Optimal. Leaf size=156 \[ \frac{\sqrt [3]{a} e^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{4/3} d}-\frac{\sqrt [3]{a} e^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{4/3} d}+\frac{\sqrt [3]{a} e^3 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{4/3} d}+\frac{e^3 x}{b} \]
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Rubi [A] time = 0.327719, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\sqrt [3]{a} e^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{4/3} d}-\frac{\sqrt [3]{a} e^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{4/3} d}+\frac{\sqrt [3]{a} e^3 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{4/3} d}+\frac{e^3 x}{b} \]
Antiderivative was successfully verified.
[In] Int[(c*e + d*e*x)^3/(a + b*(c + d*x)^3),x]
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Rubi in Sympy [A] time = 41.978, size = 156, normalized size = 1. \[ - \frac{\sqrt [3]{a} e^{3} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{3 b^{\frac{4}{3}} d} + \frac{\sqrt [3]{a} e^{3} \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{6 b^{\frac{4}{3}} d} + \frac{\sqrt{3} \sqrt [3]{a} e^{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{4}{3}} d} + \frac{e^{3} \left (c + d x\right )}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*e*x+c*e)**3/(a+b*(d*x+c)**3),x)
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Mathematica [A] time = 0.0359661, size = 145, normalized size = 0.93 \[ \frac{e^3 \left (\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )-2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )-2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )+6 \sqrt [3]{b} c+6 \sqrt [3]{b} d x\right )}{6 b^{4/3} d} \]
Antiderivative was successfully verified.
[In] Integrate[(c*e + d*e*x)^3/(a + b*(c + d*x)^3),x]
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Maple [C] time = 0.004, size = 84, normalized size = 0.5 \[{\frac{{e}^{3}x}{b}}-{\frac{{e}^{3}a}{3\,{b}^{2}d}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*e*x+c*e)^3/(a+b*(d*x+c)^3),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\frac{a e^{3} \int \frac{1}{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}\,{d x}}{b} + \frac{e^{3} x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*e*x + c*e)^3/((d*x + c)^3*b + a),x, algorithm="maxima")
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Fricas [A] time = 0.2131, size = 211, normalized size = 1.35 \[ \frac{\sqrt{3}{\left (6 \, \sqrt{3} d e^{3} x - \sqrt{3} e^{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} +{\left (d x + c\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right ) + 2 \, \sqrt{3} e^{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (d x + c - \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right ) - 6 \, e^{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3}{\left (d x + c\right )} + \sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )\right )}}{18 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*e*x + c*e)^3/((d*x + c)^3*b + a),x, algorithm="fricas")
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Sympy [A] time = 1.98354, size = 44, normalized size = 0.28 \[ \frac{e^{3} \operatorname{RootSum}{\left (27 t^{3} b^{4} + a, \left ( t \mapsto t \log{\left (x + \frac{- 3 t b e^{3} + c e^{3}}{d e^{3}} \right )} \right )\right )}}{d} + \frac{e^{3} x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*e*x+c*e)**3/(a+b*(d*x+c)**3),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d e x + c e\right )}^{3}}{{\left (d x + c\right )}^{3} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*e*x + c*e)^3/((d*x + c)^3*b + a),x, algorithm="giac")
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