Optimal. Leaf size=205 \[ \frac {a^3 (d x)^{1+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d (1+m) \left (a+b x^2\right )}+\frac {3 a^2 b (d x)^{3+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 (3+m) \left (a+b x^2\right )}+\frac {3 a b^2 (d x)^{5+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^5 (5+m) \left (a+b x^2\right )}+\frac {b^3 (d x)^{7+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^7 (7+m) \left (a+b x^2\right )} \]
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Rubi [A]
time = 0.05, antiderivative size = 205, 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 = {1126, 276}
\begin {gather*} \frac {3 a b^2 \sqrt {a^2+2 a b x^2+b^2 x^4} (d x)^{m+5}}{d^5 (m+5) \left (a+b x^2\right )}+\frac {3 a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4} (d x)^{m+3}}{d^3 (m+3) \left (a+b x^2\right )}+\frac {b^3 \sqrt {a^2+2 a b x^2+b^2 x^4} (d x)^{m+7}}{d^7 (m+7) \left (a+b x^2\right )}+\frac {a^3 \sqrt {a^2+2 a b x^2+b^2 x^4} (d x)^{m+1}}{d (m+1) \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rule 1126
Rubi steps
\begin {align*} \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int (d x)^m \left (a b+b^2 x^2\right )^3 \, dx}{b^2 \left (a b+b^2 x^2\right )}\\ &=\frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (a^3 b^3 (d x)^m+\frac {3 a^2 b^4 (d x)^{2+m}}{d^2}+\frac {3 a b^5 (d x)^{4+m}}{d^4}+\frac {b^6 (d x)^{6+m}}{d^6}\right ) \, dx}{b^2 \left (a b+b^2 x^2\right )}\\ &=\frac {a^3 (d x)^{1+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d (1+m) \left (a+b x^2\right )}+\frac {3 a^2 b (d x)^{3+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 (3+m) \left (a+b x^2\right )}+\frac {3 a b^2 (d x)^{5+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^5 (5+m) \left (a+b x^2\right )}+\frac {b^3 (d x)^{7+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^7 (7+m) \left (a+b x^2\right )}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 131, normalized size = 0.64 \begin {gather*} \frac {x (d x)^m \sqrt {\left (a+b x^2\right )^2} \left (a^3 \left (105+71 m+15 m^2+m^3\right )+3 a^2 b \left (35+47 m+13 m^2+m^3\right ) x^2+3 a b^2 \left (21+31 m+11 m^2+m^3\right ) x^4+b^3 \left (15+23 m+9 m^2+m^3\right ) x^6\right )}{(1+m) (3+m) (5+m) (7+m) \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 199, normalized size = 0.97
method | result | size |
gosper | \(\frac {x \left (b^{3} m^{3} x^{6}+9 b^{3} m^{2} x^{6}+3 a \,b^{2} m^{3} x^{4}+23 m \,x^{6} b^{3}+33 a \,b^{2} m^{2} x^{4}+15 b^{3} x^{6}+3 a^{2} b \,m^{3} x^{2}+93 m \,x^{4} a \,b^{2}+39 a^{2} b \,m^{2} x^{2}+63 a \,b^{2} x^{4}+a^{3} m^{3}+141 m \,x^{2} a^{2} b +15 a^{3} m^{2}+105 a^{2} b \,x^{2}+71 m \,a^{3}+105 a^{3}\right ) \left (d x \right )^{m} \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}}}{\left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right ) \left (b \,x^{2}+a \right )^{3}}\) | \(199\) |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (b^{3} m^{3} x^{6}+9 b^{3} m^{2} x^{6}+3 a \,b^{2} m^{3} x^{4}+23 m \,x^{6} b^{3}+33 a \,b^{2} m^{2} x^{4}+15 b^{3} x^{6}+3 a^{2} b \,m^{3} x^{2}+93 