Integrand size = 13, antiderivative size = 113 \[ \int \frac {1}{x^2 (a+b x)^{4/3}} \, dx=-\frac {4 b}{a^2 \sqrt [3]{a+b x}}-\frac {1}{a x \sqrt [3]{a+b x}}-\frac {4 b \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{7/3}}+\frac {2 b \log (x)}{3 a^{7/3}}-\frac {2 b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{7/3}} \] Output:
-4*b/a^2/(b*x+a)^(1/3)-1/a/x/(b*x+a)^(1/3)-4/3*b*arctan(1/3*(a^(1/3)+2*(b* x+a)^(1/3))*3^(1/2)/a^(1/3))*3^(1/2)/a^(7/3)+2/3*b*ln(x)/a^(7/3)-2*b*ln(a^ (1/3)-(b*x+a)^(1/3))/a^(7/3)
Time = 0.20 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^2 (a+b x)^{4/3}} \, dx=\frac {-\frac {3 \sqrt [3]{a} (a+4 b x)}{x \sqrt [3]{a+b x}}-4 \sqrt {3} b \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-4 b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )+2 b \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{3 a^{7/3}} \] Input:
Integrate[1/(x^2*(a + b*x)^(4/3)),x]
Output:
((-3*a^(1/3)*(a + 4*b*x))/(x*(a + b*x)^(1/3)) - 4*Sqrt[3]*b*ArcTan[(1 + (2 *(a + b*x)^(1/3))/a^(1/3))/Sqrt[3]] - 4*b*Log[a^(1/3) - (a + b*x)^(1/3)] + 2*b*Log[a^(2/3) + a^(1/3)*(a + b*x)^(1/3) + (a + b*x)^(2/3)])/(3*a^(7/3))
Time = 0.20 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {52, 61, 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.
| \(\displaystyle \int \frac {1}{x^2 (a+b x)^{4/3}} \, dx\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {4 b \int \frac {1}{x (a+b x)^{4/3}}dx}{3 a}-\frac {1}{a x \sqrt [3]{a+b x}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {4 b \left (\frac {\int \frac {1}{x \sqrt [3]{a+b x}}dx}{a}+\frac {3}{a \sqrt [3]{a+b x}}\right )}{3 a}-\frac {1}{a x \sqrt [3]{a+b x}}\) |
| \(\Big \downarrow \) 67 |
| \(\displaystyle -\frac {4 b \left (\frac {\frac {3}{2} \int \frac {1}{a^{2/3}+\sqrt [3]{a+b x} \sqrt [3]{a}+(a+b x)^{2/3}}d\sqrt [3]{a+b x}-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{a+b x}}d\sqrt [3]{a+b x}}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x}}\right )}{3 a}-\frac {1}{a x \sqrt [3]{a+b x}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {4 b \left (\frac {\frac {3}{2} \int \frac {1}{a^{2/3}+\sqrt [3]{a+b x} \sqrt [3]{a}+(a+b x)^{2/3}}d\sqrt [3]{a+b x}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x}}\right )}{3 a}-\frac {1}{a x \sqrt [3]{a+b x}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {4 b \left (\frac {-\frac {3 \int \frac {1}{-(a+b x)^{2/3}-3}d\left (\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x}}\right )}{3 a}-\frac {1}{a x \sqrt [3]{a+b x}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {4 b \left (\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{2 \sqrt [3]{a}}-\frac {\log (x)}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x}}\right )}{3 a}-\frac {1}{a x \sqrt [3]{a+b x}}\) |
Input:
Int[1/(x^2*(a + b*x)^(4/3)),x]
Output:
-(1/(a*x*(a + b*x)^(1/3))) - (4*b*(3/(a*(a + b*x)^(1/3)) + ((Sqrt[3]*ArcTa n[(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)))/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[((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] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
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 = 0.23 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.96
