Optimal. Leaf size=91 \[ \frac {2 b (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d^3 \left (a+b x^2\right )}-\frac {2 a \sqrt {a^2+2 a b x^2+b^2 x^4}}{d \sqrt {d x} \left (a+b x^2\right )} \]
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Rubi [A] time = 0.03, antiderivative size = 91, 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} \begin {gather*} \frac {2 b (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d^3 \left (a+b x^2\right )}-\frac {2 a \sqrt {a^2+2 a b x^2+b^2 x^4}}{d \sqrt {d x} \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 1112
Rubi steps
\begin {align*} \int \frac {\sqrt {a^2+2 a b x^2+b^2 x^4}}{(d x)^{3/2}} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \frac {a b+b^2 x^2}{(d x)^{3/2}} \, dx}{a b+b^2 x^2}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (\frac {a b}{(d x)^{3/2}}+\frac {b^2 \sqrt {d x}}{d^2}\right ) \, dx}{a b+b^2 x^2}\\ &=-\frac {2 a \sqrt {a^2+2 a b x^2+b^2 x^4}}{d \sqrt {d x} \left (a+b x^2\right )}+\frac {2 b (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d^3 \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 43, normalized size = 0.47 \begin {gather*} \frac {2 x \left (b x^2-3 a\right ) \sqrt {\left (a+b x^2\right )^2}}{3 (d x)^{3/2} \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 26.01, size = 67, normalized size = 0.74 \begin {gather*} \frac {2 \left (b d^2 x^2-3 a d^2\right ) \left (a d^2+b d^2 x^2\right )}{3 d^5 \sqrt {d x} \sqrt {\frac {\left (a d^2+b d^2 x^2\right )^2}{d^4}}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 22, normalized size = 0.24 \begin {gather*} \frac {2 \, {\left (b x^{2} - 3 \, a\right )} \sqrt {d x}}{3 \, d^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 41, normalized size = 0.45 \begin {gather*} \frac {2 \, {\left (\frac {\sqrt {d x} b x \mathrm {sgn}\left (b x^{2} + a\right )}{d} - \frac {3 \, a \mathrm {sgn}\left (b x^{2} + a\right )}{\sqrt {d x}}\right )}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 39, normalized size = 0.43 \begin {gather*} -\frac {2 \left (-b \,x^{2}+3 a \right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}\, x}{3 \left (b \,x^{2}+a \right ) \left (d x \right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 25, normalized size = 0.27 \begin {gather*} -\frac {2 \, {\left (\frac {3 \, a}{\sqrt {d x}} - \frac {\left (d x\right )^{\frac {3}{2}} b}{d^{2}}\right )}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.35, size = 52, normalized size = 0.57 \begin {gather*} \frac {\left (\frac {2\,x^2}{3\,d}-\frac {2\,a}{b\,d}\right )\,\sqrt {{\left (b\,x^2+a\right )}^2}}{x^2\,\sqrt {d\,x}+\frac {a\,\sqrt {d\,x}}{b}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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