Integrand size = 21, antiderivative size = 201 \[ \int \frac {(c+d x)^3}{\left (a+b (c+d x)^3\right )^3} \, dx=-\frac {c+d x}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac {c+d x}{18 a b d \left (a+b (c+d x)^3\right )}-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{5/3} b^{4/3} d}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{5/3} b^{4/3} d}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{54 a^{5/3} b^{4/3} d} \]
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Time = 0.11 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {379, 294, 205, 206, 31, 648, 631, 210, 642} \[ \int \frac {(c+d x)^3}{\left (a+b (c+d x)^3\right )^3} \, dx=-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{5/3} b^{4/3} d}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{5/3} b^{4/3} d}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{54 a^{5/3} b^{4/3} d}+\frac {c+d x}{18 a b d \left (a+b (c+d x)^3\right )}-\frac {c+d x}{6 b d \left (a+b (c+d x)^3\right )^2} \]
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Rule 31
Rule 205
Rule 206
Rule 210
Rule 294
Rule 379
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^3}{\left (a+b x^3\right )^3} \, dx,x,c+d x\right )}{d} \\ & = -\frac {c+d x}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac {\text {Subst}\left (\int \frac {1}{\left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{6 b d} \\ & = -\frac {c+d x}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac {c+d x}{18 a b d \left (a+b (c+d x)^3\right )}+\frac {\text {Subst}\left (\int \frac {1}{a+b x^3} \, dx,x,c+d x\right )}{9 a b d} \\ & = -\frac {c+d x}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac {c+d x}{18 a b d \left (a+b (c+d x)^3\right )}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{27 a^{5/3} b d}+\frac {\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 )}{27 a^{5/3} b d} \\ & = -\frac {c+d x}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac {c+d x}{18 a b d \left (a+b (c+d x)^3\right )}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{5/3} b^{4/3} d}-\frac {\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 )}{54 a^{5/3} b^{4/3} d}+\frac {\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 )}{18 a^{4/3} b d} \\ & = -\frac {c+d x}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac {c+d x}{18 a b d \left (a+b (c+d x)^3\right )}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{5/3} b^{4/3} d}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{54 a^{5/3} b^{4/3} d}+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{9 a^{5/3} b^{4/3} d} \\ & = -\frac {c+d x}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac {c+d x}{18 a b d \left (a+b (c+d x)^3\right )}-\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{9 \sqrt {3} a^{5/3} b^{4/3} d}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{5/3} b^{4/3} d}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{54 a^{5/3} b^{4/3} d} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.89 \[ \int \frac {(c+d x)^3}{\left (a+b (c+d x)^3\right )^3} \, dx=\frac {-\frac {9 \sqrt [3]{b} (c+d x)}{\left (a+b (c+d x)^3\right )^2}+\frac {3 \sqrt [3]{b} (c+d x)}{a \left (a+b (c+d x)^3\right )}+\frac {2 \sqrt {3} \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{5/3}}+\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{a^{5/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{a^{5/3}}}{54 b^{4/3} d} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.01 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {\frac {d^{3} x^{4}}{18 a}+\frac {2 c \,d^{2} x^{3}}{9 a}+\frac {c^{2} d \,x^{2}}{3 a}-\frac {\left (-2 c^{3} b +a \right ) x}{9 b a}-\frac {c \left (-c^{3} b +2 a \right )}{18 b d a}}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}}{27 b^{2} a d}\) | \(186\) |
risch | \(\frac {\frac {d^{3} x^{4}}{18 a}+\frac {2 c \,d^{2} x^{3}}{9 a}+\frac {c^{2} d \,x^{2}}{3 a}-\frac {\left (-2 c^{3} b +a \right ) x}{9 b a}-\frac {c \left (-c^{3} b +2 a \right )}{18 b d a}}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}}{27 b^{2} a d}\) | \(186\) |
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Leaf count of result is larger than twice the leaf count of optimal. 765 vs. \(2 (160) = 320\).
