Optimal. Leaf size=104 \[ \frac {3 a x^{1+m} \sqrt {a+b x^{2 (1+m)}}}{8 (1+m)}+\frac {x^{1+m} \left (a+b x^{2 (1+m)}\right )^{3/2}}{4 (1+m)}+\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} x^{1+m}}{\sqrt {a+b x^{2 (1+m)}}}\right )}{8 \sqrt {b} (1+m)} \]
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Rubi [A]
time = 0.03, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {352, 201, 223,
212} \begin {gather*} \frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} x^{m+1}}{\sqrt {a+b x^{2 (m+1)}}}\right )}{8 \sqrt {b} (m+1)}+\frac {x^{m+1} \left (a+b x^{2 (m+1)}\right )^{3/2}}{4 (m+1)}+\frac {3 a x^{m+1} \sqrt {a+b x^{2 (m+1)}}}{8 (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 212
Rule 223
Rule 352
Rubi steps
\begin {align*} \int x^m \left (a+b x^{2+2 m}\right )^{3/2} \, dx &=\frac {\text {Subst}\left (\int \left (a+b x^2\right )^{3/2} \, dx,x,x^{1+m}\right )}{1+m}\\ &=\frac {x^{1+m} \left (a+b x^{2 (1+m)}\right )^{3/2}}{4 (1+m)}+\frac {(3 a) \text {Subst}\left (\int \sqrt {a+b x^2} \, dx,x,x^{1+m}\right )}{4 (1+m)}\\ &=\frac {3 a x^{1+m} \sqrt {a+b x^{2 (1+m)}}}{8 (1+m)}+\frac {x^{1+m} \left (a+b x^{2 (1+m)}\right )^{3/2}}{4 (1+m)}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,x^{1+m}\right )}{8 (1+m)}\\ &=\frac {3 a x^{1+m} \sqrt {a+b x^{2 (1+m)}}}{8 (1+m)}+\frac {x^{1+m} \left (a+b x^{2 (1+m)}\right )^{3/2}}{4 (1+m)}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x^{1+m}}{\sqrt {a+b x^{2+2 m}}}\right )}{8 (1+m)}\\ &=\frac {3 a x^{1+m} \sqrt {a+b x^{2 (1+m)}}}{8 (1+m)}+\frac {x^{1+m} \left (a+b x^{2 (1+m)}\right )^{3/2}}{4 (1+m)}+\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {b} x^{1+m}}{\sqrt {a+b x^{2 (1+m)}}}\right )}{8 \sqrt {b} (1+m)}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.06, size = 86, normalized size = 0.83 \begin {gather*} \frac {a x^{1+m} \sqrt {a+b x^{2+2 m}} \, _2F_1\left (-\frac {3}{2},\frac {1+m}{2+2 m};1+\frac {1+m}{2+2 m};-\frac {b x^{2+2 m}}{a}\right )}{(1+m) \sqrt {1+\frac {b x^{2+2 m}}{a}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{m} \left (a +b \,x^{2+2 m}\right )^{\frac {3}{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 = -0.01 \begin {gather*} \int x^m\,{\left (a+b\,x^{2\,m+2}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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