| method | result | size |
| risch | \(-\frac {\left (b x +a \right )^{\frac {2}{3}}}{a^{2} x}-\frac {b \left (\frac {9}{\left (b x +a \right )^{\frac {1}{3}}}+\frac {4 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}}-\frac {2 \ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{a^{\frac {1}{3}}}+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{a^{\frac {1}{3}}}\right )}{3 a^{2}}\) | \(108\) |
| pseudoelliptic | \(-\frac {\left (b x +a \right )^{\frac {2}{3}}}{a^{2} x}-\frac {4 b \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {7}{3}}}+\frac {2 b \ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{3 a^{\frac {7}{3}}}-\frac {4 b \sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (b x +a \right )^{\frac {1}{3}}}{3 a^{\frac {1}{3}}}+\frac {\sqrt {3}}{3}\right )}{3 a^{\frac {7}{3}}}-\frac {3 b}{a^{2} \left (b x +a \right )^{\frac {1}{3}}}\) | \(110\) |
| derivativedivides | \(3 b \left (-\frac {1}{a^{2} \left (b x +a \right )^{\frac {1}{3}}}+\frac {-\frac {\left (b x +a \right )^{\frac {2}{3}}}{3 b x}-\frac {4 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 a^{\frac {1}{3}}}+\frac {2 \ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{9 a^{\frac {1}{3}}}-\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a^{\frac {1}{3}}}}{a^{2}}\right )\) | \(112\) |
| default | \(3 b \left (-\frac {1}{a^{2} \left (b x +a \right )^{\frac {1}{3}}}+\frac {-\frac {\left (b x +a \right )^{\frac {2}{3}}}{3 b x}-\frac {4 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 a^{\frac {1}{3}}}+\frac {2 \ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{9 a^{\frac {1}{3}}}-\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a^{\frac {1}{3}}}}{a^{2}}\right )\) | \(112\) |
Input:
int(1/x^2/(b*x+a)^(4/3),x,method=_RETURNVERBOSE)
Output:
-1/a^2*(b*x+a)^(2/3)/x-1/3*b/a^2*(9/(b*x+a)^(1/3)+4/a^(1/3)*ln((b*x+a)^(1/ 3)-a^(1/3))-2/a^(1/3)*ln((b*x+a)^(2/3)+a^(1/3)*(b*x+a)^(1/3)+a^(2/3))+4*3^ (1/2)/a^(1/3)*arctan(1/3*3^(1/2)*(2/a^(1/3)*(b*x+a)^(1/3)+1)))
Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (88) = 176\).
Time = 0.11 (sec) , antiderivative size = 407, normalized size of antiderivative = 3.60 \[ \int \frac {1}{x^2 (a+b x)^{4/3}} \, dx=\left [\frac {6 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x^{2} + a^{2} b x\right )} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b x - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {2}{3}} \left (-a\right )^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} a + \left (-a\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} - 3 \, {\left (b x + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {2}{3}} + 3 \, a}{x}\right ) + 2 \, {\left (b^{2} x^{2} + a b x\right )} \left (-a\right )^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) - 4 \, {\left (b^{2} x^{2} + a b x\right )} \left (-a\right )^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right ) - 3 \, {\left (4 \, a b x + a^{2}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{3 \, {\left (a^{3} b x^{2} + a^{4} x\right )}}, -\frac {12 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x^{2} + a^{2} b