∫ (−q+px4)∘ ____ q+px4 __________________________________________

Optimal. Leaf size=428 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8 a-8 \text {$\#$1}^6 a \sqrt {p} \sqrt {q}+4 \text {$\#$1}^6 b+24 \text {$\#$1}^4 a p q-16 \text {$\#$1}^4 b \sqrt {p} \sqrt {q}+16 \text {$\#$1}^4 c-32 \text {$\#$1}^2 a p^{3/2} q^{3/2}+16 \text {$\#$1}^2 b p q+16 a p^2 q^2\& ,\frac {-\text {$\#$1}^6 \log \left (-\text {$\#$1} x+\sqrt {p x^4+q}-\sqrt {p} x^2-\sqrt {q}\right )+\text {$\#$1}^6 \log (x)+2 \text {$\#$1}^4 \sqrt {p} \sqrt {q} \log \left (-\text {$\#$1} x+\sqrt {p x^4+q}-\sqrt {p} x^2-\sqrt {q}\right )-2 \text {$\#$1}^4 \sqrt {p} \sqrt {q} \log (x)+4 \text {$\#$1}^2 p q \log \left (-\text {$\#$1} x+\sqrt {p x^4+q}-\sqrt {p} x^2-\sqrt {q}\right )-4 \text {$\#$1}^2 p q \log (x)-8 p^{3/2} q^{3/2} \log \left (-\text {$\#$1} x+\sqrt {p x^4+q}-\sqrt {p} x^2-\sqrt {q}\right )+8 p^{3/2} q^{3/2} \log (x)}{\text {$\#$1}^7 (-a)+6 \text {$\#$1}^5 a \sqrt {p} \sqrt {q}-3 \text {$\#$1}^5 b-12 \text {$\#$1}^3 a p q+8 \text {$\#$1}^3 b \sqrt {p} \sqrt {q}-8 \text {$\#$1}^3 c+8 \text {$\#$1} a p^{3/2} q^{3/2}-4 \text {$\#$1} b p q}\& \right ] \]

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Rubi [F] time = 1.70, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (-q+p x^4\right ) \sqrt {q+p x^4}}{c x^4+b x^2 \left (q+p x^4\right )+a \left (q+p x^4\right )^2} \, dx \end {gather*}

\begin {align*} \int \frac {\left (-q+p x^4\right ) \sqrt {q+p x^4}}{c x^4+b x^2 \left (q+p x^4\right )+a \left (q+p x^4\right )^2} \, dx &=\int \left (\frac {q \sqrt {q+p x^4}}{-a q^2-b q x^2-c \left (1+\frac {2 a p q}{c}\right ) x^4-b p x^6-a p^2 x^8}+\frac {p x^4 \sqrt {q+p x^4}}{a q^2+b q x^2+c \left (1+\frac {2 a p q}{c}\right ) x^4+b p x^6+a p^2 x^8}\right ) \, dx\\ &=p \int \frac {x^4 \sqrt {q+p x^4}}{a q^2+b q x^2+c \left (1+\frac {2 a p q}{c}\right ) x^4+b p x^6+a p^2 x^8} \, dx+q \int \frac {\sqrt {q+p x^4}}{-a q^2-b q x^2-c \left (1+\frac {2 a p q}{c}\right ) x^4-b p x^6-a p^2 x^8} \, dx\\ &=p \int \frac {x^4 \sqrt {q+p x^4}}{c x^4+b x^2 \left (q+p x^4\right )+a \left (q+p x^4\right )^2} \, dx+q \int \frac {\sqrt {q+p x^4}}{-c x^4-b x^2 \left (q+p x^4\right )-a \left (q+p x^4\right )^2} \, dx\\ \end {align*}

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Mathematica [C] time = 6.71, size = 3958, normalized size = 9.25 \begin {gather*} \text {Result too large to show} \end {gather*}

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IntegrateAlgebraic [A] time = 3.58, size = 189, normalized size = 0.44 \begin {gather*} -\frac {\left (-b+\sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {b-\sqrt {b^2-4 a c}} x}{\sqrt {2} \sqrt {a} \sqrt {q+p x^4}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {b+\sqrt {b^2-4 a c}} \tan ^{-1}\left (\frac {\sqrt {b+\sqrt {b^2-4 a c}} x}{\sqrt {2} \sqrt {a} \sqrt {q+p x^4}}\right )}{\sqrt {2} \sqrt {a} \sqrt {b^2-4 a c}} \end {gather*}

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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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method	result	size

default	$\frac {\left (\frac {\arctan \left (\frac {\sqrt {p \,x^{4}+q}\, \sqrt {2}\, a}{x \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) a}}\right )}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) a}}+\frac {\arctan \left (\frac {\sqrt {p \,x^{4}+q}\, \sqrt {2}\, a}{x \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) a}}\right ) b}{\sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) a}}-\frac {\arctanh \left (\frac {\sqrt {p \,x^{4}+q}\, \sqrt {2}\, a}{x \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) a}}\right )}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) a}}+\frac {\arctanh \left (\frac {\sqrt {p \,x^{4}+q}\, \sqrt {2}\, a}{x \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) a}}\right ) b}{\sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) a}}\right ) \sqrt {2}}{2}$	$242$
elliptic	$\frac {\left (\frac {\arctan \left (\frac {\sqrt {p \,x^{4}+q}\, \sqrt {2}\, a}{x \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) a}}\right )}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) a}}+\frac {\arctan \left (\frac {\sqrt {p \,x^{4}+q}\, \sqrt {2}\, a}{x \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) a}}\right ) b}{\sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) a}}-\frac {\arctanh \left (\frac {\sqrt {p \,x^{4}+q}\, \sqrt {2}\, a}{x \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) a}}\right )}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) a}}+\frac {\arctanh \left (\frac {\sqrt {p \,x^{4}+q}\, \sqrt {2}\, a}{x \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) a}}\right ) b}{\sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) a}}\right ) \sqrt {2}}{2}$	$242$

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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {p x^{4} + q} {\left (p x^{4} - q\right )}}{c x^{4} + {\left (p x^{4} + q\right )} b x^{2} + {\left (p x^{4} + q\right )}^{2} a}\,{d x} \end {gather*}

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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {\sqrt {p\,x^4+q}\,\left (q-p\,x^4\right )}{a\,{\left (p\,x^4+q\right )}^2+c\,x^4+b\,x^2\,\left (p\,x^4+q\right )} \,d x \end {gather*}

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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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3.31.22 \(\int \frac {(-q+p x^4) \sqrt {q+p x^4}}{c x^4+b x^2 (q+p x^4)+a (q+p x^4)^2} \, dx\)