Optimal. Leaf size=189 \[ -\frac {\sqrt {2} \left (\sqrt {b^2-4 a c}-b\right ) \tan ^{-1}\left (\frac {x \sqrt {b-\sqrt {b^2-4 a c}}}{\sqrt {2} \sqrt {a} \sqrt {p x^3+q}}\right )}{\sqrt {a} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b} \tan ^{-1}\left (\frac {x \sqrt {\sqrt {b^2-4 a c}+b}}{\sqrt {2} \sqrt {a} \sqrt {p x^3+q}}\right )}{\sqrt {a} \sqrt {b^2-4 a c}} \]
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Rubi [F] time = 1.34, 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 (-2 q+p x^3\right ) \sqrt {q+p x^3}}{c x^4+b x^2 \left (q+p x^3\right )+a \left (q+p x^3\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {\left (-2 q+p x^3\right ) \sqrt {q+p x^3}}{c x^4+b x^2 \left (q+p x^3\right )+a \left (q+p x^3\right )^2} \, dx &=\int \left (-\frac {2 q \sqrt {q+p x^3}}{a q^2+b q x^2+2 a p q x^3+c x^4+b p x^5+a p^2 x^6}+\frac {p x^3 \sqrt {q+p x^3}}{a q^2+b q x^2+2 a p q x^3+c x^4+b p x^5+a p^2 x^6}\right ) \, dx\\ &=p \int \frac {x^3 \sqrt {q+p x^3}}{a q^2+b q x^2+2 a p q x^3+c x^4+b p x^5+a p^2 x^6} \, dx-(2 q) \int \frac {\sqrt {q+p x^3}}{a q^2+b q x^2+2 a p q x^3+c x^4+b p x^5+a p^2 x^6} \, dx\\ \end {align*}
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Mathematica [C] time = 6.68, size = 16759, normalized size = 88.67 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 1.59, size = 189, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {2} \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^3}}\right )}{\sqrt {a} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \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^3}}\right )}{\sqrt {a} \sqrt {b^2-4 a c}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 5.94, size = 1321, normalized size = 6.99
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.63, size = 1362, normalized size = 7.21
method | result | size |
default | \(-\frac {2 i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{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}}}}\, \sqrt {\frac {x -\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{p}}{-\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}}}\, \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}}}}\, \EllipticF \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}, \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 )}{3 a p \sqrt {p \,x^{3}+q}}-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,p^{2} \textit {\_Z}^{6}+b p \,\textit {\_Z}^{5}+2 a q p \,\textit {\_Z}^{3}+c \,\textit {\_Z}^{4}+b q \,\textit {\_Z}^{2}+a \,q^{2}\right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{5} b p +3 \underline {\hspace {1.25 ex}}\alpha ^{3} a p q +\underline {\hspace {1.25 ex}}\alpha ^{4} c +\underline {\hspace {1.25 ex}}\alpha ^{2} b q +3 a \,q^{2}\right ) \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 (2 q \,p^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{4} a \,p^{2}+\underline {\hspace {1.25 ex}}\alpha ^{3} b p +\underline {\hspace {1.25 ex}}\alpha a p q +\underline {\hspace {1.25 ex}}\alpha ^{2} c \right )+i \left (-q \,p^{2}\right )^{\frac {1}{3}} p^{3} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a q +i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, a \,q^{2} p^{2}+\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{5} a \,p^{3}+i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} p c +i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha q p c -i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, q c +\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} b \,p^{2}+i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b q \,p^{2}-\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{3} a \,p^{3} q +\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a \,p^{2} q +i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a q \,p^{2}+\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{3} c p +i \left (-q \,p^{2}\right )^{\frac {2}{3}} p^{3} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} a -\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} b \,p^{2} q +i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} b \,p^{2}-\left (-q \,p^{2}\right )^{\frac {1}{3}} a \,p^{2} q^{2}-\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha c p q -\left (-q \,p^{2}\right )^{\frac {2}{3}} c q \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 {i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, a p q -2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} a \,p^{3}+i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} b \,p^{3}+i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a \,p^{3} q -3 p^{4} \underline {\hspace {1.25 ex}}\alpha ^{5} a +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} a \,p^{4}+i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b p +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} c \,p^{2}+3 \left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{3} a \,p^{2}-2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} b \,p^{2}+i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha c -3 b \,p^{3} \underline {\hspace {1.25 ex}}\alpha ^{4}+i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a \,p^{2}-i \sqrt {3}\, c p q -3 p^{3} \underline {\hspace {1.25 ex}}\alpha ^{2} a q -2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a \,p^{2} q +3 \left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} b p -3 \underline {\hspace {1.25 ex}}\alpha ^{3} p^{2} c +3 \left (-q \,p^{2}\right )^{\frac {2}{3}} a q p -2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} c p +3 \left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha c +3 q c p}{2 p q c}, \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 \underline {\hspace {1.25 ex}}\alpha \left (6 \underline {\hspace {1.25 ex}}\alpha ^{4} a \,p^{2}+5 \underline {\hspace {1.25 ex}}\alpha ^{3} b p +6 \underline {\hspace {1.25 ex}}\alpha a p q +4 \underline {\hspace {1.25 ex}}\alpha ^{2} c +2 b q \right ) \sqrt {p \,x^{3}+q}}\right )}{a \,p^{2} q^{2} c}\) | \(1362\) |
