Optimal. Leaf size=81 \[ \frac {a^2 x (d x)^m}{m \sqrt {c x^2}}+\frac {2 a b x (d x)^{m+1}}{d (m+1) \sqrt {c x^2}}+\frac {b^2 x (d x)^{m+2}}{d^2 (m+2) \sqrt {c x^2}} \]
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Rubi [A] time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {15, 16, 43} \[ \frac {a^2 x (d x)^m}{m \sqrt {c x^2}}+\frac {2 a b x (d x)^{m+1}}{d (m+1) \sqrt {c x^2}}+\frac {b^2 x (d x)^{m+2}}{d^2 (m+2) \sqrt {c x^2}} \]
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)^2}{\sqrt {c x^2}} \, dx &=\frac {x \int \frac {(d x)^m (a+b x)^2}{x} \, dx}{\sqrt {c x^2}}\\ &=\frac {(d x) \int (d x)^{-1+m} (a+b x)^2 \, dx}{\sqrt {c x^2}}\\ &=\frac {(d x) \int \left (a^2 (d x)^{-1+m}+\frac {2 a b (d x)^m}{d}+\frac {b^2 (d x)^{1+m}}{d^2}\right ) \, dx}{\sqrt {c x^2}}\\ &=\frac {a^2 x (d x)^m}{m \sqrt {c x^2}}+\frac {2 a b x (d x)^{1+m}}{d (1+m) \sqrt {c x^2}}+\frac {b^2 x (d x)^{2+m}}{d^2 (2+m) \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 62, normalized size = 0.77 \[ \frac {x (d x)^m \left (a^2 \left (m^2+3 m+2\right )+2 a b m (m+2) x+b^2 m (m+1) x^2\right )}{m (m+1) (m+2) \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 85, normalized size = 1.05 \[ \frac {{\left (a^{2} m^{2} + 3 \, a^{2} m + {\left (b^{2} m^{2} + b^{2} m\right )} x^{2} + 2 \, a^{2} + 2 \, {\left (a b m^{2} + 2 \, a b m\right )} x\right )} \sqrt {c x^{2}} \left (d x\right )^{m}}{{\left (c m^{3} + 3 \, c m^{2} + 2 \, c m\right )} x} \]
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 \[ \int \frac {{\left (b x + a\right )}^{2} \left (d x\right )^{m}}{\sqrt {c x^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 79, normalized size = 0.98 \[ \frac {\left (b^{2} m^{2} x^{2}+2 a b \,m^{2} x +b^{2} m \,x^{2}+a^{2} m^{2}+4 a b m x +3 a^{2} m +2 a^{2}\right ) x \left (d x \right )^{m}}{\left (m +2\right ) \left (m +1\right ) \sqrt {c \,x^{2}}\, m} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.64, size = 57, normalized size = 0.70 \[ \frac {b^{2} d^{m} x^{2} x^{m}}{\sqrt {c} {\left (m + 2\right )}} + \frac {2 \, a b d^{m} x x^{m}}{\sqrt {c} {\left (m + 1\right )}} + \frac {a^{2} d^{m} x^{m}}{\sqrt {c} m} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 62, normalized size = 0.77 \[ \frac {{\left (d\,x\right )}^m\,\left (\frac {a^2\,x}{m}+\frac {b^2\,x^3\,\left (m+1\right )}{m^2+3\,m+2}+\frac {2\,a\,b\,x^2\,\left (m+2\right )}{m^2+3\,m+2}\right )}{\sqrt {c\,x^2}} \]
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 \[ \begin {cases} \frac {\int \frac {b^{2}}{\sqrt {c x^{2}}}\, dx + \int \frac {a^{2}}{x^{2} \sqrt {c x^{2}}}\, dx + \int \frac {2 a b}{x \sqrt {c x^{2}}}\, dx}{d^{2}} & \text {for}\: m = -2 \\\frac {\int \frac {2 a b}{\sqrt {c x^{2}}}\, dx + \int \frac {a^{2}}{x \sqrt {c x^{2}}}\, dx + \int \frac {b^{2} x}{\sqrt {c x^{2}}}\, dx}{d} & \text {for}\: m = -1 \\\int \frac {\left (a + b x\right )^{2}}{\sqrt {c x^{2}}}\, dx & \text {for}\: m = 0 \\\frac {a^{2} d^{m} m^{2} x x^{m}}{\sqrt {c} m^{3} \sqrt {x^{2}} + 3 \sqrt {c} m^{2} \sqrt {x^{2}} + 2 \sqrt {c} m \sqrt {x^{2}}} + \frac {3 a^{2} d^{m} m x x^{m}}{\sqrt {c} m^{3} \sqrt {x^{2}} + 3 \sqrt {c} m^{2} \sqrt {x^{2}} + 2 \sqrt {c} m \sqrt {x^{2}}} + \frac {2 a^{2} d^{m} x x^{m}}{\sqrt {c} m^{3} \sqrt {x^{2}} + 3 \sqrt {c} m^{2} \sqrt {x^{2}} + 2 \sqrt {c} m \sqrt {x^{2}}} + \frac {2 a b d^{m} m^{2} x^{2} x^{m}}{\sqrt {c} m^{3} \sqrt {x^{2}} + 3 \sqrt {c} m^{2} \sqrt {x^{2}} + 2 \sqrt {c} m \sqrt {x^{2}}} + \frac {4 a b d^{m} m x^{2} x^{m}}{\sqrt {c} m^{3} \sqrt {x^{2}} + 3 \sqrt {c} m^{2} \sqrt {x^{2}} + 2 \sqrt {c} m \sqrt {x^{2}}} + \frac {b^{2} d^{m} m^{2} x^{3} x^{m}}{\sqrt {c} m^{3} \sqrt {x^{2}} + 3 \sqrt {c} m^{2} \sqrt {x^{2}} + 2 \sqrt {c} m \sqrt {x^{2}}} + \frac {b^{2} d^{m} m x^{3} x^{m}}{\sqrt {c} m^{3} \sqrt {x^{2}} + 3 \sqrt {c} m^{2} \sqrt {x^{2}} + 2 \sqrt {c} m \sqrt {x^{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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