Integrand size = 24, antiderivative size = 220 \[ \int \frac {(c e+d e x)^4}{\left (a+b (c+d x)^3\right )^3} \, dx=-\frac {e^4 (c+d x)^2}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac {e^4 (c+d x)^2}{9 a b d \left (a+b (c+d x)^3\right )}-\frac {e^4 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{4/3} b^{5/3} d}-\frac {e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{4/3} b^{5/3} d}+\frac {e^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{54 a^{4/3} b^{5/3} d} \]
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Time = 0.13 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {379, 294, 296, 298, 31, 648, 631, 210, 642} \[ \int \frac {(c e+d e x)^4}{\left (a+b (c+d x)^3\right )^3} \, dx=-\frac {e^4 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{4/3} b^{5/3} d}-\frac {e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{4/3} b^{5/3} d}+\frac {e^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{54 a^{4/3} b^{5/3} d}+\frac {e^4 (c+d x)^2}{9 a b d \left (a+b (c+d x)^3\right )}-\frac {e^4 (c+d x)^2}{6 b d \left (a+b (c+d x)^3\right )^2} \]
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Rule 31
Rule 210
Rule 294
Rule 296
Rule 298
Rule 379
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \frac {e^4 \text {Subst}\left (\int \frac {x^4}{\left (a+b x^3\right )^3} \, dx,x,c+d x\right )}{d} \\ & = -\frac {e^4 (c+d x)^2}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac {e^4 \text {Subst}\left (\int \frac {x}{\left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{3 b d} \\ & = -\frac {e^4 (c+d x)^2}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac {e^4 (c+d x)^2}{9 a b d \left (a+b (c+d x)^3\right )}+\frac {e^4 \text {Subst}\left (\int \frac {x}{a+b x^3} \, dx,x,c+d x\right )}{9 a b d} \\ & = -\frac {e^4 (c+d x)^2}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac {e^4 (c+d x)^2}{9 a b d \left (a+b (c+d x)^3\right )}-\frac {e^4 \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{27 a^{4/3} b^{4/3} d}+\frac {e^4 \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 )}{27 a^{4/3} b^{4/3} d} \\ & = -\frac {e^4 (c+d x)^2}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac {e^4 (c+d x)^2}{9 a b d \left (a+b (c+d x)^3\right )}-\frac {e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{4/3} b^{5/3} d}+\frac {e^4 \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^{4/3} b^{5/3} d}+\frac {e^4 \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 b^{4/3} d} \\ & = -\frac {e^4 (c+d x)^2}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac {e^4 (c+d x)^2}{9 a b d \left (a+b (c+d x)^3\right )}-\frac {e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{4/3} b^{5/3} d}+\frac {e^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{54 a^{4/3} b^{5/3} d}+\frac {e^4 \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^{4/3} b^{5/3} d} \\ & = -\frac {e^4 (c+d x)^2}{6 b d \left (a+b (c+d x)^3\right )^2}+\frac {e^4 (c+d x)^2}{9 a b d \left (a+b (c+d x)^3\right )}-\frac {e^4 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{9 \sqrt {3} a^{4/3} b^{5/3} d}-\frac {e^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{4/3} b^{5/3} d}+\frac {e^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{54 a^{4/3} b^{5/3} d} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.84 \[ \int \frac {(c e+d e x)^4}{\left (a+b (c+d x)^3\right )^3} \, dx=\frac {e^4 \left (-\frac {9 b^{2/3} (c+d x)^2}{\left (a+b (c+d x)^3\right )^2}+\frac {6 b^{2/3} (c+d x)^2}{a \left (a+b (c+d x)^3\right )}+\frac {2 \sqrt {3} \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{4/3}}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{a^{4/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{a^{4/3}}\right )}{54 b^{5/3} d} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.06 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.99
method | result | size |
default | \(e^{4} \left (\frac {\frac {d^{4} x^{5}}{9 a}+\frac {5 c \,d^{3} x^{4}}{9 a}+\frac {10 c^{2} d^{2} x^{3}}{9 a}-\frac {d \left (-20 c^{3} b +a \right ) x^{2}}{18 b a}-\frac {c \left (-5 c^{3} b +a \right ) x}{9 b a}-\frac {c^{2} \left (-2 c^{3} b +a \right )}{18 b d a}}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b \,d^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\left (\textit {\_R} d +c \right ) \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}}{27 b^{2} a d}\right )\) | \(218\) |
risch | \(\frac {\frac {d^{4} e^{4} x^{5}}{9 a}+\frac {5 c \,d^{3} e^{4} x^{4}}{9 a}+\frac {10 c^{2} d^{2} e^{4} x^{3}}{9 a}-\frac {d \,e^{4} \left (-20 c^{3} b +a \right ) x^{2}}{18 b a}-\frac {c \,e^{4} \left (-5 c^{3} b +a \right ) x}{9 b a}-\frac {c^{2} e^{4} \left (-2 c^{3} b +a \right )}{18 b d a}}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )^{2}}+\frac {e^{4} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b \,d^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\left (\textit {\_R} d +c \right ) \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}\right )}{27 a \,b^{2} d}\) | \(235\) |
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Leaf count of result is larger than twice the leaf count of optimal. 882 vs. \(2 (179) = 358\).