elliptic | \(-\frac {2 i \sqrt {3}\, \left (-q \,p^{2}\right )^{\frac {1}{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}}}}\, \sqrt {\frac {x -\frac {\left (-q \,p^{2}\right )^{\frac {1}{3}}}{p}}{-\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}}}\, \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}}}}\, \EllipticF \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}, \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 )}{3 a p \sqrt {p \,x^{3}+q}}-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,p^{2} \textit {\_Z}^{6}+b p \,\textit {\_Z}^{5}+2 a q p \,\textit {\_Z}^{3}+c \,\textit {\_Z}^{4}+b q \,\textit {\_Z}^{2}+a \,q^{2}\right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{5} b p +3 \underline {\hspace {1.25 ex}}\alpha ^{3} a p q +\underline {\hspace {1.25 ex}}\alpha ^{4} c +\underline {\hspace {1.25 ex}}\alpha ^{2} b q +3 a \,q^{2}\right ) \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 (2 q \,p^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{4} a \,p^{2}+\underline {\hspace {1.25 ex}}\alpha ^{3} b p +\underline {\hspace {1.25 ex}}\alpha a p q +\underline {\hspace {1.25 ex}}\alpha ^{2} c \right )+i \left (-q \,p^{2}\right )^{\frac {1}{3}} p^{3} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a q +i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, a \,q^{2} p^{2}+\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{5} a \,p^{3}+i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} p c +i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha q p c -i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, q c +\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} b \,p^{2}+i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b q \,p^{2}-\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{3} a \,p^{3} q +\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a \,p^{2} q +i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a q \,p^{2}+\left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{3} c p +i \left (-q \,p^{2}\right )^{\frac {2}{3}} p^{3} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} a -\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} b \,p^{2} q +i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} b \,p^{2}-\left (-q \,p^{2}\right )^{\frac {1}{3}} a \,p^{2} q^{2}-\left (-q \,p^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha c p q -\left (-q \,p^{2}\right )^{\frac {2}{3}} c q \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 {i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, a p q -2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} a \,p^{3}+i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} b \,p^{3}+i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a \,p^{3} q -3 p^{4} \underline {\hspace {1.25 ex}}\alpha ^{5} a +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} a \,p^{4}+i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b p +i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} c \,p^{2}+3 \left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{3} a \,p^{2}-2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} b \,p^{2}+i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha c -3 b \,p^{3} \underline {\hspace {1.25 ex}}\alpha ^{4}+i \left (-q \,p^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a \,p^{2}-i \sqrt {3}\, c p q -3 p^{3} \underline {\hspace {1.25 ex}}\alpha ^{2} a q -2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha a \,p^{2} q +3 \left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} b p -3 \underline {\hspace {1.25 ex}}\alpha ^{3} p^{2} c +3 \left (-q \,p^{2}\right )^{\frac {2}{3}} a q p -2 i \left (-q \,p^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} c p +3 \left (-q \,p^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha c +3 q c p}{2 p q c}, \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 \underline {\hspace {1.25 ex}}\alpha \left (6 \underline {\hspace {1.25 ex}}\alpha ^{4} a \,p^{2}+5 \underline {\hspace {1.25 ex}}\alpha ^{3} b p +6 \underline {\hspace {1.25 ex}}\alpha a p q +4 \underline {\hspace {1.25 ex}}\alpha ^{2} c +2 b q \right ) \sqrt {p \,x^{3}+q}}\right )}{a \,p^{2} q^{2} c}\) | \(1362\) |
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 )}}{c x^{4} + {\left (p x^{3} + q\right )} b x^{2} + {\left (p x^{3} + q\right )}^{2} a}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\sqrt {p\,x^3+q}\,\left (2\,q-p\,x^3\right )}{a\,{\left (p\,x^3+q\right )}^2+c\,x^4+b\,x^2\,\left (p\,x^3+q\right )} \,d x \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 {\left (p x^{3} - 2 q\right ) \sqrt {p x^{3} + q}}{a p^{2} x^{6} + 2 a p q x^{3} + a q^{2} + b p x^{5} + b q x^{2} + c x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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