Integrand size = 17, antiderivative size = 145 \[ \int x \left (a+b \sqrt {c+d x}\right )^p \, dx=-\frac {2 a \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{1+p}}{b^4 d^2 (1+p)}+\frac {2 \left (3 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{2+p}}{b^4 d^2 (2+p)}-\frac {6 a \left (a+b \sqrt {c+d x}\right )^{3+p}}{b^4 d^2 (3+p)}+\frac {2 \left (a+b \sqrt {c+d x}\right )^{4+p}}{b^4 d^2 (4+p)} \]
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Time = 0.08 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {378, 1412, 786} \[ \int x \left (a+b \sqrt {c+d x}\right )^p \, dx=-\frac {2 a \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{p+1}}{b^4 d^2 (p+1)}+\frac {2 \left (3 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{p+2}}{b^4 d^2 (p+2)}-\frac {6 a \left (a+b \sqrt {c+d x}\right )^{p+3}}{b^4 d^2 (p+3)}+\frac {2 \left (a+b \sqrt {c+d x}\right )^{p+4}}{b^4 d^2 (p+4)} \]
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Rule 378
Rule 786
Rule 1412
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (a+b \sqrt {x}\right )^p (-c+x) \, dx,x,c+d x\right )}{d^2} \\ & = \frac {2 \text {Subst}\left (\int x (a+b x)^p \left (-c+x^2\right ) \, dx,x,\sqrt {c+d x}\right )}{d^2} \\ & = \frac {2 \text {Subst}\left (\int \left (\frac {\left (-a^3+a b^2 c\right ) (a+b x)^p}{b^3}+\frac {\left (3 a^2-b^2 c\right ) (a+b x)^{1+p}}{b^3}-\frac {3 a (a+b x)^{2+p}}{b^3}+\frac {(a+b x)^{3+p}}{b^3}\right ) \, dx,x,\sqrt {c+d x}\right )}{d^2} \\ & = -\frac {2 a \left (a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{1+p}}{b^4 d^2 (1+p)}+\frac {2 \left (3 a^2-b^2 c\right ) \left (a+b \sqrt {c+d x}\right )^{2+p}}{b^4 d^2 (2+p)}-\frac {6 a \left (a+b \sqrt {c+d x}\right )^{3+p}}{b^4 d^2 (3+p)}+\frac {2 \left (a+b \sqrt {c+d x}\right )^{4+p}}{b^4 d^2 (4+p)} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.88 \[ \int x \left (a+b \sqrt {c+d x}\right )^p \, dx=-\frac {2 \left (a+b \sqrt {c+d x}\right )^{1+p} \left (6 a^3-6 a^2 b (1+p) \sqrt {c+d x}-b^3 \left (3+4 p+p^2\right ) \sqrt {c+d x} (-2 c+d (2+p) x)+a b^2 \left (2 c \left (-3+p+p^2\right )+3 d \left (2+3 p+p^2\right ) x\right )\right )}{b^4 d^2 (1+p) (2+p) (3+p) (4+p)} \]
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\[\int x \left (a +b \sqrt {d x +c}\right )^{p}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (137) = 274\).
Time = 0.31 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.03 \[ \int x \left (a+b \sqrt {c+d x}\right )^p \, dx=-\frac {2 \, {\left (6 \, b^{4} c^{2} - 12 \, a^{2} b^{2} c + 6 \, a^{4} + 2 \, {\left (b^{4} c^{2} + a^{2} b^{2} c\right )} p^{2} - {\left (b^{4} d^{2} p^{3} + 6 \, b^{4} d^{2} p^{2} + 11 \, b^{4} d^{2} p + 6 \, b^{4} d^{2}\right )} x^{2} + 4 \, {\left (2 \, b^{4} c^{2} - a^{2} b^{2} c\right )} p - {\left (b^{4} c d p^{3} + {\left (4 \, b^{4} c - 3 \, a^{2} b^{2}\right )} d p^{2} + 3 \, {\left (b^{4} c - a^{2} b^{2}\right )} d p\right )} x + {\left (4 \, a b^{3} c p^{2} + 2 \, {\left (5 \, a b^{3} c - 3 \, a^{3} b\right )} p - {\left (a b^{3} d p^{3} + 3 \, a b^{3} d p^{2} + 2 \, a b^{3} d p\right )} x\right )} \sqrt {d x + c}\right )} {\left (\sqrt {d x + c} b + a\right )}^{p}}{b^{4} d^{2} p^{4} + 10 \, b^{4} d^{2} p^{3} + 35 \, b^{4} d^{2} p^{2} + 50 \, b^{4} d^{2} p + 24 \, b^{4} d^{2}} \]
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\[ \int x \left (a+b \sqrt {c+d x}\right )^p \, dx=\int x \left (a + b \sqrt {c + d x}\right )^{p}\, dx \]
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none
Time = 0.20 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.29 \[ \int x \left (a+b \sqrt {c+d x}\right )^p \, dx=-\frac {2 \, {\left (\frac {{\left ({\left (d x + c\right )} b^{2} {\left (p + 1\right )} + \sqrt {d x + c} a b p - a^{2}\right )} {\left (\sqrt {d x + c} b + a\right )}^{p} c}{{\left (p^{2} + 3 \, p + 2\right )} b^{2}} - \frac {{\left ({\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )} {\left (d x + c\right )}^{2} b^{4} + {\left (p^{3} + 3 \, p^{2} + 2 \, p\right )} {\left (d x + c\right )}^{\frac {3}{2}} a b^{3} - 3 \, {\left (p^{2} + p\right )} {\left (d x + c\right )} a^{2} b^{2} + 6 \, \sqrt {d x + c} a^{3} b p - 6 \, a^{4}\right )} {\left (\sqrt {d x + c} b + a\right )}^{p}}{{\left (p^{4} + 10 \, p^{3} + 35 \, p^{2} + 50 \, p + 24\right )} b^{4}}\right )}}{d^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 806 vs. \(2 (137) = 274\).
