Optimal. Leaf size=399 \[ -\frac {\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {x \left (\sqrt {\frac {2}{2-\sqrt {2}}} \sqrt [8]{a} \sqrt [8]{b}-\frac {2 \sqrt [8]{a} \sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}\right ) \sqrt {p x^3+q}}{-\sqrt [4]{a} p x^3-\sqrt [4]{a} q+\sqrt [4]{b} x^2}\right )}{4 a^{3/8} b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {p x^3+q}}{\sqrt [4]{a} p x^3+\sqrt [4]{a} q-\sqrt [4]{b} x^2}\right )}{4 a^{3/8} b^{5/8}}-\frac {\sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {p x^3+q}}{\sqrt [4]{a} p x^3+\sqrt [4]{a} q+\sqrt [4]{b} x^2}\right )}{4 a^{3/8} b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {p x^3+q}}{\sqrt [4]{a} p x^3+\sqrt [4]{a} q+\sqrt [4]{b} x^2}\right )}{4 a^{3/8} b^{5/8}} \]
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Rubi [F] time = 2.33, 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 {x^4 \left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^4 \left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx &=\int \left (-\frac {2 q x^4 \sqrt {q+p x^3}}{a q^4+4 a p q^3 x^3+6 a p^2 q^2 x^6+b x^8+4 a p^3 q x^9+a p^4 x^{12}}+\frac {p x^7 \sqrt {q+p x^3}}{a q^4+4 a p q^3 x^3+6 a p^2 q^2 x^6+b x^8+4 a p^3 q x^9+a p^4 x^{12}}\right ) \, dx\\ &=p \int \frac {x^7 \sqrt {q+p x^3}}{a q^4+4 a p q^3 x^3+6 a p^2 q^2 x^6+b x^8+4 a p^3 q x^9+a p^4 x^{12}} \, dx-(2 q) \int \frac {x^4 \sqrt {q+p x^3}}{a q^4+4 a p q^3 x^3+6 a p^2 q^2 x^6+b x^8+4 a p^3 q x^9+a p^4 x^{12}} \, dx\\ &=p \int \frac {x^7 \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx-(2 q) \int \frac {x^4 \sqrt {q+p x^3}}{b x^8+a \left (q+p x^3\right )^4} \, dx\\ \end {align*}
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Mathematica [C] time = 7.82, size = 65821, normalized size = 164.96 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 3.50, size = 367, normalized size = 0.92 \begin {gather*} -\frac {\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {q+p x^3}}{\sqrt [4]{a} q-\sqrt [4]{b} x^2+\sqrt [4]{a} p x^3}\right )}{4 a^{3/8} b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {q+p x^3}}{\sqrt [4]{a} q-\sqrt [4]{b} x^2+\sqrt [4]{a} p x^3}\right )}{4 a^{3/8} b^{5/8}}-\frac {\sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {q+p x^3}}{\sqrt [4]{a} q+\sqrt [4]{b} x^2+\sqrt [4]{a} p x^3}\right )}{4 a^{3/8} b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {q+p x^3}}{\sqrt [4]{a} q+\sqrt [4]{b} x^2+\sqrt [4]{a} p x^3}\right )}{4 a^{3/8} b^{5/8}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 15.79, size = 2875, normalized size = 7.21
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.85, size = 1596, normalized size = 4.00
method | result | size |
default | \(\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,p^{4} \textit {\_Z}^{12}+4 a \,p^{3} q \,\textit {\_Z}^{9}+6 a \,p^{2} q^{2} \textit {\_Z}^{6}+b \,\textit {\_Z}^{8}+4 a p \,q^{3} \textit {\_Z}^{3}+a \,q^{4}\right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{6} p^{2}-\underline {\hspace {1.25 ex}}\alpha ^{3} p q -2 q^{2}\right ) \underline {\hspace {1.25 ex}}\alpha ^{2} \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i p \left (2 x +\frac {-i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}+\left (-q \,p^{2}\right )^{\frac {1}{3}}}{p}\right )}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {p \left (x -\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{p}\right )}{-3 \left (-q \,p^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i p \left (2 x +\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}+\left (-q \,p^{2}\right )^{\frac {1}{3}}}{p}\right )}{2 \left (-q \,p^{2}\right )^{\frac {1}{3}}}}\, \left (\left (-q \,p^{2}\right )^{\frac {2}{3}} b \,q^{2}-3 i \left (-q \,p^{2}\right )^{\frac {1}{3}} p^{5} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} a \,q^{2}-i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} q p b -i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha \,q^{2} p b +2 q \,p^{2} \left (a \,p^{5} \underline {\hspace {1.25 ex}}\alpha ^{9}+3 a \,p^{4} q \,\underline {\hspace {1.25 ex}}\alpha ^{6}+3 a \,p^{3} q^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}+b