Optimal. Leaf size=97 \[ \frac {b \sqrt {a^2+2 a b x^3+b^2 x^6} (d x)^{m+4}}{d^4 (m+4) \left (a+b x^3\right )}+\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6} (d x)^{m+1}}{d (m+1) \left (a+b x^3\right )} \]
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Rubi [A] time = 0.04, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1355, 14} \[ \frac {b \sqrt {a^2+2 a b x^3+b^2 x^6} (d x)^{m+4}}{d^4 (m+4) \left (a+b x^3\right )}+\frac {a \sqrt {a^2+2 a b x^3+b^2 x^6} (d x)^{m+1}}{d (m+1) \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 14
Rule 1355
Rubi steps
\begin {align*} \int (d x)^m \sqrt {a^2+2 a b x^3+b^2 x^6} \, dx &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int (d x)^m \left (a b+b^2 x^3\right ) \, dx}{a b+b^2 x^3}\\ &=\frac {\sqrt {a^2+2 a b x^3+b^2 x^6} \int \left (a b (d x)^m+\frac {b^2 (d x)^{3+m}}{d^3}\right ) \, dx}{a b+b^2 x^3}\\ &=\frac {a (d x)^{1+m} \sqrt {a^2+2 a b x^3+b^2 x^6}}{d (1+m) \left (a+b x^3\right )}+\frac {b (d x)^{4+m} \sqrt {a^2+2 a b x^3+b^2 x^6}}{d^4 (4+m) \left (a+b x^3\right )}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 53, normalized size = 0.55 \[ \frac {x \sqrt {\left (a+b x^3\right )^2} (d x)^m \left (a (m+4)+b (m+1) x^3\right )}{(m+1) (m+4) \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 35, normalized size = 0.36 \[ \frac {{\left ({\left (b m + b\right )} x^{4} + {\left (a m + 4 \, a\right )} x\right )} \left (d x\right )^{m}}{m^{2} + 5 \, m + 4} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 83, normalized size = 0.86 \[ \frac {\left (d x\right )^{m} b m x^{4} \mathrm {sgn}\left (b x^{3} + a\right ) + \left (d x\right )^{m} b x^{4} \mathrm {sgn}\left (b x^{3} + a\right ) + \left (d x\right )^{m} a m x \mathrm {sgn}\left (b x^{3} + a\right ) + 4 \, \left (d x\right )^{m} a x \mathrm {sgn}\left (b x^{3} + a\right )}{m^{2} + 5 \, m + 4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 56, normalized size = 0.58 \[ \frac {\left (b m \,x^{3}+b \,x^{3}+a m +4 a \right ) \sqrt {\left (b \,x^{3}+a \right )^{2}}\, x \left (d x \right )^{m}}{\left (m +4\right ) \left (m +1\right ) \left (b \,x^{3}+a \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 35, normalized size = 0.36 \[ \frac {{\left (b d^{m} {\left (m + 1\right )} x^{4} + a d^{m} {\left (m + 4\right )} x\right )} x^{m}}{m^{2} + 5 \, m + 4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d\,x\right )}^m\,\sqrt {a^2+2\,a\,b\,x^3+b^2\,x^6} \,d x \]
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 \[ \int \left (d x\right )^{m} \sqrt {\left (a + b x^{3}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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