Optimal. Leaf size=288 \[ \frac{x \left (3 a^2 f-2 a b e+b^2 d\right )}{b^4}-\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^4 \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{18 a^{2/3} b^{13/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{9 a^{2/3} b^{13/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{3 \sqrt{3} a^{2/3} b^{13/3}}+\frac{x^4 (b e-2 a f)}{4 b^3}+\frac{f x^7}{7 b^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.679117, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{x \left (3 a^2 f-2 a b e+b^2 d\right )}{b^4}-\frac{x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^4 \left (a+b x^3\right )}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{18 a^{2/3} b^{13/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{9 a^{2/3} b^{13/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{3 \sqrt{3} a^{2/3} b^{13/3}}+\frac{x^4 (b e-2 a f)}{4 b^3}+\frac{f x^7}{7 b^2} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 141.744, size = 284, normalized size = 0.99 \[ \frac{f x^{7}}{7 b^{2}} - \frac{x^{4} \left (2 a f - b e\right )}{4 b^{3}} + \frac{x \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{b^{4}} + \frac{x \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{3 b^{4} \left (a + b x^{3}\right )} - \frac{\left (10 a^{3} f - 7 a^{2} b e + 4 a b^{2} d - b^{3} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{2}{3}} b^{\frac{13}{3}}} + \frac{\left (10 a^{3} f - 7 a^{2} b e + 4 a b^{2} d - b^{3} c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{2}{3}} b^{\frac{13}{3}}} + \frac{\sqrt{3} \left (10 a^{3} f - 7 a^{2} b e + 4 a b^{2} d - b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{2}{3}} b^{\frac{13}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.299475, size = 277, normalized size = 0.96 \[ \frac{252 \sqrt [3]{b} x \left (3 a^2 f-2 a b e+b^2 d\right )-\frac{84 \sqrt [3]{b} x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a+b x^3}+\frac{28 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-10 a^3 f+7 a^2 b e-4 a b^2 d+b^3 c\right )}{a^{2/3}}+\frac{28 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (10 a^3 f-7 a^2 b e+4 a b^2 d-b^3 c\right )}{a^{2/3}}+\frac{14 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (10 a^3 f-7 a^2 b e+4 a b^2 d-b^3 c\right )}{a^{2/3}}+63 b^{4/3} x^4 (b e-2 a f)+36 b^{7/3} f x^7}{252 b^{13/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3)^2,x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.015, size = 514, normalized size = 1.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(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^3/(b*x^3 + a)^2,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.235551, size = 531, normalized size = 1.84 \[ \frac{\sqrt{3}{\left (14 \, \sqrt{3}{\left (a b^{3} c - 4 \, a^{2} b^{2} d + 7 \, a^{3} b e - 10 \, a^{4} f +{\left (b^{4} c - 4 \, a b^{3} d + 7 \, a^{2} b^{2} e - 10 \, a^{3} b f\right )} x^{3}\right )} \log \left (\left (-a^{2} b\right )^{\frac{2}{3}} x^{2} + \left (-a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 28 \, \sqrt{3}{\left (a b^{3} c - 4 \, a^{2} b^{2} d + 7 \, a^{3} b e - 10 \, a^{4} f +{\left (b^{4} c - 4 \, a b^{3} d + 7 \, a^{2} b^{2} e - 10 \, a^{3} b f\right )} x^{3}\right )} \log \left (\left (-a^{2} b\right )^{\frac{1}{3}} x - a\right ) + 84 \,{\left (a b^{3} c - 4 \, a^{2} b^{2} d + 7 \, a^{3} b e - 10 \, a^{4} f +{\left (b^{4} c - 4 \, a b^{3} d + 7 \, a^{2} b^{2} e - 10 \, a^{3} b f\right )} x^{3}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} x + \sqrt{3} a}{3 \, a}\right ) + 3 \, \sqrt{3}{\left (12 \, b^{3} f x^{10} + 3 \,{\left (7 \, b^{3} e - 10 \, a b^{2} f\right )} x^{7} + 21 \,{\left (4 \, b^{3} d - 7 \, a b^{2} e + 10 \, a^{2} b f\right )} x^{4} - 28 \,{\left (b^{3} c - 4 \, a b^{2} d + 7 \, a^{2} b e - 10 \, a^{3} f\right )} x\right )} \left (-a^{2} b\right )^{\frac{1}{3}}\right )}}{756 \,{\left (b^{5} x^{3} + a b^{4}\right )} \left (-a^{2} b\right )^{\frac{1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^3/(b*x^3 + a)^2,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 17.6266, size = 398, normalized size = 1.38 \[ \frac{x \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{3 a b^{4} + 3 b^{5} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a^{2} b^{13} + 1000 a^{9} f^{3} - 2100 a^{8} b e f^{2} + 1200 a^{7} b^{2} d f^{2} + 1470 a^{7} b^{2} e^{2} f - 300 a^{6} b^{3} c f^{2} - 1680 a^{6} b^{3} d e f - 343 a^{6} b^{3} e^{3} + 420 a^{5} b^{4} c e f + 480 a^{5} b^{4} d^{2} f + 588 a^{5} b^{4} d e^{2} - 240 a^{4} b^{5} c d f - 147 a^{4} b^{5} c e^{2} - 336 a^{4} b^{5} d^{2} e + 30 a^{3} b^{6} c^{2} f + 168 a^{3} b^{6} c d e + 64 a^{3} b^{6} d^{3} - 21 a^{2} b^{7} c^{2} e - 48 a^{2} b^{7} c d^{2} + 12 a b^{8} c^{2} d - b^{9} c^{3}, \left ( t \mapsto t \log{\left (- \frac{9 t a b^{4}}{10 a^{3} f - 7 a^{2} b e + 4 a b^{2} d - b^{3} c} + x \right )} \right )\right )} + \frac{f x^{7}}{7 b^{2}} - \frac{x^{4} \left (2 a f - b e\right )}{4 b^{3}} + \frac{x \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.219428, size = 471, normalized size = 1.64 \[ -\frac{{\left (b^{3} c - 4 \, a b^{2} d - 10 \, a^{3} f + 7 \, a^{2} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b^{4}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 10 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 7 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a b^{5}} - \frac{b^{3} c x - a b^{2} d x - a^{3} f x + a^{2} b x e}{3 \,{\left (b x^{3} + a\right )} b^{4}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - 10 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} f + 7 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a b^{5}} + \frac{4 \, b^{12} f x^{7} - 14 \, a b^{11} f x^{4} + 7 \, b^{12} x^{4} e + 28 \, b^{12} d x + 84 \, a^{2} b^{10} f x - 56 \, a b^{11} x e}{28 \, b^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^9 + e*x^6 + d*x^3 + c)*x^3/(b*x^3 + a)^2,x, algorithm="giac")
[Out]