Optimal. Leaf size=335 \[ \frac{x^5 \left (3 a^2 f-2 a b e+b^2 d\right )}{5 b^4}+\frac{x^2 \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )}{2 b^5}+\frac{a x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^5 \left (a+b x^3\right )}-\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-14 a^3 f+11 a^2 b e-8 a b^2 d+5 b^3 c\right )}{18 b^{17/3}}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-14 a^3 f+11 a^2 b e-8 a b^2 d+5 b^3 c\right )}{9 b^{17/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-14 a^3 f+11 a^2 b e-8 a b^2 d+5 b^3 c\right )}{3 \sqrt{3} b^{17/3}}+\frac{x^8 (b e-2 a f)}{8 b^3}+\frac{f x^{11}}{11 b^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 1.42045, antiderivative size = 335, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x^5 \left (3 a^2 f-2 a b e+b^2 d\right )}{5 b^4}+\frac{x^2 \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )}{2 b^5}+\frac{a x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^5 \left (a+b x^3\right )}-\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-14 a^3 f+11 a^2 b e-8 a b^2 d+5 b^3 c\right )}{18 b^{17/3}}+\frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-14 a^3 f+11 a^2 b e-8 a b^2 d+5 b^3 c\right )}{9 b^{17/3}}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-14 a^3 f+11 a^2 b e-8 a b^2 d+5 b^3 c\right )}{3 \sqrt{3} b^{17/3}}+\frac{x^8 (b e-2 a f)}{8 b^3}+\frac{f x^{11}}{11 b^2} \]
Antiderivative was successfully verified.
[In] Int[(x^7*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**7*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.325497, size = 319, normalized size = 0.95 \[ \frac{792 b^{5/3} x^5 \left (3 a^2 f-2 a b e+b^2 d\right )+1980 b^{2/3} x^2 \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )+\frac{1320 a b^{2/3} x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a+b x^3}-440 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (14 a^3 f-11 a^2 b e+8 a b^2 d-5 b^3 c\right )-440 \sqrt{3} a^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (14 a^3 f-11 a^2 b e+8 a b^2 d-5 b^3 c\right )+220 a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (14 a^3 f-11 a^2 b e+8 a b^2 d-5 b^3 c\right )+495 b^{8/3} x^8 (b e-2 a f)+360 b^{11/3} f x^{11}}{3960 b^{17/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^7*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.017, size = 584, normalized size = 1.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^7*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a)^2,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^7/(b*x^3 + a)^2,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.264341, size = 645, normalized size = 1.93 \[ \frac{\sqrt{3}{\left (220 \, \sqrt{3}{\left (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f +{\left (5 \, b^{4} c - 8 \, a b^{3} d + 11 \, a^{2} b^{2} e - 14 \, a^{3} b f\right )} x^{3}\right )} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x^{2} - b x \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} - a \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) - 440 \, \sqrt{3}{\left (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f +{\left (5 \, b^{4} c - 8 \, a b^{3} d + 11 \, a^{2} b^{2} e - 14 \, a^{3} b f\right )} x^{3}\right )} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x + b \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}\right ) - 1320 \,{\left (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f +{\left (5 \, b^{4} c - 8 \, a b^{3} d + 11 \, a^{2} b^{2} e - 14 \, a^{3} b f\right )} x^{3}\right )} \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} a x - \sqrt{3} b \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}}{3 \, b \left (-\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}}\right ) + 3 \, \sqrt{3}{\left (120 \, b^{4} f x^{14} + 15 \,{\left (11 \, b^{4} e - 14 \, a b^{3} f\right )} x^{11} + 33 \,{\left (8 \, b^{4} d - 11 \, a b^{3} e + 14 \, a^{2} b^{2} f\right )} x^{8} + 132 \,{\left (5 \, b^{4} c - 8 \, a b^{3} d + 11 \, a^{2} b^{2} e - 14 \, a^{3} b f\right )} x^{5} + 220 \,{\left (5 \, a b^{3} c - 8 \, a^{2} b^{2} d + 11 \, a^{3} b e - 14 \, a^{4} f\right )} x^{2}\right )}\right )}}{11880 \,{\left (b^{6} x^{3} + a b^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^7/(b*x^3 + a)^2,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 54.3489, size = 530, normalized size = 1.58 \[ - \frac{x^{2} \left (a^{4} f - a^{3} b e + a^{2} b^{2} d - a b^{3} c\right )}{3 a b^{5} + 3 b^{6} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} b^{17} + 2744 a^{11} f^{3} - 6468 a^{10} b e f^{2} + 4704 a^{9} b^{2} d f^{2} + 5082 a^{9} b^{2} e^{2} f - 2940 a^{8} b^{3} c f^{2} - 7392 a^{8} b^{3} d e f - 1331 a^{8} b^{3} e^{3} + 4620 a^{7} b^{4} c e f + 2688 a^{7} b^{4} d^{2} f + 2904 a^{7} b^{4} d e^{2} - 3360 a^{6} b^{5} c d f - 1815 a^{6} b^{5} c e^{2} - 2112 a^{6} b^{5} d^{2} e + 1050 a^{5} b^{6} c^{2} f + 2640 a^{5} b^{6} c d e + 512 a^{5} b^{6} d^{3} - 825 a^{4} b^{7} c^{2} e - 960 a^{4} b^{7} c d^{2} + 600 a^{3} b^{8} c^{2} d - 125 a^{2} b^{9} c^{3}, \left ( t \mapsto t \log{\left (\frac{81 t^{2} b^{11}}{196 a^{7} f^{2} - 308 a^{6} b e f + 224 a^{5} b^{2} d f + 121 a^{5} b^{2} e^{2} - 140 a^{4} b^{3} c f - 176 a^{4} b^{3} d e + 110 a^{3} b^{4} c e + 64 a^{3} b^{4} d^{2} - 80 a^{2} b^{5} c d + 25 a b^{6} c^{2}} + x \right )} \right )\right )} + \frac{f x^{11}}{11 b^{2}} - \frac{x^{8} \left (2 a f - b e\right )}{8 b^{3}} + \frac{x^{5} \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{5 b^{4}} - \frac{x^{2} \left (4 a^{3} f - 3 a^{2} b e + 2 a b^{2} d - b^{3} c\right )}{2 b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**7*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.21984, size = 597, normalized size = 1.78 \[ \frac{{\left (5 \, a b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 8 \, a^{2} b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 14 \, a^{4} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 11 \, a^{3} b \left (-\frac{a}{b}\right )^{\frac{1}{3}} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b^{5}} + \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 8 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 11 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, b^{7}} + \frac{a b^{3} c x^{2} - a^{2} b^{2} d x^{2} - a^{4} f x^{2} + a^{3} b x^{2} e}{3 \,{\left (b x^{3} + a\right )} b^{5}} - \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{3} c - 8 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{3} f + 11 \, \left (-a b^{2}\right )^{\frac{2}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, b^{7}} + \frac{40 \, b^{20} f x^{11} - 110 \, a b^{19} f x^{8} + 55 \, b^{20} x^{8} e + 88 \, b^{20} d x^{5} + 264 \, a^{2} b^{18} f x^{5} - 176 \, a b^{19} x^{5} e + 220 \, b^{20} c x^{2} - 440 \, a b^{19} d x^{2} - 880 \, a^{3} b^{17} f x^{2} + 660 \, a^{2} b^{18} x^{2} e}{440 \, b^{22}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^7/(b*x^3 + a)^2,x, algorithm="giac")
[Out]