Integrand size = 15, antiderivative size = 119 \[ \int \sqrt [4]{b x^7+a x^8} \, dx=\frac {\left (-7 b^2+4 a b x+32 a^2 x^2\right ) \sqrt [4]{b x^7+a x^8}}{96 a^2 x}+\frac {7 b^3 \arctan \left (\frac {\sqrt [4]{b x^7+a x^8}}{\sqrt [4]{a} x^2}\right )}{64 a^{11/4}}+\frac {7 b^3 \text {arctanh}\left (\frac {\sqrt [4]{b x^7+a x^8}}{\sqrt [4]{a} x^2}\right )}{64 a^{11/4}} \]
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Time = 0.14 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.65, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {2029, 2049, 2057, 65, 338, 304, 209, 212} \[ \int \sqrt [4]{b x^7+a x^8} \, dx=-\frac {7 b^3 x^{21/4} (a x+b)^{3/4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{64 a^{11/4} \left (a x^8+b x^7\right )^{3/4}}+\frac {7 b^3 x^{21/4} (a x+b)^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{64 a^{11/4} \left (a x^8+b x^7\right )^{3/4}}-\frac {7 b^2 \sqrt [4]{a x^8+b x^7}}{96 a^2 x}+\frac {1}{3} x \sqrt [4]{a x^8+b x^7}+\frac {b \sqrt [4]{a x^8+b x^7}}{24 a} \]
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Rule 65
Rule 209
Rule 212
Rule 304
Rule 338
Rule 2029
Rule 2049
Rule 2057
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x \sqrt [4]{b x^7+a x^8}+\frac {1}{12} b \int \frac {x^7}{\left (b x^7+a x^8\right )^{3/4}} \, dx \\ & = \frac {b \sqrt [4]{b x^7+a x^8}}{24 a}+\frac {1}{3} x \sqrt [4]{b x^7+a x^8}-\frac {\left (7 b^2\right ) \int \frac {x^6}{\left (b x^7+a x^8\right )^{3/4}} \, dx}{96 a} \\ & = \frac {b \sqrt [4]{b x^7+a x^8}}{24 a}-\frac {7 b^2 \sqrt [4]{b x^7+a x^8}}{96 a^2 x}+\frac {1}{3} x \sqrt [4]{b x^7+a x^8}+\frac {\left (7 b^3\right ) \int \frac {x^5}{\left (b x^7+a x^8\right )^{3/4}} \, dx}{128 a^2} \\ & = \frac {b \sqrt [4]{b x^7+a x^8}}{24 a}-\frac {7 b^2 \sqrt [4]{b x^7+a x^8}}{96 a^2 x}+\frac {1}{3} x \sqrt [4]{b x^7+a x^8}+\frac {\left (7 b^3 x^{21/4} (b+a x)^{3/4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{128 a^2 \left (b x^7+a x^8\right )^{3/4}} \\ & = \frac {b \sqrt [4]{b x^7+a x^8}}{24 a}-\frac {7 b^2 \sqrt [4]{b x^7+a x^8}}{96 a^2 x}+\frac {1}{3} x \sqrt [4]{b x^7+a x^8}+\frac {\left (7 b^3 x^{21/4} (b+a x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{32 a^2 \left (b x^7+a x^8\right )^{3/4}} \\ & = \frac {b \sqrt [4]{b x^7+a x^8}}{24 a}-\frac {7 b^2 \sqrt [4]{b x^7+a x^8}}{96 a^2 x}+\frac {1}{3} x \sqrt [4]{b x^7+a x^8}+\frac {\left (7 b^3 x^{21/4} (b+a x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{32 a^2 \left (b x^7+a x^8\right )^{3/4}} \\ & = \frac {b \sqrt [4]{b x^7+a x^8}}{24 a}-\frac {7 b^2 \sqrt [4]{b x^7+a x^8}}{96 a^2 x}+\frac {1}{3} x \sqrt [4]{b x^7+a x^8}+\frac {\left (7 b^3 x^{21/4} (b+a x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{64 a^{5/2} \left (b x^7+a x^8\right )^{3/4}}-\frac {\left (7 b^3 x^{21/4} (b+a x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{64 a^{5/2} \left (b x^7+a x^8\right )^{3/4}} \\ & = \frac {b \sqrt [4]{b x^7+a x^8}}{24 a}-\frac {7 b^2 \sqrt [4]{b x^7+a x^8}}{96 a^2 x}+\frac {1}{3} x \sqrt [4]{b x^7+a x^8}-\frac {7 b^3 x^{21/4} (b+a x)^{3/4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{64 a^{11/4} \left (b x^7+a x^8\right )^{3/4}}+\frac {7 b^3 x^{21/4} (b+a x)^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{64 a^{11/4} \left (b x^7+a x^8\right )^{3/4}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.08 \[ \int \sqrt [4]{b x^7+a x^8} \, dx=\frac {\sqrt [4]{x^7 (b+a x)} \left (2 a^{3/4} x^{3/4} \sqrt [4]{b+a x} \left (-7 b^2+4 a b x+32 a^2 x^2\right )-21 b^3 \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )+21 b^3 \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )\right )}{192 a^{11/4} x^{7/4} \sqrt [4]{b+a x}} \]