Time = 0.30 (sec) , antiderivative size = 1886, normalized size of antiderivative = 8.57 \[ \int \frac {(c e+d e x)^4}{\left (a+b (c+d x)^3\right )^3} \, dx=\text {Too large to display} \]
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Time = 1.40 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.51 \[ \int \frac {(c e+d e x)^4}{\left (a+b (c+d x)^3\right )^3} \, dx=\frac {- a c^{2} e^{4} + 2 b c^{5} e^{4} + 20 b c^{2} d^{3} e^{4} x^{3} + 10 b c d^{4} e^{4} x^{4} + 2 b d^{5} e^{4} x^{5} + x^{2} \left (- a d^{2} e^{4} + 20 b c^{3} d^{2} e^{4}\right ) + x \left (- 2 a c d e^{4} + 10 b c^{4} d e^{4}\right )}{18 a^{3} b d + 36 a^{2} b^{2} c^{3} d + 18 a b^{3} c^{6} d + 270 a b^{3} c^{2} d^{5} x^{4} + 108 a b^{3} c d^{6} x^{5} + 18 a b^{3} d^{7} x^{6} + x^{3} \cdot \left (36 a^{2} b^{2} d^{4} + 360 a b^{3} c^{3} d^{4}\right ) + x^{2} \cdot \left (108 a^{2} b^{2} c d^{3} + 270 a b^{3} c^{4} d^{3}\right ) + x \left (108 a^{2} b^{2} c^{2} d^{2} + 108 a b^{3} c^{5} d^{2}\right )} + \frac {e^{4} \operatorname {RootSum} {\left (19683 t^{3} a^{4} b^{5} + 1, \left ( t \mapsto t \log {\left (x + \frac {729 t^{2} a^{3} b^{3} e^{8} + c e^{8}}{d e^{8}} \right )} \right )\right )}}{d} \]
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\[ \int \frac {(c e+d e x)^4}{\left (a+b (c+d x)^3\right )^3} \, dx=\int { \frac {{\left (d e x + c e\right )}^{4}}{{\left ({\left (d x + c\right )}^{3} b + a\right )}^{3}} \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.41 \[ \int \frac {(c e+d e x)^4}{\left (a+b (c+d x)^3\right )^3} \, dx=-\frac {2 \, \sqrt {3} \left (-\frac {e^{12}}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac {2}{3}}\right )}}{3 \, \left (-a^{2} b\right )^{\frac {2}{3}}}\right ) + \left (-\frac {e^{12}}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac {2}{3}}\right )}^{2} + 3 \, \left (-a^{2} b\right )^{\frac {4}{3}}\right ) - 2 \, \left (-\frac {e^{12}}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left | a b d x + a b c + \left (-a^{2} b\right )^{\frac {2}{3}} \right |}\right )}{54 \, a b} + \frac {2 \, b d^{5} e^{4} x^{5} + 10 \, b c d^{4} e^{4} x^{4} + 20 \, b c^{2} d^{3} e^{4} x^{3} + 20 \, b c^{3} d^{2} e^{4} x^{2} + 10 \, b c^{4} d e^{4} x + 2 \, b c^{5} e^{4} - a d^{2} e^{4} x^{2} - 2 \, a c d e^{4} x - a c^{2} e^{4}}{18 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}^{2} a b d} \]
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Time = 6.50 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.95 \[ \int \frac {(c e+d e x)^4}{\left (a+b (c+d x)^3\right )^3} \, dx=\frac {\frac {d^4\,e^4\,x^5}{9\,a}-\frac {a\,c^2\,e^4-2\,b\,c^5\,e^4}{18\,a\,b\,d}+\frac {10\,c^2\,d^2\,e^4\,x^3}{9\,a}+\frac {5\,c\,d^3\,e^4\,x^4}{9\,a}-\frac {c\,x\,\left (a\,e^4-5\,b\,c^3\,e^4\right )}{9\,a\,b}-\frac {d\,e^4\,x^2\,\left (a-20\,b\,c^3\right )}{18\,a\,b}}{x^3\,\left (20\,b^2\,c^3\,d^3+2\,a\,b\,d^3\right )+x^2\,\left (15\,b^2\,c^4\,d^2+6\,a\,b\,c\,d^2\right )+a^2+x\,\left (6\,d\,b^2\,c^5+6\,a\,d\,b\,c^2\right )+b^2\,c^6+b^2\,d^6\,x^6+2\,a\,b\,c^3+6\,b^2\,c\,d^5\,x^5+15\,b^2\,c^2\,d^4\,x^4}-\frac {\ln \left (2\,{\left (-b\right )}^{4/3}\,c-a^{1/3}\,b+2\,{\left (-b\right )}^{4/3}\,d\,x+\sqrt {3}\,a^{1/3}\,b\,1{}\mathrm {i}\right )\,\left (e^4+\sqrt {3}\,e^4\,1{}\mathrm {i}\right )}{54\,a^{4/3}\,{\left (-b\right )}^{5/3}\,d}+\frac {e^4\,\ln \left (a^{1/3}\,b+{\left (-b\right )}^{4/3}\,c+{\left (-b\right )}^{4/3}\,d\,x\right )}{27\,a^{4/3}\,{\left (-b\right )}^{5/3}\,d}+\frac {e^4\,\ln \left (\frac {d^4\,e^8\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{81\,a^{5/3}\,{\left (-b\right )}^{4/3}}+\frac {c\,d^4\,e^8}{81\,a^2\,b}+\frac {d^5\,e^8\,x}{81\,a^2\,b}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{27\,a^{4/3}\,{\left (-b\right )}^{5/3}\,d} \]
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