Optimal. Leaf size=40 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {b} x^3}{\sqrt {a} \left (p x^5+q\right )^{3/2}}\right )}{3 \sqrt {a} \sqrt {b}} \]
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Rubi [A] time = 0.67, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6714, 205} \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \left (p x^5+q\right )^{3/2}}{\sqrt {b} x^3}\right )}{3 \sqrt {a} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 6714
Rubi steps
\begin {align*} \int \frac {x^2 \sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{b x^6+a \left (q+p x^5\right )^3} \, dx &=\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\frac {\left (q+p x^5\right )^{3/2}}{x^3}\right )\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \left (q+p x^5\right )^{3/2}}{\sqrt {b} x^3}\right )}{3 \sqrt {a} \sqrt {b}}\\ \end {align*}
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Mathematica [F] time = 0.37, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \sqrt {q+p x^5} \left (-2 q+3 p x^5\right )}{b x^6+a \left (q+p x^5\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 3.16, size = 40, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt {b} x^3}{\sqrt {a} \left (q+p x^5\right )^{3/2}}\right )}{3 \sqrt {a} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.21, size = 465, normalized size = 11.62 \begin {gather*} \left [-\frac {\sqrt {-a b} \log \left (\frac {a^{2} p^{6} x^{30} + 6 \, a^{2} p^{5} q x^{25} + 15 \, a^{2} p^{4} q^{2} x^{20} - 6 \, a b p^{3} x^{21} + 20 \, a^{2} p^{3} q^{3} x^{15} - 18 \, a b p^{2} q x^{16} + 15 \, a^{2} p^{2} q^{4} x^{10} - 18 \, a b p q^{2} x^{11} + b^{2} x^{12} + 6 \, a^{2} p q^{5} x^{5} - 6 \, a b q^{3} x^{6} + a^{2} q^{6} - 4 \, {\left (a p^{4} x^{23} + 4 \, a p^{3} q x^{18} + 6 \, a p^{2} q^{2} x^{13} - b p x^{14} + 4 \, a p q^{3} x^{8} - b q x^{9} + a q^{4} x^{3}\right )} \sqrt {p x^{5} + q} \sqrt {-a b}}{a^{2} p^{6} x^{30} + 6 \, a^{2} p^{5} q x^{25} + 15 \, a^{2} p^{4} q^{2} x^{20} + 2 \, a b p^{3} x^{21} + 20 \, a^{2} p^{3} q^{3} x^{15} + 6 \, a b p^{2} q x^{16} + 15 \, a^{2} p^{2} q^{4} x^{10} + 6 \, a b p q^{2} x^{11} + b^{2} x^{12} + 6 \, a^{2} p q^{5} x^{5} + 2 \, a b q^{3} x^{6} + a^{2} q^{6}}\right )}{6 \, a b}, \frac {\sqrt {a b} \arctan \left (\frac {{\left (a p^{3} x^{15} + 3 \, a p^{2} q x^{10} + 3 \, a p q^{2} x^{5} - b x^{6} + a q^{3}\right )} \sqrt {p x^{5} + q} \sqrt {a b}}{2 \, {\left (a b p^{2} x^{13} + 2 \, a b p q x^{8} + a b q^{2} x^{3}\right )}}\right )}{3 \, a b}\right ] \end {gather*}
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*} \int \frac {{\left (3 \, p x^{5} - 2 \, q\right )} \sqrt {p x^{5} + q} x^{2}}{b x^{6} + {\left (p x^{5} + q\right )}^{3} a}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \sqrt {p \,x^{5}+q}\, \left (3 p \,x^{5}-2 q \right )}{b \,x^{6}+a \left (p \,x^{5}+q \right )^{3}}\, dx\]
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*} \int \frac {{\left (3 \, p x^{5} - 2 \, q\right )} \sqrt {p x^{5} + q} x^{2}}{b x^{6} + {\left (p x^{5} + q\right )}^{3} 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 = -0.02 \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^{2} \sqrt {p x^{5} + q} \left (3 p x^{5} - 2 q\right )}{a p^{3} x^{15} + 3 a p^{2} q x^{10} + 3 a p q^{2} x^{5} + a q^{3} + b x^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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