Optimal. Leaf size=126 \[ \frac {128 \sqrt {a x+b x^3}}{35 b^5 \sqrt {x}}-\frac {64 x^{5/2}}{35 b^4 \sqrt {a x+b x^3}}-\frac {16 x^{11/2}}{35 b^3 \left (a x+b x^3\right )^{3/2}}-\frac {8 x^{17/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac {x^{23/2}}{7 b \left (a x+b x^3\right )^{7/2}} \]
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Rubi [A] time = 0.20, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2015, 2014} \[ -\frac {8 x^{17/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac {16 x^{11/2}}{35 b^3 \left (a x+b x^3\right )^{3/2}}-\frac {64 x^{5/2}}{35 b^4 \sqrt {a x+b x^3}}+\frac {128 \sqrt {a x+b x^3}}{35 b^5 \sqrt {x}}-\frac {x^{23/2}}{7 b \left (a x+b x^3\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2014
Rule 2015
Rubi steps
\begin {align*} \int \frac {x^{27/2}}{\left (a x+b x^3\right )^{9/2}} \, dx &=-\frac {x^{23/2}}{7 b \left (a x+b x^3\right )^{7/2}}+\frac {8 \int \frac {x^{21/2}}{\left (a x+b x^3\right )^{7/2}} \, dx}{7 b}\\ &=-\frac {x^{23/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac {8 x^{17/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}+\frac {48 \int \frac {x^{15/2}}{\left (a x+b x^3\right )^{5/2}} \, dx}{35 b^2}\\ &=-\frac {x^{23/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac {8 x^{17/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac {16 x^{11/2}}{35 b^3 \left (a x+b x^3\right )^{3/2}}+\frac {64 \int \frac {x^{9/2}}{\left (a x+b x^3\right )^{3/2}} \, dx}{35 b^3}\\ &=-\frac {x^{23/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac {8 x^{17/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac {16 x^{11/2}}{35 b^3 \left (a x+b x^3\right )^{3/2}}-\frac {64 x^{5/2}}{35 b^4 \sqrt {a x+b x^3}}+\frac {128 \int \frac {x^{3/2}}{\sqrt {a x+b x^3}} \, dx}{35 b^4}\\ &=-\frac {x^{23/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac {8 x^{17/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac {16 x^{11/2}}{35 b^3 \left (a x+b x^3\right )^{3/2}}-\frac {64 x^{5/2}}{35 b^4 \sqrt {a x+b x^3}}+\frac {128 \sqrt {a x+b x^3}}{35 b^5 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 77, normalized size = 0.61 \[ \frac {\sqrt {x} \left (128 a^4+448 a^3 b x^2+560 a^2 b^2 x^4+280 a b^3 x^6+35 b^4 x^8\right )}{35 b^5 \left (a+b x^2\right )^3 \sqrt {x \left (a+b x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 108, normalized size = 0.86 \[ \frac {{\left (35 \, b^{4} x^{8} + 280 \, a b^{3} x^{6} + 560 \, a^{2} b^{2} x^{4} + 448 \, a^{3} b x^{2} + 128 \, a^{4}\right )} \sqrt {b x^{3} + a x} \sqrt {x}}{35 \, {\left (b^{9} x^{9} + 4 \, a b^{8} x^{7} + 6 \, a^{2} b^{7} x^{5} + 4 \, a^{3} b^{6} x^{3} + a^{4} b^{5} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 80, normalized size = 0.63 \[ \frac {\sqrt {b x^{2} + a}}{b^{5}} - \frac {128 \, \sqrt {a}}{35 \, b^{5}} + \frac {140 \, {\left (b x^{2} + a\right )}^{3} a - 70 \, {\left (b x^{2} + a\right )}^{2} a^{2} + 28 \, {\left (b x^{2} + a\right )} a^{3} - 5 \, a^{4}}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 70, normalized size = 0.56 \[ \frac {\left (b \,x^{2}+a \right ) \left (35 x^{8} b^{4}+280 a \,x^{6} b^{3}+560 a^{2} x^{4} b^{2}+448 a^{3} x^{2} b +128 a^{4}\right ) x^{\frac {9}{2}}}{35 \left (b \,x^{3}+a x \right )^{\frac {9}{2}} b^{5}} \]
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 \[ \int \frac {x^{\frac {27}{2}}}{{\left (b x^{3} + a x\right )}^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^{27/2}}{{\left (b\,x^3+a\,x\right )}^{9/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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