Optimal. Leaf size=40 \[ \frac {b \left (a+b x^2\right )^7}{112 a^2 x^{14}}-\frac {\left (a+b x^2\right )^7}{16 a x^{16}} \]
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Rubi [A] time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {28, 266, 45, 37} \[ \frac {b \left (a+b x^2\right )^7}{112 a^2 x^{14}}-\frac {\left (a+b x^2\right )^7}{16 a x^{16}} \]
Antiderivative was successfully verified.
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Rule 28
Rule 37
Rule 45
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{17}} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^6}{x^{17}} \, dx}{b^6}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^6}{x^9} \, dx,x,x^2\right )}{2 b^6}\\ &=-\frac {\left (a+b x^2\right )^7}{16 a x^{16}}-\frac {\operatorname {Subst}\left (\int \frac {\left (a b+b^2 x\right )^6}{x^8} \, dx,x,x^2\right )}{16 a b^5}\\ &=-\frac {\left (a+b x^2\right )^7}{16 a x^{16}}+\frac {b \left (a+b x^2\right )^7}{112 a^2 x^{14}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 78, normalized size = 1.95 \[ -\frac {a^6}{16 x^{16}}-\frac {3 a^5 b}{7 x^{14}}-\frac {5 a^4 b^2}{4 x^{12}}-\frac {2 a^3 b^3}{x^{10}}-\frac {15 a^2 b^4}{8 x^8}-\frac {a b^5}{x^6}-\frac {b^6}{4 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 70, normalized size = 1.75 \[ -\frac {28 \, b^{6} x^{12} + 112 \, a b^{5} x^{10} + 210 \, a^{2} b^{4} x^{8} + 224 \, a^{3} b^{3} x^{6} + 140 \, a^{4} b^{2} x^{4} + 48 \, a^{5} b x^{2} + 7 \, a^{6}}{112 \, x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 70, normalized size = 1.75 \[ -\frac {28 \, b^{6} x^{12} + 112 \, a b^{5} x^{10} + 210 \, a^{2} b^{4} x^{8} + 224 \, a^{3} b^{3} x^{6} + 140 \, a^{4} b^{2} x^{4} + 48 \, a^{5} b x^{2} + 7 \, a^{6}}{112 \, x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 69, normalized size = 1.72 \[ -\frac {b^{6}}{4 x^{4}}-\frac {a \,b^{5}}{x^{6}}-\frac {15 a^{2} b^{4}}{8 x^{8}}-\frac {2 a^{3} b^{3}}{x^{10}}-\frac {5 a^{4} b^{2}}{4 x^{12}}-\frac {3 a^{5} b}{7 x^{14}}-\frac {a^{6}}{16 x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 70, normalized size = 1.75 \[ -\frac {28 \, b^{6} x^{12} + 112 \, a b^{5} x^{10} + 210 \, a^{2} b^{4} x^{8} + 224 \, a^{3} b^{3} x^{6} + 140 \, a^{4} b^{2} x^{4} + 48 \, a^{5} b x^{2} + 7 \, a^{6}}{112 \, x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.37, size = 69, normalized size = 1.72 \[ -\frac {\frac {a^6}{16}+\frac {3\,a^5\,b\,x^2}{7}+\frac {5\,a^4\,b^2\,x^4}{4}+2\,a^3\,b^3\,x^6+\frac {15\,a^2\,b^4\,x^8}{8}+a\,b^5\,x^{10}+\frac {b^6\,x^{12}}{4}}{x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.68, size = 75, normalized size = 1.88 \[ \frac {- 7 a^{6} - 48 a^{5} b x^{2} - 140 a^{4} b^{2} x^{4} - 224 a^{3} b^{3} x^{6} - 210 a^{2} b^{4} x^{8} - 112 a b^{5} x^{10} - 28 b^{6} x^{12}}{112 x^{16}} \]
Verification of antiderivative is not currently implemented for this CAS.
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