x\right )} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \arctan \left (\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} - \left (-a\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}}\right ) - 2 \, {\left (b^{2} x^{2} + a b x\right )} \left (-a\right )^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) + 4 \, {\left (b^{2} x^{2} + a b x\right )} \left (-a\right )^{\frac {2}{3}} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right ) + 3 \, {\left (4 \, a b x + a^{2}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{3 \, {\left (a^{3} b x^{2} + a^{4} x\right )}}\right ] \] Input:
integrate(1/x^2/(b*x+a)^(4/3),x, algorithm="fricas")
Output:
[1/3*(6*sqrt(1/3)*(a*b^2*x^2 + a^2*b*x)*sqrt((-a)^(1/3)/a)*log((2*b*x - 3* sqrt(1/3)*(2*(b*x + a)^(2/3)*(-a)^(2/3) - (b*x + a)^(1/3)*a + (-a)^(1/3)*a )*sqrt((-a)^(1/3)/a) - 3*(b*x + a)^(1/3)*(-a)^(2/3) + 3*a)/x) + 2*(b^2*x^2 + a*b*x)*(-a)^(2/3)*log((b*x + a)^(2/3) - (b*x + a)^(1/3)*(-a)^(1/3) + (- a)^(2/3)) - 4*(b^2*x^2 + a*b*x)*(-a)^(2/3)*log((b*x + a)^(1/3) + (-a)^(1/3 )) - 3*(4*a*b*x + a^2)*(b*x + a)^(2/3))/(a^3*b*x^2 + a^4*x), -1/3*(12*sqrt (1/3)*(a*b^2*x^2 + a^2*b*x)*sqrt(-(-a)^(1/3)/a)*arctan(sqrt(1/3)*(2*(b*x + a)^(1/3) - (-a)^(1/3))*sqrt(-(-a)^(1/3)/a)) - 2*(b^2*x^2 + a*b*x)*(-a)^(2 /3)*log((b*x + a)^(2/3) - (b*x + a)^(1/3)*(-a)^(1/3) + (-a)^(2/3)) + 4*(b^ 2*x^2 + a*b*x)*(-a)^(2/3)*log((b*x + a)^(1/3) + (-a)^(1/3)) + 3*(4*a*b*x + a^2)*(b*x + a)^(2/3))/(a^3*b*x^2 + a^4*x)]
Result contains complex when optimal does not.
Time = 1.33 (sec) , antiderivative size = 857, normalized size of antiderivative = 7.58 \[ \int \frac {1}{x^2 (a+b x)^{4/3}} \, dx=\text {Too large to display} \] Input:
integrate(1/x**2/(b*x+a)**(4/3),x)
Output:
-9*a**(4/3)*b**(2/3)*exp(2*I*pi/3)*gamma(-1/3)/(-9*a**(10/3)*(a/b + x)**(1 /3)*exp(2*I*pi/3)*gamma(2/3) + 9*a**(7/3)*b*(a/b + x)**(4/3)*exp(2*I*pi/3) *gamma(2/3)) + 12*a**(1/3)*b**(5/3)*(a/b + x)*exp(2*I*pi/3)*gamma(-1/3)/(- 9*a**(10/3)*(a/b + x)**(1/3)*exp(2*I*pi/3)*gamma(2/3) + 9*a**(7/3)*b*(a/b + x)**(4/3)*exp(2*I*pi/3)*gamma(2/3)) - 4*a*b*(a/b + x)**(1/3)*exp(2*I*pi/ 3)*log(1 - b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gamma(-1/3)/(-9*a**(10/3)*( a/b + x)**(1/3)*exp(2*I*pi/3)*gamma(2/3) + 9*a**(7/3)*b*(a/b + x)**(4/3)*e xp(2*I*pi/3)*gamma(2/3)) - 4*a*b*(a/b + x)**(1/3)*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(-1/3)/(-9*a** (10/3)*(a/b + x)**(1/3)*exp(2*I*pi/3)*gamma(2/3) + 9*a**(7/3)*b*(a/b + x)* *(4/3)*exp(2*I*pi/3)*gamma(2/3)) - 4*a*b*(a/b + x)**(1/3)*log(1 - b**(1/3) *(a/b + x)**(1/3)*exp_polar(4*I*pi/3)/a**(1/3))*gamma(-1/3)/(-9*a**(10/3)* (a/b + x)**(1/3)*exp(2*I*pi/3)*gamma(2/3) + 9*a**(7/3)*b*(a/b + x)**(4/3)* exp(2*I*pi/3)*gamma(2/3)) + 4*b**2*(a/b + x)**(4/3)*exp(2*I*pi/3)*log(1 - b**(1/3)*(a/b + x)**(1/3)/a**(1/3))*gamma(-1/3)/(-9*a**(10/3)*(a/b + x)**( 1/3)*exp(2*I*pi/3)*gamma(2/3) + 9*a**(7/3)*b*(a/b + x)**(4/3)*exp(2*I*pi/3 )*gamma(2/3)) + 4*b**2*(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(-1/3)/(-9*a**(10/3)*(a/ b + x)**(1/3)*exp(2*I*pi/3)*gamma(2/3) + 9*a**(7/3)*b*(a/b + x)**(4/3)*exp (2*I*pi/3)*gamma(2/3)) + 4*b**2*(a/b + x)**(4/3)*log(1 - b**(1/3)*(a/b ...
