Optimal. Leaf size=52 \[ -\frac {2 x^{m+1} \left (a+b \sqrt {x}\right )^{p+1} \, _2F_1\left (1,2 m+p+3;p+2;\frac {a+b \sqrt {x}}{a}\right )}{a (p+1)} \]
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Rubi [A] time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.21, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {341, 66, 64} \[ \frac {x^{m+1} \left (a+b \sqrt {x}\right )^p \left (\frac {b \sqrt {x}}{a}+1\right )^{-p} \, _2F_1\left (2 (m+1),-p;2 m+3;-\frac {b \sqrt {x}}{a}\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 64
Rule 66
Rule 341
Rubi steps
\begin {align*} \int \left (a+b \sqrt {x}\right )^p x^m \, dx &=2 \operatorname {Subst}\left (\int x^{-1+2 (1+m)} (a+b x)^p \, dx,x,\sqrt {x}\right )\\ &=\left (2 \left (a+b \sqrt {x}\right )^p \left (1+\frac {b \sqrt {x}}{a}\right )^{-p}\right ) \operatorname {Subst}\left (\int x^{-1+2 (1+m)} \left (1+\frac {b x}{a}\right )^p \, dx,x,\sqrt {x}\right )\\ &=\frac {\left (a+b \sqrt {x}\right )^p \left (1+\frac {b \sqrt {x}}{a}\right )^{-p} x^{1+m} \, _2F_1\left (2 (1+m),-p;3+2 m;-\frac {b \sqrt {x}}{a}\right )}{1+m}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 65, normalized size = 1.25 \[ \frac {x^{m+1} \left (a+b \sqrt {x}\right )^p \left (\frac {b \sqrt {x}}{a}+1\right )^{-p} \, _2F_1\left (2 (m+1),-p;2 (m+1)+1;-\frac {b \sqrt {x}}{a}\right )}{m+1} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int {\left (b \sqrt {x} + a\right )}^{p} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.18, size = 0, normalized size = 0.00 \[ \int x^{m} \left (b \sqrt {x}+a \right )^{p}\, dx \]
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 \[ \int {\left (b \sqrt {x} + a\right )}^{p} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^m\,{\left (a+b\,\sqrt {x}\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 36.19, size = 46, normalized size = 0.88 \[ \frac {2 a^{p} x x^{m} \Gamma \left (2 m + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 2 m + 2 \\ 2 m + 3 \end {matrix}\middle | {\frac {b \sqrt {x} e^{i \pi }}{a}} \right )}}{\Gamma \left (2 m + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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