Integrand size = 21, antiderivative size = 156 \[ \int \frac {(c+d x)^4}{a+b (c+d x)^3} \, dx=\frac {(c+d x)^2}{2 b d}+\frac {a^{2/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{5/3} d}+\frac {a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{5/3} d}-\frac {a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{5/3} d} \]
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Time = 0.13 (sec) , antiderivative size = 156, 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, 298, 31, 648, 631, 210, 642} \[ \int \frac {(c+d x)^4}{a+b (c+d x)^3} \, dx=\frac {a^{2/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{5/3} d}+\frac {a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{5/3} d}-\frac {a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{5/3} d}+\frac {(c+d x)^2}{2 b d} \]
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Rule 31
Rule 210
Rule 298
Rule 327
Rule 379
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^4}{a+b x^3} \, dx,x,c+d x\right )}{d} \\ & = \frac {(c+d x)^2}{2 b d}-\frac {a \text {Subst}\left (\int \frac {x}{a+b x^3} \, dx,x,c+d x\right )}{b d} \\ & = \frac {(c+d x)^2}{2 b d}+\frac {a^{2/3} \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{3 b^{4/3} d}-\frac {a^{2/3} \text {Subst}\left (\int \frac {\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^{4/3} d} \\ & = \frac {(c+d x)^2}{2 b d}+\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} \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^{5/3} d}-\frac {a \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^{4/3} d} \\ & = \frac {(c+d x)^2}{2 b d}+\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} \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^{5/3} d} \\ & = \frac {(c+d x)^2}{2 b d}+\frac {a^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} b^{5/3} d}+\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} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.02 \[ \int \frac {(c+d x)^4}{a+b (c+d x)^3} \, dx=\frac {(c+d x)^2}{2 b d}-\frac {a^{2/3} \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{5/3} d}+\frac {a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 b^{5/3} d}-\frac {a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 b^{5/3} d} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.00 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.59
method | result | size |
default | \(\frac {c x +\frac {1}{2} d \,x^{2}}{b}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\left (\textit {\_R} d +c \right ) \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}\right ) a}{3 b^{2} d}\) | \(92\) |
risch | \(\frac {c x}{b}+\frac {d \,x^{2}}{2 b}+\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\left (-\textit {\_R} d -c \right ) \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}\right )}{3 b^{2} d}\) | \(96\) |
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Time = 0.25 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.05 \[ \int \frac {(c+d x)^4}{a+b (c+d x)^3} \, dx=\frac {3 \, d^{2} x^{2} + 6 \, c d x - 2 \, \sqrt {3} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (b d x + b c\right )} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} - \sqrt {3} a}{3 \, a}\right ) - \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a d^{2} x^{2} + 2 \, a c d x + a c^{2} - {\left (b d x + b c\right )} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} + a \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}}\right ) + 2 \, \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a d x + a c + b \left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}}\right )}{6 \, b d} \]
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Time = 0.17 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.29 \[ \int \frac {(c+d x)^4}{a+b (c+d x)^3} \, dx=\frac {\operatorname {RootSum} {\left (27 t^{3} b^{5} - a^{2}, \left ( t \mapsto t \log {\left (x + \frac {9 t^{2} b^{3} + a c}{a d} \right )} \right )\right )}}{d} + \frac {c x}{b} + \frac {d x^{2}}{2 b} \]
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\[ \int \frac {(c+d x)^4}{a+b (c+d x)^3} \, dx=\int { \frac {{\left (d x + c\right )}^{4}}{{\left (d x + c\right )}^{3} b + a} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.03 \[ \int \frac {(c+d x)^4}{a+b (c+d x)^3} \, dx=\frac {b d^{7} x^{2} + 2 \, b c d^{6} x}{2 \, b^{2} d^{6}} - \frac {2 \, \sqrt {3} \left (a^{2} b d^{15}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, a b d x + 2 \, a b c - \left (a^{2} b\right )^{\frac {2}{3}}\right )}}{3 \, \left (a^{2} b\right )^{\frac {2}{3}}}\right ) + \left (a^{2} b d^{15}\right )^{\frac {1}{3}} \log \left ({\left (2 \, a b d x + 2 \, a b c - \left (a^{2} b\right )^{\frac {2}{3}}\right )}^{2} + 3 \, \left (a^{2} b\right )^{\frac {4}{3}}\right ) - 2 \, \left (a^{2} b d^{15}\right )^{\frac {1}{3}} \log \left ({\left | a b d x + a b c + \left (a^{2} b\right )^{\frac {2}{3}} \right |}\right )}{6 \, b^{2} d^{6}} \]
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Time = 0.27 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.08 \[ \int \frac {(c+d x)^4}{a+b (c+d x)^3} \, dx=\frac {d\,x^2}{2\,b}+\frac {c\,x}{b}+\frac {a^{2/3}\,\ln \left (b^{1/3}\,c+a^{1/3}+b^{1/3}\,d\,x\right )}{3\,b^{5/3}\,d}-\frac {a^{2/3}\,\ln \left (\frac {a^2\,c\,d^4}{b}+\frac {a^2\,d^5\,x}{b}+\frac {a^{7/3}\,d^4\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{b^{4/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,b^{5/3}\,d}+\frac {a^{2/3}\,\ln \left (\frac {a^2\,c\,d^4}{b}+\frac {a^2\,d^5\,x}{b}+\frac {9\,a^{7/3}\,d^4\,{\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2}{b^{4/3}}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{b^{5/3}\,d} \]
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