Optimal. Leaf size=136 \[ \frac{5 a^2 x^{m+1} \sqrt{a+b x^{2 (m+1)}}}{16 (m+1)}+\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a+b x^{2 (m+1)}}}\right )}{16 \sqrt{b} (m+1)}+\frac{x^{m+1} \left (a+b x^{2 (m+1)}\right )^{5/2}}{6 (m+1)}+\frac{5 a x^{m+1} \left (a+b x^{2 (m+1)}\right )^{3/2}}{24 (m+1)} \]
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Rubi [A] time = 0.0539171, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {345, 195, 217, 206} \[ \frac{5 a^2 x^{m+1} \sqrt{a+b x^{2 (m+1)}}}{16 (m+1)}+\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a+b x^{2 (m+1)}}}\right )}{16 \sqrt{b} (m+1)}+\frac{x^{m+1} \left (a+b x^{2 (m+1)}\right )^{5/2}}{6 (m+1)}+\frac{5 a x^{m+1} \left (a+b x^{2 (m+1)}\right )^{3/2}}{24 (m+1)} \]
Antiderivative was successfully verified.
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Rule 345
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int x^m \left (a+b x^{2+2 m}\right )^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b x^2\right )^{5/2} \, dx,x,x^{1+m}\right )}{1+m}\\ &=\frac{x^{1+m} \left (a+b x^{2 (1+m)}\right )^{5/2}}{6 (1+m)}+\frac{(5 a) \operatorname{Subst}\left (\int \left (a+b x^2\right )^{3/2} \, dx,x,x^{1+m}\right )}{6 (1+m)}\\ &=\frac{5 a x^{1+m} \left (a+b x^{2 (1+m)}\right )^{3/2}}{24 (1+m)}+\frac{x^{1+m} \left (a+b x^{2 (1+m)}\right )^{5/2}}{6 (1+m)}+\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \sqrt{a+b x^2} \, dx,x,x^{1+m}\right )}{8 (1+m)}\\ &=\frac{5 a^2 x^{1+m} \sqrt{a+b x^{2 (1+m)}}}{16 (1+m)}+\frac{5 a x^{1+m} \left (a+b x^{2 (1+m)}\right )^{3/2}}{24 (1+m)}+\frac{x^{1+m} \left (a+b x^{2 (1+m)}\right )^{5/2}}{6 (1+m)}+\frac{\left (5 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,x^{1+m}\right )}{16 (1+m)}\\ &=\frac{5 a^2 x^{1+m} \sqrt{a+b x^{2 (1+m)}}}{16 (1+m)}+\frac{5 a x^{1+m} \left (a+b x^{2 (1+m)}\right )^{3/2}}{24 (1+m)}+\frac{x^{1+m} \left (a+b x^{2 (1+m)}\right )^{5/2}}{6 (1+m)}+\frac{\left (5 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x^{1+m}}{\sqrt{a+b x^{2+2 m}}}\right )}{16 (1+m)}\\ &=\frac{5 a^2 x^{1+m} \sqrt{a+b x^{2 (1+m)}}}{16 (1+m)}+\frac{5 a x^{1+m} \left (a+b x^{2 (1+m)}\right )^{3/2}}{24 (1+m)}+\frac{x^{1+m} \left (a+b x^{2 (1+m)}\right )^{5/2}}{6 (1+m)}+\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x^{1+m}}{\sqrt{a+b x^{2 (1+m)}}}\right )}{16 \sqrt{b} (1+m)}\\ \end{align*}
Mathematica [C] time = 0.0667992, size = 88, normalized size = 0.65 \[ \frac{a^2 x^{m+1} \sqrt{a+b x^{2 m+2}} \, _2F_1\left (-\frac{5}{2},\frac{m+1}{2 m+2};\frac{m+1}{2 m+2}+1;-\frac{b x^{2 m+2}}{a}\right )}{(m+1) \sqrt{\frac{b x^{2 m+2}}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.089, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( a+b{x}^{2+2\,m} \right ) ^{{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{2 \, m + 2} + a\right )}^{\frac{5}{2}} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{2 \, m + 2} + a\right )}^{\frac{5}{2}} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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