Optimal. Leaf size=156 \[ -\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{5/3} d}+\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{5/3} d}+\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3} d}-\frac{1}{2 a d (c+d x)^2} \]
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Rubi [A] time = 0.294321, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ -\frac{b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{5/3} d}+\frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{5/3} d}+\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3} d}-\frac{1}{2 a d (c+d x)^2} \]
Antiderivative was successfully verified.
[In] Int[1/((c + d*x)^3*(a + b*(c + d*x)^3)),x]
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Rubi in Sympy [A] time = 39.342, size = 148, normalized size = 0.95 \[ - \frac{1}{2 a d \left (c + d x\right )^{2}} - \frac{b^{\frac{2}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{3 a^{\frac{5}{3}} d} + \frac{b^{\frac{2}{3}} \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{6 a^{\frac{5}{3}} d} + \frac{\sqrt{3} b^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{5}{3}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(d*x+c)**3/(a+b*(d*x+c)**3),x)
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Mathematica [A] time = 0.067749, size = 139, normalized size = 0.89 \[ \frac{b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )-\frac{3 a^{2/3}}{(c+d x)^2}-2 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )-2 \sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{6 a^{5/3} d} \]
Antiderivative was successfully verified.
[In] Integrate[1/((c + d*x)^3*(a + b*(c + d*x)^3)),x]
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Maple [C] time = 0.007, size = 87, normalized size = 0.6 \[ -{\frac{1}{3\,ad}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}}-{\frac{1}{2\,ad \left ( dx+c \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(d*x+c)^3/(a+b*(d*x+c)^3),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\frac{b \int \frac{1}{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}\,{d x}}{a} - \frac{1}{2 \,{\left (a d^{3} x^{2} + 2 \, a c d^{2} x + a c^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((d*x + c)^3*b + a)*(d*x + c)^3),x, algorithm="maxima")
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Fricas [A] time = 0.217644, size = 343, normalized size = 2.2 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + a^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}} +{\left (a b d x + a b c\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) - 2 \, \sqrt{3}{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b d x + b c - a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 6 \,{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3} a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + 2 \, \sqrt{3}{\left (b d x + b c\right )}}{3 \, a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}\right )}}{18 \,{\left (a d^{3} x^{2} + 2 \, a c d^{2} x + a c^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((d*x + c)^3*b + a)*(d*x + c)^3),x, algorithm="fricas")
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Sympy [A] time = 4.37909, size = 61, normalized size = 0.39 \[ - \frac{1}{2 a c^{2} d + 4 a c d^{2} x + 2 a d^{3} x^{2}} + \frac{\operatorname{RootSum}{\left (27 t^{3} a^{5} + b^{2}, \left ( t \mapsto t \log{\left (x + \frac{- 3 t a^{2} + b c}{b d} \right )} \right )\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(d*x+c)**3/(a+b*(d*x+c)**3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left ({\left (d x + c\right )}^{3} b + a\right )}{\left (d x + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((d*x + c)^3*b + a)*(d*x + c)^3),x, algorithm="giac")
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