Time = 0.35 (sec) , antiderivative size = 806, normalized size of antiderivative = 5.56 \[ \int x \left (a+b \sqrt {c+d x}\right )^p \, dx=-\frac {2 \, {\left ({\left (\sqrt {d x + c} b + a\right )}^{2} {\left (\sqrt {d x + c} b + a\right )}^{p} b^{2} c p^{3} - {\left (\sqrt {d x + c} b + a\right )} {\left (\sqrt {d x + c} b + a\right )}^{p} a b^{2} c p^{3} + 8 \, {\left (\sqrt {d x + c} b + a\right )}^{2} {\left (\sqrt {d x + c} b + a\right )}^{p} b^{2} c p^{2} - 9 \, {\left (\sqrt {d x + c} b + a\right )} {\left (\sqrt {d x + c} b + a\right )}^{p} a b^{2} c p^{2} - {\left (\sqrt {d x + c} b + a\right )}^{4} {\left (\sqrt {d x + c} b + a\right )}^{p} p^{3} + 3 \, {\left (\sqrt {d x + c} b + a\right )}^{3} {\left (\sqrt {d x + c} b + a\right )}^{p} a p^{3} - 3 \, {\left (\sqrt {d x + c} b + a\right )}^{2} {\left (\sqrt {d x + c} b + a\right )}^{p} a^{2} p^{3} + {\left (\sqrt {d x + c} b + a\right )} {\left (\sqrt {d x + c} b + a\right )}^{p} a^{3} p^{3} + 19 \, {\left (\sqrt {d x + c} b + a\right )}^{2} {\left (\sqrt {d x + c} b + a\right )}^{p} b^{2} c p - 26 \, {\left (\sqrt {d x + c} b + a\right )} {\left (\sqrt {d x + c} b + a\right )}^{p} a b^{2} c p - 6 \, {\left (\sqrt {d x + c} b + a\right )}^{4} {\left (\sqrt {d x + c} b + a\right )}^{p} p^{2} + 21 \, {\left (\sqrt {d x + c} b + a\right )}^{3} {\left (\sqrt {d x + c} b + a\right )}^{p} a p^{2} - 24 \, {\left (\sqrt {d x + c} b + a\right )}^{2} {\left (\sqrt {d x + c} b + a\right )}^{p} a^{2} p^{2} + 9 \, {\left (\sqrt {d x + c} b + a\right )} {\left (\sqrt {d x + c} b + a\right )}^{p} a^{3} p^{2} + 12 \, {\left (\sqrt {d x + c} b + a\right )}^{2} {\left (\sqrt {d x + c} b + a\right )}^{p} b^{2} c - 24 \, {\left (\sqrt {d x + c} b + a\right )} {\left (\sqrt {d x + c} b + a\right )}^{p} a b^{2} c - 11 \, {\left (\sqrt {d x + c} b + a\right )}^{4} {\left (\sqrt {d x + c} b + a\right )}^{p} p + 42 \, {\left (\sqrt {d x + c} b + a\right )}^{3} {\left (\sqrt {d x + c} b + a\right )}^{p} a p - 57 \, {\left (\sqrt {d x + c} b + a\right )}^{2} {\left (\sqrt {d x + c} b + a\right )}^{p} a^{2} p + 26 \, {\left (\sqrt {d x + c} b + a\right )} {\left (\sqrt {d x + c} b + a\right )}^{p} a^{3} p - 6 \, {\left (\sqrt {d x + c} b + a\right )}^{4} {\left (\sqrt {d x + c} b + a\right )}^{p} + 24 \, {\left (\sqrt {d x + c} b + a\right )}^{3} {\left (\sqrt {d x + c} b + a\right )}^{p} a - 36 \, {\left (\sqrt {d x + c} b + a\right )}^{2} {\left (\sqrt {d x + c} b + a\right )}^{p} a^{2} + 24 \, {\left (\sqrt {d x + c} b + a\right )} {\left (\sqrt {d x + c} b + a\right )}^{p} a^{3}\right )}}{{\left (b^{2} p^{4} + 10 \, b^{2} p^{3} + 35 \, b^{2} p^{2} + 50 \, b^{2} p + 24 \, b^{2}\right )} b^{2} d^{2}} \]
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Timed out. \[ \int x \left (a+b \sqrt {c+d x}\right )^p \, dx=\int x\,{\left (a+b\,\sqrt {c+d\,x}\right )}^p \,d x \]
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