Optimal. Leaf size=314 \[ -\frac{\sqrt [3]{b} d \left (\sqrt [3]{a} \left (a-2 b c^3\right )-\sqrt [3]{b} c \left (2 a-b c^3\right )\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} \left (a+b c^3\right )^2}+\frac{\sqrt [3]{b} d \left (\sqrt [3]{a} \left (a-2 b c^3\right )-\sqrt [3]{b} c \left (2 a-b c^3\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \left (a+b c^3\right )^2}+\frac{\sqrt [3]{b} d \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} c\right )^3 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \left (a+b c^3\right )^2}-\frac{1}{x \left (a+b c^3\right )}-\frac{3 b c^2 d \log (x)}{\left (a+b c^3\right )^2}+\frac{b c^2 d \log \left (a+b (c+d x)^3\right )}{\left (a+b c^3\right )^2} \]
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Rubi [A] time = 1.09058, antiderivative size = 312, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529 \[ \frac{b^{2/3} d \left (-\frac{\sqrt [3]{a} \left (a-2 b c^3\right )}{\sqrt [3]{b}}+2 a c-b c^4\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} \left (a+b c^3\right )^2}+\frac{\sqrt [3]{b} d \left (\sqrt [3]{a} \left (a-2 b c^3\right )-\sqrt [3]{b} c \left (2 a-b c^3\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} \left (a+b c^3\right )^2}+\frac{\sqrt [3]{b} d \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} c\right )^3 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \left (a+b c^3\right )^2}-\frac{1}{x \left (a+b c^3\right )}-\frac{3 b c^2 d \log (x)}{\left (a+b c^3\right )^2}+\frac{b c^2 d \log \left (a+b (c+d x)^3\right )}{\left (a+b c^3\right )^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(a + b*(c + d*x)^3)),x]
[Out]
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Rubi in Sympy [A] time = 133.834, size = 314, normalized size = 1. \[ - \frac{3 b c^{2} d \log{\left (- d x \right )}}{\left (a + b c^{3}\right )^{2}} + \frac{b c^{2} d \log{\left (a + b \left (c + d x\right )^{3} \right )}}{\left (a + b c^{3}\right )^{2}} - \frac{1}{x \left (a + b c^{3}\right )} + \frac{\sqrt [3]{b} d \left (\sqrt [3]{a} \left (a - 2 b c^{3}\right ) - \sqrt [3]{b} c \left (2 a - b c^{3}\right )\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{3 a^{\frac{2}{3}} \left (a + b c^{3}\right )^{2}} - \frac{\sqrt [3]{b} d \left (\sqrt [3]{a} \left (a - 2 b c^{3}\right ) - \sqrt [3]{b} c \left (2 a - b c^{3}\right )\right ) \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{6 a^{\frac{2}{3}} \left (a + b c^{3}\right )^{2}} + \frac{\sqrt{3} \sqrt [3]{b} d \left (a^{\frac{4}{3}} - 2 \sqrt [3]{a} b c^{3} + 2 a \sqrt [3]{b} c - b^{\frac{4}{3}} c^{4}\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{2}{3}} \left (a + b c^{3}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(a+b*(d*x+c)**3),x)
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Mathematica [C] time = 0.141238, size = 173, normalized size = 0.55 \[ \frac{d x \text{RootSum}\left [\text{$\#$1}^3 b d^3+3 \text{$\#$1}^2 b c d^2+3 \text{$\#$1} b c^2 d+a+b c^3\&,\frac{3 \text{$\#$1}^2 b c^2 d^2 \log (x-\text{$\#$1})-3 a c \log (x-\text{$\#$1})-\text{$\#$1} a d \log (x-\text{$\#$1})+6 b c^4 \log (x-\text{$\#$1})+8 \text{$\#$1} b c^3 d \log (x-\text{$\#$1})}{\text{$\#$1}^2 d^2+2 \text{$\#$1} c d+c^2}\&\right ]-3 \left (a+b c^3+3 b c^2 d x \log (x)\right )}{3 x \left (a+b c^3\right )^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*(a + b*(c + d*x)^3)),x]
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Maple [C] time = 0.014, size = 144, normalized size = 0.5 \[ -{\frac{1}{ \left ( b{c}^{3}+a \right ) x}}-3\,{\frac{{c}^{2}db\ln \left ( x \right ) }{ \left ( b{c}^{3}+a \right ) ^{2}}}+{\frac{d}{3\, \left ( b{c}^{3}+a \right ) ^{2}}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{ \left ( 3\,{{\it \_R}}^{2}b{c}^{2}{d}^{2}+8\,{\it \_R}\,b{c}^{3}d+6\,b{c}^{4}-{\it \_R}\,ad-3\,ac \right ) \ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(a+b*(d*x+c)^3),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\frac{3 \, b c^{2} d \log \left (x\right )}{b^{2} c^{6} + 2 \, a b c^{3} + a^{2}} + \frac{b d^{2} \int \frac{3 \, b c^{2} d^{2} x^{2} + 6 \, b c^{4} +{\left (8 \, b c^{3} - a\right )} d x - 3 \, a c}{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}\,{d x}}{b^{2} c^{6} + 2 \, a b c^{3} + a^{2}} - \frac{1}{{\left (b c^{3} + a\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((d*x + c)^3*b + a)*x^2),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(1/(((d*x + c)^3*b + a)*x^2),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(a+b*(d*x+c)**3),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left ({\left (d x + c\right )}^{3} b + a\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((d*x + c)^3*b + a)*x^2),x, algorithm="giac")
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