Time = 0.11 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^2 (a+b x)^{4/3}} \, dx=-\frac {4 \, \sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {7}{3}}} - \frac {4 \, {\left (b x + a\right )} b - 3 \, a b}{{\left (b x + a\right )}^{\frac {4}{3}} a^{2} - {\left (b x + a\right )}^{\frac {1}{3}} a^{3}} + \frac {2 \, b \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{3 \, a^{\frac {7}{3}}} - \frac {4 \, b \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{3 \, a^{\frac {7}{3}}} \] Input:
integrate(1/x^2/(b*x+a)^(4/3),x, algorithm="maxima")
Output:
-4/3*sqrt(3)*b*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3))/a ^(7/3) - (4*(b*x + a)*b - 3*a*b)/((b*x + a)^(4/3)*a^2 - (b*x + a)^(1/3)*a^ 3) + 2/3*b*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3))/a^(7/3 ) - 4/3*b*log((b*x + a)^(1/3) - a^(1/3))/a^(7/3)
Time = 0.43 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^2 (a+b x)^{4/3}} \, dx=-\frac {4 \, \sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {7}{3}}} + \frac {2 \, b \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{3 \, a^{\frac {7}{3}}} - \frac {4 \, b \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{3 \, a^{\frac {7}{3}}} - \frac {4 \, {\left (b x + a\right )} b - 3 \, a b}{{\left ({\left (b x + a\right )}^{\frac {4}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} a\right )} a^{2}} \] Input:
integrate(1/x^2/(b*x+a)^(4/3),x, algorithm="giac")
Output:
-4/3*sqrt(3)*b*arctan(1/3*sqrt(3)*(2*(b*x + a)^(1/3) + a^(1/3))/a^(1/3))/a ^(7/3) + 2/3*b*log((b*x + a)^(2/3) + (b*x + a)^(1/3)*a^(1/3) + a^(2/3))/a^ (7/3) - 4/3*b*log(abs((b*x + a)^(1/3) - a^(1/3)))/a^(7/3) - (4*(b*x + a)*b - 3*a*b)/(((b*x + a)^(4/3) - (b*x + a)^(1/3)*a)*a^2)
Time = 0.09 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.53 \[ \int \frac {1}{x^2 (a+b x)^{4/3}} \, dx=-\frac {\frac {3\,b}{a}-\frac {4\,b\,\left (a+b\,x\right )}{a^2}}{a\,{\left (a+b\,x\right )}^{1/3}-{\left (a+b\,x\right )}^{4/3}}+\frac {\ln \left (a^{7/3}\,{\left (2\,b-\sqrt {3}\,b\,2{}\mathrm {i}\right )}^2-16\,a^2\,b^2\,{\left (a+b\,x\right )}^{1/3}\right )\,\left (2\,b-\sqrt {3}\,b\,2{}\mathrm {i}\right )}{3\,a^{7/3}}+\frac {\ln \left (a^{7/3}\,{\left (2\,b+\sqrt {3}\,b\,2{}\mathrm {i}\right )}^2-16\,a^2\,b^2\,{\left (a+b\,x\right )}^{1/3}\right )\,\left (2\,b+\sqrt {3}\,b\,2{}\mathrm {i}\right )}{3\,a^{7/3}}-\frac {4\,b\,\ln \left (16\,a^{7/3}\,b^2-16\,a^2\,b^2\,{\left (a+b\,x\right )}^{1/3}\right )}{3\,a^{7/3}} \] Input:
int(1/(x^2*(a + b*x)^(4/3)),x)
Output:
(log(a^(7/3)*(2*b - 3^(1/2)*b*2i)^2 - 16*a^2*b^2*(a + b*x)^(1/3))*(2*b - 3 ^(1/2)*b*2i))/(3*a^(7/3)) - ((3*b)/a - (4*b*(a + b*x))/a^2)/(a*(a + b*x)^( 1/3) - (a + b*x)^(4/3)) + (log(a^(7/3)*(2*b + 3^(1/2)*b*2i)^2 - 16*a^2*b^2 *(a + b*x)^(1/3))*(2*b + 3^(1/2)*b*2i))/(3*a^(7/3)) - (4*b*log(16*a^(7/3)* b^2 - 16*a^2*b^2*(a + b*x)^(1/3)))/(3*a^(7/3))
Time = 0.18 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.92 \[ \int \frac {1}{x^2 (a+b x)^{4/3}} \, dx=\frac {4 \left (b x +a \right )^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {2 \left (b x +a \right )^{\frac {1}{6}}+a^{\frac {1}{6}}}{a^{\frac {1}{6}} \sqrt {3}}\right ) b x -4 \left (b x +a \right )^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {2 \left (b x +a \right )^{\frac {1}{6}}-a^{\frac {1}{6}}}{a^{\frac {1}{6}} \sqrt {3}}\right ) b x -4 \left (b x +a \right )^{\frac {1}{3}} \mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}+a^{\frac {1}{6}}\right ) b x -4 \left (b x +a \right )^{\frac {1}{3}} \mathrm {log}\left (\left (b x +a \right )^{\frac {1}{6}}-a^{\frac {1}{6}}\right ) b x +2 \left (b x +a \right )^{\frac {1}{3}} \mathrm {log}\left (-a^{\frac {1}{6}} \left (b x +a \right )^{\frac {1}{6}}+\left (b x +a \right )^{\frac {1}{3}}+a^{\frac {1}{3}}\right ) b x +2 \left (b x +a \right )^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{6}} \left (b x +a \right )^{\frac {1}{6}}+\left (b x +a \right )^{\frac {1}{3}}+a^{\frac {1}{3}}\right ) b x -3 a^{\frac {4}{3}}-12 a^{\frac {1}{3}} b x}{3 a^{\frac {7}{3}} \left (b x +a \right )^{\frac {1}{3}} x} \] Input:
int(1/x^2/(b*x+a)^(4/3),x)
Output:
(4*(a + b*x)**(1/3)*sqrt(3)*atan((2*(a + b*x)**(1/6) + a**(1/6))/(a**(1/6) *sqrt(3)))*b*x - 4*(a + b*x)**(1/3)*sqrt(3)*atan((2*(a + b*x)**(1/6) - a** (1/6))/(a**(1/6)*sqrt(3)))*b*x - 4*(a + b*x)**(1/3)*log((a + b*x)**(1/6) + a**(1/6))*b*x - 4*(a + b*x)**(1/3)*log((a + b*x)**(1/6) - a**(1/6))*b*x + 2*(a + b*x)**(1/3)*log( - a**(1/6)*(a + b*x)**(1/6) + (a + b*x)**(1/3) + a**(1/3))*b*x + 2*(a + b*x)**(1/3)*log(a**(1/6)*(a + b*x)**(1/6) + (a + b* x)**(1/3) + a**(1/3))*b*x - 3*a**(1/3)*a - 12*a**(1/3)*b*x)/(3*a**(1/3)*(a + b*x)**(1/3)*a**2*x)