Optimal. Leaf size=93 \[ \frac {2 b (d x)^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 d^3 \left (a+b x^2\right )}+\frac {2 a (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 d \left (a+b x^2\right )} \]
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Rubi [A] time = 0.03, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1112, 14} \[ \frac {2 b (d x)^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 d^3 \left (a+b x^2\right )}+\frac {2 a (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 d \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 14
Rule 1112
Rubi steps
\begin {align*} \int (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int (d x)^{3/2} \left (a b+b^2 x^2\right ) \, dx}{a b+b^2 x^2}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (a b (d x)^{3/2}+\frac {b^2 (d x)^{7/2}}{d^2}\right ) \, dx}{a b+b^2 x^2}\\ &=\frac {2 a (d x)^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{5 d \left (a+b x^2\right )}+\frac {2 b (d x)^{9/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{9 d^3 \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 44, normalized size = 0.47 \[ \frac {2 x (d x)^{3/2} \sqrt {\left (a+b x^2\right )^2} \left (9 a+5 b x^2\right )}{45 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 22, normalized size = 0.24 \[ \frac {2}{45} \, {\left (5 \, b d x^{4} + 9 \, a d x^{2}\right )} \sqrt {d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 42, normalized size = 0.45 \[ \frac {2}{45} \, {\left (5 \, \sqrt {d x} b x^{4} \mathrm {sgn}\left (b x^{2} + a\right ) + 9 \, \sqrt {d x} a x^{2} \mathrm {sgn}\left (b x^{2} + a\right )\right )} d \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 39, normalized size = 0.42 \[ \frac {2 \left (5 b \,x^{2}+9 a \right ) \left (d x \right )^{\frac {3}{2}} \sqrt {\left (b \,x^{2}+a \right )^{2}}\, x}{45 \left (b \,x^{2}+a \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.25, size = 25, normalized size = 0.27 \[ \frac {2 \, {\left (5 \, \left (d x\right )^{\frac {9}{2}} b + 9 \, \left (d x\right )^{\frac {5}{2}} a d^{2}\right )}}{45 \, d^{3}} \]
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 \[ \int {\left (d\,x\right )}^{3/2}\,\sqrt {{\left (b\,x^2+a\right )}^2} \,d x \]
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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