Integrand size = 20, antiderivative size = 65 \[ \int \frac {x^4 (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx=-\frac {a x (a+b x)^{1+n}}{b^2 c (1+n) \sqrt {c x^2}}+\frac {x (a+b x)^{2+n}}{b^2 c (2+n) \sqrt {c x^2}} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 65, 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} \[ \int \frac {x^4 (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx=\frac {x (a+b x)^{n+2}}{b^2 c (n+2) \sqrt {c x^2}}-\frac {a x (a+b x)^{n+1}}{b^2 c (n+1) \sqrt {c x^2}} \]
[In]
[Out]
Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {x \int x (a+b x)^n \, dx}{c \sqrt {c x^2}} \\ & = \frac {x \int \left (-\frac {a (a+b x)^n}{b}+\frac {(a+b x)^{1+n}}{b}\right ) \, dx}{c \sqrt {c x^2}} \\ & = -\frac {a x (a+b x)^{1+n}}{b^2 c (1+n) \sqrt {c x^2}}+\frac {x (a+b x)^{2+n}}{b^2 c (2+n) \sqrt {c x^2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.69 \[ \int \frac {x^4 (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx=\frac {x^3 (a+b x)^{1+n} (-a+b (1+n) x)}{b^2 (1+n) (2+n) \left (c x^2\right )^{3/2}} \]
[In]
[Out]
Time = 0.44 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.71
method | result | size |
gosper | \(-\frac {x^{3} \left (b x +a \right )^{1+n} \left (-b n x -b x +a \right )}{b^{2} \left (c \,x^{2}\right )^{\frac {3}{2}} \left (n^{2}+3 n +2\right )}\) | \(46\) |
risch | \(-\frac {x \left (-b^{2} n \,x^{2}-a b n x -b^{2} x^{2}+a^{2}\right ) \left (b x +a \right )^{n}}{c \sqrt {c \,x^{2}}\, b^{2} \left (2+n \right ) \left (1+n \right )}\) | \(61\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.11 \[ \int \frac {x^4 (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx=\frac {{\left (a b n x + {\left (b^{2} n + b^{2}\right )} x^{2} - a^{2}\right )} \sqrt {c x^{2}} {\left (b x + a\right )}^{n}}{{\left (b^{2} c^{2} n^{2} + 3 \, b^{2} c^{2} n + 2 \, b^{2} c^{2}\right )} x} \]
[In]
[Out]
\[ \int \frac {x^4 (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx=\begin {cases} \frac {a^{n} x^{5}}{2 \left (c x^{2}\right )^{\frac {3}{2}}} & \text {for}\: b = 0 \\\int \frac {x^{4}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )^{2}}\, dx & \text {for}\: n = -2 \\\int \frac {x^{4}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )}\, dx & \text {for}\: n = -1 \\- \frac {a^{2} x^{3} \left (a + b x\right )^{n}}{b^{2} n^{2} \left (c x^{2}\right )^{\frac {3}{2}} + 3 b^{2} n \left (c x^{2}\right )^{\frac {3}{2}} + 2 b^{2} \left (c x^{2}\right )^{\frac {3}{2}}} + \frac {a b n x^{4} \left (a + b x\right )^{n}}{b^{2} n^{2} \left (c x^{2}\right )^{\frac {3}{2}} + 3 b^{2} n \left (c x^{2}\right )^{\frac {3}{2}} + 2 b^{2} \left (c x^{2}\right )^{\frac {3}{2}}} + \frac {b^{2} n x^{5} \left (a + b x\right )^{n}}{b^{2} n^{2} \left (c x^{2}\right )^{\frac {3}{2}} + 3 b^{2} n \left (c x^{2}\right )^{\frac {3}{2}} + 2 b^{2} \left (c x^{2}\right )^{\frac {3}{2}}} + \frac {b^{2} x^{5} \left (a + b x\right )^{n}}{b^{2} n^{2} \left (c x^{2}\right )^{\frac {3}{2}} + 3 b^{2} n \left (c x^{2}\right )^{\frac {3}{2}} + 2 b^{2} \left (c x^{2}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.69 \[ \int \frac {x^4 (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx=\frac {{\left (b^{2} {\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )} {\left (b x + a\right )}^{n}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2} c^{\frac {3}{2}}} \]
[In]
[Out]
Exception generated. \[ \int \frac {x^4 (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Time = 0.43 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.23 \[ \int \frac {x^4 (a+b x)^n}{\left (c x^2\right )^{3/2}} \, dx=\frac {{\left (a+b\,x\right )}^n\,\left (\frac {x^3\,\left (n+1\right )}{c\,\left (n^2+3\,n+2\right )}-\frac {a^2\,x}{b^2\,c\,\left (n^2+3\,n+2\right )}+\frac {a\,n\,x^2}{b\,c\,\left (n^2+3\,n+2\right )}\right )}{\sqrt {c\,x^2}} \]
[In]
[Out]