\(\int \frac {(a+b x)^{4/3}}{\sqrt [3]{a-b x}} \, dx\) [185]

Optimal result

Integrand size = 20, antiderivative size = 151 \[ \int \frac {(a+b x)^{4/3}}{\sqrt [3]{a-b x}} \, dx=-\frac {4 a (a-b x)^{2/3} \sqrt [3]{a+b x}}{3 b}-\frac {(a-b x)^{2/3} (a+b x)^{4/3}}{2 b}+\frac {8 a^2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{a-b x}}{\sqrt {3} \sqrt [3]{a+b x}}\right )}{3 \sqrt {3} b}+\frac {4 a^2 \log (a+b x)}{9 b}+\frac {4 a^2 \log \left (1+\frac {\sqrt [3]{a-b x}}{\sqrt [3]{a+b x}}\right )}{3 b} \] Output:

-4/3*a*(-b*x+a)^(2/3)*(b*x+a)^(1/3)/b-1/2*(-b*x+a)^(2/3)*(b*x+a)^(4/3)/b-8 
/9*a^2*arctan(-1/3*3^(1/2)+2/3*(-b*x+a)^(1/3)*3^(1/2)/(b*x+a)^(1/3))*3^(1/ 
2)/b+4/9*a^2*ln(b*x+a)/b+4/3*a^2*ln(1+(-b*x+a)^(1/3)/(b*x+a)^(1/3))/b
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.07 \[ \int \frac {(a+b x)^{4/3}}{\sqrt [3]{a-b x}} \, dx=-\frac {3 (a-b x)^{2/3} \sqrt [3]{a+b x} (11 a+3 b x)+16 \sqrt {3} a^2 \arctan \left (\frac {\sqrt {3} \sqrt [3]{a+b x}}{-2 \sqrt [3]{a-b x}+\sqrt [3]{a+b x}}\right )-16 a^2 \log \left (b \left (\sqrt [3]{a-b x}+\sqrt [3]{a+b x}\right )\right )+8 a^2 \log \left ((a-b x)^{2/3}-\sqrt [3]{a-b x} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{18 b} \] Input:

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

Output:

-1/18*(3*(a - b*x)^(2/3)*(a + b*x)^(1/3)*(11*a + 3*b*x) + 16*Sqrt[3]*a^2*A 
rcTan[(Sqrt[3]*(a + b*x)^(1/3))/(-2*(a - b*x)^(1/3) + (a + b*x)^(1/3))] - 
16*a^2*Log[b*((a - b*x)^(1/3) + (a + b*x)^(1/3))] + 8*a^2*Log[(a - b*x)^(2 
/3) - (a - b*x)^(1/3)*(a + b*x)^(1/3) + (a + b*x)^(2/3)])/b
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {60, 60, 72}

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.

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 72

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

Input:

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

Output:

-1/2*((a - b*x)^(2/3)*(a + b*x)^(4/3))/b + (4*a*(-(((a - b*x)^(2/3)*(a + b 
*x)^(1/3))/b) + (2*a*((Sqrt[3]*ArcTan[1/Sqrt[3] - (2*(a - b*x)^(1/3))/(Sqr 
t[3]*(a + b*x)^(1/3))])/b + Log[a + b*x]/(2*b) + (3*Log[1 + (a - b*x)^(1/3 
)/(a + b*x)^(1/3)])/(2*b)))/3))/3
 

Defintions of rubi rules used

rule 60

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

rule 72

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

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.19 \[ \int \frac {(a+b x)^{4/3}}{\sqrt [3]{a-b x}} \, dx=-\frac {16 \, \sqrt {3} a^{2} \arctan \left (\frac {\sqrt {3} {\left (b x - a\right )} + 2 \, \sqrt {3} {\left (b x + a\right )}^{\frac {1}{3}} {\left (-b x + a\right )}^{\frac {2}{3}}}{3 \, {\left (b x - a\right )}}\right ) + 8 \, a^{2} \log \left (\frac {b x - {\left (b x + a\right )}^{\frac {2}{3}} {\left (-b x + a\right )}^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} {\left (-b x + a\right )}^{\frac {2}{3}} - a}{b x - a}\right ) - 16 \, a^{2} \log \left (-\frac {b x - {\left (b x + a\right )}^{\frac {1}{3}} {\left (-b x + a\right )}^{\frac {2}{3}} - a}{b x - a}\right ) + 3 \, {\left (3 \, b x + 11 \, a\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (-b x + a\right )}^{\frac {2}{3}}}{18 \, b} \] Input:

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

Output:

-1/18*(16*sqrt(3)*a^2*arctan(1/3*(sqrt(3)*(b*x - a) + 2*sqrt(3)*(b*x + a)^ 
(1/3)*(-b*x + a)^(2/3))/(b*x - a)) + 8*a^2*log((b*x - (b*x + a)^(2/3)*(-b* 
x + a)^(1/3) + (b*x + a)^(1/3)*(-b*x + a)^(2/3) - a)/(b*x - a)) - 16*a^2*l 
og(-(b*x - (b*x + a)^(1/3)*(-b*x + a)^(2/3) - a)/(b*x - a)) + 3*(3*b*x + 1 
1*a)*(b*x + a)^(1/3)*(-b*x + a)^(2/3))/b
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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