Optimal. Leaf size=252 \[ -\frac{(b c-a d)^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{13/3}}+\frac{(b c-a d)^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{13/3}}-\frac{(b c-a d)^4 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{13/3}}+\frac{d x (2 b c-a d) \left (a^2 d^2-2 a b c d+2 b^2 c^2\right )}{b^4}+\frac{d^2 x^4 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )}{4 b^3}+\frac{d^3 x^7 (4 b c-a d)}{7 b^2}+\frac{d^4 x^{10}}{10 b} \]
[Out]
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Rubi [A] time = 0.415438, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ -\frac{(b c-a d)^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{13/3}}+\frac{(b c-a d)^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{13/3}}-\frac{(b c-a d)^4 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{13/3}}+\frac{d x (2 b c-a d) \left (a^2 d^2-2 a b c d+2 b^2 c^2\right )}{b^4}+\frac{d^2 x^4 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )}{4 b^3}+\frac{d^3 x^7 (4 b c-a d)}{7 b^2}+\frac{d^4 x^{10}}{10 b} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^3)^4/(a + b*x^3),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{d^{4} x^{10}}{10 b} - \frac{d^{3} x^{7} \left (a d - 4 b c\right )}{7 b^{2}} + \frac{d^{2} x^{4} \left (a^{2} d^{2} - 4 a b c d + 6 b^{2} c^{2}\right )}{4 b^{3}} - \frac{\left (a d - 2 b c\right ) \left (a^{2} d^{2} - 2 a b c d + 2 b^{2} c^{2}\right ) \int d\, dx}{b^{4}} + \frac{\left (a d - b c\right )^{4} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{2}{3}} b^{\frac{13}{3}}} - \frac{\left (a d - b c\right )^{4} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{2}{3}} b^{\frac{13}{3}}} - \frac{\sqrt{3} \left (a d - b c\right )^{4} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}} b^{\frac{13}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**3+c)**4/(b*x**3+a),x)
[Out]
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Mathematica [A] time = 0.199474, size = 253, normalized size = 1. \[ \frac{-\frac{70 (b c-a d)^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}+\frac{140 (b c-a d)^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}+\frac{140 \sqrt{3} (b c-a d)^4 \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{2/3}}+105 b^{4/3} d^2 x^4 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )+420 \sqrt [3]{b} d x \left (-a^3 d^3+4 a^2 b c d^2-6 a b^2 c^2 d+4 b^3 c^3\right )+60 b^{7/3} d^3 x^7 (4 b c-a d)+42 b^{10/3} d^4 x^{10}}{420 b^{13/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^3)^4/(a + b*x^3),x]
[Out]
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Maple [B] time = 0.006, size = 661, normalized size = 2.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^3+c)^4/(b*x^3+a),x)
[Out]
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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((d*x^3 + c)^4/(b*x^3 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.216267, size = 494, normalized size = 1.96 \[ -\frac{\sqrt{3}{\left (70 \, \sqrt{3}{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (\left (a^{2} b\right )^{\frac{2}{3}} x^{2} - \left (a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 140 \, \sqrt{3}{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (\left (a^{2} b\right )^{\frac{1}{3}} x + a\right ) - 420 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} x - \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3}{\left (14 \, b^{3} d^{4} x^{10} + 20 \,{\left (4 \, b^{3} c d^{3} - a b^{2} d^{4}\right )} x^{7} + 35 \,{\left (6 \, b^{3} c^{2} d^{2} - 4 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{4} + 140 \,{\left (4 \, b^{3} c^{3} d - 6 \, a b^{2} c^{2} d^{2} + 4 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} x\right )} \left (a^{2} b\right )^{\frac{1}{3}}\right )}}{1260 \, \left (a^{2} b\right )^{\frac{1}{3}} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^4/(b*x^3 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.40425, size = 369, normalized size = 1.46 \[ \operatorname{RootSum}{\left (27 t^{3} a^{2} b^{13} - a^{12} d^{12} + 12 a^{11} b c d^{11} - 66 a^{10} b^{2} c^{2} d^{10} + 220 a^{9} b^{3} c^{3} d^{9} - 495 a^{8} b^{4} c^{4} d^{8} + 792 a^{7} b^{5} c^{5} d^{7} - 924 a^{6} b^{6} c^{6} d^{6} + 792 a^{5} b^{7} c^{7} d^{5} - 495 a^{4} b^{8} c^{8} d^{4} + 220 a^{3} b^{9} c^{9} d^{3} - 66 a^{2} b^{10} c^{10} d^{2} + 12 a b^{11} c^{11} d - b^{12} c^{12}, \left ( t \mapsto t \log{\left (\frac{3 t a b^{4}}{a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}} + x \right )} \right )\right )} + \frac{d^{4} x^{10}}{10 b} - \frac{x^{7} \left (a d^{4} - 4 b c d^{3}\right )}{7 b^{2}} + \frac{x^{4} \left (a^{2} d^{4} - 4 a b c d^{3} + 6 b^{2} c^{2} d^{2}\right )}{4 b^{3}} - \frac{x \left (a^{3} d^{4} - 4 a^{2} b c d^{3} + 6 a b^{2} c^{2} d^{2} - 4 b^{3} c^{3} d\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**3+c)**4/(b*x**3+a),x)
[Out]
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GIAC/XCAS [A] time = 0.220205, size = 622, normalized size = 2.47 \[ \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{4} c^{4} - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c^{3} d + 6 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{2} c^{2} d^{2} - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} b c d^{3} + \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} d^{4}\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{5}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{4} c^{4} - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c^{3} d + 6 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b^{2} c^{2} d^{2} - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} b c d^{3} + \left (-a b^{2}\right )^{\frac{1}{3}} a^{4} d^{4}\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{5}} - \frac{{\left (b^{10} c^{4} - 4 \, a b^{9} c^{3} d + 6 \, a^{2} b^{8} c^{2} d^{2} - 4 \, a^{3} b^{7} c d^{3} + a^{4} b^{6} d^{4}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{10}} + \frac{14 \, b^{9} d^{4} x^{10} + 80 \, b^{9} c d^{3} x^{7} - 20 \, a b^{8} d^{4} x^{7} + 210 \, b^{9} c^{2} d^{2} x^{4} - 140 \, a b^{8} c d^{3} x^{4} + 35 \, a^{2} b^{7} d^{4} x^{4} + 560 \, b^{9} c^{3} d x - 840 \, a b^{8} c^{2} d^{2} x + 560 \, a^{2} b^{7} c d^{3} x - 140 \, a^{3} b^{6} d^{4} x}{140 \, b^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^4/(b*x^3 + a),x, algorithm="giac")
[Out]