Optimal. Leaf size=189 \[ -\frac {\sqrt {2} \left (\sqrt {b^2-4 a c}-b\right ) \tan ^{-1}\left (\frac {x \sqrt {b-\sqrt {b^2-4 a c}}}{\sqrt {2} \sqrt {a} \sqrt {p x^5+q}}\right )}{\sqrt {a} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b} \tan ^{-1}\left (\frac {x \sqrt {\sqrt {b^2-4 a c}+b}}{\sqrt {2} \sqrt {a} \sqrt {p x^5+q}}\right )}{\sqrt {a} \sqrt {b^2-4 a c}} \]
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Rubi [F] time = 1.37, 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 {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{c x^4+b x^2 \left (q+p x^5\right )+a \left (q+p x^5\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{c x^4+b x^2 \left (q+p x^5\right )+a \left (q+p x^5\right )^2} \, dx &=\int \left (-\frac {2 q \sqrt {q+p x^5}}{a q^2+b q x^2+c x^4+2 a p q x^5+b p x^7+a p^2 x^{10}}+\frac {3 p x^5 \sqrt {q+p x^5}}{a q^2+b q x^2+c x^4+2 a p q x^5+b p x^7+a p^2 x^{10}}\right ) \, dx\\ &=(3 p) \int \frac {x^5 \sqrt {q+p x^5}}{a q^2+b q x^2+c x^4+2 a p q x^5+b p x^7+a p^2 x^{10}} \, dx-(2 q) \int \frac {\sqrt {q+p x^5}}{a q^2+b q x^2+c x^4+2 a p q x^5+b p x^7+a p^2 x^{10}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.41, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{c x^4+b x^2 \left (q+p x^5\right )+a \left (q+p x^5\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 13.42, size = 189, normalized size = 1.00 \begin {gather*} -\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^5+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^5+q}}\right )}{\sqrt {a} \sqrt {b^2-4 a c}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 86.73, 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(-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 [F] time = 8.88, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {p \,x^{5}+q}\, \left (3 p \,x^{5}-2 q \right )}{c \,x^{4}+b \,x^{2} \left (p \,x^{5}+q \right )+a \left (p \,x^{5}+q \right )^{2}}\, dx \end {gather*}
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 {{\left (3 \, p x^{5} - 2 \, q\right )} \sqrt {p x^{5} + q}}{c x^{4} + {\left (p x^{5} + q\right )} b x^{2} + {\left (p x^{5} + q\right )}^{2} 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 = -0.01 \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 {\sqrt {p x^{5} + q} \left (3 p x^{5} - 2 q\right )}{a p^{2} x^{10} + 2 a p q x^{5} + a q^{2} + b p x^{7} + b q x^{2} + c x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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