Optimal. Leaf size=375 \[ \frac {2 b c-a d}{10 a^3 x^{10}}-\frac {c}{13 a^2 x^{13}}-\frac {a^2 e-2 a b d+3 b^2 c}{7 a^4 x^7}-\frac {b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right )}{18 a^{19/3}}+\frac {b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right )}{9 a^{19/3}}+\frac {b^{4/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right )}{3 \sqrt {3} a^{19/3}}-\frac {b^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^6 \left (a+b x^3\right )}-\frac {b \left (-2 a^3 f+3 a^2 b e-4 a b^2 d+5 b^3 c\right )}{a^6 x}+\frac {a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{4 a^5 x^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.53, antiderivative size = 375, 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, 1834, 292, 31, 634, 617, 204, 628} \[ -\frac {b^2 x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^6 \left (a+b x^3\right )}+\frac {2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c}{4 a^5 x^4}-\frac {b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (10 a^2 b e-7 a^3 f-13 a b^2 d+16 b^3 c\right )}{18 a^{19/3}}-\frac {b \left (3 a^2 b e-2 a^3 f-4 a b^2 d+5 b^3 c\right )}{a^6 x}+\frac {b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (10 a^2 b e-7 a^3 f-13 a b^2 d+16 b^3 c\right )}{9 a^{19/3}}+\frac {b^{4/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (10 a^2 b e-7 a^3 f-13 a b^2 d+16 b^3 c\right )}{3 \sqrt {3} a^{19/3}}-\frac {a^2 e-2 a b d+3 b^2 c}{7 a^4 x^7}+\frac {2 b c-a d}{10 a^3 x^{10}}-\frac {c}{13 a^2 x^{13}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 204
Rule 292
Rule 617
Rule 628
Rule 634
Rule 1829
Rule 1834
Rubi steps
\begin {align*} \int \frac {c+d x^3+e x^6+f x^9}{x^{14} \left (a+b x^3\right )^2} \, dx &=-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^6 \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-\frac {3 b^3 \left (b^2 c-a b d+a^2 e\right ) x^6}{a^2}+\frac {3 b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^9}{a^3}-\frac {3 b^4 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^{12}}{a^4}+\frac {b^5 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^{15}}{a^5}}{x^{14} \left (a+b x^3\right )} \, dx}{3 a b^3}\\ &=-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^6 \left (a+b x^3\right )}-\frac {\int \left (-\frac {3 b^3 c}{a x^{14}}-\frac {3 b^3 (-2 b c+a d)}{a^2 x^{11}}-\frac {3 b^3 \left (3 b^2 c-2 a b d+a^2 e\right )}{a^3 x^8}-\frac {3 b^3 \left (-4 b^3 c+3 a b^2 d-2 a^2 b e+a^3 f\right )}{a^4 x^5}+\frac {3 b^4 \left (-5 b^3 c+4 a b^2 d-3 a^2 b e+2 a^3 f\right )}{a^5 x^2}-\frac {b^5 \left (-16 b^3 c+13 a b^2 d-10 a^2 b e+7 a^3 f\right ) x}{a^5 \left (a+b x^3\right )}\right ) \, dx}{3 a b^3}\\ &=-\frac {c}{13 a^2 x^{13}}+\frac {2 b c-a d}{10 a^3 x^{10}}-\frac {3 b^2 c-2 a b d+a^2 e}{7 a^4 x^7}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{4 a^5 x^4}-\frac {b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right )}{a^6 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^6 \left (a+b x^3\right )}-\frac {\left (b^2 \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right )\right ) \int \frac {x}{a+b x^3} \, dx}{3 a^6}\\ &=-\frac {c}{13 a^2 x^{13}}+\frac {2 b c-a d}{10 a^3 x^{10}}-\frac {3 b^2 c-2 a b d+a^2 e}{7 a^4 x^7}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{4 a^5 x^4}-\frac {b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right )}{a^6 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^6 \left (a+b x^3\right )}+\frac {\left (b^{5/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{19/3}}-\frac {\left (b^{5/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right )\right ) \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^{19/3}}\\ &=-\frac {c}{13 a^2 x^{13}}+\frac {2 b c-a d}{10 a^3 x^{10}}-\frac {3 b^2 c-2 a b d+a^2 e}{7 a^4 x^7}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{4 a^5 x^4}-\frac {b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right )}{a^6 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^6 \left (a+b x^3\right )}+\frac {b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{19/3}}-\frac {\left (b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right )\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^{19/3}}-\frac {\left (b^{5/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right )\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^6}\\ &=-\frac {c}{13 a^2 x^{13}}+\frac {2 b c-a d}{10 a^3 x^{10}}-\frac {3 b^2 c-2 a b d+a^2 e}{7 a^4 x^7}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{4 a^5 x^4}-\frac {b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right )}{a^6 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^6 \left (a+b x^3\right )}+\frac {b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{19/3}}-\frac {b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{19/3}}-\frac {\left (b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right )\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^{19/3}}\\ &=-\frac {c}{13 a^2 x^{13}}+\frac {2 b c-a d}{10 a^3 x^{10}}-\frac {3 b^2 c-2 a b d+a^2 e}{7 a^4 x^7}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{4 a^5 x^4}-\frac {b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right )}{a^6 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^6 \left (a+b x^3\right )}+\frac {b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 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^{19/3}}+\frac {b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{19/3}}-\frac {b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{19/3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.41, size = 370, normalized size = 0.99 \[ \frac {2 b c-a d}{10 a^3 x^{10}}-\frac {c}{13 a^2 x^{13}}-\frac {a^2 e-2 a b d+3 b^2 c}{7 a^4 x^7}+\frac {b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (7 a^3 f-10 a^2 b e+13 a b^2 d-16 b^3 c\right )}{18 a^{19/3}}+\frac {b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right )}{9 a^{19/3}}+\frac {b^{4/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right )}{3 \sqrt {3} a^{19/3}}+\frac {b^2 x^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{3 a^6 \left (a+b x^3\right )}+\frac {b \left (2 a^3 f-3 a^2 b e+4 a b^2 d-5 b^3 c\right )}{a^6 x}+\frac {a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{4 a^5 x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.81, size = 507, normalized size = 1.35 \[ -\frac {5460 \, {\left (16 \, b^{5} c - 13 \, a b^{4} d + 10 \, a^{2} b^{3} e - 7 \, a^{3} b^{2} f\right )} x^{15} + 4095 \, {\left (16 \, a b^{4} c - 13 \, a^{2} b^{3} d + 10 \, a^{3} b^{2} e - 7 \, a^{4} b f\right )} x^{12} - 585 \, {\left (16 \, a^{2} b^{3} c - 13 \, a^{3} b^{2} d + 10 \, a^{4} b e - 7 \, a^{5} f\right )} x^{9} + 234 \, {\left (16 \, a^{3} b^{2} c - 13 \, a^{4} b d + 10 \, a^{5} e\right )} x^{6} + 1260 \, a^{5} c - 126 \, {\left (16 \, a^{4} b c - 13 \, a^{5} d\right )} x^{3} + 1820 \, \sqrt {3} {\left ({\left (16 \, b^{5} c - 13 \, a b^{4} d + 10 \, a^{2} b^{3} e - 7 \, a^{3} b^{2} f\right )} x^{16} + {\left (16 \, a b^{4} c - 13 \, a^{2} b^{3} d + 10 \, a^{3} b^{2} e - 7 \, a^{4} b f\right )} x^{13}\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - 910 \, {\left ({\left (16 \, b^{5} c - 13 \, a b^{4} d + 10 \, a^{2} b^{3} e - 7 \, a^{3} b^{2} f\right )} x^{16} + {\left (16 \, a b^{4} c - 13 \, a^{2} b^{3} d + 10 \, a^{3} b^{2} e - 7 \, a^{4} b f\right )} x^{13}\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (-\frac {b}{a}\right )^{\frac {2}{3}} - a \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right ) + 1820 \, {\left ({\left (16 \, b^{5} c - 13 \, a b^{4} d + 10 \, a^{2} b^{3} e - 7 \, a^{3} b^{2} f\right )} x^{16} + {\left (16 \, a b^{4} c - 13 \, a^{2} b^{3} d + 10 \, a^{3} b^{2} e - 7 \, a^{4} b f\right )} x^{13}\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{16380 \, {\left (a^{6} b x^{16} + a^{7} x^{13}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 482, normalized size = 1.29 \[ \frac {\sqrt {3} {\left (16 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 13 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d - 7 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f + 10 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{7}} + \frac {{\left (16 \, b^{5} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 13 \, a b^{4} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 7 \, a^{3} b^{2} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 10 \, a^{2} b^{3} \left (-\frac {a}{b}\right )^{\frac {1}{3}} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{7}} - \frac {{\left (16 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 13 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d - 7 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f + 10 