Integrand size = 17, antiderivative size = 234 \[ \int \frac {x^3}{a+b (c+d x)^3} \, dx=\frac {x}{b d^3}+\frac {\left (a-3 \sqrt [3]{a} b^{2/3} c^2+b c^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{4/3} d^4}-\frac {\left (a+3 \sqrt [3]{a} b^{2/3} c^2+b c^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{4/3} d^4}+\frac {\left (a+3 \sqrt [3]{a} b^{2/3} c^2+b c^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} b^{4/3} d^4}-\frac {c \log \left (a+b (c+d x)^3\right )}{b d^4} \]
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Time = 0.27 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {378, 1901, 1885, 1874, 31, 648, 631, 210, 642, 266} \[ \int \frac {x^3}{a+b (c+d x)^3} \, dx=\frac {\left (-3 \sqrt [3]{a} b^{2/3} c^2+a+b c^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{4/3} d^4}-\frac {\left (3 \sqrt [3]{a} b^{2/3} c^2+a+b c^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{4/3} d^4}+\frac {\left (3 \sqrt [3]{a} b^{2/3} c^2+a+b c^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} b^{4/3} d^4}-\frac {c \log \left (a+b (c+d x)^3\right )}{b d^4}+\frac {x}{b d^3} \]
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Rule 31
Rule 210
Rule 266
Rule 378
Rule 631
Rule 642
Rule 648
Rule 1874
Rule 1885
Rule 1901
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(-c+x)^3}{a+b x^3} \, dx,x,c+d x\right )}{d^4} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{b}-\frac {a+b c^3-3 b c^2 x+3 b c x^2}{b \left (a+b x^3\right )}\right ) \, dx,x,c+d x\right )}{d^4} \\ & = \frac {x}{b d^3}-\frac {\text {Subst}\left (\int \frac {a+b c^3-3 b c^2 x+3 b c x^2}{a+b x^3} \, dx,x,c+d x\right )}{b d^4} \\ & = \frac {x}{b d^3}-\frac {\text {Subst}\left (\int \frac {a+b c^3-3 b c^2 x}{a+b x^3} \, dx,x,c+d x\right )}{b d^4}-\frac {(3 c) \text {Subst}\left (\int \frac {x^2}{a+b x^3} \, dx,x,c+d x\right )}{d^4} \\ & = \frac {x}{b d^3}-\frac {c \log \left (a+b (c+d x)^3\right )}{b d^4}-\frac {\text {Subst}\left (\int \frac {\sqrt [3]{a} \left (-3 \sqrt [3]{a} b c^2+2 \sqrt [3]{b} \left (a+b c^3\right )\right )+\sqrt [3]{b} \left (-3 \sqrt [3]{a} b c^2-\sqrt [3]{b} \left (a+b c^3\right )\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{3 a^{2/3} b^{4/3} d^4}-\frac {\left (a+3 \sqrt [3]{a} b^{2/3} c^2+b c^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{3 a^{2/3} b d^4} \\ & = \frac {x}{b d^3}-\frac {\left (a+3 \sqrt [3]{a} b^{2/3} c^2+b c^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{4/3} d^4}-\frac {c \log \left (a+b (c+d x)^3\right )}{b d^4}-\frac {\left (a-3 \sqrt [3]{a} b^{2/3} c^2+b c^3\right ) \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 \sqrt [3]{a} b d^4}+\frac {\left (a+3 \sqrt [3]{a} b^{2/3} c^2+b c^3\right ) \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 a^{2/3} b^{4/3} d^4} \\ & = \frac {x}{b d^3}-\frac {\left (a+3 \sqrt [3]{a} b^{2/3} c^2+b c^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{4/3} d^4}+\frac {\left (a+3 \sqrt [3]{a} b^{2/3} c^2+b c^3\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} b^{4/3} d^4}-\frac {c \log \left (a+b (c+d x)^3\right )}{b d^4}-\frac {\left (a-3 \sqrt [3]{a} b^{2/3} c^2+b c^3\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{a^{2/3} b^{4/3} d^4} \\ & = \frac {x}{b d^3}+\frac {\left (a-3 \sqrt [3]{a} b^{2/3} c^2+b c^3\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{2/3} b^{4/3} d^4}-\frac {\left (a+3 \sqrt [3]{a} b^{2/3} c^2+b c^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{4/3} d^4}+\frac {\left (a+3 \sqrt [3]{a} b^{2/3} c^2+b c^3\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} b^{4/3} d^4}-\frac {c \log \left (a+b (c+d x)^3\right )}{b d^4} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.56 \[ \int \frac {x^3}{a+b (c+d x)^3} \, dx=-\frac {-3 b d x+\text {RootSum}\left [a+b c^3+3 b c^2 d \text {$\#$1}+3 b c d^2 \text {$\#$1}^2+b d^3 \text {$\#$1}^3\&,\frac {a \log (x-\text {$\#$1})+b c^3 \log (x-\text {$\#$1})+3 b c^2 d \log (x-\text {$\#$1}) \text {$\#$1}+3 b c d^2 \log (x-\text {$\#$1}) \text {$\#$1}^2}{c^2+2 c d \text {$\#$1}+d^2 \text {$\#$1}^2}\&\right ]}{3 b^2 d^4} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.46
method | result | size |
default | \(\frac {x}{b \,d^{3}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +b \,c^{3}+a \right )}{\sum }\frac {\left (-3 \textit {\_R}^{2} b c \,d^{2}-3 \textit {\_R} b \,c^{2} d -b \,c^{3}-a \right ) \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}}{3 b^{2} d^{4}}\) | \(108\) |
risch | \(\frac {x}{b \,d^{3}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +b \,c^{3}+a \right )}{\sum }\frac {\left (-3 \textit {\_R}^{2} b c \,d^{2}-3 \textit {\_R} b \,c^{2} d -b \,c^{3}-a \right ) \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}}{3 b^{2} d^{4}}\) | \(108\) |
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Result contains complex when optimal does not.
