Optimal. Leaf size=270 \[ \frac {2 b c-a d}{2 a^3 x^2}-\frac {c}{5 a^2 x^5}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^3 b \left (a+b x^3\right )}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right )}{18 a^{11/3} b^{4/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right )}{9 a^{11/3} b^{4/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right )}{3 \sqrt {3} a^{11/3} b^{4/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.27, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1829, 1488, 200, 31, 634, 617, 204, 628} \[ \frac {x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^3 b \left (a+b x^3\right )}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (2 a^2 b e+a^3 f-5 a b^2 d+8 b^3 c\right )}{18 a^{11/3} b^{4/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (2 a^2 b e+a^3 f-5 a b^2 d+8 b^3 c\right )}{9 a^{11/3} b^{4/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (2 a^2 b e+a^3 f-5 a b^2 d+8 b^3 c\right )}{3 \sqrt {3} a^{11/3} b^{4/3}}+\frac {2 b c-a d}{2 a^3 x^2}-\frac {c}{5 a^2 x^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 200
Rule 204
Rule 617
Rule 628
Rule 634
Rule 1488
Rule 1829
Rubi steps
\begin {align*} \int \frac {c+d x^3+e x^6+f x^9}{x^6 \left (a+b x^3\right )^2} \, dx &=\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^3 b \left (a+b x^3\right )}-\frac {\int \frac {-3 b^3 c+3 b^3 \left (\frac {b c}{a}-d\right ) x^3-b^2 \left (\frac {2 b^3 c}{a^2}-\frac {2 b^2 d}{a}+2 b e+a f\right ) x^6}{x^6 \left (a+b x^3\right )} \, dx}{3 a b^3}\\ &=\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^3 b \left (a+b x^3\right )}-\frac {\int \left (-\frac {3 b^3 c}{a x^6}-\frac {3 b^3 (-2 b c+a d)}{a^2 x^3}-\frac {b^2 \left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right )}{a^2 \left (a+b x^3\right )}\right ) \, dx}{3 a b^3}\\ &=-\frac {c}{5 a^2 x^5}+\frac {2 b c-a d}{2 a^3 x^2}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^3 b \left (a+b x^3\right )}+\frac {\left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) \int \frac {1}{a+b x^3} \, dx}{3 a^3 b}\\ &=-\frac {c}{5 a^2 x^5}+\frac {2 b c-a d}{2 a^3 x^2}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^3 b \left (a+b x^3\right )}+\frac {\left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{11/3} b}+\frac {\left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) \int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^{11/3} b}\\ &=-\frac {c}{5 a^2 x^5}+\frac {2 b c-a d}{2 a^3 x^2}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^3 b \left (a+b x^3\right )}+\frac {\left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{11/3} b^{4/3}}-\frac {\left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{11/3} b^{4/3}}+\frac {\left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{10/3} b}\\ &=-\frac {c}{5 a^2 x^5}+\frac {2 b c-a d}{2 a^3 x^2}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^3 b \left (a+b x^3\right )}+\frac {\left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{11/3} b^{4/3}}-\frac {\left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{11/3} b^{4/3}}+\frac {\left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a^{11/3} b^{4/3}}\\ &=-\frac {c}{5 a^2 x^5}+\frac {2 b c-a d}{2 a^3 x^2}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{3 a^3 b \left (a+b x^3\right )}-\frac {\left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{11/3} b^{4/3}}+\frac {\left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{11/3} b^{4/3}}-\frac {\left (8 b^3 c-5 a b^2 d+2 a^2 b e+a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{11/3} b^{4/3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.19, size = 253, normalized size = 0.94 \[ \frac {-\frac {45 a^{2/3} (a d-2 b c)}{x^2}-\frac {18 a^{5/3} c}{x^5}+\frac {10 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right )}{b^{4/3}}-\frac {10 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right )}{b^{4/3}}-\frac {30 a^{2/3} x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{b \left (a+b x^3\right )}-\frac {5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f+2 a^2 b e-5 a b^2 d+8 b^3 c\right )}{b^{4/3}}}{90 a^{11/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.62, size = 897, normalized size = 3.32 \[ \left [-\frac {18 \, a^{4} b^{2} c - 15 \, {\left (8 \, a^{2} b^{4} c - 5 \, a^{3} b^{3} d + 2 \, a^{4} b^{2} e - 2 \, a^{5} b f\right )} x^{6} - 9 \, {\left (8 \, a^{3} b^{3} c - 5 \, a^{4} b^{2} d\right )} x^{3} - 15 \, \sqrt {\frac {1}{3}} {\left ({\left (8 \, a b^{5} c - 5 \, a^{2} b^{4} d + 2 \, a^{3} b^{3} e + a^{4} b^{2} f\right )} x^{8} + {\left (8 \, a^{2} b^{4} c - 5 \, a^{3} b^{3} d + 2 \, a^{4} b^{2} e + a^{5} b f\right )} x^{5}\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{3} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{3} + a}\right ) + 5 \, {\left ({\left (8 \, b^{4} c - 5 \, a b^{3} d + 2 \, a^{2} b^{2} e + a^{3} b f\right )} x^{8} + {\left (8 \, a b^{3} c - 5 \, a^{2} b^{2} d + 2 \, a^{3} b e + a^{4} f\right )} x^{5}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) - 10 \, {\left ({\left (8 \, b^{4} c - 5 \, a b^{3} d + 2 \, a^{2} b^{2} e + a^{3} b f\right )} x^{8} + {\left (8 \, a b^{3} c - 5 \, a^{2} b^{2} d + 2 \, a^{3} b e + a^{4} f\right )} x^{5}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{90 \, {\left (a^{5} b^{3} x^{8} + a^{6} b^{2} x^{5}\right )}}, -\frac {18 \, a^{4} b^{2} c - 15 \, {\left (8 \, a^{2} b^{4} c - 5 \, a^{3} b^{3} d + 2 \, a^{4} b^{2} e - 2 \, a^{5} b f\right )} x^{6} - 9 \, {\left (8 \, a^{3} b^{3} c - 5 \, a^{4} b^{2} d\right )} x^{3} - 30 \, \sqrt {\frac {1}{3}} {\left ({\left (8 \, a b^{5} c - 5 \, a^{2} b^{4} d + 2 \, a^{3} b^{3} e + a^{4} b^{2} f\right )} x^{8} + {\left (8 \, a^{2} b^{4} c - 5 \, a^{3} b^{3} d + 2 \, a^{4} b^{2} e + a^{5} b f\right )} x^{5}\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) + 5 \, {\left ({\left (8 \, b^{4} c - 5 \, a b^{3} d + 2 \, a^{2} b^{2} e + a^{3} b f\right )} x^{8} + {\left (8 \, a b^{3} c - 5 \, a^{2} b^{2} d + 2 \, a^{3} b e + a^{4} f\right )} x^{5}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) - 10 \, {\left ({\left (8 \, b^{4} c - 5 \, a b^{3} d + 2 \, a^{2} b^{2} e + a^{3} b f\right )} x^{8} + {\left (8 \, a b^{3} c - 5 \, a^{2} b^{2} d + 2 \, a^{3} b e + a^{4} f\right )} x^{5}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{90 \, {\left (a^{5} b^{3} x^{8} + a^{6} b^{2} x^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 264, normalized size = 0.98 \[ -\frac {\sqrt {3} {\left (8 \, b^{3} c - 5 \, a b^{2} d + a^{3} f + 2 \, a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3}} - \frac {{\left (8 \, b^{3} c - 5 \, a b^{2} d + a^{3} f + 2 \, a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3}} - \frac {{\left (8 \, b^{3} c - 5 \, a b^{2} d + a^{3} f + 2 \, a^{2} b e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{4} b} + \frac {b^{3} c x - a b^{2} d x - a^{3} f x + a^{2} b x e}{3 \, {\left (b x^{3} + a\right )} a^{3} b} + \frac {10 \, b c x^{3} - 5 \, a d x^{3} - 2 \, a c}{10 \, a^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 477, normalized size = 1.77 \[ \frac {e x}{3 \left (b \,x^{3}+a \right ) a}-\frac {b d x}{3 \left (b \,x^{3}+a \right ) a^{2}}+\frac {b^{2} c x}{3 \left (b \,x^{3}+a \right ) a^{3}}-\frac {f x}{3 \left (b \,x^{3}+a \right ) b}+\frac {2 \sqrt {3}\, e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a b}+\frac {2 e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a b}-\frac {e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a b}-\frac {5 \sqrt {3}\, d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2}}-\frac {5 d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2}}+\frac {5 d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2}}+\frac {8 \sqrt {3}\, b c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3}}+\frac {8 b c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3}}-\frac {4 b c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3}}+\frac {\sqrt {3}\, f \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}+\frac {f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}-\frac {f \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}-\frac {d}{2 a^{2} x^{2}}+\frac {b c}{a^{3} x^{2}}-\frac {c}{5 a^{2} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.93, size = 268, normalized size = 0.99 \[ \frac {5 \, {\left (8 \, b^{3} c - 5 \, a b^{2} d + 2 \, a^{2} b e - 2 \, a^{3} f\right )} x^{6} - 6 \, a^{2} b c + 3 \, {\left (8 \, a b^{2} c - 5 \, a^{2} b d\right )} x^{3}}{30 \, {\left (a^{3} b^{2} x^{8} + a^{4} b x^{5}\right )}} + \frac {\sqrt {3} {\left (8 \, b^{3} c - 5 \, a b^{2} d + 2 \, a^{2} b e + a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (8 \, b^{3} c - 5 \, a b^{2} d + 2 \, a^{2} b e + a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (8 \, b^{3} c - 5 \, a b^{2} d + 2 \, a^{2} b e + a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 5.13, size = 248, normalized size = 0.92 \[ \frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (f\,a^3+2\,e\,a^2\,b-5\,d\,a\,b^2+8\,c\,b^3\right )}{9\,a^{11/3}\,b^{4/3}}-\frac {\frac {c}{5\,a}+\frac {x^3\,\left (5\,a\,d-8\,b\,c\right )}{10\,a^2}-\frac {x^6\,\left (-2\,f\,a^3+2\,e\,a^2\,b-5\,d\,a\,b^2+8\,c\,b^3\right )}{6\,a^3\,b}}{b\,x^8+a\,x^5}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (f\,a^3+2\,e\,a^2\,b-5\,d\,a\,b^2+8\,c\,b^3\right )}{9\,a^{11/3}\,b^{4/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (f\,a^3+2\,e\,a^2\,b-5\,d\,a\,b^2+8\,c\,b^3\right )}{9\,a^{11/3}\,b^{4/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________