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{7}} - \frac {b^{5} c x^{2} - a b^{4} d x^{2} - a^{3} b^{2} f x^{2} + a^{2} b^{3} x^{2} e}{3 \, {\left (b x^{3} + a\right )} a^{6}} - \frac {9100 \, b^{4} c x^{12} - 7280 \, a b^{3} d x^{12} - 3640 \, a^{3} b f x^{12} + 5460 \, a^{2} b^{2} x^{12} e - 1820 \, a b^{3} c x^{9} + 1365 \, a^{2} b^{2} d x^{9} + 455 \, a^{4} f x^{9} - 910 \, a^{3} b x^{9} e + 780 \, a^{2} b^{2} c x^{6} - 520 \, a^{3} b d x^{6} + 260 \, a^{4} x^{6} e - 364 \, a^{3} b c x^{3} + 182 \, a^{4} d x^{3} + 140 \, a^{4} c}{1820 \, a^{6} x^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 631, normalized size = 1.68 \[ \frac {b^{2} f \,x^{2}}{3 \left (b \,x^{3}+a \right ) a^{3}}-\frac {b^{3} e \,x^{2}}{3 \left (b \,x^{3}+a \right ) a^{4}}+\frac {b^{4} d \,x^{2}}{3 \left (b \,x^{3}+a \right ) a^{5}}-\frac {b^{5} c \,x^{2}}{3 \left (b \,x^{3}+a \right ) a^{6}}+\frac {7 \sqrt {3}\, b 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 {1}{3}} a^{3}}-\frac {7 b f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}+\frac {7 b 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 {1}{3}} a^{3}}-\frac {10 \sqrt {3}\, b^{2} 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 {1}{3}} a^{4}}+\frac {10 b^{2} e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{4}}-\frac {5 b^{2} 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 {1}{3}} a^{4}}+\frac {13 \sqrt {3}\, b^{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 {1}{3}} a^{5}}-\frac {13 b^{3} d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{5}}+\frac {13 b^{3} 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 {1}{3}} a^{5}}-\frac {16 \sqrt {3}\, b^{4} 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 {1}{3}} a^{6}}+\frac {16 b^{4} c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{6}}-\frac {8 b^{4} 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 {1}{3}} a^{6}}+\frac {2 b f}{a^{3} x}-\frac {3 b^{2} e}{a^{4} x}+\frac {4 b^{3} d}{a^{5} x}-\frac {5 b^{4} c}{a^{6} x}-\frac {f}{4 a^{2} x^{4}}+\frac {b e}{2 a^{3} x^{4}}-\frac {3 b^{2} d}{4 a^{4} x^{4}}+\frac {b^{3} c}{a^{5} x^{4}}-\frac {e}{7 a^{2} x^{7}}+\frac {2 b d}{7 a^{3} x^{7}}-\frac {3 b^{2} c}{7 a^{4} x^{7}}-\frac {d}{10 a^{2} x^{10}}+\frac {b c}{5 a^{3} x^{10}}-\frac {c}{13 a^{2} x^{13}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.97, size = 374, normalized size = 1.00 \[ -\frac {1820 \, {\left (16 \, b^{5} c - 13 \, a b^{4} d + 10 \, a^{2} b^{3} e - 7 \, a^{3} b^{2} f\right )} x^{15} + 1365 \, {\left (16 \, a b^{4} c - 13 \, a^{2} b^{3} d + 10 \, a^{3} b^{2} e - 7 \, a^{4} b f\right )} x^{12} - 195 \, {\left (16 \, a^{2} b^{3} c - 13 \, a^{3} b^{2} d + 10 \, a^{4} b e - 7 \, a^{5} f\right )} x^{9} + 78 \, {\left (16 \, a^{3} b^{2} c - 13 \, a^{4} b d + 10 \, a^{5} e\right )} x^{6} + 420 \, a^{5} c - 42 \, {\left (16 \, a^{4} b c - 13 \, a^{5} d\right )} x^{3}}{5460 \, {\left (a^{6} b x^{16} + a^{7} x^{13}\right )}} - \frac {\sqrt {3} {\left (16 \, b^{4} c - 13 \, a b^{3} d + 10 \, a^{2} b^{2} e - 7 \, a^{3} b f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (16 \, b^{4} c - 13 \, a b^{3} d + 10 \, a^{2} b^{2} e - 7 \, a^{3} b f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (16 \, b^{4} c - 13 \, a b^{3} d + 10 \, a^{2} b^{2} e - 7 \, a^{3} b f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 5.12, size = 348, normalized size = 0.93 \[ \frac {b^{4/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-7\,f\,a^3+10\,e\,a^2\,b-13\,d\,a\,b^2+16\,c\,b^3\right )}{9\,a^{19/3}}-\frac {\frac {c}{13\,a}-\frac {x^9\,\left (-7\,f\,a^3+10\,e\,a^2\,b-13\,d\,a\,b^2+16\,c\,b^3\right )}{28\,a^4}+\frac {x^3\,\left (13\,a\,d-16\,b\,c\right )}{130\,a^2}+\frac {x^6\,\left (10\,e\,a^2-13\,d\,a\,b+16\,c\,b^2\right )}{70\,a^3}+\frac {b\,x^{12}\,\left (-7\,f\,a^3+10\,e\,a^2\,b-13\,d\,a\,b^2+16\,c\,b^3\right )}{4\,a^5}+\frac {b^2\,x^{15}\,\left (-7\,f\,a^3+10\,e\,a^2\,b-13\,d\,a\,b^2+16\,c\,b^3\right )}{3\,a^6}}{b\,x^{16}+a\,x^{13}}-\frac {b^{4/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-7\,f\,a^3+10\,e\,a^2\,b-13\,d\,a\,b^2+16\,c\,b^3\right )}{9\,a^{19/3}}+\frac {b^{4/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-7\,f\,a^3+10\,e\,a^2\,b-13\,d\,a\,b^2+16\,c\,b^3\right )}{9\,a^{19/3}} \]
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]
________________________________________________________________________________________