Optimal. Leaf size=210 \[ \frac{c \left (2 \sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{2/3} d^3}-\frac{c \left (2 \sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} b^{2/3} d^3}+\frac{c \left (2 \sqrt [3]{a}-\sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{2/3} d^3}+\frac{\log \left (a+b (c+d x)^3\right )}{3 b d^3} \]
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Rubi [A] time = 0.4734, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471 \[ \frac{c \left (2 \sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{2/3} d^3}-\frac{c \left (2 \sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} b^{2/3} d^3}+\frac{c \left (2 \sqrt [3]{a}-\sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{2/3} d^3}+\frac{\log \left (a+b (c+d x)^3\right )}{3 b d^3} \]
Antiderivative was successfully verified.
[In] Int[x^2/(a + b*(c + d*x)^3),x]
[Out]
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Rubi in Sympy [A] time = 52.2893, size = 204, normalized size = 0.97 \[ \frac{\log{\left (a + b \left (c + d x\right )^{3} \right )}}{3 b d^{3}} + \frac{\sqrt{3} c \left (2 \sqrt [3]{a} - \sqrt [3]{b} c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}} b^{\frac{2}{3}} d^{3}} + \frac{c \left (2 \sqrt [3]{a} + \sqrt [3]{b} c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{3 a^{\frac{2}{3}} b^{\frac{2}{3}} d^{3}} - \frac{c \left (2 \sqrt [3]{a} + \sqrt [3]{b} c\right ) \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{6 a^{\frac{2}{3}} b^{\frac{2}{3}} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(a+b*(d*x+c)**3),x)
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Mathematica [C] time = 0.0452165, size = 81, normalized size = 0.39 \[ \frac{\text{RootSum}\left [\text{$\#$1}^3 b d^3+3 \text{$\#$1}^2 b c d^2+3 \text{$\#$1} b c^2 d+a+b c^3\&,\frac{\text{$\#$1}^2 \log (x-\text{$\#$1})}{\text{$\#$1}^2 d^2+2 \text{$\#$1} c d+c^2}\&\right ]}{3 b d} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(a + b*(c + d*x)^3),x]
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Maple [C] time = 0.006, size = 74, normalized size = 0.4 \[{\frac{1}{3\,bd}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{{{\it \_R}}^{2}\ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(a+b*(d*x+c)^3),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (d x + c\right )}^{3} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((d*x + c)^3*b + a),x, algorithm="maxima")
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((d*x + c)^3*b + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.16654, size = 158, normalized size = 0.75 \[ \operatorname{RootSum}{\left (27 t^{3} a^{2} b^{3} d^{9} - 27 t^{2} a^{2} b^{2} d^{6} + t \left (9 a^{2} b d^{3} - 18 a b^{2} c^{3} d^{3}\right ) - a^{2} - 2 a b c^{3} - b^{2} c^{6}, \left ( t \mapsto t \log{\left (x + \frac{18 t^{2} a^{2} b^{2} d^{6} - 12 t a^{2} b d^{3} - 3 t a b^{2} c^{3} d^{3} + 2 a^{2} + a b c^{3} - b^{2} c^{6}}{8 a b c^{2} d - b^{2} c^{5} d} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(a+b*(d*x+c)**3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (d x + c\right )}^{3} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((d*x + c)^3*b + a),x, algorithm="giac")
[Out]