m \,x^{4} a \,b^{2}+39 a^{2} b \,m^{2} x^{2}+63 a \,b^{2} x^{4}+a^{3} m^{3}+141 m \,x^{2} a^{2} b +15 a^{3} m^{2}+105 a^{2} b \,x^{2}+71 m \,a^{3}+105 a^{3}\right ) x \left (d x \right )^{m}}{\left (b \,x^{2}+a \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(199\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 119, normalized size = 0.58 \begin {gather*} \frac {{\left ({\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )} b^{3} d^{m} x^{7} + 3 \, {\left (m^{3} + 11 \, m^{2} + 31 \, m + 21\right )} a b^{2} d^{m} x^{5} + 3 \, {\left (m^{3} + 13 \, m^{2} + 47 \, m + 35\right )} a^{2} b d^{m} x^{3} + {\left (m^{3} + 15 \, m^{2} + 71 \, m + 105\right )} a^{3} d^{m} x\right )} x^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 159, normalized size = 0.78 \begin {gather*} \frac {{\left ({\left (b^{3} m^{3} + 9 \, b^{3} m^{2} + 23 \, b^{3} m + 15 \, b^{3}\right )} x^{7} + 3 \, {\left (a b^{2} m^{3} + 11 \, a b^{2} m^{2} + 31 \, a b^{2} m + 21 \, a b^{2}\right )} x^{5} + 3 \, {\left (a^{2} b m^{3} + 13 \, a^{2} b m^{2} + 47 \, a^{2} b m + 35 \, a^{2} b\right )} x^{3} + {\left (a^{3} m^{3} + 15 \, a^{3} m^{2} + 71 \, a^{3} m + 105 \, a^{3}\right )} x\right )} \left (d x\right )^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \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*} \int \left (d x\right )^{m} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 384 vs.
\(2 (161) = 322\).
time = 2.73, size = 384, normalized size = 1.87 \begin {gather*} \frac {\left (d x\right )^{m} b^{3} m^{3} x^{7} \mathrm {sgn}\left (b x^{2} + a\right ) + 9 \, \left (d x\right )^{m} b^{3} m^{2} x^{7} \mathrm {sgn}\left (b x^{2} + a\right ) + 3 \, \left (d x\right )^{m} a b^{2} m^{3} x^{5} \mathrm {sgn}\left (b x^{2} + a\right ) + 23 \, \left (d x\right )^{m} b^{3} m x^{7} \mathrm {sgn}\left (b x^{2} + a\right ) + 33 \, \left (d x\right )^{m} a b^{2} m^{2} x^{5} \mathrm {sgn}\left (b x^{2} + a\right ) + 15 \, \left (d x\right )^{m} b^{3} x^{7} \mathrm {sgn}\left (b x^{2} + a\right ) + 3 \, \left (d x\right )^{m} a^{2} b m^{3} x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) + 93 \, \left (d x\right )^{m} a b^{2} m x^{5} \mathrm {sgn}\left (b x^{2} + a\right ) + 39 \, \left (d x\right )^{m} a^{2} b m^{2} x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) + 63 \, \left (d x\right )^{m} a b^{2} x^{5} \mathrm {sgn}\left (b x^{2} + a\right ) + \left (d x\right )^{m} a^{3} m^{3} x \mathrm {sgn}\left (b x^{2} + a\right ) + 141 \, \left (d x\right )^{m} a^{2} b m x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) + 15 \, \left (d x\right )^{m} a^{3} m^{2} x \mathrm {sgn}\left (b x^{2} + a\right ) + 105 \, \left (d x\right )^{m} a^{2} b x^{3} \mathrm {sgn}\left (b x^{2} + a\right ) + 71 \, \left (d x\right )^{m} a^{3} m x \mathrm {sgn}\left (b x^{2} + a\right ) + 105 \, \left (d x\right )^{m} a^{3} x \mathrm {sgn}\left (b x^{2} + a\right )}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (d\,x\right )}^m\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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