Optimal. Leaf size=180 \[ -\frac {1024 b^2 \sqrt {a x+b x^3}}{35 a^7 x^{3/2}}+\frac {512 b \sqrt {a x+b x^3}}{35 a^6 x^{7/2}}-\frac {384 \sqrt {a x+b x^3}}{35 a^5 x^{11/2}}+\frac {64}{7 a^4 x^{9/2} \sqrt {a x+b x^3}}+\frac {8}{7 a^3 x^{7/2} \left (a x+b x^3\right )^{3/2}}+\frac {12}{35 a^2 x^{5/2} \left (a x+b x^3\right )^{5/2}}+\frac {1}{7 a x^{3/2} \left (a x+b x^3\right )^{7/2}} \]
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Rubi [A] time = 0.28, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2015, 2016, 2014} \[ -\frac {1024 b^2 \sqrt {a x+b x^3}}{35 a^7 x^{3/2}}+\frac {512 b \sqrt {a x+b x^3}}{35 a^6 x^{7/2}}-\frac {384 \sqrt {a x+b x^3}}{35 a^5 x^{11/2}}+\frac {64}{7 a^4 x^{9/2} \sqrt {a x+b x^3}}+\frac {8}{7 a^3 x^{7/2} \left (a x+b x^3\right )^{3/2}}+\frac {12}{35 a^2 x^{5/2} \left (a x+b x^3\right )^{5/2}}+\frac {1}{7 a x^{3/2} \left (a x+b x^3\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2014
Rule 2015
Rule 2016
Rubi steps
\begin {align*} \int \frac {1}{x^{3/2} \left (a x+b x^3\right )^{9/2}} \, dx &=\frac {1}{7 a x^{3/2} \left (a x+b x^3\right )^{7/2}}+\frac {12 \int \frac {1}{x^{5/2} \left (a x+b x^3\right )^{7/2}} \, dx}{7 a}\\ &=\frac {1}{7 a x^{3/2} \left (a x+b x^3\right )^{7/2}}+\frac {12}{35 a^2 x^{5/2} \left (a x+b x^3\right )^{5/2}}+\frac {24 \int \frac {1}{x^{7/2} \left (a x+b x^3\right )^{5/2}} \, dx}{7 a^2}\\ &=\frac {1}{7 a x^{3/2} \left (a x+b x^3\right )^{7/2}}+\frac {12}{35 a^2 x^{5/2} \left (a x+b x^3\right )^{5/2}}+\frac {8}{7 a^3 x^{7/2} \left (a x+b x^3\right )^{3/2}}+\frac {64 \int \frac {1}{x^{9/2} \left (a x+b x^3\right )^{3/2}} \, dx}{7 a^3}\\ &=\frac {1}{7 a x^{3/2} \left (a x+b x^3\right )^{7/2}}+\frac {12}{35 a^2 x^{5/2} \left (a x+b x^3\right )^{5/2}}+\frac {8}{7 a^3 x^{7/2} \left (a x+b x^3\right )^{3/2}}+\frac {64}{7 a^4 x^{9/2} \sqrt {a x+b x^3}}+\frac {384 \int \frac {1}{x^{11/2} \sqrt {a x+b x^3}} \, dx}{7 a^4}\\ &=\frac {1}{7 a x^{3/2} \left (a x+b x^3\right )^{7/2}}+\frac {12}{35 a^2 x^{5/2} \left (a x+b x^3\right )^{5/2}}+\frac {8}{7 a^3 x^{7/2} \left (a x+b x^3\right )^{3/2}}+\frac {64}{7 a^4 x^{9/2} \sqrt {a x+b x^3}}-\frac {384 \sqrt {a x+b x^3}}{35 a^5 x^{11/2}}-\frac {(1536 b) \int \frac {1}{x^{7/2} \sqrt {a x+b x^3}} \, dx}{35 a^5}\\ &=\frac {1}{7 a x^{3/2} \left (a x+b x^3\right )^{7/2}}+\frac {12}{35 a^2 x^{5/2} \left (a x+b x^3\right )^{5/2}}+\frac {8}{7 a^3 x^{7/2} \left (a x+b x^3\right )^{3/2}}+\frac {64}{7 a^4 x^{9/2} \sqrt {a x+b x^3}}-\frac {384 \sqrt {a x+b x^3}}{35 a^5 x^{11/2}}+\frac {512 b \sqrt {a x+b x^3}}{35 a^6 x^{7/2}}+\frac {\left (1024 b^2\right ) \int \frac {1}{x^{3/2} \sqrt {a x+b