Optimal. Leaf size=43 \[ 2 a A \sqrt {x}+\frac {2}{3} a B x^{3/2}+\frac {2}{5} A c x^{5/2}+\frac {2}{7} B c x^{7/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {780}
\begin {gather*} 2 a A \sqrt {x}+\frac {2}{3} a B x^{3/2}+\frac {2}{5} A c x^{5/2}+\frac {2}{7} B c x^{7/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 780
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+c x^2\right )}{\sqrt {x}} \, dx &=\int \left (\frac {a A}{\sqrt {x}}+a B \sqrt {x}+A c x^{3/2}+B c x^{5/2}\right ) \, dx\\ &=2 a A \sqrt {x}+\frac {2}{3} a B x^{3/2}+\frac {2}{5} A c x^{5/2}+\frac {2}{7} B c x^{7/2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 34, normalized size = 0.79 \begin {gather*} \frac {2}{105} \sqrt {x} \left (35 a (3 A+B x)+3 c x^2 (7 A+5 B x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 30, normalized size = 0.70
method | result | size |
trager | \(\left (\frac {2}{7} B c \,x^{3}+\frac {2}{5} A c \,x^{2}+\frac {2}{3} B a x +2 A a \right ) \sqrt {x}\) | \(29\) |
gosper | \(\frac {2 \sqrt {x}\, \left (15 B c \,x^{3}+21 A c \,x^{2}+35 B a x +105 A a \right )}{105}\) | \(30\) |
derivativedivides | \(\frac {2 a B \,x^{\frac {3}{2}}}{3}+\frac {2 A c \,x^{\frac {5}{2}}}{5}+\frac {2 B c \,x^{\frac {7}{2}}}{7}+2 a A \sqrt {x}\) | \(30\) |
default | \(\frac {2 a B \,x^{\frac {3}{2}}}{3}+\frac {2 A c \,x^{\frac {5}{2}}}{5}+\frac {2 B c \,x^{\frac {7}{2}}}{7}+2 a A \sqrt {x}\) | \(30\) |
risch | \(\frac {2 \sqrt {x}\, \left (15 B c \,x^{3}+21 A c \,x^{2}+35 B a x +105 A a \right )}{105}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 29, normalized size = 0.67 \begin {gather*} \frac {2}{7} \, B c x^{\frac {7}{2}} + \frac {2}{5} \, A c x^{\frac {5}{2}} + \frac {2}{3} \, B a x^{\frac {3}{2}} + 2 \, A a \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.26, size = 29, normalized size = 0.67 \begin {gather*} \frac {2}{105} \, {\left (15 \, B c x^{3} + 21 \, A c x^{2} + 35 \, B a x + 105 \, A a\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 44, normalized size = 1.02 \begin {gather*} 2 A a \sqrt {x} + \frac {2 A c x^{\frac {5}{2}}}{5} + \frac {2 B a x^{\frac {3}{2}}}{3} + \frac {2 B c x^{\frac {7}{2}}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.56, size = 29, normalized size = 0.67 \begin {gather*} \frac {2}{7} \, B c x^{\frac {7}{2}} + \frac {2}{5} \, A c x^{\frac {5}{2}} + \frac {2}{3} \, B a x^{\frac {3}{2}} + 2 \, A a \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 29, normalized size = 0.67 \begin {gather*} 2\,A\,a\,\sqrt {x}+\frac {2\,B\,a\,x^{3/2}}{3}+\frac {2\,A\,c\,x^{5/2}}{5}+\frac {2\,B\,c\,x^{7/2}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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