Optimal. Leaf size=59 \[ \frac {a \sqrt {c x^2} (d x)^{m+2}}{d^2 (m+2) x}+\frac {b \sqrt {c x^2} (d x)^{m+3}}{d^3 (m+3) x} \]
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Rubi [A] time = 0.03, antiderivative size = 59, 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 \sqrt {c x^2} (d x)^{m+2}}{d^2 (m+2) x}+\frac {b \sqrt {c x^2} (d x)^{m+3}}{d^3 (m+3) x} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 16
Rule 43
Rubi steps
\begin {align*} \int (d x)^m \sqrt {c x^2} (a+b x) \, dx &=\frac {\sqrt {c x^2} \int x (d x)^m (a+b x) \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int (d x)^{1+m} (a+b x) \, dx}{d x}\\ &=\frac {\sqrt {c x^2} \int \left (a (d x)^{1+m}+\frac {b (d x)^{2+m}}{d}\right ) \, dx}{d x}\\ &=\frac {a (d x)^{2+m} \sqrt {c x^2}}{d^2 (2+m) x}+\frac {b (d x)^{3+m} \sqrt {c x^2}}{d^3 (3+m) x}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 38, normalized size = 0.64 \begin {gather*} \frac {x \sqrt {c x^2} (d x)^m (a (m+3)+b (m+2) x)}{(m+2) (m+3)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.35, size = 0, normalized size = 0.00 \begin {gather*} \int (d x)^m \sqrt {c x^2} (a+b x) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.52, size = 44, normalized size = 0.75 \begin {gather*} \frac {{\left ({\left (b m + 2 \, b\right )} x^{2} + {\left (a m + 3 \, a\right )} x\right )} \sqrt {c x^{2}} \left (d x\right )^{m}}{m^{2} + 5 \, m + 6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 40, normalized size = 0.68 \begin {gather*} \frac {\left (b m x +a m +2 b x +3 a \right ) \sqrt {c \,x^{2}}\, x \left (d x \right )^{m}}{\left (m +3\right ) \left (m +2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.52, size = 39, normalized size = 0.66 \begin {gather*} \frac {b \sqrt {c} d^{m} x^{3} x^{m}}{m + 3} + \frac {a \sqrt {c} d^{m} x^{2} x^{m}}{m + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 39, normalized size = 0.66 \begin {gather*} \frac {x\,{\left (d\,x\right )}^m\,\sqrt {c\,x^2}\,\left (3\,a+a\,m+2\,b\,x+b\,m\,x\right )}{m^2+5\,m+6} \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 {a \sqrt {c x^{2}}}{x^{3}}\, dx + \int \frac {b \sqrt {c x^{2}}}{x^{2}}\, dx}{d^{3}} & \text {for}\: m = -3 \\\frac {\int \frac {a \sqrt {c x^{2}}}{x^{2}}\, dx + \int \frac {b \sqrt {c x^{2}}}{x}\, dx}{d^{2}} & \text {for}\: m = -2 \\\frac {a \sqrt {c} d^{m} m x x^{m} \sqrt {x^{2}}}{m^{2} + 5 m + 6} + \frac {3 a \sqrt {c} d^{m} x x^{m} \sqrt {x^{2}}}{m^{2} + 5 m + 6} + \frac {b \sqrt {c} d^{m} m x^{2} x^{m} \sqrt {x^{2}}}{m^{2} + 5 m + 6} + \frac {2 b \sqrt {c} d^{m} x^{2} x^{m} \sqrt {x^{2}}}{m^{2} + 5 m + 6} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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