Integrand size = 20, antiderivative size = 123 \[ \int \frac {x^4 (a+b x)^n}{\sqrt {c x^2}} \, dx=-\frac {a^3 x (a+b x)^{1+n}}{b^4 (1+n) \sqrt {c x^2}}+\frac {3 a^2 x (a+b x)^{2+n}}{b^4 (2+n) \sqrt {c x^2}}-\frac {3 a x (a+b x)^{3+n}}{b^4 (3+n) \sqrt {c x^2}}+\frac {x (a+b x)^{4+n}}{b^4 (4+n) \sqrt {c x^2}} \]
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Time = 0.03 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 45} \[ \int \frac {x^4 (a+b x)^n}{\sqrt {c x^2}} \, dx=-\frac {a^3 x (a+b x)^{n+1}}{b^4 (n+1) \sqrt {c x^2}}+\frac {3 a^2 x (a+b x)^{n+2}}{b^4 (n+2) \sqrt {c x^2}}-\frac {3 a x (a+b x)^{n+3}}{b^4 (n+3) \sqrt {c x^2}}+\frac {x (a+b x)^{n+4}}{b^4 (n+4) \sqrt {c x^2}} \]
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Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {x \int x^3 (a+b x)^n \, dx}{\sqrt {c x^2}} \\ & = \frac {x \int \left (-\frac {a^3 (a+b x)^n}{b^3}+\frac {3 a^2 (a+b x)^{1+n}}{b^3}-\frac {3 a (a+b x)^{2+n}}{b^3}+\frac {(a+b x)^{3+n}}{b^3}\right ) \, dx}{\sqrt {c x^2}} \\ & = -\frac {a^3 x (a+b x)^{1+n}}{b^4 (1+n) \sqrt {c x^2}}+\frac {3 a^2 x (a+b x)^{2+n}}{b^4 (2+n) \sqrt {c x^2}}-\frac {3 a x (a+b x)^{3+n}}{b^4 (3+n) \sqrt {c x^2}}+\frac {x (a+b x)^{4+n}}{b^4 (4+n) \sqrt {c x^2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.78 \[ \int \frac {x^4 (a+b x)^n}{\sqrt {c x^2}} \, dx=\frac {x (a+b x)^{1+n} \left (-6 a^3+6 a^2 b (1+n) x-3 a b^2 \left (2+3 n+n^2\right ) x^2+b^3 \left (6+11 n+6 n^2+n^3\right ) x^3\right )}{b^4 (1+n) (2+n) (3+n) (4+n) \sqrt {c x^2}} \]
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Time = 0.14 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.09
method | result | size |
gosper | \(-\frac {x \left (b x +a \right )^{1+n} \left (-b^{3} n^{3} x^{3}-6 b^{3} n^{2} x^{3}+3 a \,b^{2} n^{2} x^{2}-11 b^{3} n \,x^{3}+9 a \,b^{2} n \,x^{2}-6 b^{3} x^{3}-6 a^{2} b n x +6 a \,b^{2} x^{2}-6 a^{2} b x +6 a^{3}\right )}{b^{4} \sqrt {c \,x^{2}}\, \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}\) | \(134\) |
risch | \(-\frac {x \left (-b^{4} n^{3} x^{4}-a \,b^{3} n^{3} x^{3}-6 b^{4} n^{2} x^{4}-3 a \,b^{3} n^{2} x^{3}-11 b^{4} n \,x^{4}+3 a^{2} b^{2} n^{2} x^{2}-2 x^{3} a n \,b^{3}-6 b^{4} x^{4}+3 a^{2} n \,x^{2} b^{2}-6 x \,a^{3} n b +6 a^{4}\right ) \left (b x +a \right )^{n}}{\sqrt {c \,x^{2}}\, \left (3+n \right ) \left (4+n \right ) \left (2+n \right ) \left (1+n \right ) b^{4}}\) | \(154\) |
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Time = 0.23 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.28 \[ \int \frac {x^4 (a+b x)^n}{\sqrt {c x^2}} \, dx=\frac {{\left (6 \, a^{3} b n x + {\left (b^{4} n^{3} + 6 \, b^{4} n^{2} + 11 \, b^{4} n + 6 \, b^{4}\right )} x^{4} - 6 \, a^{4} + {\left (a b^{3} n^{3} + 3 \, a b^{3} n^{2} + 2 \, a b^{3} n\right )} x^{3} - 3 \, {\left (a^{2} b^{2} n^{2} + a^{2} b^{2} n\right )} x^{2}\right )} \sqrt {c x^{2}} {\left (b x + a\right )}^{n}}{{\left (b^{4} c n^{4} + 10 \, b^{4} c n^{3} + 35 \, b^{4} c n^{2} + 50 \, b^{4} c n + 24 \, b^{4} c\right )} x} \]
