Optimal. Leaf size=77 \[ \frac {16 x}{35 a^4 \sqrt {a+b x^2}}+\frac {8 x}{35 a^3 \left (a+b x^2\right )^{3/2}}+\frac {6 x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {192, 191} \[ \frac {16 x}{35 a^4 \sqrt {a+b x^2}}+\frac {8 x}{35 a^3 \left (a+b x^2\right )^{3/2}}+\frac {6 x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {x}{7 a \left (a+b x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^2\right )^{9/2}} \, dx &=\frac {x}{7 a \left (a+b x^2\right )^{7/2}}+\frac {6 \int \frac {1}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a}\\ &=\frac {x}{7 a \left (a+b x^2\right )^{7/2}}+\frac {6 x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {24 \int \frac {1}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^2}\\ &=\frac {x}{7 a \left (a+b x^2\right )^{7/2}}+\frac {6 x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {8 x}{35 a^3 \left (a+b x^2\right )^{3/2}}+\frac {16 \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{35 a^3}\\ &=\frac {x}{7 a \left (a+b x^2\right )^{7/2}}+\frac {6 x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {8 x}{35 a^3 \left (a+b x^2\right )^{3/2}}+\frac {16 x}{35 a^4 \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 51, normalized size = 0.66 \[ \frac {x \left (35 a^3+70 a^2 b x^2+56 a b^2 x^4+16 b^3 x^6\right )}{35 a^4 \left (a+b x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 91, normalized size = 1.18 \[ \frac {{\left (16 \, b^{3} x^{7} + 56 \, a b^{2} x^{5} + 70 \, a^{2} b x^{3} + 35 \, a^{3} x\right )} \sqrt {b x^{2} + a}}{35 \, {\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.12, size = 55, normalized size = 0.71 \[ \frac {{\left (2 \, {\left (4 \, x^{2} {\left (\frac {2 \, b^{3} x^{2}}{a^{4}} + \frac {7 \, b^{2}}{a^{3}}\right )} + \frac {35 \, b}{a^{2}}\right )} x^{2} + \frac {35}{a}\right )} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 48, normalized size = 0.62 \[ \frac {\left (16 b^{3} x^{6}+56 a \,b^{2} x^{4}+70 a^{2} b \,x^{2}+35 a^{3}\right ) x}{35 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 61, normalized size = 0.79 \[ \frac {16 \, x}{35 \, \sqrt {b x^{2} + a} a^{4}} + \frac {8 \, x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {6 \, x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.60, size = 61, normalized size = 0.79 \[ \frac {16\,x}{35\,a^4\,\sqrt {b\,x^2+a}}+\frac {8\,x}{35\,a^3\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {6\,x}{35\,a^2\,{\left (b\,x^2+a\right )}^{5/2}}+\frac {x}{7\,a\,{\left (b\,x^2+a\right )}^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.19, size = 1265, normalized size = 16.43 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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