Optimal. Leaf size=156 \[ \frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{5/3} d}-\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{5/3} d}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3} d}+\frac{(c+d x)^2}{2 b d} \]
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Rubi [A] time = 0.369284, 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 \[ \frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{5/3} d}-\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{5/3} d}+\frac{a^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3} d}+\frac{(c+d x)^2}{2 b d} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)^4/(a + b*(c + d*x)^3),x]
[Out]
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Rubi in Sympy [A] time = 38.3999, size = 146, normalized size = 0.94 \[ \frac{a^{\frac{2}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{3 b^{\frac{5}{3}} d} - \frac{a^{\frac{2}{3}} \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{6 b^{\frac{5}{3}} d} + \frac{\sqrt{3} a^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{5}{3}} d} + \frac{\left (c + d x\right )^{2}}{2 b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)**4/(a+b*(d*x+c)**3),x)
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Mathematica [A] time = 0.0817617, size = 159, normalized size = 1.02 \[ \frac{a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{5/3} d}-\frac{a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{5/3} d}-\frac{a^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3} d}+\frac{(c+d x)^2}{2 b d} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)^4/(a + b*(c + d*x)^3),x]
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Maple [C] time = 0.009, size = 93, normalized size = 0.6 \[{\frac{d{x}^{2}}{2\,b}}+{\frac{cx}{b}}-{\frac{a}{3\,{b}^{2}d}\sum _{{\it \_R}={\it RootOf} \left ( b{d}^{3}{{\it \_Z}}^{3}+3\,bc{d}^{2}{{\it \_Z}}^{2}+3\,b{c}^{2}d{\it \_Z}+b{c}^{3}+a \right ) }{\frac{ \left ({\it \_R}\,d+c \right ) \ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)^4/(a+b*(d*x+c)^3),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\frac{a \int \frac{d x + c}{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}\,{d x}}{b} + \frac{d x^{2} + 2 \, c x}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^4/((d*x + c)^3*b + a),x, algorithm="maxima")
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Fricas [A] time = 0.217276, size = 246, normalized size = 1.58 \[ -\frac{\sqrt{3}{\left (\sqrt{3} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a d^{2} x^{2} + 2 \, a c d x + a c^{2} -{\left (b d x + b c\right )} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} + a \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) - 2 \, \sqrt{3} \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a d x + a c + b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}\right ) - 3 \, \sqrt{3}{\left (d^{2} x^{2} + 2 \, c d x\right )} - 6 \, \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3} b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} - 2 \, \sqrt{3}{\left (a d x + a c\right )}}{3 \, b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}}\right )\right )}}{18 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^4/((d*x + c)^3*b + a),x, algorithm="fricas")
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Sympy [A] time = 1.96544, size = 46, normalized size = 0.29 \[ \frac{\operatorname{RootSum}{\left (27 t^{3} b^{5} - a^{2}, \left ( t \mapsto t \log{\left (x + \frac{9 t^{2} b^{3} + a c}{a d} \right )} \right )\right )}}{d} + \frac{c x}{b} + \frac{d x^{2}}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)**4/(a+b*(d*x+c)**3),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x + c\right )}^{4}}{{\left (d x + c\right )}^{3} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x + c)^4/((d*x + c)^3*b + a),x, algorithm="giac")
[Out]