Integrand size = 19, antiderivative size = 267 \[ \int \frac {\left (c+d x^3\right )^4}{\left (a+b x^3\right )^2} \, dx=\frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac {d^3 (2 b c-a d) x^4}{2 b^3}+\frac {d^4 x^7}{7 b^2}+\frac {(b c-a d)^4 x}{3 a b^4 \left (a+b x^3\right )}-\frac {2 (b c-a d)^3 (b c+5 a d) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{13/3}}+\frac {2 (b c-a d)^3 (b c+5 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{13/3}}-\frac {(b c-a d)^3 (b c+5 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} b^{13/3}} \]
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Time = 0.18 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {398, 393, 206, 31, 648, 631, 210, 642} \[ \int \frac {\left (c+d x^3\right )^4}{\left (a+b x^3\right )^2} \, dx=-\frac {2 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) (b c-a d)^3 (5 a d+b c)}{3 \sqrt {3} a^{5/3} b^{13/3}}-\frac {(b c-a d)^3 (5 a d+b c) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} b^{13/3}}+\frac {2 (b c-a d)^3 (5 a d+b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{13/3}}+\frac {d^2 x \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )}{b^4}+\frac {x (b c-a d)^4}{3 a b^4 \left (a+b x^3\right )}+\frac {d^3 x^4 (2 b c-a d)}{2 b^3}+\frac {d^4 x^7}{7 b^2} \]
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Rule 31
Rule 206
Rule 210
Rule 393
Rule 398
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right )}{b^4}+\frac {2 d^3 (2 b c-a d) x^3}{b^3}+\frac {d^4 x^6}{b^2}+\frac {(b c-a d)^3 (b c+3 a d)+4 b d (b c-a d)^3 x^3}{b^4 \left (a+b x^3\right )^2}\right ) \, dx \\ & = \frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac {d^3 (2 b c-a d) x^4}{2 b^3}+\frac {d^4 x^7}{7 b^2}+\frac {\int \frac {(b c-a d)^3 (b c+3 a d)+4 b d (b c-a d)^3 x^3}{\left (a+b x^3\right )^2} \, dx}{b^4} \\ & = \frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac {d^3 (2 b c-a d) x^4}{2 b^3}+\frac {d^4 x^7}{7 b^2}+\frac {(b c-a d)^4 x}{3 a b^4 \left (a+b x^3\right )}+\frac {\left (2 (b c-a d)^3 (b c+5 a d)\right ) \int \frac {1}{a+b x^3} \, dx}{3 a b^4} \\ & = \frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac {d^3 (2 b c-a d) x^4}{2 b^3}+\frac {d^4 x^7}{7 b^2}+\frac {(b c-a d)^4 x}{3 a b^4 \left (a+b x^3\right )}+\frac {\left (2 (b c-a d)^3 (b c+5 a d)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{5/3} b^4}+\frac {\left (2 (b c-a d)^3 (b c+5 a d)\right ) \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}{9 a^{5/3} b^4} \\ & = \frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac {d^3 (2 b c-a d) x^4}{2 b^3}+\frac {d^4 x^7}{7 b^2}+\frac {(b c-a d)^4 x}{3 a b^4 \left (a+b x^3\right )}+\frac {2 (b c-a d)^3 (b c+5 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{13/3}}-\frac {\left ((b c-a d)^3 (b c+5 a d)\right ) \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}{9 a^{5/3} b^{13/3}}+\frac {\left ((b c-a d)^3 (b c+5 a d)\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{4/3} b^4} \\ & = \frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac {d^3 (2 b c-a d) x^4}{2 b^3}+\frac {d^4 x^7}{7 b^2}+\frac {(b c-a d)^4 x}{3 a b^4 \left (a+b x^3\right )}+\frac {2 (b c-a d)^3 (b c+5 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{13/3}}-\frac {(b c-a d)^3 (b c+5 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} b^{13/3}}+\frac {\left (2 (b c-a d)^3 (b c+5 a d)\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a^{5/3} b^{13/3}} \\ & = \frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac {d^3 (2 b c-a d) x^4}{2 b^3}+\frac {d^4 x^7}{7 b^2}+\frac {(b c-a d)^4 x}{3 a b^4 \left (a+b x^3\right )}-\frac {2 (b c-a d)^3 (b c+5 a d) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{13/3}}+\frac {2 (b c-a d)^3 (b c+5 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{13/3}}-\frac {(b c-a d)^3 (b c+5 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} b^{13/3}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.97 \[ \int \frac {\left (c+d x^3\right )^4}{\left (a+b x^3\right )^2} \, dx=\frac {126 \sqrt [3]{b} d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x+63 b^{4/3} d^3 (2 b c-a d) x^4+18 b^{7/3} d^4 x^7+\frac {42 \sqrt [3]{b} (b c-a d)^4 x}{a \left (a+b x^3\right )}+\frac {28 \sqrt {3} (b c-a d)^3 (b c+5 a d) \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{5/3}}+\frac {28 (b c-a d)^3 (b c+5 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}+\frac {14 (-b c+a d)^3 (b c+5 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}}{126 b^{13/3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.93 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.82
method | result | size |
risch | \(\frac {d^{4} x^{7}}{7 b^{2}}-\frac {d^{4} a \,x^{4}}{2 b^{3}}+\frac {d^{3} c \,x^{4}}{b^{2}}+\frac {3 d^{4} a^{2} x}{b^{4}}-\frac {8 d^{3} a c x}{b^{3}}+\frac {6 d^{2} c^{2} x}{b^{2}}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) x}{3 a \,b^{4} \left (b \,x^{3}+a \right )}-\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (5 a^{4} d^{4}-14 a^{3} b c \,d^{3}+12 a^{2} b^{2} c^{2} d^{2}-2 a \,b^{3} c^{3} d -b^{4} c^{4}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{9 b^{5} a}\) | \(218\) |
default | \(\frac {d^{2} \left (\frac {1}{7} b^{2} d^{2} x^{7}-\frac {1}{2} a b \,d^{2} x^{4}+b^{2} c d \,x^{4}+3 a^{2} d^{2} x -8 a b c d x +6 b^{2} c^{2} x \right )}{b^{4}}-\frac {-\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) x}{3 a \left (b \,x^{3}+a \right )}+\frac {2 \left (5 a^{4} d^{4}-14 a^{3} b c \,d^{3}+12 a^{2} b^{2} c^{2} d^{2}-2 a \,b^{3} c^{3} d -b^{4} c^{4}\right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3 a}}{b^{4}}\) | \(281\) |
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Leaf count of result is larger than twice the leaf count of optimal. 639 vs. \(2 (224) = 448\).
Time = 0.31 (sec) , antiderivative size = 1316, normalized size of antiderivative = 4.93 \[ \int \frac {\left (c+d x^3\right )^4}{\left (a+b x^3\right )^2} \, dx=\text {Too large to display} \]
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Time = 11.47 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.52 \[ \int \frac {\left (c+d x^3\right )^4}{\left (a+b x^3\right )^2} \, dx=x^{4} \left (- \frac {a d^{4}}{2 b^{3}} + \frac {c d^{3}}{b^{2}}\right ) + x \left (\frac {3 a^{2} d^{4}}{b^{4}} - \frac {8 a c d^{3}}{b^{3}} + \frac {6 c^{2} d^{2}}{b^{2}}\right ) + \frac {x \left (a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}\right )}{3 a^{2} b^{4} + 3 a b^{5} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} a^{5} b^{13} + 1000 a^{12} d^{12} - 8400 a^{11} b c d^{11} + 30720 a^{10} b^{2} c^{2} d^{10} - 63472 a^{9} b^{3} c^{3} d^{9} + 79848 a^{8} b^{4} c^{4} d^{8} - 60192 a^{7} b^{5} c^{5} d^{7} + 22848 a^{6} b^{6} c^{6} d^{6} + 288 a^{5} b^{7} c^{7} d^{5} - 3528 a^{4} b^{8} c^{8} d^{4} + 752 a^{3} b^{9} c^{9} d^{3} + 192 