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\[ \int \frac {x^4 (a+b x)^n}{\sqrt {c x^2}} \, dx=\begin {cases} \frac {a^{n} x^{5}}{4 \sqrt {c x^{2}}} & \text {for}\: b = 0 \\\int \frac {x^{4}}{\sqrt {c x^{2}} \left (a + b x\right )^{4}}\, dx & \text {for}\: n = -4 \\\int \frac {x^{4}}{\sqrt {c x^{2}} \left (a + b x\right )^{3}}\, dx & \text {for}\: n = -3 \\\int \frac {x^{4}}{\sqrt {c x^{2}} \left (a + b x\right )^{2}}\, dx & \text {for}\: n = -2 \\\int \frac {x^{4}}{\sqrt {c x^{2}} \left (a + b x\right )}\, dx & \text {for}\: n = -1 \\- \frac {6 a^{4} x \left (a + b x\right )^{n}}{b^{4} n^{4} \sqrt {c x^{2}} + 10 b^{4} n^{3} \sqrt {c x^{2}} + 35 b^{4} n^{2} \sqrt {c x^{2}} + 50 b^{4} n \sqrt {c x^{2}} + 24 b^{4} \sqrt {c x^{2}}} + \frac {6 a^{3} b n x^{2} \left (a + b x\right )^{n}}{b^{4} n^{4} \sqrt {c x^{2}} + 10 b^{4} n^{3} \sqrt {c x^{2}} + 35 b^{4} n^{2} \sqrt {c x^{2}} + 50 b^{4} n \sqrt {c x^{2}} + 24 b^{4} \sqrt {c x^{2}}} - \frac {3 a^{2} b^{2} n^{2} x^{3} \left (a + b x\right )^{n}}{b^{4} n^{4} \sqrt {c x^{2}} + 10 b^{4} n^{3} \sqrt {c x^{2}} + 35 b^{4} n^{2} \sqrt {c x^{2}} + 50 b^{4} n \sqrt {c x^{2}} + 24 b^{4} \sqrt {c x^{2}}} - \frac {3 a^{2} b^{2} n x^{3} \left (a + b x\right )^{n}}{b^{4} n^{4} \sqrt {c x^{2}} + 10 b^{4} n^{3} \sqrt {c x^{2}} + 35 b^{4} n^{2} \sqrt {c x^{2}} + 50 b^{4} n \sqrt {c x^{2}} + 24 b^{4} \sqrt {c x^{2}}} + \frac {a b^{3} n^{3} x^{4} \left (a + b x\right )^{n}}{b^{4} n^{4} \sqrt {c x^{2}} + 10 b^{4} n^{3} \sqrt {c x^{2}} + 35 b^{4} n^{2} \sqrt {c x^{2}} + 50 b^{4} n \sqrt {c x^{2}} + 24 b^{4} \sqrt {c x^{2}}} + \frac {3 a b^{3} n^{2} x^{4} \left (a + b x\right )^{n}}{b^{4} n^{4} \sqrt {c x^{2}} + 10 b^{4} n^{3} \sqrt {c x^{2}} + 35 b^{4} n^{2} \sqrt {c x^{2}} + 50 b^{4} n \sqrt {c x^{2}} + 24 b^{4} \sqrt {c x^{2}}} + \frac {2 a b^{3} n x^{4} \left (a + b x\right )^{n}}{b^{4} n^{4} \sqrt {c x^{2}} + 10 b^{4} n^{3} \sqrt {c x^{2}} + 35 b^{4} n^{2} \sqrt {c x^{2}} + 50 b^{4} n \sqrt {c x^{2}} + 24 b^{4} \sqrt {c x^{2}}} + \frac {b^{4} n^{3} x^{5} \left (a + b x\right )^{n}}{b^{4} n^{4} \sqrt {c x^{2}} + 10 b^{4} n^{3} \sqrt {c x^{2}} + 35 b^{4} n^{2} \sqrt {c x^{2}} + 50 b^{4} n \sqrt {c x^{2}} + 24 b^{4} \sqrt {c x^{2}}} + \frac {6 b^{4} n^{2} x^{5} \left (a + b x\right )^{n}}{b^{4} n^{4} \sqrt {c x^{2}} + 10 b^{4} n^{3} \sqrt {c x^{2}} + 35 b^{4} n^{2} \sqrt {c x^{2}} + 50 b^{4} n \sqrt {c x^{2}} + 24 b^{4} \sqrt {c x^{2}}} + \frac {11 b^{4} n x^{5} \left (a + b x\right )^{n}}{b^{4} n^{4} \sqrt {c x^{2}} + 10 b^{4} n^{3} \sqrt {c x^{2}} + 35 b^{4} n^{2} \sqrt {c x^{2}} + 50 b^{4} n \sqrt {c x^{2}} + 24 b^{4} \sqrt {c x^{2}}} + \frac {6 b^{4} x^{5} \left (a + b x\right )^{n}}{b^{4} n^{4} \sqrt {c x^{2}} + 10 b^{4} n^{3} \sqrt {c x^{2}} + 35 b^{4} n^{2} \sqrt {c x^{2}} + 50 b^{4} n \sqrt {c x^{2}} + 24 b^{4} \sqrt {c x^{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.85 \[ \int \frac {x^4 (a+b x)^n}{\sqrt {c x^2}} \, dx=\frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{n}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4} \sqrt {c}} \]
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Exception generated. \[ \int \frac {x^4 (a+b x)^n}{\sqrt {c x^2}} \, dx=\text {Exception raised: TypeError} \]
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Time = 0.59 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.51 \[ \int \frac {x^4 (a+b x)^n}{\sqrt {c x^2}} \, dx=\frac {{\left (a+b\,x\right )}^n\,\left (\frac {x^5\,\left (n^3+6\,n^2+11\,n+6\right )}{n^4+10\,n^3+35\,n^2+50\,n+24}-\frac {6\,a^4\,x}{b^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {6\,a^3\,n\,x^2}{b^3\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {a\,n\,x^4\,\left (n^2+3\,n+2\right )}{b\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}-\frac {3\,a^2\,n\,x^3\,\left (n+1\right )}{b^2\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}\right )}{\sqrt {c\,x^2}} \]
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