Integrand size = 13, antiderivative size = 130 \[ \int \frac {1}{x^3 \sqrt [3]{a+b x}} \, dx=-\frac {(a+b x)^{2/3}}{2 a x^2}+\frac {2 b (a+b x)^{2/3}}{3 a^2 x}+\frac {2 b^2 \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3}}-\frac {b^2 \log (x)}{9 a^{7/3}}+\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{3 a^{7/3}} \] Output:
-1/2*(b*x+a)^(2/3)/a/x^2+2/3*b*(b*x+a)^(2/3)/a^2/x+2/9*b^2*arctan(1/3*(a^( 1/3)+2*(b*x+a)^(1/3))*3^(1/2)/a^(1/3))*3^(1/2)/a^(7/3)-1/9*b^2*ln(x)/a^(7/ 3)+1/3*b^2*ln(a^(1/3)-(b*x+a)^(1/3))/a^(7/3)
Time = 0.12 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x^3 \sqrt [3]{a+b x}} \, dx=-\frac {(a+b x)^{2/3} (7 a-4 (a+b x))}{6 a^2 x^2}+\frac {2 b^2 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3}}+\frac {2 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{9 a^{7/3}}-\frac {b^2 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{9 a^{7/3}} \] Input:
Integrate[1/(x^3*(a + b*x)^(1/3)),x]
Output:
-1/6*((a + b*x)^(2/3)*(7*a - 4*(a + b*x)))/(a^2*x^2) + (2*b^2*ArcTan[1/Sqr t[3] + (2*(a + b*x)^(1/3))/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(7/3)) + (2*b^ 2*Log[a^(1/3) - (a + b*x)^(1/3)])/(9*a^(7/3)) - (b^2*Log[a^(2/3) + a^(1/3) *(a + b*x)^(1/3) + (a + b*x)^(2/3)])/(9*a^(7/3))
Time = 0.20 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {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.
| \(\displaystyle \int \frac {1}{x^3 \sqrt [3]{a+b x}} \, dx\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {2 b \int \frac {1}{x^2 \sqrt [3]{a+b x}}dx}{3 a}-\frac {(a+b x)^{2/3}}{2 a x^2}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {2 b \left (-\frac {b \int \frac {1}{x \sqrt [3]{a+b x}}dx}{3 a}-\frac {(a+b x)^{2/3}}{a x}\right )}{3 a}-\frac {(a+b x)^{2/3}}{2 a x^2}\) |
| \(\Big \downarrow \) 67 |
| \(\displaystyle -\frac {2 b \left (-\frac {b \left (\frac {3}{2} \int \frac {1}{a^{2/3}+\sqrt [3]{a+b x} \sqrt [3]{a}+(a+b x)^{2/3}}d\sqrt [3]{a+b x}-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{a+b x}}d\sqrt [3]{a+b x}}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}\right )}{3 a}-\frac {(a+b x)^{2/3}}{a x}\right )}{3 a}-\frac {(a+b x)^{2/3}}{2 a x^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {2 b \left (-\frac {b \left (\frac {3}{2} \int \frac {1}{a^{2/3}+\sqrt [3]{a+b x} \sqrt [3]{a}+(a+b x)^{2/3}}d\sqrt [3]{a+b x}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}\right )}{3 a}-\frac {(a+b x)^{2/3}}{a x}\right )}{3 a}-\frac {(a+b x)^{2/3}}{2 a x^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {2 b \left (-\frac {b \left (-\frac {3 \int \frac {1}{-(a+b x)^{2/3}-3}d\left (\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}\right )}{3 a}-\frac {(a+b x)^{2/3}}{a x}\right )}{3 a}-\frac {(a+b x)^{2/3}}{2 a x^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {2 b \left (-\frac {b \left (\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}\right )}{3 a}-\frac {(a+b x)^{2/3}}{a x}\right )}{3 a}-\frac {(a+b x)^{2/3}}{2 a x^2}\) |
Input:
Int[1/(x^3*(a + b*x)^(1/3)),x]
Output:
-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)
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
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]
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]
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])
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]
Time = 0.20 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.84
| method | result | size |
| risch | \(-\frac {\left (b x +a \right )^{\frac {2}{3}} \left (-4 b x +3 a \right )}{6 a^{2} x^{2}}+\frac {2 b^{2} \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 a^{\frac {7}{3}}}-\frac {b^{2} \ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{9 a^{\frac {7}{3}}}+\frac {2 b^{2} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a^{\frac {7}{3}}}\) | \(109\) |
