Optimal. Leaf size=93 \[ \frac {2 a (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )}+\frac {2 b (d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 d^3 \left (a+b x^2\right )} \]
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Rubi [A]
time = 0.02, 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 = {1126, 14}
\begin {gather*} \frac {2 b (d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 d^3 \left (a+b x^2\right )}+\frac {2 a (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 1126
Rubi steps
\begin {align*} \int \sqrt {d x} \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 \sqrt {d x} \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 \sqrt {d x}+\frac {b^2 (d x)^{5/2}}{d^2}\right ) \, dx}{a b+b^2 x^2}\\ &=\frac {2 a (d x)^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{3 d \left (a+b x^2\right )}+\frac {2 b (d x)^{7/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}{7 d^3 \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 44, normalized size = 0.47 \begin {gather*} \frac {2 x \sqrt {d x} \sqrt {\left (a+b x^2\right )^2} \left (7 a+3 b x^2\right )}{21 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 41, normalized size = 0.44
method | result | size |
gosper | \(\frac {2 x \left (3 b \,x^{2}+7 a \right ) \sqrt {d x}\, \sqrt {\left (b \,x^{2}+a \right )^{2}}}{21 \left (b \,x^{2}+a \right )}\) | \(39\) |
default | \(\frac {2 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (d x \right )^{\frac {3}{2}} \left (3 b \,x^{2}+7 a \right )}{21 d \left (b \,x^{2}+a \right )}\) | \(41\) |
risch | \(\frac {2 d \sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (3 b \,x^{2}+7 a \right ) x^{2}}{21 \left (b \,x^{2}+a \right ) \sqrt {d x}}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 25, normalized size = 0.27 \begin {gather*} \frac {2 \, {\left (3 \, \left (d x\right )^{\frac {7}{2}} b + 7 \, \left (d x\right )^{\frac {3}{2}} a d^{2}\right )}}{21 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 18, normalized size = 0.19 \begin {gather*} \frac {2}{21} \, {\left (3 \, b x^{3} + 7 \, a x\right )} \sqrt {d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 39.24, size = 27, normalized size = 0.29 \begin {gather*} \frac {2 a \left (d x\right )^{\frac {3}{2}}}{3 d} + \frac {2 b \left (d x\right )^{\frac {7}{2}}}{7 d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.58, size = 37, normalized size = 0.40 \begin {gather*} \frac {2}{7} \, \sqrt {d x} b x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) + \frac {2}{3} \, \sqrt {d x} a x \mathrm {sgn}\left (b x^{2} + a\right ) \end {gather*}
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 \begin {gather*} \int \sqrt {d\,x}\,\sqrt {{\left (b\,x^2+a\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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