x^3}} \, dx}{35 a^6}\\ &=\frac {1}{7 a x^{3/2} \left (a x+b x^3\right )^{7/2}}+\frac {12}{35 a^2 x^{5/2} \left (a x+b x^3\right )^{5/2}}+\frac {8}{7 a^3 x^{7/2} \left (a x+b x^3\right )^{3/2}}+\frac {64}{7 a^4 x^{9/2} \sqrt {a x+b x^3}}-\frac {384 \sqrt {a x+b x^3}}{35 a^5 x^{11/2}}+\frac {512 b \sqrt {a x+b x^3}}{35 a^6 x^{7/2}}-\frac {1024 b^2 \sqrt {a x+b x^3}}{35 a^7 x^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 99, normalized size = 0.55 \[ -\frac {\sqrt {x \left (a+b x^2\right )} \left (7 a^6-28 a^5 b x^2+280 a^4 b^2 x^4+2240 a^3 b^3 x^6+4480 a^2 b^4 x^8+3584 a b^5 x^{10}+1024 b^6 x^{12}\right )}{35 a^7 x^{11/2} \left (a+b x^2\right )^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.11, size = 132, normalized size = 0.73 \[ -\frac {{\left (1024 \, b^{6} x^{12} + 3584 \, a b^{5} x^{10} + 4480 \, a^{2} b^{4} x^{8} + 2240 \, a^{3} b^{3} x^{6} + 280 \, a^{4} b^{2} x^{4} - 28 \, a^{5} b x^{2} + 7 \, a^{6}\right )} \sqrt {b x^{3} + a x} \sqrt {x}}{35 \, {\left (a^{7} b^{4} x^{14} + 4 \, a^{8} b^{3} x^{12} + 6 \, a^{9} b^{2} x^{10} + 4 \, a^{10} b x^{8} + a^{11} x^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 202, normalized size = 1.12 \[ -\frac {{\left ({\left (2 \, x^{2} {\left (\frac {281 \, b^{6} x^{2}}{a^{7}} + \frac {896 \, b^{5}}{a^{6}}\right )} + \frac {1925 \, b^{4}}{a^{5}}\right )} x^{2} + \frac {700 \, b^{3}}{a^{4}}\right )} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {4 \, {\left (25 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} b^{\frac {5}{2}} - 120 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b^{\frac {5}{2}} + 210 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {5}{2}} - 140 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b^{\frac {5}{2}} + 33 \, a^{4} b^{\frac {5}{2}}\right )}}{5 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{5} a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 92, normalized size = 0.51 \[ -\frac {\left (b \,x^{2}+a \right ) \left (1024 b^{6} x^{12}+3584 b^{5} x^{10} a +4480 x^{8} b^{4} a^{2}+2240 b^{3} x^{6} a^{3}+280 b^{2} x^{4} a^{4}-28 b \,x^{2} a^{5}+7 a^{6}\right )}{35 \left (b \,x^{3}+a x \right )^{\frac {9}{2}} a^{7} \sqrt {x}} \]
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 \[ \int \frac {1}{{\left (b x^{3} + a x\right )}^{\frac {9}{2}} x^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{x^{3/2}\,{\left (b\,x^3+a\,x\right )}^{9/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{\frac {3}{2}} \left (x \left (a + b x^{2}\right )\right )^{\frac {9}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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