Optimal. Leaf size=266 \[ \frac {x^9 \left (6 a^2 f-3 a b e+b^2 d\right )}{9 b^5}+\frac {a^3 \left (-7 a^3 f+6 a^2 b e-5 a b^2 d+4 b^3 c\right )}{3 b^8 \left (a+b x^3\right )}+\frac {a^2 \log \left (a+b x^3\right ) \left (-21 a^3 f+15 a^2 b e-10 a b^2 d+6 b^3 c\right )}{3 b^8}-\frac {a x^3 \left (-15 a^3 f+10 a^2 b e-6 a b^2 d+3 b^3 c\right )}{3 b^7}+\frac {x^6 \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )}{6 b^6}-\frac {a^4 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^8 \left (a+b x^3\right )^2}+\frac {x^{12} (b e-3 a f)}{12 b^4}+\frac {f x^{15}}{15 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.44, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1821, 1620} \[ \frac {x^6 \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{6 b^6}-\frac {a x^3 \left (10 a^2 b e-15 a^3 f-6 a b^2 d+3 b^3 c\right )}{3 b^7}+\frac {a^3 \left (6 a^2 b e-7 a^3 f-5 a b^2 d+4 b^3 c\right )}{3 b^8 \left (a+b x^3\right )}-\frac {a^4 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^8 \left (a+b x^3\right )^2}+\frac {a^2 \log \left (a+b x^3\right ) \left (15 a^2 b e-21 a^3 f-10 a b^2 d+6 b^3 c\right )}{3 b^8}+\frac {x^9 \left (6 a^2 f-3 a b e+b^2 d\right )}{9 b^5}+\frac {x^{12} (b e-3 a f)}{12 b^4}+\frac {f x^{15}}{15 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1620
Rule 1821
Rubi steps
\begin {align*} \int \frac {x^{14} \left (c+d x^3+e x^6+f x^9\right )}{\left (a+b x^3\right )^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^4 \left (c+d x+e x^2+f x^3\right )}{(a+b x)^3} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {a \left (-3 b^3 c+6 a b^2 d-10 a^2 b e+15 a^3 f\right )}{b^7}+\frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x}{b^6}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x^2}{b^5}+\frac {(b e-3 a f) x^3}{b^4}+\frac {f x^4}{b^3}-\frac {a^4 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{b^7 (a+b x)^3}+\frac {a^3 \left (-4 b^3 c+5 a b^2 d-6 a^2 b e+7 a^3 f\right )}{b^7 (a+b x)^2}-\frac {a^2 \left (-6 b^3 c+10 a b^2 d-15 a^2 b e+21 a^3 f\right )}{b^7 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {a \left (3 b^3 c-6 a b^2 d+10 a^2 b e-15 a^3 f\right ) x^3}{3 b^7}+\frac {\left (b^3 c-3 a b^2 d+6 a^2 b e-10 a^3 f\right ) x^6}{6 b^6}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) x^9}{9 b^5}+\frac {(b e-3 a f) x^{12}}{12 b^4}+\frac {f x^{15}}{15 b^3}-\frac {a^4 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{6 b^8 \left (a+b x^3\right )^2}+\frac {a^3 \left (4 b^3 c-5 a b^2 d+6 a^2 b e-7 a^3 f\right )}{3 b^8 \left (a+b x^3\right )}+\frac {a^2 \left (6 b^3 c-10 a b^2 d+15 a^2 b e-21 a^3 f\right ) \log \left (a+b x^3\right )}{3 b^8}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.19, size = 246, normalized size = 0.92 \[ \frac {20 b^3 x^9 \left (6 a^2 f-3 a b e+b^2 d\right )+30 b^2 x^6 \left (-10 a^3 f+6 a^2 b e-3 a b^2 d+b^3 c\right )+60 a b x^3 \left (15 a^3 f-10 a^2 b e+6 a b^2 d-3 b^3 c\right )-\frac {60 a^3 \left (7 a^3 