Optimal. Leaf size=195 \[ -\frac {(4 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{7/3} b^{2/3}}+\frac {(4 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{7/3} b^{2/3}}+\frac {(4 A b-a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3} b^{2/3}}+\frac {a B-4 A b}{3 a^2 b x}+\frac {A b-a B}{3 a b x \left (a+b x^3\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 196, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {457, 325, 292, 31, 634, 617, 204, 628} \[ -\frac {(4 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{7/3} b^{2/3}}+\frac {(4 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{7/3} b^{2/3}}+\frac {(4 A b-a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3} b^{2/3}}-\frac {4 A b-a B}{3 a^2 b x}+\frac {A b-a B}{3 a b x \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 204
Rule 292
Rule 325
Rule 457
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {A+B x^3}{x^2 \left (a+b x^3\right )^2} \, dx &=\frac {A b-a B}{3 a b x \left (a+b x^3\right )}+\frac {(4 A b-a B) \int \frac {1}{x^2 \left (a+b x^3\right )} \, dx}{3 a b}\\ &=-\frac {4 A b-a B}{3 a^2 b x}+\frac {A b-a B}{3 a b x \left (a+b x^3\right )}-\frac {(4 A b-a B) \int \frac {x}{a+b x^3} \, dx}{3 a^2}\\ &=-\frac {4 A b-a B}{3 a^2 b x}+\frac {A b-a B}{3 a b x \left (a+b x^3\right )}+\frac {(4 A b-a B) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{7/3} \sqrt [3]{b}}-\frac {(4 A b-a B) \int \frac {\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^{7/3} \sqrt [3]{b}}\\ &=-\frac {4 A b-a B}{3 a^2 b x}+\frac {A b-a B}{3 a b x \left (a+b x^3\right )}+\frac {(4 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{7/3} b^{2/3}}-\frac {(4 A b-a B) \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^{7/3} b^{2/3}}-\frac {(4 A b-a B) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^2 \sqrt [3]{b}}\\ &=-\frac {4 A b-a B}{3 a^2 b x}+\frac {A b-a B}{3 a b x \left (a+b x^3\right )}+\frac {(4 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{7/3} b^{2/3}}-\frac {(4 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{7/3} b^{2/3}}-\frac {(4 A b-a B) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a^{7/3} b^{2/3}}\\ &=-\frac {4 A b-a B}{3 a^2 b x}+\frac {A b-a B}{3 a b x \left (a+b x^3\right )}+\frac {(4 A b-a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3} b^{2/3}}+\frac {(4 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{7/3} b^{2/3}}-\frac {(4 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{7/3} b^{2/3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 164, normalized size = 0.84 \[ \frac {\frac {(a B-4 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+\frac {2 (4 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac {2 \sqrt {3} (4 A b-a B) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}+\frac {6 \sqrt [3]{a} x^2 (a B-A b)}{a+b x^3}-\frac {18 \sqrt [3]{a} A}{x}}{18 a^{7/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.95, size = 570, normalized size = 2.92 \[ \left [-\frac {18 \, A a^{2} b^{2} - 6 \, {\left (B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} + 3 \, \sqrt {\frac {1}{3}} {\left ({\left (B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{4} + {\left (B a^{3} b - 4 \, A a^{2} b^{2}\right )} x\right )} \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x^{3} - a b - 3 \, \sqrt {\frac {1}{3}} {\left (a b x + 2 \, \left (a b^{2}\right )^{\frac {2}{3}} x^{2} - \left (a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (a b^{2}\right )^{\frac {2}{3}} x}{b x^{3} + a}\right ) - {\left ({\left (B a b - 4 \, A b^{2}\right )} x^{4} + {\left (B a^{2} - 4 \, A a b\right )} x\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} - \left (a b^{2}\right )^{\frac {1}{3}} b x + \left (a b^{2}\right )^{\frac {2}{3}}\right ) + 2 \, {\left ({\left (B a b - 4 \, A b^{2}\right )} x^{4} + {\left (B a^{2} - 4 \, A a b\right )} x\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (b x + \left (a b^{2}\right )^{\frac {1}{3}}\right )}{18 \, {\left (a^{3} b^{3} x^{4} + a^{4} b^{2} x\right )}}, -\frac {18 \, A a^{2} b^{2} - 6 \, {\left (B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} + 6 \, \sqrt {\frac {1}{3}} {\left ({\left (B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{4} + {\left (B a^{3} b - 4 \, A a^{2} b^{2}\right )} x\right )} \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x - \left (a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) - {\left ({\left (B a b - 4 \, A b^{2}\right )} x^{4} + {\left (B a^{2} - 4 \, A a b\right )} x\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} - \left (a b^{2}\right )^{\frac {1}{3}} b x + \left (a b^{2}\right )^{\frac {2}{3}}\right ) + 2 \, {\left ({\left (B a b - 4 \, A b^{2}\right )} x^{4} + {\left (B a^{2} - 4 \, A a b\right )} x\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (b x + \left (a b^{2}\right )^{\frac {1}{3}}\right )}{18 \, {\left (a^{3} b^{3} x^{4} + a^{4} b^{2} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 180, normalized size = 0.92 \[ \frac {\sqrt {3} {\left (B a - 4 \, A b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2}} - \frac {{\left (B a - 4 \, A b\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2}} - \frac {{\left (B a \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 4 \, A b \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{3}} + \frac {B a x^{3} - 4 \, A b x^{3} - 3 \, A a}{3 \, {\left (b x^{4} + a x\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 241, normalized size = 1.24 \[ -\frac {A b \,x^{2}}{3 \left (b \,x^{3}+a \right ) a^{2}}+\frac {B \,x^{2}}{3 \left (b \,x^{3}+a \right ) a}-\frac {4 \sqrt {3}\, A \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2}}+\frac {4 A \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2}}-\frac {2 A \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2}}+\frac {\sqrt {3}\, B \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a b}-\frac {B \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a b}+\frac {B \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {1}{3}} a b}-\frac {A}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.08, size = 166, normalized size = 0.85 \[ \frac {{\left (B a - 4 \, A b\right )} x^{3} - 3 \, A a}{3 \, {\left (a^{2} b x^{4} + a^{3} x\right )}} + \frac {\sqrt {3} {\left (B a - 4 \, A b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (B a - 4 \, A b\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (B a - 4 \, A b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.57, size = 156, normalized size = 0.80 \[ \frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (4\,A\,b-B\,a\right )}{9\,a^{7/3}\,b^{2/3}}-\frac {\frac {A}{a}+\frac {x^3\,\left (4\,A\,b-B\,a\right )}{3\,a^2}}{b\,x^4+a\,x}+\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (4\,A\,b-B\,a\right )}{9\,a^{7/3}\,b^{2/3}}-\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (4\,A\,b-B\,a\right )}{9\,a^{7/3}\,b^{2/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.42, size = 122, normalized size = 0.63 \[ \frac {- 3 A a + x^{3} \left (- 4 A b + B a\right )}{3 a^{3} x + 3 a^{2} b x^{4}} + \operatorname {RootSum} {\left (729 t^{3} a^{7} b^{2} - 64 A^{3} b^{3} + 48 A^{2} B a b^{2} - 12 A B^{2} a^{2} b + B^{3} a^{3}, \left (t \mapsto t \log {\left (\frac {81 t^{2} a^{5} b}{16 A^{2} b^{2} - 8 A B a b + B^{2} a^{2}} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________