Integrand size = 17, antiderivative size = 217 \[ \int \left (c x^2\right )^{5/2} (a+b x)^n \, dx=-\frac {a^5 c^2 \sqrt {c x^2} (a+b x)^{1+n}}{b^6 (1+n) x}+\frac {5 a^4 c^2 \sqrt {c x^2} (a+b x)^{2+n}}{b^6 (2+n) x}-\frac {10 a^3 c^2 \sqrt {c x^2} (a+b x)^{3+n}}{b^6 (3+n) x}+\frac {10 a^2 c^2 \sqrt {c x^2} (a+b x)^{4+n}}{b^6 (4+n) x}-\frac {5 a c^2 \sqrt {c x^2} (a+b x)^{5+n}}{b^6 (5+n) x}+\frac {c^2 \sqrt {c x^2} (a+b x)^{6+n}}{b^6 (6+n) x} \]
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Time = 0.06 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {15, 45} \[ \int \left (c x^2\right )^{5/2} (a+b x)^n \, dx=-\frac {a^5 c^2 \sqrt {c x^2} (a+b x)^{n+1}}{b^6 (n+1) x}+\frac {5 a^4 c^2 \sqrt {c x^2} (a+b x)^{n+2}}{b^6 (n+2) x}-\frac {10 a^3 c^2 \sqrt {c x^2} (a+b x)^{n+3}}{b^6 (n+3) x}+\frac {10 a^2 c^2 \sqrt {c x^2} (a+b x)^{n+4}}{b^6 (n+4) x}-\frac {5 a c^2 \sqrt {c x^2} (a+b x)^{n+5}}{b^6 (n+5) x}+\frac {c^2 \sqrt {c x^2} (a+b x)^{n+6}}{b^6 (n+6) x} \]
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Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c^2 \sqrt {c x^2}\right ) \int x^5 (a+b x)^n \, dx}{x} \\ & = \frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (-\frac {a^5 (a+b x)^n}{b^5}+\frac {5 a^4 (a+b x)^{1+n}}{b^5}-\frac {10 a^3 (a+b x)^{2+n}}{b^5}+\frac {10 a^2 (a+b x)^{3+n}}{b^5}-\frac {5 a (a+b x)^{4+n}}{b^5}+\frac {(a+b x)^{5+n}}{b^5}\right ) \, dx}{x} \\ & = -\frac {a^5 c^2 \sqrt {c x^2} (a+b x)^{1+n}}{b^6 (1+n) x}+\frac {5 a^4 c^2 \sqrt {c x^2} (a+b x)^{2+n}}{b^6 (2+n) x}-\frac {10 a^3 c^2 \sqrt {c x^2} (a+b x)^{3+n}}{b^6 (3+n) x}+\frac {10 a^2 c^2 \sqrt {c x^2} (a+b x)^{4+n}}{b^6 (4+n) x}-\frac {5 a c^2 \sqrt {c x^2} (a+b x)^{5+n}}{b^6 (5+n) x}+\frac {c^2 \sqrt {c x^2} (a+b x)^{6+n}}{b^6 (6+n) x} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.79 \[ \int \left (c x^2\right )^{5/2} (a+b x)^n \, dx=\frac {c^3 x (a+b x)^{1+n} \left (-120 a^5+120 a^4 b (1+n) x-60 a^3 b^2 \left (2+3 n+n^2\right ) x^2+20 a^2 b^3 \left (6+11 n+6 n^2+n^3\right ) x^3-5 a b^4 \left (24+50 n+35 n^2+10 n^3+n^4\right ) x^4+b^5 \left (120+274 n+225 n^2+85 n^3+15 n^4+n^5\right ) x^5\right )}{b^6 (1+n) (2+n) (3+n) (4+n) (5+n) (6+n) \sqrt {c x^2}} \]
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Time = 0.17 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.29
method | result | size |
gosper | \(-\frac {\left (c \,x^{2}\right )^{\frac {5}{2}} \left (b x +a \right )^{1+n} \left (-b^{5} n^{5} x^{5}-15 b^{5} n^{4} x^{5}+5 a \,b^{4} n^{4} x^{4}-85 b^{5} n^{3} x^{5}+50 a \,b^{4} n^{3} x^{4}-225 b^{5} n^{2} x^{5}-20 a^{2} b^{3} n^{3} x^{3}+175 a \,b^{4} n^{2} x^{4}-274 b^{5} n \,x^{5}-120 a^{2} b^{3} n^{2} x^{3}+250 x^{4} a n \,b^{4}-120 b^{5} x^{5}+60 a^{3} b^{2} n^{2} x^{2}-220 a^{2} n \,x^{3} b^{3}+120 a \,b^{4} x^{4}+180 x^{2} a^{3} n \,b^{2}-120 a^{2} b^{3} x^{3}-120 x \,a^{4} n b +120 a^{3} b^{2} x^{2}-120 a^{4} b x +120 a^{5}\right )}{b^{6} x^{5} \left (n^{6}+21 n^{5}+175 n^{4}+735 n^{3}+1624 n^{2}+1764 n +720\right )}\) | \(280\) |
