Optimal. Leaf size=65 \[ -\frac {a d^2 x (d x)^{-2+m}}{c (2-m) \sqrt {c x^2}}-\frac {b d x (d x)^{-1+m}}{c (1-m) \sqrt {c x^2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {15, 16, 45}
\begin {gather*} -\frac {a d^2 x (d x)^{m-2}}{c (2-m) \sqrt {c x^2}}-\frac {b d x (d x)^{m-1}}{c (1-m) \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 16
Rule 45
Rubi steps
\begin {align*} \int \frac {(d x)^m (a+b x)}{\left (c x^2\right )^{3/2}} \, dx &=\frac {x \int \frac {(d x)^m (a+b x)}{x^3} \, dx}{c \sqrt {c x^2}}\\ &=\frac {\left (d^3 x\right ) \int (d x)^{-3+m} (a+b x) \, dx}{c \sqrt {c x^2}}\\ &=\frac {\left (d^3 x\right ) \int \left (a (d x)^{-3+m}+\frac {b (d x)^{-2+m}}{d}\right ) \, dx}{c \sqrt {c x^2}}\\ &=-\frac {a d^2 x (d x)^{-2+m}}{c (2-m) \sqrt {c x^2}}-\frac {b d x (d x)^{-1+m}}{c (1-m) \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 38, normalized size = 0.58 \begin {gather*} \frac {x (d x)^m (a (-1+m)+b (-2+m) x)}{(-2+m) (-1+m) \left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 40, normalized size = 0.62
method | result | size |
gosper | \(\frac {x \left (b m x +a m -2 b x -a \right ) \left (d x \right )^{m}}{\left (-1+m \right ) \left (-2+m \right ) \left (c \,x^{2}\right )^{\frac {3}{2}}}\) | \(40\) |
risch | \(\frac {\left (b m x +a m -2 b x -a \right ) \left (d x \right )^{m}}{c x \sqrt {c \,x^{2}}\, \left (-1+m \right ) \left (-2+m \right )}\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 39, normalized size = 0.60 \begin {gather*} \frac {b d^{m} x^{m}}{c^{\frac {3}{2}} {\left (m - 1\right )} x} + \frac {a d^{m} x^{m}}{c^{\frac {3}{2}} {\left (m - 2\right )} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.49, size = 53, normalized size = 0.82 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (a m + {\left (b m - 2 \, b\right )} x - a\right )} \left (d x\right )^{m}}{{\left (c^{2} m^{2} - 3 \, c^{2} m + 2 \, c^{2}\right )} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} d \left (\int \frac {a x}{\left (c x^{2}\right )^{\frac {3}{2}}}\, dx + \int \frac {b x^{2}}{\left (c x^{2}\right )^{\frac {3}{2}}}\, dx\right ) & \text {for}\: m = 1 \\d^{2} \left (\int \frac {a x^{2}}{\left (c x^{2}\right )^{\frac {3}{2}}}\, dx + \int \frac {b x^{3}}{\left (c x^{2}\right )^{\frac {3}{2}}}\, dx\right ) & \text {for}\: m = 2 \\\frac {a m x \left (d x\right )^{m}}{m^{2} \left (c x^{2}\right )^{\frac {3}{2}} - 3 m \left (c x^{2}\right )^{\frac {3}{2}} + 2 \left (c x^{2}\right )^{\frac {3}{2}}} - \frac {a x \left (d x\right )^{m}}{m^{2} \left (c x^{2}\right )^{\frac {3}{2}} - 3 m \left (c x^{2}\right )^{\frac {3}{2}} + 2 \left (c x^{2}\right )^{\frac {3}{2}}} + \frac {b m x^{2} \left (d x\right )^{m}}{m^{2} \left (c x^{2}\right )^{\frac {3}{2}} - 3 m \left (c x^{2}\right )^{\frac {3}{2}} + 2 \left (c x^{2}\right )^{\frac {3}{2}}} - \frac {2 b x^{2} \left (d x\right )^{m}}{m^{2} \left (c x^{2}\right )^{\frac {3}{2}} - 3 m \left (c x^{2}\right )^{\frac {3}{2}} + 2 \left (c x^{2}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.26, size = 48, normalized size = 0.74 \begin {gather*} \frac {b\,{\left (d\,x\right )}^m}{c\,\sqrt {c\,x^2}\,\left (m-1\right )}+\frac {a\,{\left (d\,x\right )}^m}{c\,x\,\sqrt {c\,x^2}\,\left (m-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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