Integrand size = 22, antiderivative size = 238 \[ \int \frac {x^6 \sqrt {x+x^4}}{b+a x^6} \, dx=\frac {x \sqrt {x+x^4}}{3 a}+\frac {\sqrt [4]{-1} \sqrt {\left (\sqrt {a}-i \sqrt {b}\right ) \sqrt {b}} \arctan \left (\frac {(1+i) \sqrt {\sqrt {a} \sqrt {b}-i b} \sqrt {x+x^4}}{\sqrt {2} \left (\sqrt {a}-i \sqrt {b}\right ) x^2}\right )}{3 a^{3/2}}+\frac {(-1)^{3/4} \sqrt {\left (\sqrt {a}+i \sqrt {b}\right ) \sqrt {b}} \arctan \left (\frac {(1+i) \sqrt {\sqrt {a} \sqrt {b}+i b} x \sqrt {x+x^4}}{\sqrt {2} \sqrt {b} (1+x) \left (1-x+x^2\right )}\right )}{3 a^{3/2}}+\frac {\text {arctanh}\left (\frac {x^2}{\sqrt {x+x^4}}\right )}{3 a} \]
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Time = 0.42 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.06, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {2081, 1507, 1505, 1306, 201, 221, 1189, 399, 385, 214} \[ \int \frac {x^6 \sqrt {x+x^4}}{b+a x^6} \, dx=\frac {\sqrt {x^4+x} \text {arcsinh}\left (x^{3/2}\right )}{3 a \sqrt {x^3+1} \sqrt {x}}+\frac {\sqrt [4]{b} \sqrt {x^4+x} \sqrt {\sqrt {-a}+\sqrt {b}} \text {arctanh}\left (\frac {x^{3/2} \sqrt {\sqrt {-a}+\sqrt {b}}}{\sqrt [4]{b} \sqrt {x^3+1}}\right )}{3 (-a)^{3/2} \sqrt {x^3+1} \sqrt {x}}-\frac {\sqrt [4]{b} \sqrt {x^4+x} \sqrt {\sqrt {-a} \sqrt {b}+a} \text {arctanh}\left (\frac {x^{3/2} \sqrt {\sqrt {-a} \sqrt {b}+a}}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {x^3+1}}\right )}{3 (-a)^{7/4} \sqrt {x^3+1} \sqrt {x}}+\frac {\sqrt {x^4+x} x}{3 a} \]
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Rule 201
Rule 214
Rule 221
Rule 385
Rule 399
Rule 1189
Rule 1306
Rule 1505
Rule 1507
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {x+x^4} \int \frac {x^{13/2} \sqrt {1+x^3}}{b+a x^6} \, dx}{\sqrt {x} \sqrt {1+x^3}} \\ & = \frac {\left (2 \sqrt {x+x^4}\right ) \text {Subst}\left (\int \frac {x^{14} \sqrt {1+x^6}}{b+a x^{12}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3}} \\ & = \frac {\left (2 \sqrt {x+x^4}\right ) \text {Subst}\left (\int \frac {x^4 \sqrt {1+x^2}}{b+a x^4} \, dx,x,x^{3/2}\right )}{3 \sqrt {x} \sqrt {1+x^3}} \\ & = \frac {\left (2 \sqrt {x+x^4}\right ) \text {Subst}\left (\int \sqrt {1+x^2} \, dx,x,x^{3/2}\right )}{3 a \sqrt {x} \sqrt {1+x^3}}-\frac {\left (2 b \sqrt {x+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt {1+x^2}}{b+a x^4} \, dx,x,x^{3/2}\right )}{3 a \sqrt {x} \sqrt {1+x^3}} \\ & = \frac {x \sqrt {x+x^4}}{3 a}+\frac {\sqrt {x+x^4} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^{3/2}\right )}{3 a \sqrt {x} \sqrt {1+x^3}}-\frac {\left (\sqrt {-a} \sqrt {b} \sqrt {x+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt {1+x^2}}{\sqrt {-a} \sqrt {b}-a x^2} \, dx,x,x^{3/2}\right )}{3 a \sqrt {x} \sqrt {1+x^3}}-\frac {\left (\sqrt {-a} \sqrt {b} \sqrt {x+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt {1+x^2}}{\sqrt {-a} \sqrt {b}+a x^2} \, dx,x,x^{3/2}\right )}{3 a \sqrt {x} \sqrt {1+x^3}} \\ & = \frac {x \sqrt {x+x^4}}{3 a}+\frac {\sqrt {x+x^4} \text {arcsinh}\left (x^{3/2}\right )}{3 a \sqrt {x} \sqrt {1+x^3}}+\frac {\left (\sqrt {-a} \left (-a+\sqrt {-a} \sqrt {b}\right ) \sqrt {b} \sqrt {x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2} \left (\sqrt {-a} \sqrt {b}+a x^2\right )} \, dx,x,x^{3/2}\right )}{3 a^2 \sqrt {x} \sqrt {1+x^3}}-\frac {\left (\sqrt {-a} \left (a+\sqrt {-a} \sqrt {b}\right ) \sqrt {b} \sqrt {x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2} \left (\sqrt {-a} \sqrt {b}-a x^2\right )} \, dx,x,x^{3/2}\right )}{3 a^2 \sqrt {x} \sqrt {1+x^3}} \\ & = \frac {x \sqrt {x+x^4}}{3 a}+\frac {\sqrt {x+x^4} \text {arcsinh}\left (x^{3/2}\right )}{3 a \sqrt {x} \sqrt {1+x^3}}+\frac {\left (\sqrt {-a} \left (-a+\sqrt {-a} \sqrt {b}\right ) \sqrt {b} \sqrt {x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a} \sqrt {b}-\left (-a+\sqrt {-a} \sqrt {b}\right ) x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {1+x^3}}\right )}{3 a^2 \sqrt {x} \sqrt {1+x^3}}-\frac {\left (\sqrt {-a} \left (a+\sqrt {-a} \sqrt {b}\right ) \sqrt {b} \sqrt {x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a} \sqrt {b}-\left (a+\sqrt {-a} \sqrt {b}\right ) x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {1+x^3}}\right )}{3 a^2 \sqrt {x} \sqrt {1+x^3}} \\ & = \frac {x \sqrt {x+x^4}}{3 a}+\frac {\sqrt {x+x^4} \text {arcsinh}\left (x^{3/2}\right )}{3 a \sqrt {x} \sqrt {1+x^3}}+\frac {\sqrt {\sqrt {-a}+\sqrt {b}} \sqrt [4]{b} \sqrt {x+x^4} \text {arctanh}\left (\frac {\sqrt {\sqrt {-a}+\sqrt {b}} x^{3/2}}{\sqrt [4]{b} \sqrt {1+x^3}}\right )}{3 (-a)^{3/2} \sqrt {x} \sqrt {1+x^3}}-\frac {\sqrt {a+\sqrt {-a} \sqrt {b}} \sqrt [4]{b} \sqrt {x+x^4} \text {arctanh}\left (\frac {\sqrt {a+\sqrt {-a} \sqrt {b}} x^{3/2}}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {1+x^3}}\right )}{3 (-a)^{7/4} \sqrt {x} \sqrt {1+x^3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.26 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.74 \[ \int \frac {x^6 \sqrt {x+x^4}}{b+a x^6} \, dx=\frac {\sqrt {x+x^4} \left (x^{3/2} \sqrt {1+x^3}+\log \left (x^{3/2}+\sqrt {1+x^3}\right )-b \text {RootSum}\left [16 a+16 b-32 a \text {$\#$1}-32 b \text {$\#$1}+24 a \text {$\#$1}^2+16 b \text {$\#$1}^2-8 a \text {$\#$1}^3+a \text {$\#$1}^4\&,\frac {\log \left (2+2 x^3+2 x^{3/2} \sqrt {1+x^3}-\text {$\#$1}\right ) \text {$\#$1}^2}{-8 a-8 b+12 a \text {$\#$1}+8 b \text {$\#$1}-6 a \text {$\#$1}^2+a \text {$\#$1}^3}\&\right ]\right )}{3 a \sqrt {x} \sqrt {1+x^3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(374\) vs. \(2(171)=342\).