risch | \(-\frac {c^{2} \sqrt {c \,x^{2}}\, \left (-b^{6} n^{5} x^{6}-a \,b^{5} n^{5} x^{5}-15 b^{6} n^{4} x^{6}-10 a \,b^{5} n^{4} x^{5}-85 b^{6} n^{3} x^{6}+5 a^{2} b^{4} n^{4} x^{4}-35 a \,b^{5} n^{3} x^{5}-225 b^{6} n^{2} x^{6}+30 a^{2} b^{4} n^{3} x^{4}-50 a \,b^{5} n^{2} x^{5}-274 b^{6} n \,x^{6}-20 a^{3} b^{3} n^{3} x^{3}+55 a^{2} b^{4} n^{2} x^{4}-24 x^{5} a n \,b^{5}-120 b^{6} x^{6}-60 a^{3} b^{3} n^{2} x^{3}+30 a^{2} n \,x^{4} b^{4}+60 a^{4} b^{2} n^{2} x^{2}-40 x^{3} a^{3} n \,b^{3}+60 x^{2} a^{4} n \,b^{2}-120 x \,a^{5} n b +120 a^{6}\right ) \left (b x +a \right )^{n}}{x \left (5+n \right ) \left (6+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) b^{6}}\) | \(309\) |
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Time = 0.23 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.62 \[ \int \left (c x^2\right )^{5/2} (a+b x)^n \, dx=\frac {{\left (120 \, a^{5} b c^{2} n x - 120 \, a^{6} c^{2} + {\left (b^{6} c^{2} n^{5} + 15 \, b^{6} c^{2} n^{4} + 85 \, b^{6} c^{2} n^{3} + 225 \, b^{6} c^{2} n^{2} + 274 \, b^{6} c^{2} n + 120 \, b^{6} c^{2}\right )} x^{6} + {\left (a b^{5} c^{2} n^{5} + 10 \, a b^{5} c^{2} n^{4} + 35 \, a b^{5} c^{2} n^{3} + 50 \, a b^{5} c^{2} n^{2} + 24 \, a b^{5} c^{2} n\right )} x^{5} - 5 \, {\left (a^{2} b^{4} c^{2} n^{4} + 6 \, a^{2} b^{4} c^{2} n^{3} + 11 \, a^{2} b^{4} c^{2} n^{2} + 6 \, a^{2} b^{4} c^{2} n\right )} x^{4} + 20 \, {\left (a^{3} b^{3} c^{2} n^{3} + 3 \, a^{3} b^{3} c^{2} n^{2} + 2 \, a^{3} b^{3} c^{2} n\right )} x^{3} - 60 \, {\left (a^{4} b^{2} c^{2} n^{2} + a^{4} b^{2} c^{2} n\right )} x^{2}\right )} \sqrt {c x^{2}} {\left (b x + a\right )}^{n}}{{\left (b^{6} n^{6} + 21 \, b^{6} n^{5} + 175 \, b^{6} n^{4} + 735 \, b^{6} n^{3} + 1624 \, b^{6} n^{2} + 1764 \, b^{6} n + 720 \, b^{6}\right )} x} \]
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\[ \int \left (c x^2\right )^{5/2} (a+b x)^n \, dx=\int \left (c x^{2}\right )^{\frac {5}{2}} \left (a + b x\right )^{n}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.94 \[ \int \left (c x^2\right )^{5/2} (a+b x)^n \, dx=\frac {{\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{6} c^{\frac {5}{2}} x^{6} + {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a b^{5} c^{\frac {5}{2}} x^{5} - 5 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{2} b^{4} c^{\frac {5}{2}} x^{4} + 20 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{3} b^{3} c^{\frac {5}{2}} x^{3} - 60 \, {\left (n^{2} + n\right )} a^{4} b^{2} c^{\frac {5}{2}} x^{2} + 120 \, a^{5} b c^{\frac {5}{2}} n x - 120 \, a^{6} c^{\frac {5}{2}}\right )} {\left (b x + a\right )}^{n}}{{\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 640 vs. \(2 (205) = 410\).