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Time = 0.97 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.21
method | result | size |
pseudoelliptic | \(\frac {128 a^{\frac {11}{4}} \left (x^{7} \left (a x +b \right )\right )^{\frac {1}{4}} x^{2}+16 a^{\frac {7}{4}} b x \left (x^{7} \left (a x +b \right )\right )^{\frac {1}{4}}-28 b^{2} a^{\frac {3}{4}} \left (x^{7} \left (a x +b \right )\right )^{\frac {1}{4}}+21 \ln \left (\frac {-a^{\frac {1}{4}} x^{2}-\left (x^{7} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x^{2}-\left (x^{7} \left (a x +b \right )\right )^{\frac {1}{4}}}\right ) x \,b^{3}+42 \arctan \left (\frac {\left (x^{7} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x^{2}}\right ) x \,b^{3}}{384 a^{\frac {11}{4}} x}\) | \(144\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.25 \[ \int \sqrt [4]{b x^7+a x^8} \, dx=\frac {21 \, a^{2} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} x \log \left (\frac {7 \, {\left (a^{3} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} x^{2} + {\left (a x^{8} + b x^{7}\right )}^{\frac {1}{4}} b^{3}\right )}}{x^{2}}\right ) - 21 \, a^{2} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} x \log \left (-\frac {7 \, {\left (a^{3} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} x^{2} - {\left (a x^{8} + b x^{7}\right )}^{\frac {1}{4}} b^{3}\right )}}{x^{2}}\right ) - 21 i \, a^{2} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} x \log \left (-\frac {7 \, {\left (i \, a^{3} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} x^{2} - {\left (a x^{8} + b x^{7}\right )}^{\frac {1}{4}} b^{3}\right )}}{x^{2}}\right ) + 21 i \, a^{2} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} x \log \left (-\frac {7 \, {\left (-i \, a^{3} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} x^{2} - {\left (a x^{8} + b x^{7}\right )}^{\frac {1}{4}} b^{3}\right )}}{x^{2}}\right ) + 4 \, {\left (a x^{8} + b x^{7}\right )}^{\frac {1}{4}} {\left (32 \, a^{2} x^{2} + 4 \, a b x - 7 \, b^{2}\right )}}{384 \, a^{2} x} \]
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\[ \int \sqrt [4]{b x^7+a x^8} \, dx=\int \sqrt [4]{a x^{8} + b x^{7}}\, dx \]
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\[ \int \sqrt [4]{b x^7+a x^8} \, dx=\int { {\left (a x^{8} + b x^{7}\right )}^{\frac {1}{4}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 261 vs. \(2 (99) = 198\).
Time = 0.28 (sec) , antiderivative size = 261, normalized size of antiderivative = 2.19 \[ \int \sqrt [4]{b x^7+a x^8} \, dx=\frac {\frac {42 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{4} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a^{3}} + \frac {42 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{4} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a^{3}} + \frac {21 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{4} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{a^{3}} + \frac {21 \, \sqrt {2} b^{4} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{\left (-a\right )^{\frac {3}{4}} a^{2}} - \frac {8 \, {\left (7 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{4}} b^{4} - 18 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} a b^{4} - 21 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} a^{2} b^{4}\right )} x^{3}}{a^{2} b^{3}}}{768 \, b} \]
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Time = 5.62 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.32 \[ \int \sqrt [4]{b x^7+a x^8} \, dx=\frac {4\,x\,{\left (a\,x^8+b\,x^7\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {11}{4};\ \frac {15}{4};\ -\frac {a\,x}{b}\right )}{11\,{\left (\frac {a\,x}{b}+1\right )}^{1/4}} \]
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