Optimal. Leaf size=99 \[ -\frac{a^4}{b^5 \sqrt [4]{a+b x^4}}-\frac{4 a^3 \left (a+b x^4\right )^{3/4}}{3 b^5}+\frac{6 a^2 \left (a+b x^4\right )^{7/4}}{7 b^5}-\frac{4 a \left (a+b x^4\right )^{11/4}}{11 b^5}+\frac{\left (a+b x^4\right )^{15/4}}{15 b^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0540024, antiderivative size = 99, 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} \[ -\frac{a^4}{b^5 \sqrt [4]{a+b x^4}}-\frac{4 a^3 \left (a+b x^4\right )^{3/4}}{3 b^5}+\frac{6 a^2 \left (a+b x^4\right )^{7/4}}{7 b^5}-\frac{4 a \left (a+b x^4\right )^{11/4}}{11 b^5}+\frac{\left (a+b x^4\right )^{15/4}}{15 b^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{19}}{\left (a+b x^4\right )^{5/4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^4}{(a+b x)^{5/4}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{a^4}{b^4 (a+b x)^{5/4}}-\frac{4 a^3}{b^4 \sqrt [4]{a+b x}}+\frac{6 a^2 (a+b x)^{3/4}}{b^4}-\frac{4 a (a+b x)^{7/4}}{b^4}+\frac{(a+b x)^{11/4}}{b^4}\right ) \, dx,x,x^4\right )\\ &=-\frac{a^4}{b^5 \sqrt [4]{a+b x^4}}-\frac{4 a^3 \left (a+b x^4\right )^{3/4}}{3 b^5}+\frac{6 a^2 \left (a+b x^4\right )^{7/4}}{7 b^5}-\frac{4 a \left (a+b x^4\right )^{11/4}}{11 b^5}+\frac{\left (a+b x^4\right )^{15/4}}{15 b^5}\\ \end{align*}
Mathematica [A] time = 0.0266705, size = 61, normalized size = 0.62 \[ \frac{192 a^2 b^2 x^8-512 a^3 b x^4-2048 a^4-112 a b^3 x^{12}+77 b^4 x^{16}}{1155 b^5 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 58, normalized size = 0.6 \begin{align*} -{\frac{-77\,{x}^{16}{b}^{4}+112\,a{x}^{12}{b}^{3}-192\,{a}^{2}{x}^{8}{b}^{2}+512\,{a}^{3}{x}^{4}b+2048\,{a}^{4}}{1155\,{b}^{5}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.96882, size = 109, normalized size = 1.1 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{15}{4}}}{15 \, b^{5}} - \frac{4 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} a}{11 \, b^{5}} + \frac{6 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} a^{2}}{7 \, b^{5}} - \frac{4 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} a^{3}}{3 \, b^{5}} - \frac{a^{4}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.46559, size = 162, normalized size = 1.64 \begin{align*} \frac{{\left (77 \, b^{4} x^{16} - 112 \, a b^{3} x^{12} + 192 \, a^{2} b^{2} x^{8} - 512 \, a^{3} b x^{4} - 2048 \, a^{4}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{1155 \,{\left (b^{6} x^{4} + a b^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 19.8584, size = 116, normalized size = 1.17 \begin{align*} \begin{cases} - \frac{2048 a^{4}}{1155 b^{5} \sqrt [4]{a + b x^{4}}} - \frac{512 a^{3} x^{4}}{1155 b^{4} \sqrt [4]{a + b x^{4}}} + \frac{64 a^{2} x^{8}}{385 b^{3} \sqrt [4]{a + b x^{4}}} - \frac{16 a x^{12}}{165 b^{2} \sqrt [4]{a + b x^{4}}} + \frac{x^{16}}{15 b \sqrt [4]{a + b x^{4}}} & \text{for}\: b \neq 0 \\\frac{x^{20}}{20 a^{\frac{5}{4}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.09859, size = 96, normalized size = 0.97 \begin{align*} \frac{77 \,{\left (b x^{4} + a\right )}^{\frac{15}{4}} - 420 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} a + 990 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} a^{2} - 1540 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} a^{3} - \frac{1155 \, a^{4}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}}{1155 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]