Time = 3.02 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.58
method | result | size |
pseudoelliptic | \(-\frac {\left (-\frac {\left (-\ln \left (\frac {\sqrt {b}\, x^{3}+\sqrt {a +b}\, x^{3}-\sqrt {x^{4}+x}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x +\sqrt {b}}{x^{3}}\right )+\ln \left (\frac {\sqrt {b}\, x^{3}+\sqrt {a +b}\, x^{3}+\sqrt {x^{4}+x}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x +\sqrt {b}}{x^{3}}\right )\right ) \left (-\sqrt {b \left (a +b \right )}+b \right ) \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}}{2}+\left (-2 x \sqrt {x^{4}+x}+\ln \left (\frac {-x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )-\ln \left (\frac {x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )\right ) a \sqrt {b}\right ) \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}+2 a b \left (\arctan \left (\frac {\sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x^{2}-2 \sqrt {b}\, \sqrt {x^{4}+x}}{\sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}\, x^{2}}\right )-\arctan \left (\frac {\sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x^{2}+2 \sqrt {b}\, \sqrt {x^{4}+x}}{x^{2} \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}}\right )\right )}{6 \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}\, \sqrt {b}\, a^{2}}\) | \(375\) |
default | \(\frac {\frac {x \sqrt {x^{4}+x}}{3}-\frac {\ln \left (\frac {-x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )}{6}+\frac {\ln \left (\frac {x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )}{6}}{a}+\frac {\left (-\sqrt {b \left (a +b \right )}+b \right ) \left (\frac {\sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, \left (-\ln \left (\frac {\sqrt {b}\, x^{3}+\sqrt {a +b}\, x^{3}-\sqrt {x^{4}+x}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x +\sqrt {b}}{x^{3}}\right )+\ln \left (\frac {\sqrt {b}\, x^{3}+\sqrt {a +b}\, x^{3}+\sqrt {x^{4}+x}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x +\sqrt {b}}{x^{3}}\right )\right ) \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}}{4}+\left (\sqrt {b \left (a +b \right )}+b \right ) \left (\arctan \left (\frac {\sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x^{2}-2 \sqrt {b}\, \sqrt {x^{4}+x}}{\sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}\, x^{2}}\right )-\arctan \left (\frac {\sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x^{2}+2 \sqrt {b}\, \sqrt {x^{4}+x}}{x^{2} \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}}\right )\right )\right )}{3 \sqrt {b}\, a^{2} \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}}\) | \(381\) |
risch | \(\frac {x^{2} \left (x^{3}+1\right )}{3 a \sqrt {x \left (x^{3}+1\right )}}+\frac {\ln \left (-2 x^{3}-2 x \sqrt {x^{4}+x}-1\right )}{6 a}+\frac {\sqrt {b \left (a +b \right )}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, \ln \left (\frac {\sqrt {b}\, x^{3}+\sqrt {a +b}\, x^{3}-\sqrt {x^{4}+x}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x +\sqrt {b}}{x^{3}}\right )}{12 a^{2} \sqrt {b}}-\frac {\sqrt {b \left (a +b \right )}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, \ln \left (\frac {\sqrt {b}\, x^{3}+\sqrt {a +b}\, x^{3}+\sqrt {x^{4}+x}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x +\sqrt {b}}{x^{3}}\right )}{12 a^{2} \sqrt {b}}+\frac {\sqrt {b}\, \arctan \left (\frac {\sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x^{2}+2 \sqrt {b}\, \sqrt {x^{4}+x}}{x^{2} \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}}\right )}{3 \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}\, a}-\frac {\sqrt {b}\, \arctan \left (\frac {\sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x^{2}-2 \sqrt {b}\, \sqrt {x^{4}+x}}{\sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}\, x^{2}}\right )}{3 \sqrt {4 \sqrt {a +b}\, \sqrt {b}-2 \sqrt {b \left (a +b \right )}-2 b}\, a}-\frac {\sqrt {b}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, \ln \left (\frac {\sqrt {b}\, x^{3}+\sqrt {a +b}\, x^{3}-\sqrt {x^{4}+x}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x +\sqrt {b}}{x^{3}}\right )}{12 a^{2}}+\frac {\sqrt {b}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, \ln \left (\frac {\sqrt {b}\, x^{3}+\sqrt {a +b}\, x^{3}+\sqrt {x^{4}+x}\, \sqrt {2 \sqrt {b \left (a +b \right )}+2 b}\, x +\sqrt {b}}{x^{3}}\right )}{12 a^{2}}\) | \(541\) |
elliptic | \(\text {Expression too large to display}\) | \(675\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1784 vs. \(2 (162) = 324\).
