Integrand size = 21, antiderivative size = 144 \[ \int \frac {(c+d x)^3}{a+b (c+d x)^3} \, dx=\frac {x}{b}+\frac {\sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{4/3} d}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{4/3} d}+\frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{4/3} d} \]
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Time = 0.10 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {379, 327, 206, 31, 648, 631, 210, 642} \[ \int \frac {(c+d x)^3}{a+b (c+d x)^3} \, dx=\frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{4/3} d}+\frac {\sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{4/3} d}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{4/3} d}+\frac {x}{b} \]
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Rule 31
Rule 206
Rule 210
Rule 327
Rule 379
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^3}{a+b x^3} \, dx,x,c+d x\right )}{d} \\ & = \frac {x}{b}-\frac {a \text {Subst}\left (\int \frac {1}{a+b x^3} \, dx,x,c+d x\right )}{b d} \\ & = \frac {x}{b}-\frac {\sqrt [3]{a} \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{3 b d}-\frac {\sqrt [3]{a} \text {Subst}\left (\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{3 b d} \\ & = \frac {x}{b}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{4/3} d}+\frac {\sqrt [3]{a} \text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{6 b^{4/3} d}-\frac {a^{2/3} \text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{2 b d} \\ & = \frac {x}{b}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{4/3} d}+\frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{4/3} d}-\frac {\sqrt [3]{a} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{b^{4/3} d} \\ & = \frac {x}{b}+\frac {\sqrt [3]{a} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} b^{4/3} d}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{4/3} d}+\frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{4/3} d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.99 \[ \int \frac {(c+d x)^3}{a+b (c+d x)^3} \, dx=\frac {6 \sqrt [3]{b} c+6 \sqrt [3]{b} d x-2 \sqrt {3} \sqrt [3]{a} \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )-2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )+\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{4/3} d} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.00 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.54
method | result | size |
default | \(\frac {x}{b}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}\right ) a}{3 b^{2} d}\) | \(78\) |
risch | \(\frac {x}{b}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}\right ) a}{3 b^{2} d}\) | \(78\) |
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Time = 0.26 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.94 \[ \int \frac {(c+d x)^3}{a+b (c+d x)^3} \, dx=\frac {6 \, d x + 2 \, \sqrt {3} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (b d x + b c\right )} \left (-\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + {\left (d x + c\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ) + 2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (d x + c - \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 \, b d} \]
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Time = 0.14 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.19 \[ \int \frac {(c+d x)^3}{a+b (c+d x)^3} \, dx=\frac {\operatorname {RootSum} {\left (27 t^{3} b^{4} + a, \left ( t \mapsto t \log {\left (x + \frac {- 3 t b + c}{d} \right )} \right )\right )}}{d} + \frac {x}{b} \]
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\[ \int \frac {(c+d x)^3}{a+b (c+d x)^3} \, dx=\int { \frac {{\left (d x + c\right )}^{3}}{{\left (d x + c\right )}^{3} b + a} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.25 \[ \int \frac {(c+d x)^3}{a+b (c+d x)^3} \, dx=\frac {x}{b} + \frac {2 \, \sqrt {3} \left (-\frac {a d^{6}}{b}\right )^{\frac {1}{3}} \arctan \left (-\frac {b d x + b c - \left (-a b^{2}\right )^{\frac {1}{3}}}{\sqrt {3} b d x + \sqrt {3} b c + \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}}}\right ) - \left (-\frac {a d^{6}}{b}\right )^{\frac {1}{3}} \log \left (4 \, {\left (\sqrt {3} b d x + \sqrt {3} b c + \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}}\right )}^{2} + 4 \, {\left (b d x + b c - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}^{2}\right ) + 2 \, \left (-\frac {a d^{6}}{b}\right )^{\frac {1}{3}} \log \left ({\left | -b d x - b c + \left (-a b^{2}\right )^{\frac {1}{3}} \right |}\right )}{6 \, b d^{3}} \]
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Time = 0.25 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.10 \[ \int \frac {(c+d x)^3}{a+b (c+d x)^3} \, dx=\frac {x}{b}+\frac {{\left (-a\right )}^{1/3}\,\ln \left ({\left (-a\right )}^{4/3}+a\,b^{1/3}\,c+a\,b^{1/3}\,d\,x\right )}{3\,b^{4/3}\,d}-\frac {{\left (-a\right )}^{1/3}\,\ln \left (3\,a\,b\,c\,d^5-3\,{\left (-a\right )}^{4/3}\,b^{2/3}\,d^5\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )+3\,a\,b\,d^6\,x\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,b^{4/3}\,d}+\frac {{\left (-a\right )}^{1/3}\,\ln \left (9\,{\left (-a\right )}^{4/3}\,b^{2/3}\,d^5\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+3\,a\,b\,c\,d^5+3\,a\,b\,d^6\,x\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{b^{4/3}\,d} \]
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