Integrand size = 15, antiderivative size = 35 \[ \int \frac {x^7}{\left (a+b x^4\right )^{5/4}} \, dx=\frac {a}{b^2 \sqrt [4]{a+b x^4}}+\frac {\left (a+b x^4\right )^{3/4}}{3 b^2} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {x^7}{\left (a+b x^4\right )^{5/4}} \, dx=\frac {a}{b^2 \sqrt [4]{a+b x^4}}+\frac {\left (a+b x^4\right )^{3/4}}{3 b^2} \]
[In]
[Out]
Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {x}{(a+b x)^{5/4}} \, dx,x,x^4\right ) \\ & = \frac {1}{4} \text {Subst}\left (\int \left (-\frac {a}{b (a+b x)^{5/4}}+\frac {1}{b \sqrt [4]{a+b x}}\right ) \, dx,x,x^4\right ) \\ & = \frac {a}{b^2 \sqrt [4]{a+b x^4}}+\frac {\left (a+b x^4\right )^{3/4}}{3 b^2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int \frac {x^7}{\left (a+b x^4\right )^{5/4}} \, dx=\frac {4 a+b x^4}{3 b^2 \sqrt [4]{a+b x^4}} \]
[In]
[Out]
Time = 4.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.69
method | result | size |
gosper | \(\frac {b \,x^{4}+4 a}{3 \left (b \,x^{4}+a \right )^{\frac {1}{4}} b^{2}}\) | \(24\) |
trager | \(\frac {b \,x^{4}+4 a}{3 \left (b \,x^{4}+a \right )^{\frac {1}{4}} b^{2}}\) | \(24\) |
pseudoelliptic | \(\frac {b \,x^{4}+4 a}{3 \left (b \,x^{4}+a \right )^{\frac {1}{4}} b^{2}}\) | \(24\) |
risch | \(\frac {a}{b^{2} \left (b \,x^{4}+a \right )^{\frac {1}{4}}}+\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}}}{3 b^{2}}\) | \(30\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {x^7}{\left (a+b x^4\right )^{5/4}} \, dx=\frac {{\left (b x^{4} + 4 \, a\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{3 \, {\left (b^{3} x^{4} + a b^{2}\right )}} \]
[In]
[Out]
Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.26 \[ \int \frac {x^7}{\left (a+b x^4\right )^{5/4}} \, dx=\begin {cases} \frac {4 a}{3 b^{2} \sqrt [4]{a + b x^{4}}} + \frac {x^{4}}{3 b \sqrt [4]{a + b x^{4}}} & \text {for}\: b \neq 0 \\\frac {x^{8}}{8 a^{\frac {5}{4}}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {x^7}{\left (a+b x^4\right )^{5/4}} \, dx=\frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{3 \, b^{2}} + \frac {a}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{2}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int \frac {x^7}{\left (a+b x^4\right )^{5/4}} \, dx=\frac {\frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{b} + \frac {3 \, a}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} b}}{3 \, b} \]
[In]
[Out]
Time = 5.77 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.66 \[ \int \frac {x^7}{\left (a+b x^4\right )^{5/4}} \, dx=\frac {b\,x^4+4\,a}{3\,b^2\,{\left (b\,x^4+a\right )}^{1/4}} \]
[In]
[Out]