| pseudoelliptic | \(\frac {4 b^{2} \sqrt {3}\, \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b x +a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) x^{2}+4 b^{2} \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right ) x^{2}-2 b^{2} \ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right ) x^{2}+12 b x \,a^{\frac {1}{3}} \left (b x +a \right )^{\frac {2}{3}}-9 \left (b x +a \right )^{\frac {2}{3}} a^{\frac {4}{3}}}{18 a^{\frac {7}{3}} x^{2}}\) | \(122\) |
| derivativedivides | \(3 b^{2} \left (-\frac {\left (b x +a \right )^{\frac {2}{3}}}{6 a \,b^{2} x^{2}}-\frac {2 \left (-\frac {\left (b x +a \right )^{\frac {2}{3}}}{3 a b x}+\frac {-\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {1}{3}}}+\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {1}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {1}{3}}}}{3 a}\right )}{3 a}\right )\) | \(130\) |
| default | \(3 b^{2} \left (-\frac {\left (b x +a \right )^{\frac {2}{3}}}{6 a \,b^{2} x^{2}}-\frac {2 \left (-\frac {\left (b x +a \right )^{\frac {2}{3}}}{3 a b x}+\frac {-\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {1}{3}}}+\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {1}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {1}{3}}}}{3 a}\right )}{3 a}\right )\) | \(130\) |
Input:
int(1/x^3/(b*x+a)^(1/3),x,method=_RETURNVERBOSE)
Output:
-1/6*(b*x+a)^(2/3)*(-4*b*x+3*a)/a^2/x^2+2/9*b^2/a^(7/3)*ln((b*x+a)^(1/3)-a ^(1/3))-1/9*b^2/a^(7/3)*ln((b*x+a)^(2/3)+a^(1/3)*(b*x+a)^(1/3)+a^(2/3))+2/ 9*b^2/a^(7/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/a^(1/3)*(b*x+a)^(1/3)+1))
Time = 0.09 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.28 \[ \int \frac {1}{x^3 \sqrt [3]{a+b x}} \, dx=\left [\frac {6 \, \sqrt {\frac {1}{3}} a b^{2} x^{2} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} \log \left (\frac {2 \, b x + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {2}{3}} a^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} a - a^{\frac {4}{3}}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} - 3 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + 3 \, a}{x}\right ) - 2 \, a^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 4 \, a^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + 3 \, {\left (4 \, a b x - 3 \, a^{2}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{18 \, a^{3} x^{2}}, \frac {12 \, \sqrt {\frac {1}{3}} a^{\frac {2}{3}} b^{2} x^{2} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{a^{\frac {1}{3}}}\right ) - 2 \, a^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 4 \, a^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + 3 \, {\left (4 \, a b x - 3 \, a^{2}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{18 \, a^{3} x^{2}}\right ] \] Input:
integrate(1/x^3/(b*x+a)^(1/3),x, algorithm="fricas")
Output:
[1/18*(6*sqrt(1/3)*a*b^2*x^2*sqrt(-1/a^(2/3))*log((2*b*x + 3*sqrt(1/3)*(2* (b*x + a)^(2/3)*a^(2/3) - (b*x + a)^(1/3)*a - a^(4/3))*sqrt(-1/a^(2/3)) - 3*(b*x + a)^(1/3)*a^(2/3) + 3*a)/x) - 2*a^(2/3)*b^2*x^2*log((b*x + a)^(2/3 ) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3)) + 4*a^(2/3)*b^2*x^2*log((b*x + a)^( 1/3) - a^(1/3)) + 3*(4*a*b*x - 3*a^2)*(b*x + a)^(2/3))/(a^3*x^2), 1/18*(12 *sqrt(1/3)*a^(2/3)*b^2*x^2*arctan(sqrt(1/3)*(2*(b*x + a)^(1/3) + a^(1/3))/ a^(1/3)) - 2*a^(2/3)*b^2*x^2*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3)) + 4*a^(2/3)*b^2*x^2*log((b*x + a)^(1/3) - a^(1/3)) + 3*(4*a*b* x - 3*a^2)*(b*x + a)^(2/3))/(a^3*x^2)]
Result contains complex when optimal does not.