p \,\underline {\hspace {1.25 ex}}\alpha ^{5}+a \,p^{2} q^{3}-b q \,\underline {\hspace {1.25 ex}}\alpha ^{2}\right )-i \left (-q \,p^{2}\right )^{\frac {1}{3}} p^{3} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{7} b +i \left (-q \,p^{2}\right )^{\frac {2}{3}} p^{6} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{10} a +i \left (-q \,p^{2}\right )^{\frac {2}{3}} p^{3} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a \,q^{3}-\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{3} b p q +\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b p \,q^{2}+3 \left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{8} a \,p^{6} q +3 \left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{7} a \,p^{5} q +3 \left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{5} a \,p^{5} q^{2}+3 \left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} a \,p^{4} q^{2}+i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, q^{2} b -3 i \left (-q \,p^{2}\right )^{\frac {1}{3}} p^{6} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{8} a q +3 i \left (-q \,p^{2}\right )^{\frac {2}{3}} p^{5} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{7} a q +3 i \left (-q \,p^{2}\right )^{\frac {2}{3}} p^{4} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} a \,q^{2}-i \left (-q \,p^{2}\right )^{\frac {1}{3}} p^{4} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a \,q^{3}+i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} q \,p^{2} b +\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{11} a \,p^{7}+\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{10} a \,p^{6}+\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{7} b \,p^{3}+\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{6} b \,p^{2}+\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a \,p^{4} q^{3}+\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha a \,p^{3} q^{3}-\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} b \,p^{2} q +i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{6} p^{2} b -i \left (-q \,p^{2}\right )^{\frac {1}{3}} p^{7} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{11} a \right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}-\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right ) \sqrt {3}\, p}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {3 i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{7} a \,p^{6} q^{2}+3 i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} a \,p^{5} q^{3}-i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{7} b \,p^{2}-3 q^{3} b p -2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, a \,p^{3} q^{4}+i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{10} a \,p^{7} q +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{6} b \,p^{3} q +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a \,p^{4} q^{4}-i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{11} a \,p^{6}+i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} b p q -3 i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{8} a \,p^{5} q -6 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{6} a \,p^{5} q^{2}-3 i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} a \,p^{4} q^{2}-6 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a \,p^{4} q^{3}-i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a \,p^{3} q^{3}-2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} b \,p^{2} q +2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b p \,q^{2}-2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{9} a \,p^{6} q -3 p^{3} \underline {\hspace {1.25 ex}}\alpha ^{6} q b -3 p^{4} \underline {\hspace {1.25 ex}}\alpha a \,q^{4}+3 \underline {\hspace {1.25 ex}}\alpha ^{3} q^{2} p^{2} b -3 p^{7} \underline {\hspace {1.25 ex}}\alpha ^{10} a q -9 p^{6} \underline {\hspace {1.25 ex}}\alpha ^{7} a \,q^{2}-9 p^{5} \underline {\hspace {1.25 ex}}\alpha ^{4} a \,q^{3}-3 \left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha \,q^{2} b -3 \underline {\hspace {1.25 ex}}\alpha ^{11} p^{6} \left (-q \,p^{2}\right )^{\frac {2}{3}} a -3 \underline {\hspace {1.25 ex}}\alpha ^{7} \left (-q \,p^{2}\right )^{\frac {2}{3}} p^{2} b -9 \underline {\hspace {1.25 ex}}\alpha ^{5} p^{4} \left (-q \,p^{2}\right )^{\frac {2}{3}} a \,q^{2}-3 \underline {\hspace {1.25 ex}}\alpha ^{2} p^{3} \left (-q \,p^{2}\right )^{\frac {2}{3}} a \,q^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{4} \left (-q \,p^{2}\right )^{\frac {2}{3}} q p b -9 \underline {\hspace {1.25 ex}}\alpha ^{8} p^{5} \left (-q \,p^{2}\right )^{\frac {2}{3}} a q +i \sqrt {3}\, b p \,q^{3}-i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} b \,p^{2} q^{2}-i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha b \,q^{2}}{2 p \,q^{3} b}, \sqrt {\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{p \left (-\frac {3 \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}+\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right )}}\right )}{2 \left (-3 \underline {\hspace {1.25 ex}}\alpha ^{9} a \,p^{4}-9 \underline {\hspace {1.25 ex}}\alpha ^{6} a \,p^{3} q -9 \underline {\hspace {1.25 ex}}\alpha ^{3} a \,p^{2} q^{2}-2 \underline {\hspace {1.25 ex}}\alpha ^{5} b -3 a p \,q^{3}\right ) \sqrt {p \,x^{3}+q}}\right )}{4 p^{2} q^{3} b}\) | \(1596\) |
elliptic | \(\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,p^{4} \textit {\_Z}^{12}+4 a \,p^{3} q \,\textit {\_Z}^{9}+6 a \,p^{2} q^{2} \textit {\_Z}^{6}+b \,\textit {\_Z}^{8}+4 a p \,q^{3} \textit {\_Z}^{3}+a \,q^{4}\right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{6} p^{2}-\underline {\hspace {1.25 ex}}\alpha ^{3} p q -2 q^{2}\right ) \underline {\hspace {1.25 ex}}\alpha ^{2} \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i p \left (2 x +\frac {-i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}+\left (-q \,p^{2}\right )^{\frac {1}{3}}}{p}\right )}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {p \left (x -\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{p}\right )}{-3 \left (-q \,p^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i p \left (2 x +\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}+\left (-q \,p^{2}\right )^{\frac {1}{3}}}{p}\right )}{2 \left (-q \,p^{2}\right )^{\frac {1}{3}}}}\, \left (\left (-q \,p^{2}\right )^{\frac {2}{3}} b \,q^{2}-3 i \left (-q \,p^{2}\right )^{\frac {1}{3}} p^{5} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} a \,q^{2}-i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} q p b -i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha \,q^{2} p b +2 q \,p^{2} \left (a \,p^{5} \underline {\hspace {1.25 ex}}\alpha ^{9}+3 a \,p^{4} q \,\underline {\hspace {1.25 ex}}\alpha ^{6}+3 a \,p^{3} q^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}+b p \,\underline {\hspace {1.25 ex}}\alpha ^{5}+a \,p^{2} q^{3}-b q \,\underline {\hspace {1.25 ex}}\alpha ^{2}\right )-i \left (-q \,p^{2}\right )^{\frac {1}{3}} p^{3} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{7} b +i \left (-q \,p^{2}\right )^{\frac {2}{3}} p^{6} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{10} a +i \left (-q \,p^{2}\right )^{\frac {2}{3}} p^{3} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a \,q^{3}-\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{3} b p q +\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b p \,q^{2}+3 \left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{8} a \,p^{6} q +3 \left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{7} a \,p^{5} q +3 \left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{5} a \,p^{5} q^{2}+3 \left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} a \,p^{4} q^{2}+i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, q^{2} b -3 i \left (-q \,p^{2}\right )^{\frac {1}{3}} p^{6} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{8} a q +3 i \left (-q \,p^{2}\right )^{\frac {2}{3}} p^{5} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{7} a q +3 i \left (-q \,p^{2}\right )^{\frac {2}{3}} p^{4} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} a \,q^{2}-i \left (-q \,p^{2}\right )^{\frac {1}{3}} p^{4} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a \,q^{3}+i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} q \,p^{2} b +\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{11} a \,p^{7}+\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{10} a \,p^{6}+\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{7} b \,p^{3}+\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{6} b \,p^{2}+\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a \,p^{4} q^{3}+\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha a \,p^{3} q^{3}-\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} b \,p^{2} q +i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{6} p^{2} b -i \left (-q \,p^{2}\right )^{\frac {1}{3}} p^{7} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{11} a \right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}-\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right ) \sqrt {3}\, p}{\left (-q \,p^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {3 i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{7} a \,p^{6} q^{2}+3 i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} a \,p^{5} q^{3}-i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{7} b \,p^{2}-3 q^{3} b p -2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, a \,p^{3} q^{4}+i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{10} a \,p^{7} q +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{6} b \,p^{3} q +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a \,p^{4} q^{4}-i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{11} a \,p^{6}+i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} b p q -3 i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{8} a \,p^{5} q -6 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{6} a \,p^{5} q^{2}-3 i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} a \,p^{4} q^{2}-6 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a \,p^{4} q^{3}-i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a \,p^{3} q^{3}-2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} b \,p^{2} q +2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b p \,q^{2}-2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{9} a \,p^{6} q -3 p^{3} \underline {\hspace {1.25 ex}}\alpha ^{6} q b -3 p^{4} \underline {\hspace {1.25 ex}}\alpha a \,q^{4}+3 \underline {\hspace {1.25 ex}}\alpha ^{3} q^{2} p^{2} b -3 p^{7} \underline {\hspace {1.25 ex}}\alpha ^{10} a q -9 p^{6} \underline {\hspace {1.25 ex}}\alpha ^{7} a \,q^{2}-9 p^{5} \underline {\hspace {1.25 ex}}\alpha ^{4} a \,q^{3}-3 \left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha \,q^{2} b -3 \underline {\hspace {1.25 ex}}\alpha ^{11} p^{6} \left (-q \,p^{2}\right )^{\frac {2}{3}} a -3 \underline {\hspace {1.25 ex}}\alpha ^{7} \left (-q \,p^{2}\right )^{\frac {2}{3}} p^{2} b -9 \underline {\hspace {1.25 ex}}\alpha ^{5} p^{4} \left (-q \,p^{2}\right )^{\frac {2}{3}} a \,q^{2}-3 \underline {\hspace {1.25 ex}}\alpha ^{2} p^{3} \left (-q \,p^{2}\right )^{\frac {2}{3}} a \,q^{3}+3 \underline {\hspace {1.25 ex}}\alpha ^{4} \left (-q \,p^{2}\right )^{\frac {2}{3}} q p b -9 \underline {\hspace {1.25 ex}}\alpha ^{8} p^{5} \left (-q \,p^{2}\right )^{\frac {2}{3}} a q +i \sqrt {3}\, b p \,q^{3}-i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} b \,p^{2} q^{2}-i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha b \,q^{2}}{2 p \,q^{3} b}, \sqrt {\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{p \left (-\frac {3 \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}+\frac {i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{3}}}{2 p}\right )}}\right )}{2 \left (-3 \underline {\hspace {1.25 ex}}\alpha ^{9} a \,p^{4}-9 \underline {\hspace {1.25 ex}}\alpha ^{6} a \,p^{3} q -9 \underline {\hspace {1.25 ex}}\alpha ^{3} a \,p^{2} q^{2}-2 \underline {\hspace {1.25 ex}}\alpha ^{5} b -3 a p \,q^{3}\right ) \sqrt {p \,x^{3}+q}}\right )}{4 p^{2} q^{3} b}\) | \(1596\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {p x^{3} + q} {\left (p x^{3} - 2 \, q\right )} x^{4}}{b x^{8} + {\left (p x^{3} + q\right )}^{4} a}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \left (p x^{3} - 2 q\right ) \sqrt {p x^{3} + q}}{a p^{4} x^{12} + 4 a p^{3} q x^{9} + 6 a p^{2} q^{2} x^{6} + 4 a p q^{3} x^{3} + a q^{4} + b x^{8}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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