Optimal. Leaf size=189 \[ -\frac {\sqrt {2} \left (-b+\sqrt {b^2-4 a c}\right ) \text {ArcTan}\left (\frac {\sqrt {b-\sqrt {b^2-4 a c}} x}{\sqrt {2} \sqrt {a} \sqrt {q+p x^5}}\right )}{\sqrt {a} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {b+\sqrt {b^2-4 a c}} \text {ArcTan}\left (\frac {\sqrt {b+\sqrt {b^2-4 a c}} x}{\sqrt {2} \sqrt {a} \sqrt {q+p x^5}}\right )}{\sqrt {a} \sqrt {b^2-4 a c}} \]
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Rubi [F]
time = 0.94, 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 [A]
time = 6.91, size = 151, normalized size = 0.80 \begin {gather*} \frac {\sqrt {2} \left (\sqrt {b-\sqrt {b^2-4 a c}} \text {ArcTan}\left (\frac {\sqrt {b-\sqrt {b^2-4 a c}} x}{\sqrt {2} \sqrt {a} \sqrt {q+p x^5}}\right )-\sqrt {b+\sqrt {b^2-4 a c}} \text {ArcTan}\left (\frac {\sqrt {b+\sqrt {b^2-4 a c}} x}{\sqrt {2} \sqrt {a} \sqrt {q+p x^5}}\right )\right )}{\sqrt {a} \sqrt {b^2-4 a c}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 8.34, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {p \,x^{5}+q}\, \left (3 p \,x^{5}-2 q \right )}{c \,x^{4}+b \,x^{2} \left (p \,x^{5}+q \right )+a \left (p \,x^{5}+q \right )^{2}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1321 vs.
\(2 (134) = 268\).
time = 50.72, size = 1321, normalized size = 6.99 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \sqrt {-\frac {b + \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (\frac {2 \, a p^{2} x^{10} + 4 \, a p q x^{5} - 2 \, c x^{4} + 2 \, a q^{2} + \sqrt {2} \sqrt {p x^{5} + q} {\left ({\left (b^{2} - 4 \, a c\right )} x^{3} - \frac {2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} p x^{6} + {\left (a b^{3} - 4 \, a^{2} b c\right )} x^{3} + 2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} q x}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}\right )} \sqrt {-\frac {b + \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} - \frac {2 \, {\left ({\left (a b^{2} - 4 \, a^{2} c\right )} p x^{7} + {\left (a b^{2} - 4 \, a^{2} c\right )} q x^{2}\right )}}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a p^{2} x^{10} + b p x^{7} + 2 \, a p q x^{5} + c x^{4} + b q x^{2} + a q^{2}}\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {-\frac {b + \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (\frac {2 \, a p^{2} x^{10} + 4 \, a p q x^{5} - 2 \, c x^{4} + 2 \, a q^{2} - \sqrt {2} \sqrt {p x^{5} + q} {\left ({\left (b^{2} - 4 \, a c\right )} x^{3} - \frac {2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} p x^{6} + {\left (a b^{3} - 4 \, a^{2} b c\right )} x^{3} + 2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} q x}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}\right )} \sqrt {-\frac {b + \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} - \frac {2 \, {\left ({\left (a b^{2} - 4 \, a^{2} c\right )} p x^{7} + {\left (a b^{2} - 4 \, a^{2} c\right )} q x^{2}\right )}}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a p^{2} x^{10} + b p x^{7} + 2 \, a p q x^{5} + c x^{4} + b q x^{2} + a q^{2}}\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {-\frac {b - \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (\frac {2 \, a p^{2} x^{10} + 4 \, a p q x^{5} - 2 \, c x^{4} + 2 \, a q^{2} + \sqrt {2} \sqrt {p x^{5} + q} {\left ({\left (b^{2} - 4 \, a c\right )} x^{3} + \frac {2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} p x^{6} + {\left (a b^{3} - 4 \, a^{2} b c\right )} x^{3} + 2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} q x}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}\right )} \sqrt {-\frac {b - \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} + \frac {2 \, {\left ({\left (a b^{2} - 4 \, a^{2} c\right )} p x^{7} + {\left (a b^{2} - 4 \, a^{2} c\right )} q x^{2}\right )}}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a p^{2} x^{10} + b p x^{7} + 2 \, a p q x^{5} + c x^{4} + b q x^{2} + a q^{2}}\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {-\frac {b - \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (\frac {2 \, a p^{2} x^{10} + 4 \, a p q x^{5} - 2 \, c x^{4} + 2 \, a q^{2} - \sqrt {2} \sqrt {p x^{5} + q} {\left ({\left (b^{2} - 4 \, a c\right )} x^{3} + \frac {2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} p x^{6} + {\left (a b^{3} - 4 \, a^{2} b c\right )} x^{3} + 2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} q x}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}\right )} \sqrt {-\frac {b - \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} + \frac {2 \, {\left ({\left (a b^{2} - 4 \, a^{2} c\right )} p x^{7} + {\left (a b^{2} - 4 \, a^{2} c\right )} q x^{2}\right )}}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a p^{2} x^{10} + b p x^{7} + 2 \, a p q x^{5} + c x^{4} + b q x^{2} + a q^{2}}\right ) \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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Giac [F(-1)] Timed out
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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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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