Optimal. Leaf size=38 \[ \frac{\left (a+b x^4\right )^{11/4}}{11 b^2}-\frac{a \left (a+b x^4\right )^{7/4}}{7 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0234202, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{\left (a+b x^4\right )^{11/4}}{11 b^2}-\frac{a \left (a+b x^4\right )^{7/4}}{7 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^7 \left (a+b x^4\right )^{3/4} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int x (a+b x)^{3/4} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^{3/4}}{b}+\frac{(a+b x)^{7/4}}{b}\right ) \, dx,x,x^4\right )\\ &=-\frac{a \left (a+b x^4\right )^{7/4}}{7 b^2}+\frac{\left (a+b x^4\right )^{11/4}}{11 b^2}\\ \end{align*}
Mathematica [A] time = 0.0133862, size = 28, normalized size = 0.74 \[ \frac{\left (a+b x^4\right )^{7/4} \left (7 b x^4-4 a\right )}{77 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 25, normalized size = 0.7 \begin{align*} -{\frac{-7\,b{x}^{4}+4\,a}{77\,{b}^{2}} \left ( b{x}^{4}+a \right ) ^{{\frac{7}{4}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.00025, size = 41, normalized size = 1.08 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{11}{4}}}{11 \, b^{2}} - \frac{{\left (b x^{4} + a\right )}^{\frac{7}{4}} a}{7 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.69814, size = 81, normalized size = 2.13 \begin{align*} \frac{{\left (7 \, b^{2} x^{8} + 3 \, a b x^{4} - 4 \, a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{77 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 5.84741, size = 65, normalized size = 1.71 \begin{align*} \begin{cases} - \frac{4 a^{2} \left (a + b x^{4}\right )^{\frac{3}{4}}}{77 b^{2}} + \frac{3 a x^{4} \left (a + b x^{4}\right )^{\frac{3}{4}}}{77 b} + \frac{x^{8} \left (a + b x^{4}\right )^{\frac{3}{4}}}{11} & \text{for}\: b \neq 0 \\\frac{a^{\frac{3}{4}} x^{8}}{8} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.64867, size = 39, normalized size = 1.03 \begin{align*} \frac{7 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} - 11 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} a}{77 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]