f-6 a^2 b e+5 a b^2 d-4 b^3 c\right )}{a+b x^3}+60 a^2 \log \left (a+b x^3\right ) \left (-21 a^3 f+15 a^2 b e-10 a b^2 d+6 b^3 c\right )+\frac {30 a^4 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{\left (a+b x^3\right )^2}+15 b^4 x^{12} (b e-3 a f)+12 b^5 f x^{15}}{180 b^8} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.59, size = 396, normalized size = 1.49 \[ \frac {12 \, b^{7} f x^{21} + 3 \, {\left (5 \, b^{7} e - 7 \, a b^{6} f\right )} x^{18} + 2 \, {\left (10 \, b^{7} d - 15 \, a b^{6} e + 21 \, a^{2} b^{5} f\right )} x^{15} + 5 \, {\left (6 \, b^{7} c - 10 \, a b^{6} d + 15 \, a^{2} b^{5} e - 21 \, a^{3} b^{4} f\right )} x^{12} - 20 \, {\left (6 \, a b^{6} c - 10 \, a^{2} b^{5} d + 15 \, a^{3} b^{4} e - 21 \, a^{4} b^{3} f\right )} x^{9} + 210 \, a^{4} b^{3} c - 270 \, a^{5} b^{2} d + 330 \, a^{6} b e - 390 \, a^{7} f - 30 \, {\left (11 \, a^{2} b^{5} c - 21 \, a^{3} b^{4} d + 34 \, a^{4} b^{3} e - 50 \, a^{5} b^{2} f\right )} x^{6} + 60 \, {\left (a^{3} b^{4} c + a^{4} b^{3} d - 4 \, a^{5} b^{2} e + 8 \, a^{6} b f\right )} x^{3} + 60 \, {\left (6 \, a^{4} b^{3} c - 10 \, a^{5} b^{2} d + 15 \, a^{6} b e - 21 \, a^{7} f + {\left (6 \, a^{2} b^{5} c - 10 \, a^{3} b^{4} d + 15 \, a^{4} b^{3} e - 21 \, a^{5} b^{2} f\right )} x^{6} + 2 \, {\left (6 \, a^{3} b^{4} c - 10 \, a^{4} b^{3} d + 15 \, a^{5} b^{2} e - 21 \, a^{6} b f\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{180 \, {\left (b^{10} x^{6} + 2 \, a b^{9} x^{3} + a^{2} b^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 349, normalized size = 1.31 \[ \frac {{\left (6 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d - 21 \, a^{5} f + 15 \, a^{4} b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{8}} - \frac {18 \, a^{2} b^{5} c x^{6} - 30 \, a^{3} b^{4} d x^{6} - 63 \, a^{5} b^{2} f x^{6} + 45 \, a^{4} b^{3} x^{6} e + 28 \, a^{3} b^{4} c x^{3} - 50 \, a^{4} b^{3} d x^{3} - 112 \, a^{6} b f x^{3} + 78 \, a^{5} b^{2} x^{3} e + 11 \, a^{4} b^{3} c - 21 \, a^{5} b^{2} d - 50 \, a^{7} f + 34 \, a^{6} b e}{6 \, {\left (b x^{3} + a\right )}^{2} b^{8}} + \frac {12 \, b^{12} f x^{15} - 45 \, a b^{11} f x^{12} + 15 \, b^{12} x^{12} e + 20 \, b^{12} d x^{9} + 120 \, a^{2} b^{10} f x^{9} - 60 \, a b^{11} x^{9} e + 30 \, b^{12} c x^{6} - 90 \, a b^{11} d x^{6} - 300 \, a^{3} b^{9} f x^{6} + 180 \, a^{2} b^{10} x^{6} e - 180 \, a b^{11} c x^{3} + 360 \, a^{2} b^{10} d x^{3} + 900 \, a^{4} b^{8} f x^{3} - 600 \, a^{3} b^{9} x^{3} e}{180 \, b^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 361, normalized size = 1.36 \[ \frac {f \,x^{15}}{15 b^{3}}-\frac {a f \,x^{12}}{4 b^{4}}+\frac {e \,x^{12}}{12 b^{3}}+\frac {2 a^{2} f \,x^{9}}{3 b^{5}}-\frac {a e \,x^{9}}{3 b^{4}}+\frac {d \,x^{9}}{9 b^{3}}-\frac {5 a^{3} f \,x^{6}}{3 b^{6}}+\frac {a^{2} e \,x^{6}}{b^{5}}-\frac {a d \,x^{6}}{2 b^{4}}+\frac {c \,x^{6}}{6 b^{3}}+\frac {5 a^{4} f \,x^{3}}{b^{7}}-\frac {10 a^{3} e \,x^{3}}{3 b^{6}}+\frac {2 a^{2} d \,x^{3}}{b^{5}}-\frac {a c \,x^{3}}{b^{4}}+\frac {a^{7} f}{6 \left (b \,x^{3}+a \right )^{2} b^{8}}-\frac {a^{6} e}{6 \left (b \,x^{3}+a \right )^{2} b^{7}}+\frac {a^{5} d}{6 \left (b \,x^{3}+a \right )^{2} b^{6}}-\frac {a^{4} c}{6 \left (b \,x^{3}+a \right )^{2} b^{5}}-\frac {7 a^{6} f}{3 \left (b \,x^{3}+a \right ) b^{8}}+\frac {2 a^{5} e}{\left (b \,x^{3}+a \right ) b^{7}}-\frac {7 a^{5} f \ln \left (b \,x^{3}+a \right )}{b^{8}}-\frac {5 a^{4} d}{3 \left (b \,x^{3}+a \right ) b^{6}}+\frac {5 a^{4} e \ln \left (b \,x^{3}+a \right )}{b^{7}}+\frac {4 a^{3} c}{3 \left (b \,x^{3}+a \right ) b^{5}}-\frac {10 a^{3} d \ln \left (b \,x^{3}+a \right )}{3 b^{6}}+\frac {2 a^{2} c \ln \left (b \,x^{3}+a \right )}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.54, size = 275, normalized size = 1.03 \[ \frac {7 \, a^{4} b^{3} c - 9 \, a^{5} b^{2} d + 11 \, a^{6} b e - 13 \, a^{7} f + 2 \, {\left (4 \, a^{3} b^{4} c - 5 \, a^{4} b^{3} d + 6 \, a^{5} b^{2} e - 7 \, a^{6} b f\right )} x^{3}}{6 \, {\left (b^{10} x^{6} + 2 \, a b^{9} x^{3} + a^{2} b^{8}\right )}} + \frac {12 \, b^{4} f x^{15} + 15 \, {\left (b^{4} e - 3 \, a b^{3} f\right )} x^{12} + 20 \, {\left (b^{4} d - 3 \, a b^{3} e + 6 \, a^{2} b^{2} f\right )} x^{9} + 30 \, {\left (b^{4} c - 3 \, a b^{3} d + 6 \, a^{2} b^{2} e - 10 \, a^{3} b f\right )} x^{6} - 60 \, {\left (3 \, a b^{3} c - 6 \, a^{2} b^{2} d + 10 \, a^{3} b e - 15 \, a^{4} f\right )} x^{3}}{180 \, b^{7}} + \frac {{\left (6 \, a^{2} b^{3} c - 10 \, a^{3} b^{2} d + 15 \, a^{4} b e - 21 \, a^{5} f\right )} \log \left (b x^{3} + a\right )}{3 \, b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.96, size = 449, normalized size = 1.69 \[ x^{12}\,\left (\frac {e}{12\,b^3}-\frac {a\,f}{4\,b^4}\right )+x^6\,\left (\frac {c}{6\,b^3}-\frac {a^3\,f}{6\,b^6}-\frac {a^2\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{2\,b^2}+\frac {a\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )}{2\,b}\right )-x^9\,\left (\frac {a^2\,f}{3\,b^5}-\frac {d}{9\,b^3}+\frac {a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{3\,b}\right )-\frac {\frac {13\,f\,a^7-11\,e\,a^6\,b+9\,d\,a^5\,b^2-7\,c\,a^4\,b^3}{6\,b}+x^3\,\left (\frac {7\,f\,a^6}{3}-2\,e\,a^5\,b+\frac {5\,d\,a^4\,b^2}{3}-\frac {4\,c\,a^3\,b^3}{3}\right )}{a^2\,b^7+2\,a\,b^8\,x^3+b^9\,x^6}-x^3\,\left (\frac {a\,\left (\frac {c}{b^3}-\frac {a^3\,f}{b^6}-\frac {3\,a^2\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b^2}+\frac {3\,a\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )}{b}\right )}{b}-\frac {a^2\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )}{b^2}+\frac {a^3\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{3\,b^3}\right )-\frac {\ln \left (b\,x^3+a\right )\,\left (21\,f\,a^5-15\,e\,a^4\,b+10\,d\,a^3\,b^2-6\,c\,a^2\,b^3\right )}{3\,b^8}+\frac {f\,x^{15}}{15\,b^3} \]
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]
________________________________________________________________________________________