Optimal. Leaf size=48 \[ \frac {a x (d x)^m}{m \sqrt {c x^2}}+\frac {b x (d x)^{m+1}}{d (m+1) \sqrt {c x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 48, 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, 43} \begin {gather*} \frac {a x (d x)^m}{m \sqrt {c x^2}}+\frac {b x (d x)^{m+1}}{d (m+1) \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 16
Rule 43
Rubi steps
\begin {align*} \int \frac {(d x)^m (a+b x)}{\sqrt {c x^2}} \, dx &=\frac {x \int \frac {(d x)^m (a+b x)}{x} \, dx}{\sqrt {c x^2}}\\ &=\frac {(d x) \int (d x)^{-1+m} (a+b x) \, dx}{\sqrt {c x^2}}\\ &=\frac {(d x) \int \left (a (d x)^{-1+m}+\frac {b (d x)^m}{d}\right ) \, dx}{\sqrt {c x^2}}\\ &=\frac {a x (d x)^m}{m \sqrt {c x^2}}+\frac {b x (d x)^{1+m}}{d (1+m) \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 0.69 \begin {gather*} \frac {x (d x)^m (a m+a+b m x)}{m (m+1) \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.40, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d x)^m (a+b x)}{\sqrt {c x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.75, size = 36, normalized size = 0.75 \begin {gather*} \frac {{\left (b m x + a m + a\right )} \sqrt {c x^{2}} \left (d x\right )^{m}}{{\left (c m^{2} + c m\right )} x} \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*} \int \frac {{\left (b x + a\right )} \left (d x\right )^{m}}{\sqrt {c x^{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 32, normalized size = 0.67 \begin {gather*} \frac {\left (b m x +a m +a \right ) x \left (d x \right )^{m}}{\left (m +1\right ) \sqrt {c \,x^{2}}\, m} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.48, size = 32, normalized size = 0.67 \begin {gather*} \frac {b d^{m} x x^{m}}{\sqrt {c} {\left (m + 1\right )}} + \frac {a d^{m} x^{m}}{\sqrt {c} m} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 30, normalized size = 0.62 \begin {gather*} \frac {\left (\frac {a\,x}{m}+\frac {b\,x^2}{m+1}\right )\,{\left (d\,x\right )}^m}{\sqrt {c\,x^2}} \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} \frac {\int \frac {b}{\sqrt {c x^{2}}}\, dx + \int \frac {a}{x \sqrt {c x^{2}}}\, dx}{d} & \text {for}\: m = -1 \\\int \frac {a + b x}{\sqrt {c x^{2}}}\, dx & \text {for}\: m = 0 \\\frac {a d^{m} m x x^{m}}{\sqrt {c} m^{2} \sqrt {x^{2}} + \sqrt {c} m \sqrt {x^{2}}} + \frac {a d^{m} x x^{m}}{\sqrt {c} m^{2} \sqrt {x^{2}} + \sqrt {c} m \sqrt {x^{2}}} + \frac {b d^{m} m x^{2} x^{m}}{\sqrt {c} m^{2} \sqrt {x^{2}} + \sqrt {c} m \sqrt {x^{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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