x)^(1/ 
3))/(a^3*c^4*x^3)]
 

Sympy [F]

\[ \int \frac {1}{(c x)^{8/3} \sqrt [3]{a x+b x^2}} \, dx=\int \frac {1}{\left (c x\right )^{\frac {8}{3}} \sqrt [3]{x \left (a + b x\right )}}\, dx \] Input:

integrate(1/(c*x)**(8/3)/(b*x**2+a*x)**(1/3),x)
 

Output:

Integral(1/((c*x)**(8/3)*(x*(a + b*x))**(1/3)), x)
 

Maxima [F]

\[ \int \frac {1}{(c x)^{8/3} \sqrt [3]{a x+b x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a x\right )}^{\frac {1}{3}} \left (c x\right )^{\frac {8}{3}}} \,d x } \] Input:

integrate(1/(c*x)^(8/3)/(b*x^2+a*x)^(1/3),x, algorithm="maxima")
 

Output:

integrate(1/((b*x^2 + a*x)^(1/3)*(c*x)^(8/3)), x)
 

Giac [F]

\[ \int \frac {1}{(c x)^{8/3} \sqrt [3]{a x+b x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a x\right )}^{\frac {1}{3}} \left (c x\right )^{\frac {8}{3}}} \,d x } \] Input:

integrate(1/(c*x)^(8/3)/(b*x^2+a*x)^(1/3),x, algorithm="giac")
 

Output:

integrate(1/((b*x^2 + a*x)^(1/3)*(c*x)^(8/3)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(c x)^{8/3} \sqrt [3]{a x+b x^2}} \, dx=\int \frac {1}{{\left (b\,x^2+a\,x\right )}^{1/3}\,{\left (c\,x\right )}^{8/3}} \,d x \] Input:

int(1/((a*x + b*x^2)^(1/3)*(c*x)^(8/3)),x)
 

Output:

int(1/((a*x + b*x^2)^(1/3)*(c*x)^(8/3)), x)
 

Reduce [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(c x)^{8/3} \sqrt [3]{a x+b x^2}} \, dx=\frac {-4 \sqrt {3}\, \mathit {atan} \left (\frac {2 \left (b x +a \right )^{\frac {1}{6}}+a^{\frac {1}{6}}}{a^{\frac {1}{6}} \sqrt {3}}\right ) b^{2} x^{2}+4 \sqrt {3}\, \mathit {atan} \left (\frac {2 \left (b x +a \right )^{\frac {1}{6}}-a^{\frac {1}{6}}}{a^{\frac {1}{6}} \sqrt {3}}\right ) b^{2} x^{2}-9 a^{\frac {4}{3}} \left (b x +a \right )^{\frac {2}{3}}+12 a^{\frac {1}{3}} \left (b x +a \right )^{\frac {2}{3}} b x +4 \,\mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}+a^{\frac {1}{6}}\right ) b^{2} x^{2}+4 \,\mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}-a^{\frac {1}{6}}\right ) b^{2} x^{2}-2 \,\mathrm {log}\left (-a^{\frac {1}{6}} \left (b x +a \right )^{\frac {1}{6}}+\left (b x +a \right )^{\frac {1}{3}}+a^{\frac {1}{3}}\right ) b^{2} x^{2}-2 \,\mathrm {log}\left (a^{\frac {1}{6}} \left (b x +a \right )^{\frac {1}{6}}+\left (b x +a \right )^{\frac {1}{3}}+a^{\frac {1}{3}}\right ) b^{2} x^{2}}{18 c^{\frac {8}{3}} a^{\frac {7}{3}} x^{2}} \] Input:

int(1/(c*x)^(8/3)/(b*x^2+a*x)^(1/3),x)
 

Output:

( - 4*sqrt(3)*atan((2*(a + b*x)**(1/6) + a**(1/6))/(a**(1/6)*sqrt(3)))*b** 
2*x**2 + 4*sqrt(3)*atan((2*(a + b*x)**(1/6) - a**(1/6))/(a**(1/6)*sqrt(3)) 
)*b**2*x**2 - 9*a**(1/3)*(a + b*x)**(2/3)*a + 12*a**(1/3)*(a + b*x)**(2/3) 
*b*x + 4*log((a + b*x)**(1/6) + a**(1/6))*b**2*x**2 + 4*log((a + b*x)**(1/ 
6) - a**(1/6))*b**2*x**2 - 2*log( - a**(1/6)*(a + b*x)**(1/6) + (a + b*x)* 
*(1/3) + a**(1/3))*b**2*x**2 - 2*log(a**(1/6)*(a + b*x)**(1/6) + (a + b*x) 
**(1/3) + a**(1/3))*b**2*x**2)/(18*c**(2/3)*a**(1/3)*a**2*c**2*x**2)