Time = 0.30 (sec) , antiderivative size = 1650, normalized size of antiderivative = 8.21 \[ \int \frac {(c+d x)^3}{\left (a+b (c+d x)^3\right )^3} \, dx=\text {Too large to display} \]
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Time = 1.27 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.29 \[ \int \frac {(c+d x)^3}{\left (a+b (c+d x)^3\right )^3} \, dx=\frac {- 2 a c + b c^{4} + 6 b c^{2} d^{2} x^{2} + 4 b c d^{3} x^{3} + b d^{4} x^{4} + x \left (- 2 a d + 4 b c^{3} d\right )}{18 a^{3} b d + 36 a^{2} b^{2} c^{3} d + 18 a b^{3} c^{6} d + 270 a b^{3} c^{2} d^{5} x^{4} + 108 a b^{3} c d^{6} x^{5} + 18 a b^{3} d^{7} x^{6} + x^{3} \cdot \left (36 a^{2} b^{2} d^{4} + 360 a b^{3} c^{3} d^{4}\right ) + x^{2} \cdot \left (108 a^{2} b^{2} c d^{3} + 270 a b^{3} c^{4} d^{3}\right ) + x \left (108 a^{2} b^{2} c^{2} d^{2} + 108 a b^{3} c^{5} d^{2}\right )} + \frac {\operatorname {RootSum} {\left (19683 t^{3} a^{5} b^{4} - 1, \left ( t \mapsto t \log {\left (x + \frac {27 t a^{2} b + c}{d} \right )} \right )\right )}}{d} \]
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\[ \int \frac {(c+d x)^3}{\left (a+b (c+d x)^3\right )^3} \, dx=\int { \frac {{\left (d x + c\right )}^{3}}{{\left ({\left (d x + c\right )}^{3} b + a\right )}^{3}} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.33 \[ \int \frac {(c+d x)^3}{\left (a+b (c+d x)^3\right )^3} \, dx=\frac {2 \, \sqrt {3} \left (\frac {1}{a^{2} b d^{3}}\right )^{\frac {1}{3}} \arctan \left (-\frac {b d x + b c + \left (a b^{2}\right )^{\frac {1}{3}}}{\sqrt {3} b d x + \sqrt {3} b c - \sqrt {3} \left (a b^{2}\right )^{\frac {1}{3}}}\right ) - \left (\frac {1}{a^{2} b d^{3}}\right )^{\frac {1}{3}} \log \left (4 \, {\left (\sqrt {3} b d x + \sqrt {3} b c - \sqrt {3} \left (a b^{2}\right )^{\frac {1}{3}}\right )}^{2} + 4 \, {\left (b d x + b c + \left (a b^{2}\right )^{\frac {1}{3}}\right )}^{2}\right ) + 2 \, \left (\frac {1}{a^{2} b d^{3}}\right )^{\frac {1}{3}} \log \left ({\left | b d x + b c + \left (a b^{2}\right )^{\frac {1}{3}} \right |}\right )}{54 \, a b} + \frac {b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} - 2 \, a d x - 2 \, a c}{18 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}^{2} a b d} \]
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Time = 6.01 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.73 \[ \int \frac {(c+d x)^3}{\left (a+b (c+d x)^3\right )^3} \, dx=\frac {\frac {d^3\,x^4}{18\,a}+\frac {c^2\,d\,x^2}{3\,a}+\frac {2\,c\,d^2\,x^3}{9\,a}-\frac {x\,\left (a-2\,b\,c^3\right )}{9\,a\,b}-\frac {2\,a\,c-b\,c^4}{18\,a\,b\,d}}{x^3\,\left (20\,b^2\,c^3\,d^3+2\,a\,b\,d^3\right )+x^2\,\left (15\,b^2\,c^4\,d^2+6\,a\,b\,c\,d^2\right )+a^2+x\,\left (6\,d\,b^2\,c^5+6\,a\,d\,b\,c^2\right )+b^2\,c^6+b^2\,d^6\,x^6+2\,a\,b\,c^3+6\,b^2\,c\,d^5\,x^5+15\,b^2\,c^2\,d^4\,x^4}+\frac {\ln \left (b^{1/3}\,c+a^{1/3}+b^{1/3}\,d\,x\right )}{27\,a^{5/3}\,b^{4/3}\,d}+\frac {\ln \left (\frac {b\,c\,d^5}{3\,a}+\frac {b\,d^6\,x}{3\,a}+\frac {b^{2/3}\,d^5\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{54\,a^{5/3}\,b^{4/3}\,d}-\frac {\ln \left (\frac {b\,c\,d^5}{3\,a}+\frac {b\,d^6\,x}{3\,a}-\frac {b^{2/3}\,d^5\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{54\,a^{5/3}\,b^{4/3}\,d} \]
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