Integrand size = 20, antiderivative size = 8 \[ \int \frac {(a+b x)^3}{\left (\frac {a d}{b}+d x\right )^3} \, dx=\frac {b^3 x}{d^3} \]
[Out]
Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {21, 8} \[ \int \frac {(a+b x)^3}{\left (\frac {a d}{b}+d x\right )^3} \, dx=\frac {b^3 x}{d^3} \]
[In]
[Out]
Rule 8
Rule 21
Rubi steps \begin{align*} \text {integral}& = \frac {b^3 \int 1 \, dx}{d^3} \\ & = \frac {b^3 x}{d^3} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^3}{\left (\frac {a d}{b}+d x\right )^3} \, dx=\frac {b^3 x}{d^3} \]
[In]
[Out]
Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.12
method | result | size |
default | \(\frac {b^{3} x}{d^{3}}\) | \(9\) |
risch | \(\frac {b^{3} x}{d^{3}}\) | \(9\) |
norman | \(\frac {\frac {b^{5} x^{3}}{d}-\frac {2 a^{3} b^{2}}{d}-\frac {3 a^{2} b^{3} x}{d}}{d^{2} \left (b x +a \right )^{2}}\) | \(46\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^3}{\left (\frac {a d}{b}+d x\right )^3} \, dx=\frac {b^{3} x}{d^{3}} \]
[In]
[Out]
Time = 0.13 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^3}{\left (\frac {a d}{b}+d x\right )^3} \, dx=\frac {b^{3} x}{d^{3}} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^3}{\left (\frac {a d}{b}+d x\right )^3} \, dx=\frac {b^{3} x}{d^{3}} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^3}{\left (\frac {a d}{b}+d x\right )^3} \, dx=\frac {b^{3} x}{d^{3}} \]
[In]
[Out]
Time = 0.01 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^3}{\left (\frac {a d}{b}+d x\right )^3} \, dx=\frac {b^3\,x}{d^3} \]
[In]
[Out]