Optimal. Leaf size=208 \[ -\frac{(b c-a d)^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{10/3}}+\frac{(b c-a d)^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{10/3}}-\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{10/3}}+\frac{d x \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}+\frac{d^2 x^4 (3 b c-a d)}{4 b^2}+\frac{d^3 x^7}{7 b} \]
[Out]
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Rubi [A] time = 0.31675, antiderivative size = 208, 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 \[ -\frac{(b c-a d)^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{10/3}}+\frac{(b c-a d)^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{10/3}}-\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{10/3}}+\frac{d x \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}+\frac{d^2 x^4 (3 b c-a d)}{4 b^2}+\frac{d^3 x^7}{7 b} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^3)^3/(a + b*x^3),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{d^{3} x^{7}}{7 b} - \frac{d^{2} x^{4} \left (a d - 3 b c\right )}{4 b^{2}} + \frac{\left (a^{2} d^{2} - 3 a b c d + 3 b^{2} c^{2}\right ) \int d\, dx}{b^{3}} - \frac{\left (a d - b c\right )^{3} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{2}{3}} b^{\frac{10}{3}}} + \frac{\left (a d - b c\right )^{3} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{2}{3}} b^{\frac{10}{3}}} + \frac{\sqrt{3} \left (a d - b c\right )^{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}} b^{\frac{10}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**3+c)**3/(b*x**3+a),x)
[Out]
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Mathematica [A] time = 0.155733, size = 203, normalized size = 0.98 \[ \frac{\frac{14 (a d-b c)^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}+\frac{28 (b c-a d)^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}+\frac{28 \sqrt{3} (b c-a d)^3 \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{a^{2/3}}+84 \sqrt [3]{b} d x \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )+21 b^{4/3} d^2 x^4 (3 b c-a d)+12 b^{7/3} d^3 x^7}{84 b^{10/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^3)^3/(a + b*x^3),x]
[Out]
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Maple [B] time = 0.004, size = 486, normalized size = 2.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^3+c)^3/(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)^3/(b*x^3 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.215725, size = 379, normalized size = 1.82 \[ \frac{\sqrt{3}{\left (14 \, \sqrt{3}{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (\left (-a^{2} b\right )^{\frac{2}{3}} x^{2} + \left (-a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 28 \, \sqrt{3}{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (\left (-a^{2} b\right )^{\frac{1}{3}} x - a\right ) + 84 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} x + \sqrt{3} a}{3 \, a}\right ) + 3 \, \sqrt{3}{\left (4 \, b^{2} d^{3} x^{7} + 7 \,{\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{4} + 28 \,{\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x\right )} \left (-a^{2} b\right )^{\frac{1}{3}}\right )}}{252 \, \left (-a^{2} b\right )^{\frac{1}{3}} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^3/(b*x^3 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.16369, size = 255, normalized size = 1.23 \[ \operatorname{RootSum}{\left (27 t^{3} a^{2} b^{10} + a^{9} d^{9} - 9 a^{8} b c d^{8} + 36 a^{7} b^{2} c^{2} d^{7} - 84 a^{6} b^{3} c^{3} d^{6} + 126 a^{5} b^{4} c^{4} d^{5} - 126 a^{4} b^{5} c^{5} d^{4} + 84 a^{3} b^{6} c^{6} d^{3} - 36 a^{2} b^{7} c^{7} d^{2} + 9 a b^{8} c^{8} d - b^{9} c^{9}, \left ( t \mapsto t \log{\left (- \frac{3 t a b^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )} \right )\right )} + \frac{d^{3} x^{7}}{7 b} - \frac{x^{4} \left (a d^{3} - 3 b c d^{2}\right )}{4 b^{2}} + \frac{x \left (a^{2} d^{3} - 3 a b c d^{2} + 3 b^{2} c^{2} d\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**3+c)**3/(b*x**3+a),x)
[Out]
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GIAC/XCAS [A] time = 0.21963, size = 473, normalized size = 2.27 \[ \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c^{3} - 3 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} c^{2} d + 3 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b c d^{2} - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{4}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b^{3} c^{3} - 3 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} c^{2} d + 3 \, \left (-a b^{2}\right )^{\frac{1}{3}} a^{2} b c d^{2} - \left (-a b^{2}\right )^{\frac{1}{3}} a^{3} d^{3}\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{4}} - \frac{{\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{7}} + \frac{4 \, b^{6} d^{3} x^{7} + 21 \, b^{6} c d^{2} x^{4} - 7 \, a b^{5} d^{3} x^{4} + 84 \, b^{6} c^{2} d x - 84 \, a b^{5} c d^{2} x + 28 \, a^{2} b^{4} d^{3} x}{28 \, b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^3/(b*x^3 + a),x, algorithm="giac")
[Out]