Time = 54.75 (sec) , antiderivative size = 1784, normalized size of antiderivative = 7.50 \[ \int \frac {x^6 \sqrt {x+x^4}}{b+a x^6} \, dx=\text {Too large to display} \]
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\[ \int \frac {x^6 \sqrt {x+x^4}}{b+a x^6} \, dx=\int \frac {x^{6} \sqrt {x \left (x + 1\right ) \left (x^{2} - x + 1\right )}}{a x^{6} + b}\, dx \]
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\[ \int \frac {x^6 \sqrt {x+x^4}}{b+a x^6} \, dx=\int { \frac {\sqrt {x^{4} + x} x^{6}}{a x^{6} + b} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 441 vs. \(2 (162) = 324\).
Time = 1.00 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.85 \[ \int \frac {x^6 \sqrt {x+x^4}}{b+a x^6} \, dx=\frac {\sqrt {x^{4} + x} x}{3 \, a} + \frac {{\left ({\left (4 \, \sqrt {-a b} \sqrt {-b^{2} - \sqrt {-a b} b} a - 5 \, \sqrt {-a b} \sqrt {-b^{2} - \sqrt {-a b} b} b\right )} a^{2} {\left | b \right |} + {\left (4 \, \sqrt {-a b} \sqrt {-b^{2} - \sqrt {-a b} b} a^{2} b - 5 \, \sqrt {-a b} \sqrt {-b^{2} - \sqrt {-a b} b} a b^{2}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {\frac {1}{x^{3}} + 1}}{\sqrt {-\frac {a b + \sqrt {a^{2} b^{2} - {\left (a^{2} + a b\right )} a b}}{a b}}}\right )}{3 \, {\left (4 \, a^{4} b^{2} - a^{3} b^{3} - 5 \, a^{2} b^{4}\right )} {\left | a \right |}} - \frac {{\left ({\left (4 \, \sqrt {-a b} \sqrt {-b^{2} + \sqrt {-a b} b} a - 5 \, \sqrt {-a b} \sqrt {-b^{2} + \sqrt {-a b} b} b\right )} a^{2} {\left | b \right |} + {\left (4 \, \sqrt {-a b} \sqrt {-b^{2} + \sqrt {-a b} b} a^{2} b - 5 \, \sqrt {-a b} \sqrt {-b^{2} + \sqrt {-a b} b} a b^{2}\right )} {\left | b \right |}\right )} \arctan \left (\frac {\sqrt {\frac {1}{x^{3}} + 1}}{\sqrt {-\frac {a b - \sqrt {a^{2} b^{2} - {\left (a^{2} + a b\right )} a b}}{a b}}}\right )}{3 \, {\left (4 \, a^{4} b^{2} - a^{3} b^{3} - 5 \, a^{2} b^{4}\right )} {\left | a \right |}} + \frac {\log \left (\sqrt {\frac {1}{x^{3}} + 1} + 1\right )}{6 \, a} - \frac {\log \left ({\left | \sqrt {\frac {1}{x^{3}} + 1} - 1 \right |}\right )}{6 \, a} \]
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Timed out. \[ \int \frac {x^6 \sqrt {x+x^4}}{b+a x^6} \, dx=\int \frac {x^6\,\sqrt {x^4+x}}{a\,x^6+b} \,d x \]
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