Optimal. Leaf size=172 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{9 \sqrt [3]{a} b^{5/3} d}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 \sqrt [3]{a} b^{5/3} d}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} \sqrt [3]{a} b^{5/3} d}-\frac{(c+d x)^2}{3 b d \left (a+b (c+d x)^3\right )} \]
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Rubi [A] time = 0.329584, antiderivative size = 172, 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 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{9 \sqrt [3]{a} b^{5/3} d}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 \sqrt [3]{a} b^{5/3} d}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} \sqrt [3]{a} b^{5/3} d}-\frac{(c+d x)^2}{3 b d \left (a+b (c+d x)^3\right )} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^4/(a + b*(c + d*x)^3)^2,x]
[Out]
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Rubi in Sympy [A] time = 39.585, size = 160, normalized size = 0.93 \[ - \frac{\left (c + d x\right )^{2}}{3 b d \left (a + b \left (c + d x\right )^{3}\right )} - \frac{2 \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{9 \sqrt [3]{a} b^{\frac{5}{3}} d} + \frac{\log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{9 \sqrt [3]{a} b^{\frac{5}{3}} d} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{9 \sqrt [3]{a} b^{\frac{5}{3}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**4/(a+b*(d*x+c)**3)**2,x)
[Out]
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Mathematica [A] time = 0.174987, size = 152, normalized size = 0.88 \[ \frac{\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{\sqrt [3]{a}}-\frac{3 b^{2/3} (c+d x)^2}{a+b (c+d x)^3}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{\sqrt [3]{a}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt [3]{a}}}{9 b^{5/3} d} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^4/(a + b*(c + d*x)^3)^2,x]
[Out]
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Maple [C] time = 0.018, size = 141, normalized size = 0.8 \[{\frac{1}{b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a} \left ( -{\frac{d{x}^{2}}{3\,b}}-{\frac{2\,cx}{3\,b}}-{\frac{{c}^{2}}{3\,bd}} \right ) }+{\frac{2}{9\,{b}^{2}d}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{ \left ({\it \_R}\,d+c \right ) \ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^4/(a+b*(d*x+c)^3)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\frac{d^{2} x^{2} + 2 \, c d x + c^{2}}{3 \,{\left (b^{2} d^{4} x^{3} + 3 \, b^{2} c d^{3} x^{2} + 3 \, b^{2} c^{2} d^{2} x +{\left (b^{2} c^{3} + a b\right )} d\right )}} + \frac{2 \, \int \frac{d x + c}{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}\,{d x}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^4/((d*x + c)^3*b + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.218562, size = 413, normalized size = 2.4 \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}}{\left (d x + c\right )}\right ) - \sqrt{3}{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} \log \left (-a b + \left (-a b^{2}\right )^{\frac{2}{3}}{\left (d x + c\right )} +{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right ) - 6 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}}{\left (d x + c\right )}}{3 \, a b}\right ) - 3 \, \sqrt{3}{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right )}}{27 \,{\left (b^{2} d^{4} x^{3} + 3 \, b^{2} c d^{3} x^{2} + 3 \, b^{2} c^{2} d^{2} x +{\left (b^{2} c^{3} + a b\right )} d\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^4/((d*x + c)^3*b + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 9.89165, size = 107, normalized size = 0.62 \[ - \frac{c^{2} + 2 c d x + d^{2} x^{2}}{3 a b d + 3 b^{2} c^{3} d + 9 b^{2} c^{2} d^{2} x + 9 b^{2} c d^{3} x^{2} + 3 b^{2} d^{4} x^{3}} + \frac{\operatorname{RootSum}{\left (729 t^{3} a b^{5} + 8, \left ( t \mapsto t \log{\left (x + \frac{81 t^{2} a b^{3} + 4 c}{4 d} \right )} \right )\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**4/(a+b*(d*x+c)**3)**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{4}}{{\left ({\left (d x + c\right )}^{3} b + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^4/((d*x + c)^3*b + a)^2,x, algorithm="giac")
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