Optimal. Leaf size=140 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} \sqrt [3]{b} d}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b} d}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b} d} \]
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Rubi [A] time = 0.249906, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} \sqrt [3]{b} d}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b} d}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b} d} \]
Antiderivative was successfully verified.
[In] Int[(a + b*(c + d*x)^3)^(-1),x]
[Out]
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Rubi in Sympy [A] time = 30.4873, size = 134, normalized size = 0.96 \[ \frac{\log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{3 a^{\frac{2}{3}} \sqrt [3]{b} d} - \frac{\log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{6 a^{\frac{2}{3}} \sqrt [3]{b} d} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}} \sqrt [3]{b} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b*(d*x+c)**3),x)
[Out]
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Mathematica [A] time = 0.0218152, size = 116, normalized size = 0.83 \[ \frac{-\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )+2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{6 a^{2/3} \sqrt [3]{b} d} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*(c + d*x)^3)^(-1),x]
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Maple [C] time = 0.002, size = 71, normalized size = 0.5 \[{\frac{1}{3\,bd}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b*(d*x+c)^3),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x + c\right )}^{3} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((d*x + c)^3*b + a),x, algorithm="maxima")
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Fricas [A] time = 0.210618, size = 161, normalized size = 1.15 \[ -\frac{\sqrt{3}{\left (\sqrt{3} \log \left (a^{2} +{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \left (a^{2} b\right )^{\frac{2}{3}} - \left (a^{2} b\right )^{\frac{1}{3}}{\left (a d x + a c\right )}\right ) - 2 \, \sqrt{3} \log \left (\left (a^{2} b\right )^{\frac{1}{3}}{\left (d x + c\right )} + a\right ) - 6 \, \arctan \left (\frac{2 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}}{\left (d x + c\right )} - \sqrt{3} a}{3 \, a}\right )\right )}}{18 \, \left (a^{2} b\right )^{\frac{1}{3}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((d*x + c)^3*b + a),x, algorithm="fricas")
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Sympy [A] time = 0.714936, size = 26, normalized size = 0.19 \[ \frac{\operatorname{RootSum}{\left (27 t^{3} a^{2} b - 1, \left ( t \mapsto t \log{\left (x + \frac{3 t a + c}{d} \right )} \right )\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b*(d*x+c)**3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x + c\right )}^{3} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((d*x + c)^3*b + a),x, algorithm="giac")
[Out]