a^{2} b^{10} c^{10} d^{2} - 48 a b^{11} c^{11} d - 8 b^{12} c^{12}, \left ( t \mapsto t \log {\left (- \frac {9 t a^{2} b^{4}}{10 a^{4} d^{4} - 28 a^{3} b c d^{3} + 24 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d - 2 b^{4} c^{4}} + x \right )} \right )\right )} + \frac {d^{4} x^{7}}{7 b^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.49 \[ \int \frac {\left (c+d x^3\right )^4}{\left (a+b x^3\right )^2} \, dx=\frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} x}{3 \, {\left (a b^{5} x^{3} + a^{2} b^{4}\right )}} + \frac {2 \, b^{2} d^{4} x^{7} + 7 \, {\left (2 \, b^{2} c d^{3} - a b d^{4}\right )} x^{4} + 14 \, {\left (6 \, b^{2} c^{2} d^{2} - 8 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x}{14 \, b^{4}} + \frac {2 \, \sqrt {3} {\left (b^{4} c^{4} + 2 \, a b^{3} c^{3} d - 12 \, a^{2} b^{2} c^{2} d^{2} + 14 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (b^{4} c^{4} + 2 \, a b^{3} c^{3} d - 12 \, a^{2} b^{2} c^{2} d^{2} + 14 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4}\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, a b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {2 \, {\left (b^{4} c^{4} + 2 \, a b^{3} c^{3} d - 12 \, a^{2} b^{2} c^{2} d^{2} + 14 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4}\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
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Time = 0.29 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.54 \[ \int \frac {\left (c+d x^3\right )^4}{\left (a+b x^3\right )^2} \, dx=-\frac {2 \, \sqrt {3} {\left (b^{4} c^{4} + 2 \, a b^{3} c^{3} d - 12 \, a^{2} b^{2} c^{2} d^{2} + 14 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{3}} - \frac {{\left (b^{4} c^{4} + 2 \, a b^{3} c^{3} d - 12 \, a^{2} b^{2} c^{2} d^{2} + 14 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4}\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{3}} - \frac {2 \, {\left (b^{4} c^{4} + 2 \, a b^{3} c^{3} d - 12 \, a^{2} b^{2} c^{2} d^{2} + 14 \, a^{3} b c d^{3} - 5 \, a^{4} d^{4}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{2} b^{4}} + \frac {b^{4} c^{4} x - 4 \, a b^{3} c^{3} d x + 6 \, a^{2} b^{2} c^{2} d^{2} x - 4 \, a^{3} b c d^{3} x + a^{4} d^{4} x}{3 \, {\left (b x^{3} + a\right )} a b^{4}} + \frac {2 \, b^{12} d^{4} x^{7} + 14 \, b^{12} c d^{3} x^{4} - 7 \, a b^{11} d^{4} x^{4} + 84 \, b^{12} c^{2} d^{2} x - 112 \, a b^{11} c d^{3} x + 42 \, a^{2} b^{10} d^{4} x}{14 \, b^{14}} \]
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Time = 5.74 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.13 \[ \int \frac {\left (c+d x^3\right )^4}{\left (a+b x^3\right )^2} \, dx=x\,\left (\frac {2\,a\,\left (\frac {2\,a\,d^4}{b^3}-\frac {4\,c\,d^3}{b^2}\right )}{b}-\frac {a^2\,d^4}{b^4}+\frac {6\,c^2\,d^2}{b^2}\right )-x^4\,\left (\frac {a\,d^4}{2\,b^3}-\frac {c\,d^3}{b^2}\right )+\frac {d^4\,x^7}{7\,b^2}+\frac {x\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{3\,a\,\left (b^5\,x^3+a\,b^4\right )}-\frac {2\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,{\left (a\,d-b\,c\right )}^3\,\left (5\,a\,d+b\,c\right )}{9\,a^{5/3}\,b^{13/3}}+\frac {2\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^3\,\left (5\,a\,d+b\,c\right )}{9\,a^{5/3}\,b^{13/3}}-\frac {2\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^3\,\left (5\,a\,d+b\,c\right )}{9\,a^{5/3}\,b^{13/3}} \]
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