Optimal. Leaf size=49 \[ -\frac {2 a^2}{3 d (d x)^{3/2}}+\frac {4 a b \sqrt {d x}}{d^3}+\frac {2 b^2 (d x)^{5/2}}{5 d^5} \]
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Rubi [A]
time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {14}
\begin {gather*} -\frac {2 a^2}{3 d (d x)^{3/2}}+\frac {4 a b \sqrt {d x}}{d^3}+\frac {2 b^2 (d x)^{5/2}}{5 d^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rubi steps
\begin {align*} \int \frac {a^2+2 a b x^2+b^2 x^4}{(d x)^{5/2}} \, dx &=\int \left (\frac {a^2}{(d x)^{5/2}}+\frac {2 a b}{d^2 \sqrt {d x}}+\frac {b^2 (d x)^{3/2}}{d^4}\right ) \, dx\\ &=-\frac {2 a^2}{3 d (d x)^{3/2}}+\frac {4 a b \sqrt {d x}}{d^3}+\frac {2 b^2 (d x)^{5/2}}{5 d^5}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 33, normalized size = 0.67 \begin {gather*} -\frac {2 x \left (5 a^2-30 a b x^2-3 b^2 x^4\right )}{15 (d x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 42, normalized size = 0.86
method | result | size |
gosper | \(-\frac {2 \left (-3 b^{2} x^{4}-30 a b \,x^{2}+5 a^{2}\right ) x}{15 \left (d x \right )^{\frac {5}{2}}}\) | \(30\) |
trager | \(-\frac {2 \left (-3 b^{2} x^{4}-30 a b \,x^{2}+5 a^{2}\right ) \sqrt {d x}}{15 d^{3} x^{2}}\) | \(35\) |
risch | \(-\frac {2 \left (-3 b^{2} x^{4}-30 a b \,x^{2}+5 a^{2}\right )}{15 d^{2} x \sqrt {d x}}\) | \(35\) |
derivativedivides | \(\frac {\frac {2 b^{2} \left (d x \right )^{\frac {5}{2}}}{5}+4 a b \,d^{2} \sqrt {d x}-\frac {2 a^{2} d^{4}}{3 \left (d x \right )^{\frac {3}{2}}}}{d^{5}}\) | \(42\) |
default | \(\frac {\frac {2 b^{2} \left (d x \right )^{\frac {5}{2}}}{5}+4 a b \,d^{2} \sqrt {d x}-\frac {2 a^{2} d^{4}}{3 \left (d x \right )^{\frac {3}{2}}}}{d^{5}}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 43, normalized size = 0.88 \begin {gather*} -\frac {2 \, {\left (\frac {5 \, a^{2}}{\left (d x\right )^{\frac {3}{2}}} - \frac {3 \, {\left (\left (d x\right )^{\frac {5}{2}} b^{2} + 10 \, \sqrt {d x} a b d^{2}\right )}}{d^{4}}\right )}}{15 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 34, normalized size = 0.69 \begin {gather*} \frac {2 \, {\left (3 \, b^{2} x^{4} + 30 \, a b x^{2} - 5 \, a^{2}\right )} \sqrt {d x}}{15 \, d^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.29, size = 46, normalized size = 0.94 \begin {gather*} - \frac {2 a^{2} x}{3 \left (d x\right )^{\frac {5}{2}}} + \frac {4 a b x^{3}}{\left (d x\right )^{\frac {5}{2}}} + \frac {2 b^{2} x^{5}}{5 \left (d x\right )^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.61, size = 53, normalized size = 1.08 \begin {gather*} -\frac {2 \, {\left (\frac {5 \, a^{2} d}{\sqrt {d x} x} - \frac {3 \, {\left (\sqrt {d x} b^{2} d^{10} x^{2} + 10 \, \sqrt {d x} a b d^{10}\right )}}{d^{10}}\right )}}{15 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.23, size = 34, normalized size = 0.69 \begin {gather*} \frac {-10\,a^2+60\,a\,b\,x^2+6\,b^2\,x^4}{15\,d^2\,x\,\sqrt {d\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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