Time = 10.97 (sec) , antiderivative size = 6315, normalized size of antiderivative = 26.99 \[ \int \frac {x^3}{a+b (c+d x)^3} \, dx=\text {Too large to display} \]
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Time = 1.64 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.02 \[ \int \frac {x^3}{a+b (c+d x)^3} \, dx=\operatorname {RootSum} {\left (27 t^{3} a^{2} b^{4} d^{12} + 81 t^{2} a^{2} b^{3} c d^{8} + t \left (54 a^{2} b^{2} c^{2} d^{4} - 27 a b^{3} c^{5} d^{4}\right ) + a^{3} + 3 a^{2} b c^{3} + 3 a b^{2} c^{6} + b^{3} c^{9}, \left ( t \mapsto t \log {\left (x + \frac {- 27 t^{2} a^{2} b^{3} c^{2} d^{8} - 3 t a^{3} b d^{4} - 60 t a^{2} b^{2} c^{3} d^{4} - 3 t a b^{3} c^{6} d^{4} - 2 a^{3} c - 12 a^{2} b c^{4} - 9 a b^{2} c^{7} + b^{3} c^{10}}{a^{3} d + 3 a^{2} b c^{3} d - 24 a b^{2} c^{6} d + b^{3} c^{9} d} \right )} \right )\right )} + \frac {x}{b d^{3}} \]
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\[ \int \frac {x^3}{a+b (c+d x)^3} \, dx=\int { \frac {x^{3}}{{\left (d x + c\right )}^{3} b + a} \,d x } \]
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\[ \int \frac {x^3}{a+b (c+d x)^3} \, dx=\int { \frac {x^{3}}{{\left (d x + c\right )}^{3} b + a} \,d x } \]
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Time = 9.62 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.60 \[ \int \frac {x^3}{a+b (c+d x)^3} \, dx=\left (\sum _{k=1}^3\ln \left (\frac {3\,\left (b\,c^5+a\,c^2\right )}{d^2}-\mathrm {root}\left (27\,a^2\,b^4\,d^{12}\,z^3+81\,a^2\,b^3\,c\,d^8\,z^2+54\,a^2\,b^2\,c^2\,d^4\,z-27\,a\,b^3\,c^5\,d^4\,z+3\,a\,b^2\,c^6+3\,a^2\,b\,c^3+b^3\,c^9+a^3,z,k\right )\,\left (\frac {3\,\left (b^2\,c^4\,d^4-5\,a\,b\,c\,d^4\right )}{d^2}+\frac {3\,x\,\left (b^2\,c^3\,d^4+a\,b\,d^4\right )}{d}-\mathrm {root}\left (27\,a^2\,b^4\,d^{12}\,z^3+81\,a^2\,b^3\,c\,d^8\,z^2+54\,a^2\,b^2\,c^2\,d^4\,z-27\,a\,b^3\,c^5\,d^4\,z+3\,a\,b^2\,c^6+3\,a^2\,b\,c^3+b^3\,c^9+a^3,z,k\right )\,a\,b^2\,d^6\,9\right )-\frac {3\,x\,\left (a\,c-2\,b\,c^4\right )}{d}\right )\,\mathrm {root}\left (27\,a^2\,b^4\,d^{12}\,z^3+81\,a^2\,b^3\,c\,d^8\,z^2+54\,a^2\,b^2\,c^2\,d^4\,z-27\,a\,b^3\,c^5\,d^4\,z+3\,a\,b^2\,c^6+3\,a^2\,b\,c^3+b^3\,c^9+a^3,z,k\right )\right )+\frac {x}{b\,d^3} \]
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