Optimal. Leaf size=59 \[ \frac{a^2 \left (a+b x^4\right )^{7/4}}{7 b^3}+\frac{\left (a+b x^4\right )^{15/4}}{15 b^3}-\frac{2 a \left (a+b x^4\right )^{11/4}}{11 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0340583, antiderivative size = 59, 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} \[ \frac{a^2 \left (a+b x^4\right )^{7/4}}{7 b^3}+\frac{\left (a+b x^4\right )^{15/4}}{15 b^3}-\frac{2 a \left (a+b x^4\right )^{11/4}}{11 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{11} \left (a+b x^4\right )^{3/4} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int x^2 (a+b x)^{3/4} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{a^2 (a+b x)^{3/4}}{b^2}-\frac{2 a (a+b x)^{7/4}}{b^2}+\frac{(a+b x)^{11/4}}{b^2}\right ) \, dx,x,x^4\right )\\ &=\frac{a^2 \left (a+b x^4\right )^{7/4}}{7 b^3}-\frac{2 a \left (a+b x^4\right )^{11/4}}{11 b^3}+\frac{\left (a+b x^4\right )^{15/4}}{15 b^3}\\ \end{align*}
Mathematica [A] time = 0.0178629, size = 39, normalized size = 0.66 \[ \frac{\left (a+b x^4\right )^{7/4} \left (32 a^2-56 a b x^4+77 b^2 x^8\right )}{1155 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 36, normalized size = 0.6 \begin{align*}{\frac{77\,{b}^{2}{x}^{8}-56\,ab{x}^{4}+32\,{a}^{2}}{1155\,{b}^{3}} \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 = 0.992419, size = 63, normalized size = 1.07 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{15}{4}}}{15 \, b^{3}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} a}{11 \, b^{3}} + \frac{{\left (b x^{4} + a\right )}^{\frac{7}{4}} a^{2}}{7 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.63973, size = 112, normalized size = 1.9 \begin{align*} \frac{{\left (77 \, b^{3} x^{12} + 21 \, a b^{2} x^{8} - 24 \, a^{2} b x^{4} + 32 \, a^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{1155 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 13.9351, size = 87, normalized size = 1.47 \begin{align*} \begin{cases} \frac{32 a^{3} \left (a + b x^{4}\right )^{\frac{3}{4}}}{1155 b^{3}} - \frac{8 a^{2} x^{4} \left (a + b x^{4}\right )^{\frac{3}{4}}}{385 b^{2}} + \frac{a x^{8} \left (a + b x^{4}\right )^{\frac{3}{4}}}{55 b} + \frac{x^{12} \left (a + b x^{4}\right )^{\frac{3}{4}}}{15} & \text{for}\: b \neq 0 \\\frac{a^{\frac{3}{4}} x^{12}}{12} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.36871, size = 58, normalized size = 0.98 \begin{align*} \frac{77 \,{\left (b x^{4} + a\right )}^{\frac{15}{4}} - 210 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} a + 165 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} a^{2}}{1155 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]