Optimal. Leaf size=108 \[ \frac {b^2 x^{n+1} (d x)^m \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(m+n+1) \left (a b+b^2 x^n\right )}+\frac {a (d x)^{m+1} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{d (m+1) \left (a+b x^n\right )} \]
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Rubi [A] time = 0.04, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1355, 14, 20, 30} \[ \frac {b^2 x^{n+1} (d x)^m \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(m+n+1) \left (a b+b^2 x^n\right )}+\frac {a (d x)^{m+1} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{d (m+1) \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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Rule 14
Rule 20
Rule 30
Rule 1355
Rubi steps
\begin {align*} \int (d x)^m \sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \, dx &=\frac {\sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \int (d x)^m \left (a b+b^2 x^n\right ) \, dx}{a b+b^2 x^n}\\ &=\frac {\sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \int \left (a b (d x)^m+b^2 x^n (d x)^m\right ) \, dx}{a b+b^2 x^n}\\ &=\frac {a (d x)^{1+m} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{d (1+m) \left (a+b x^n\right )}+\frac {\left (b^2 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}\right ) \int x^n (d x)^m \, dx}{a b+b^2 x^n}\\ &=\frac {a (d x)^{1+m} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{d (1+m) \left (a+b x^n\right )}+\frac {\left (b^2 x^{-m} (d x)^m \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}\right ) \int x^{m+n} \, dx}{a b+b^2 x^n}\\ &=\frac {a (d x)^{1+m} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{d (1+m) \left (a+b x^n\right )}+\frac {b^2 x^{1+n} (d x)^m \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(1+m+n) \left (a b+b^2 x^n\right )}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 55, normalized size = 0.51 \[ \frac {x (d x)^m \sqrt {\left (a+b x^n\right )^2} \left (a (m+n+1)+b (m+1) x^n\right )}{(m+1) (m+n+1) \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 57, normalized size = 0.53 \[ \frac {{\left (b m + b\right )} x x^{n} e^{\left (m \log \relax (d) + m \log \relax (x)\right )} + {\left (a m + a n + a\right )} x e^{\left (m \log \relax (d) + m \log \relax (x)\right )}}{m^{2} + {\left (m + 1\right )} n + 2 \, m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 173, normalized size = 1.60 \[ \frac {b m x x^{n} e^{\left (m \log \relax (d) + m \log \relax (x)\right )} \mathrm {sgn}\left (b x^{n} + a\right ) + a m x e^{\left (m \log \relax (d) + m \log \relax (x)\right )} \mathrm {sgn}\left (b x^{n} + a\right ) + b m x e^{\left (m \log \relax (d) + m \log \relax (x)\right )} \mathrm {sgn}\left (b x^{n} + a\right ) + a n x e^{\left (m \log \relax (d) + m \log \relax (x)\right )} \mathrm {sgn}\left (b x^{n} + a\right ) + b x x^{n} e^{\left (m \log \relax (d) + m \log \relax (x)\right )} \mathrm {sgn}\left (b x^{n} + a\right ) + a x e^{\left (m \log \relax (d) + m \log \relax (x)\right )} \mathrm {sgn}\left (b x^{n} + a\right ) + b x e^{\left (m \log \relax (d) + m \log \relax (x)\right )} \mathrm {sgn}\left (b x^{n} + a\right )}{m^{2} + m n + 2 \, m + n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.04, size = 132, normalized size = 1.22 \[ \frac {\sqrt {\left (b \,x^{n}+a \right )^{2}}\, \left (b m \,x^{n}+a m +a n +b \,x^{n}+a \right ) x \,{\mathrm e}^{\frac {\left (-i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )+i \pi \,\mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i d x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i d x \right )^{2}-i \pi \mathrm {csgn}\left (i d x \right )^{3}+2 \ln \relax (d )+2 \ln \relax (x )\right ) m}{2}}}{\left (b \,x^{n}+a \right ) \left (m +1\right ) \left (m +n +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.94, size = 47, normalized size = 0.44 \[ \frac {a d^{m} {\left (m + n + 1\right )} x x^{m} + b d^{m} {\left (m + 1\right )} x e^{\left (m \log \relax (x) + n \log \relax (x)\right )}}{m^{2} + m {\left (n + 2\right )} + n + 1} \]
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 \[ \int {\left (d\,x\right )}^m\,\sqrt {a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n} \,d x \]
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 \[ \int \left (d x\right )^{m} \sqrt {\left (a + b x^{n}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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