Time = 1.91 (sec) , antiderivative size = 2730, normalized size of antiderivative = 21.00 \[ \int \frac {1}{x^3 \sqrt [3]{a+b x}} \, dx=\text {Too large to display} \] Input:
integrate(1/x**3/(b*x+a)**(1/3),x)
Output:
4*a**(14/3)*b**(10/3)*(a/b + x)**(4/3)*exp(2*I*pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gamma(2/3)/(27*a**7*b**(4/3)*(a/b + x)**(4/3)*exp(2 *I*pi/3)*gamma(5/3) - 81*a**6*b**(7/3)*(a/b + x)**(7/3)*exp(2*I*pi/3)*gamm a(5/3) + 81*a**5*b**(10/3)*(a/b + x)**(10/3)*exp(2*I*pi/3)*gamma(5/3) - 27 *a**4*b**(13/3)*(a/b + x)**(13/3)*exp(2*I*pi/3)*gamma(5/3)) + 4*a**(14/3)* b**(10/3)*(a/b + x)**(4/3)*exp(-2*I*pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3 )*exp_polar(2*I*pi/3)/a**(1/3))*gamma(2/3)/(27*a**7*b**(4/3)*(a/b + x)**(4 /3)*exp(2*I*pi/3)*gamma(5/3) - 81*a**6*b**(7/3)*(a/b + x)**(7/3)*exp(2*I*p i/3)*gamma(5/3) + 81*a**5*b**(10/3)*(a/b + x)**(10/3)*exp(2*I*pi/3)*gamma( 5/3) - 27*a**4*b**(13/3)*(a/b + x)**(13/3)*exp(2*I*pi/3)*gamma(5/3)) + 4*a **(14/3)*b**(10/3)*(a/b + x)**(4/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)*exp_ polar(4*I*pi/3)/a**(1/3))*gamma(2/3)/(27*a**7*b**(4/3)*(a/b + x)**(4/3)*ex p(2*I*pi/3)*gamma(5/3) - 81*a**6*b**(7/3)*(a/b + x)**(7/3)*exp(2*I*pi/3)*g amma(5/3) + 81*a**5*b**(10/3)*(a/b + x)**(10/3)*exp(2*I*pi/3)*gamma(5/3) - 27*a**4*b**(13/3)*(a/b + x)**(13/3)*exp(2*I*pi/3)*gamma(5/3)) - 12*a**(11 /3)*b**(13/3)*(a/b + x)**(7/3)*exp(2*I*pi/3)*log(1 - b**(1/3)*(a/b + x)**( 1/3)/a**(1/3))*gamma(2/3)/(27*a**7*b**(4/3)*(a/b + x)**(4/3)*exp(2*I*pi/3) *gamma(5/3) - 81*a**6*b**(7/3)*(a/b + x)**(7/3)*exp(2*I*pi/3)*gamma(5/3) + 81*a**5*b**(10/3)*(a/b + x)**(10/3)*exp(2*I*pi/3)*gamma(5/3) - 27*a**4*b* *(13/3)*(a/b + x)**(13/3)*exp(2*I*pi/3)*gamma(5/3)) - 12*a**(11/3)*b**(...
Time = 0.13 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^3 \sqrt [3]{a+b x}} \, dx=\frac {2 \, \sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{9 \, a^{\frac {7}{3}}} - \frac {b^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{9 \, a^{\frac {7}{3}}} + \frac {2 \, b^{2} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{9 \, a^{\frac {7}{3}}} + \frac {4 \, {\left (b x + a\right )}^{\frac {5}{3}} b^{2} - 7 \, {\left (b x + a\right )}^{\frac {2}{3}} a b^{2}}{6 \, {\left ({\left (b x + a\right )}^{2} a^{2} - 2 \, {\left (b x + a\right )} a^{3} + a^{4}\right )}} \] Input:
integrate(1/x^3/(b*x+a)^(1/3),x, algorithm="maxima")
Output:
2/9*sqrt(3)*b^2*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3))/ a^(7/3) - 1/9*b^2*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3)) /a^(7/3) + 2/9*b^2*log((b*x + a)^(1/3) - a^(1/3))/a^(7/3) + 1/6*(4*(b*x + a)^(5/3)*b^2 - 7*(b*x + a)^(2/3)*a*b^2)/((b*x + a)^2*a^2 - 2*(b*x + a)*a^3 + a^4)
Time = 0.43 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 \sqrt [3]{a+b x}} \, dx=\frac {\frac {4 \, \sqrt {3} b^{3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {7}{3}}} - \frac {2 \, b^{3} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {7}{3}}} + \frac {4 \, b^{3} \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {7}{3}}} + \frac {3 \, {\left (4 \, {\left (b x + a\right )}^{\frac {5}{3}} b^{3} - 7 \, {\left (b x + a\right )}^{\frac {2}{3}} a b^{3}\right )}}{a^{2} b^{2} x^{2}}}{18 \, b} \] Input:
integrate(1/x^3/(b*x+a)^(1/3),x, algorithm="giac")
Output:
1/18*(4*sqrt(3)*b^3*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/ 3))/a^(7/3) - 2*b^3*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3 ))/a^(7/3) + 4*b^3*log(abs((b*x + a)^(1/3) - a^(1/3)))/a^(7/3) + 3*(4*(b*x + a)^(5/3)*b^3 - 7*(b*x + a)^(2/3)*a*b^3)/(a^2*b^2*x^2))/b
Time = 0.17 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.40 \[ \int \frac {1}{x^3 \sqrt [3]{a+b x}} \, dx=\frac {2\,b^2\,\ln \left ({\left (a+b\,x\right )}^{1/3}-a^{1/3}\right )}{9\,a^{7/3}}-\frac {\frac {7\,b^2\,{\left (a+b\,x\right )}^{2/3}}{6\,a}-\frac {2\,b^2\,{\left (a+b\,x\right )}^{5/3}}{3\,a^2}}{{\left (a+b\,x\right )}^2-2\,a\,\left (a+b\,x\right )+a^2}-\frac {\ln \left (\frac {4\,b^4\,{\left (a+b\,x\right )}^{1/3}}{9\,a^4}-\frac {{\left (b^2+\sqrt {3}\,b^2\,1{}\mathrm {i}\right )}^2}{9\,a^{11/3}}\right )\,\left (b^2+\sqrt {3}\,b^2\,1{}\mathrm {i}\right )}{9\,a^{7/3}}+\frac {b^2\,\ln \left (\frac {4\,b^4\,{\left (a+b\,x\right )}^{1/3}}{9\,a^4}-\frac {9\,b^4\,{\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2}{a^{11/3}}\right )\,\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}{a^{7/3}} \] Input:
int(1/(x^3*(a + b*x)^(1/3)),x)
Output:
(2*b^2*log((a + b*x)^(1/3) - a^(1/3)))/(9*a^(7/3)) - ((7*b^2*(a + b*x)^(2/ 3))/(6*a) - (2*b^2*(a + b*x)^(5/3))/(3*a^2))/((a + b*x)^2 - 2*a*(a + b*x) + a^2) - (log((4*b^4*(a + b*x)^(1/3))/(9*a^4) - (3^(1/2)*b^2*1i + b^2)^2/( 9*a^(11/3)))*(3^(1/2)*b^2*1i + b^2))/(9*a^(7/3)) + (b^2*log((4*b^4*(a + b* x)^(1/3))/(9*a^4) - (9*b^4*((3^(1/2)*1i)/9 - 1/9)^2)/a^(11/3))*((3^(1/2)*1 i)/9 - 1/9))/a^(7/3)
Time = 0.17 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.58 \[ \int \frac {1}{x^3 \sqrt [3]{a+b x}} \, dx=\frac {-4 \sqrt {3}\, \mathit {atan} \left (\frac {2 \left (b x +a \right )^{\frac {1}{6}}+a^{\frac {1}{6}}}{a^{\frac {1}{6}} \sqrt {3}}\right ) b^{2} x^{2}+4 \sqrt {3}\, \mathit {atan} \left (\frac {2 \left (b x +a \right )^{\frac {1}{6}}-a^{\frac {1}{6}}}{a^{\frac {1}{6}} \sqrt {3}}\right ) b^{2} x^{2}-9 a^{\frac {4}{3}} \left (b x +a \right )^{\frac {2}{3}}+12 a^{\frac {1}{3}} \left (b x +a \right )^{\frac {2}{3}} b x +4 \,\mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}+a^{\frac {1}{6}}\right ) b^{2} x^{2}+4 \,\mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}-a^{\frac {1}{6}}\right ) b^{2} x^{2}-2 \,\mathrm {log}\left (-a^{\frac {1}{6}} \left (b x +a \right )^{\frac {1}{6}}+\left (b x +a \right )^{\frac {1}{3}}+a^{\frac {1}{3}}\right ) b^{2} x^{2}-2 \,\mathrm {log}\left (a^{\frac {1}{6}} \left (b x +a \right )^{\frac {1}{6}}+\left (b x +a \right )^{\frac {1}{3}}+a^{\frac {1}{3}}\right ) b^{2} x^{2}}{18 a^{\frac {7}{3}} x^{2}} \] Input:
int(1/x^3/(b*x+a)^(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*a**(1/3)*a**2*x**2)