Optimal. Leaf size=352 \[ \frac {1}{4} \log (x) \left (5 a p^4 q^4+8 b p q\right )+\frac {1}{8} \left (-5 a p^4 q^4-8 b p q\right ) \log \left (\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}+p x^3+q\right )+\frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2} \left (6 a p^7 x^{21}-2 a p^6 q x^{19}+42 a p^6 q x^{18}-5 a p^5 q^2 x^{17}-10 a p^5 q^2 x^{16}+126 a p^5 q^2 x^{15}-15 a p^4 q^3 x^{15}-15 a p^4 q^3 x^{14}-20 a p^4 q^3 x^{13}+210 a p^4 q^3 x^{12}-15 a p^3 q^4 x^{12}-15 a p^3 q^4 x^{11}-20 a p^3 q^4 x^{10}+210 a p^3 q^4 x^9-5 a p^2 q^5 x^8-10 a p^2 q^5 x^7+126 a p^2 q^5 x^6-2 a p q^6 x^4+42 a p q^6 x^3+6 a q^7+24 b p x^{15}+24 b q x^{12}\right )}{48 x^{16}} \]
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Rubi [F] time = 1.82, 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^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^{12}+a \left (q+p x^3\right )^6\right )}{x^{17}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^{12}+a \left (q+p x^3\right )^6\right )}{x^{17}} \, dx &=\int \left (-\frac {2 a q^7 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^{17}}-\frac {11 a p q^6 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^{14}}-\frac {24 a p^2 q^5 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^{11}}-\frac {25 a p^3 q^4 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^8}-\frac {2 q \left (b+5 a p^4 q^2\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5}+\frac {p \left (b+3 a p^4 q^2\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^2}+4 a p^6 q x \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}+a p^7 x^4 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}\right ) \, dx\\ &=\left (a p^7\right ) \int x^4 \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \, dx+\left (4 a p^6 q\right ) \int x \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \, dx-\left (25 a p^3 q^4\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^8} \, dx-\left (24 a p^2 q^5\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^{11}} \, dx-\left (11 a p q^6\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^{14}} \, dx-\left (2 a q^7\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^{17}} \, dx+\left (p \left (b+3 a p^4 q^2\right )\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^2} \, dx-\left (2 q \left (b+5 a p^4 q^2\right )\right ) \int \frac {\sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6}}{x^5} \, dx\\ \end {align*}
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Mathematica [F] time = 1.28, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-2 q+p x^3\right ) \sqrt {q^2+2 p q x^3-2 p q x^4+p^2 x^6} \left (b x^{12}+a \left (q+p x^3\right )^6\right )}{x^{17}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.61, size = 352, normalized size = 1.00 \begin {gather*} \frac {1}{4} \log (x) \left (5 a p^4 q^4+8 b p q\right )+\frac {1}{8} \left (-5 a p^4 q^4-8 b p q\right ) \log \left (\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2}+p x^3+q\right )+\frac {\sqrt {p^2 x^6-2 p q x^4+2 p q x^3+q^2} \left (6 a p^7 x^{21}-2 a p^6 q x^{19}+42 a p^6 q x^{18}-5 a p^5 q^2 x^{17}-10 a p^5 q^2 x^{16}+126 a p^5 q^2 x^{15}-15 a p^4 q^3 x^{15}-15 a p^4 q^3 x^{14}-20 a p^4 q^3 x^{13}+210 a p^4 q^3 x^{12}-15 a p^3 q^4 x^{12}-15 a p^3 q^4 x^{11}-20 a p^3 q^4 x^{10}+210 a p^3 q^4 x^9-5 a p^2 q^5 x^8-10 a p^2 q^5 x^7+126 a p^2 q^5 x^6-2 a p q^6 x^4+42 a p q^6 x^3+6 a q^7+24 b p x^{15}+24 b q x^{12}\right )}{48 x^{16}} \end {gather*}
Antiderivative was successfully verified.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (b x^{12} + {\left (p x^{3} + q\right )}^{6} a\right )} \sqrt {p^{2} x^{6} - 2 \, p q x^{4} + 2 \, p q x^{3} + q^{2}} {\left (p x^{3} - 2 \, q\right )}}{x^{17}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.27, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (p \,x^{3}-2 q \right ) \sqrt {p^{2} x^{6}-2 x^{4} p q +2 p q \,x^{3}+q^{2}}\, \left (b \,x^{12}+a \left (p \,x^{3}+q \right )^{6}\right )}{x^{17}}\, 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 (b x^{12} + {\left (p x^{3} + q\right )}^{6} a\right )} \sqrt {p^{2} x^{6} - 2 \, p q x^{4} + 2 \, p q x^{3} + q^{2}} {\left (p x^{3} - 2 \, q\right )}}{x^{17}}\,{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 {\left (p x^{3} - 2 q\right ) \sqrt {p^{2} x^{6} - 2 p q x^{4} + 2 p q x^{3} + q^{2}} \left (a p^{6} x^{18} + 6 a p^{5} q x^{15} + 15 a p^{4} q^{2} x^{12} + 20 a p^{3} q^{3} x^{9} + 15 a p^{2} q^{4} x^{6} + 6 a p q^{5} x^{3} + a q^{6} + b x^{12}\right )}{x^{17}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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