Optimal. Leaf size=94 \[ \frac {a^2 \sqrt {c x^2} (d x)^{m+2}}{d^2 (m+2) x}+\frac {2 a b \sqrt {c x^2} (d x)^{m+3}}{d^3 (m+3) x}+\frac {b^2 \sqrt {c x^2} (d x)^{m+4}}{d^4 (m+4) x} \]
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Rubi [A] time = 0.04, antiderivative size = 94, 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} \[ \frac {a^2 \sqrt {c x^2} (d x)^{m+2}}{d^2 (m+2) x}+\frac {2 a b \sqrt {c x^2} (d x)^{m+3}}{d^3 (m+3) x}+\frac {b^2 \sqrt {c x^2} (d x)^{m+4}}{d^4 (m+4) x} \]
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)^2 \, dx &=\frac {\sqrt {c x^2} \int x (d x)^m (a+b x)^2 \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int (d x)^{1+m} (a+b x)^2 \, dx}{d x}\\ &=\frac {\sqrt {c x^2} \int \left (a^2 (d x)^{1+m}+\frac {2 a b (d x)^{2+m}}{d}+\frac {b^2 (d x)^{3+m}}{d^2}\right ) \, dx}{d x}\\ &=\frac {a^2 (d x)^{2+m} \sqrt {c x^2}}{d^2 (2+m) x}+\frac {2 a b (d x)^{3+m} \sqrt {c x^2}}{d^3 (3+m) x}+\frac {b^2 (d x)^{4+m} \sqrt {c x^2}}{d^4 (4+m) x}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 72, normalized size = 0.77 \[ \frac {x \sqrt {c x^2} (d x)^m \left (a^2 \left (m^2+7 m+12\right )+2 a b \left (m^2+6 m+8\right ) x+b^2 \left (m^2+5 m+6\right ) x^2\right )}{(m+2) (m+3) (m+4)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 94, normalized size = 1.00 \[ \frac {{\left ({\left (b^{2} m^{2} + 5 \, b^{2} m + 6 \, b^{2}\right )} x^{3} + 2 \, {\left (a b m^{2} + 6 \, a b m + 8 \, a b\right )} x^{2} + {\left (a^{2} m^{2} + 7 \, a^{2} m + 12 \, a^{2}\right )} x\right )} \sqrt {c x^{2}} \left (d x\right )^{m}}{m^{3} + 9 \, m^{2} + 26 \, m + 24} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 95, normalized size = 1.01 \[ \frac {\left (b^{2} m^{2} x^{2}+2 a b \,m^{2} x +5 b^{2} m \,x^{2}+a^{2} m^{2}+12 a b m x +6 b^{2} x^{2}+7 a^{2} m +16 a b x +12 a^{2}\right ) \sqrt {c \,x^{2}}\, x \left (d x \right )^{m}}{\left (m +4\right ) \left (m +3\right ) \left (m +2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.52, size = 64, normalized size = 0.68 \[ \frac {b^{2} \sqrt {c} d^{m} x^{4} x^{m}}{m + 4} + \frac {2 \, a b \sqrt {c} d^{m} x^{3} x^{m}}{m + 3} + \frac {a^{2} \sqrt {c} d^{m} x^{2} x^{m}}{m + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 116, normalized size = 1.23 \[ {\left (d\,x\right )}^m\,\left (\frac {a^2\,x\,\sqrt {c\,x^2}\,\left (m^2+7\,m+12\right )}{m^3+9\,m^2+26\,m+24}+\frac {b^2\,x^3\,\sqrt {c\,x^2}\,\left (m^2+5\,m+6\right )}{m^3+9\,m^2+26\,m+24}+\frac {2\,a\,b\,x^2\,\sqrt {c\,x^2}\,\left (m^2+6\,m+8\right )}{m^3+9\,m^2+26\,m+24}\right ) \]
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 \[ \begin {cases} \frac {\int \frac {a^{2} \sqrt {c x^{2}}}{x^{4}}\, dx + \int \frac {b^{2} \sqrt {c x^{2}}}{x^{2}}\, dx + \int \frac {2 a b \sqrt {c x^{2}}}{x^{3}}\, dx}{d^{4}} & \text {for}\: m = -4 \\\frac {\int \frac {a^{2} \sqrt {c x^{2}}}{x^{3}}\, dx + \int \frac {b^{2} \sqrt {c x^{2}}}{x}\, dx + \int \frac {2 a b \sqrt {c x^{2}}}{x^{2}}\, dx}{d^{3}} & \text {for}\: m = -3 \\\frac {\int b^{2} \sqrt {c x^{2}}\, dx + \int \frac {a^{2} \sqrt {c x^{2}}}{x^{2}}\, dx + \int \frac {2 a b \sqrt {c x^{2}}}{x}\, dx}{d^{2}} & \text {for}\: m = -2 \\\frac {a^{2} \sqrt {c} d^{m} m^{2} x x^{m} \sqrt {x^{2}}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {7 a^{2} \sqrt {c} d^{m} m x x^{m} \sqrt {x^{2}}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {12 a^{2} \sqrt {c} d^{m} x x^{m} \sqrt {x^{2}}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {2 a b \sqrt {c} d^{m} m^{2} x^{2} x^{m} \sqrt {x^{2}}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {12 a b \sqrt {c} d^{m} m x^{2} x^{m} \sqrt {x^{2}}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {16 a b \sqrt {c} d^{m} x^{2} x^{m} \sqrt {x^{2}}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {b^{2} \sqrt {c} d^{m} m^{2} x^{3} x^{m} \sqrt {x^{2}}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {5 b^{2} \sqrt {c} d^{m} m x^{3} x^{m} \sqrt {x^{2}}}{m^{3} + 9 m^{2} + 26 m + 24} + \frac {6 b^{2} \sqrt {c} d^{m} x^{3} x^{m} \sqrt {x^{2}}}{m^{3} + 9 m^{2} + 26 m + 24} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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