Time = 0.29 (sec) , antiderivative size = 640, normalized size of antiderivative = 2.95 \[ \int \left (c x^2\right )^{5/2} (a+b x)^n \, dx={\left (\frac {120 \, a^{6} a^{n} c^{2} \mathrm {sgn}\left (x\right )}{b^{6} n^{6} + 21 \, b^{6} n^{5} + 175 \, b^{6} n^{4} + 735 \, b^{6} n^{3} + 1624 \, b^{6} n^{2} + 1764 \, b^{6} n + 720 \, b^{6}} + \frac {{\left (b x + a\right )}^{n} b^{6} c^{2} n^{5} x^{6} \mathrm {sgn}\left (x\right ) + {\left (b x + a\right )}^{n} a b^{5} c^{2} n^{5} x^{5} \mathrm {sgn}\left (x\right ) + 15 \, {\left (b x + a\right )}^{n} b^{6} c^{2} n^{4} x^{6} \mathrm {sgn}\left (x\right ) + 10 \, {\left (b x + a\right )}^{n} a b^{5} c^{2} n^{4} x^{5} \mathrm {sgn}\left (x\right ) + 85 \, {\left (b x + a\right )}^{n} b^{6} c^{2} n^{3} x^{6} \mathrm {sgn}\left (x\right ) - 5 \, {\left (b x + a\right )}^{n} a^{2} b^{4} c^{2} n^{4} x^{4} \mathrm {sgn}\left (x\right ) + 35 \, {\left (b x + a\right )}^{n} a b^{5} c^{2} n^{3} x^{5} \mathrm {sgn}\left (x\right ) + 225 \, {\left (b x + a\right )}^{n} b^{6} c^{2} n^{2} x^{6} \mathrm {sgn}\left (x\right ) - 30 \, {\left (b x + a\right )}^{n} a^{2} b^{4} c^{2} n^{3} x^{4} \mathrm {sgn}\left (x\right ) + 50 \, {\left (b x + a\right )}^{n} a b^{5} c^{2} n^{2} x^{5} \mathrm {sgn}\left (x\right ) + 274 \, {\left (b x + a\right )}^{n} b^{6} c^{2} n x^{6} \mathrm {sgn}\left (x\right ) + 20 \, {\left (b x + a\right )}^{n} a^{3} b^{3} c^{2} n^{3} x^{3} \mathrm {sgn}\left (x\right ) - 55 \, {\left (b x + a\right )}^{n} a^{2} b^{4} c^{2} n^{2} x^{4} \mathrm {sgn}\left (x\right ) + 24 \, {\left (b x + a\right )}^{n} a b^{5} c^{2} n x^{5} \mathrm {sgn}\left (x\right ) + 120 \, {\left (b x + a\right )}^{n} b^{6} c^{2} x^{6} \mathrm {sgn}\left (x\right ) + 60 \, {\left (b x + a\right )}^{n} a^{3} b^{3} c^{2} n^{2} x^{3} \mathrm {sgn}\left (x\right ) - 30 \, {\left (b x + a\right )}^{n} a^{2} b^{4} c^{2} n x^{4} \mathrm {sgn}\left (x\right ) - 60 \, {\left (b x + a\right )}^{n} a^{4} b^{2} c^{2} n^{2} x^{2} \mathrm {sgn}\left (x\right ) + 40 \, {\left (b x + a\right )}^{n} a^{3} b^{3} c^{2} n x^{3} \mathrm {sgn}\left (x\right ) - 60 \, {\left (b x + a\right )}^{n} a^{4} b^{2} c^{2} n x^{2} \mathrm {sgn}\left (x\right ) + 120 \, {\left (b x + a\right )}^{n} a^{5} b c^{2} n x \mathrm {sgn}\left (x\right ) - 120 \, {\left (b x + a\right )}^{n} a^{6} c^{2} \mathrm {sgn}\left (x\right )}{b^{6} n^{6} + 21 \, b^{6} n^{5} + 175 \, b^{6} n^{4} + 735 \, b^{6} n^{3} + 1624 \, b^{6} n^{2} + 1764 \, b^{6} n + 720 \, b^{6}}\right )} \sqrt {c} \]
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Time = 0.78 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.95 \[ \int \left (c x^2\right )^{5/2} (a+b x)^n \, dx=\frac {{\left (a+b\,x\right )}^n\,\left (\frac {c^2\,x^6\,\sqrt {c\,x^2}\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}{n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720}-\frac {120\,a^6\,c^2\,\sqrt {c\,x^2}}{b^6\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {120\,a^5\,c^2\,n\,x\,\sqrt {c\,x^2}}{b^5\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}-\frac {5\,a^2\,c^2\,n\,x^4\,\sqrt {c\,x^2}\,\left (n^3+6\,n^2+11\,n+6\right )}{b^2\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}-\frac {60\,a^4\,c^2\,n\,x^2\,\sqrt {c\,x^2}\,\left (n+1\right )}{b^4\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {a\,c^2\,n\,x^5\,\sqrt {c\,x^2}\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{b\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {20\,a^3\,c^2\,n\,x^3\,\sqrt {c\,x^2}\,\left (n^2+3\,n+2\right )}{b^3\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}\right )}{x} \]
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