Optimal. Leaf size=106 \[ -\frac{6 a^2 \left (a-b x^4\right )^{13/4}}{13 b^5}+\frac{4 a^3 \left (a-b x^4\right )^{9/4}}{9 b^5}-\frac{a^4 \left (a-b x^4\right )^{5/4}}{5 b^5}-\frac{\left (a-b x^4\right )^{21/4}}{21 b^5}+\frac{4 a \left (a-b x^4\right )^{17/4}}{17 b^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0569475, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {266, 43} \[ -\frac{6 a^2 \left (a-b x^4\right )^{13/4}}{13 b^5}+\frac{4 a^3 \left (a-b x^4\right )^{9/4}}{9 b^5}-\frac{a^4 \left (a-b x^4\right )^{5/4}}{5 b^5}-\frac{\left (a-b x^4\right )^{21/4}}{21 b^5}+\frac{4 a \left (a-b x^4\right )^{17/4}}{17 b^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{19} \sqrt [4]{a-b x^4} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int x^4 \sqrt [4]{a-b x} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{a^4 \sqrt [4]{a-b x}}{b^4}-\frac{4 a^3 (a-b x)^{5/4}}{b^4}+\frac{6 a^2 (a-b x)^{9/4}}{b^4}-\frac{4 a (a-b x)^{13/4}}{b^4}+\frac{(a-b x)^{17/4}}{b^4}\right ) \, dx,x,x^4\right )\\ &=-\frac{a^4 \left (a-b x^4\right )^{5/4}}{5 b^5}+\frac{4 a^3 \left (a-b x^4\right )^{9/4}}{9 b^5}-\frac{6 a^2 \left (a-b x^4\right )^{13/4}}{13 b^5}+\frac{4 a \left (a-b x^4\right )^{17/4}}{17 b^5}-\frac{\left (a-b x^4\right )^{21/4}}{21 b^5}\\ \end{align*}
Mathematica [A] time = 0.0306596, size = 62, normalized size = 0.58 \[ -\frac{\left (a-b x^4\right )^{5/4} \left (2880 a^2 b^2 x^8+2560 a^3 b x^4+2048 a^4+3120 a b^3 x^{12}+3315 b^4 x^{16}\right )}{69615 b^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 59, normalized size = 0.6 \begin{align*} -{\frac{3315\,{x}^{16}{b}^{4}+3120\,a{x}^{12}{b}^{3}+2880\,{a}^{2}{x}^{8}{b}^{2}+2560\,{a}^{3}{x}^{4}b+2048\,{a}^{4}}{69615\,{b}^{5}} \left ( -b{x}^{4}+a \right ) ^{{\frac{5}{4}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.0157, size = 116, normalized size = 1.09 \begin{align*} -\frac{{\left (-b x^{4} + a\right )}^{\frac{21}{4}}}{21 \, b^{5}} + \frac{4 \,{\left (-b x^{4} + a\right )}^{\frac{17}{4}} a}{17 \, b^{5}} - \frac{6 \,{\left (-b x^{4} + a\right )}^{\frac{13}{4}} a^{2}}{13 \, b^{5}} + \frac{4 \,{\left (-b x^{4} + a\right )}^{\frac{9}{4}} a^{3}}{9 \, b^{5}} - \frac{{\left (-b x^{4} + a\right )}^{\frac{5}{4}} a^{4}}{5 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.42569, size = 174, normalized size = 1.64 \begin{align*} \frac{{\left (3315 \, b^{5} x^{20} - 195 \, a b^{4} x^{16} - 240 \, a^{2} b^{3} x^{12} - 320 \, a^{3} b^{2} x^{8} - 512 \, a^{4} b x^{4} - 2048 \, a^{5}\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{69615 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 26.6008, size = 134, normalized size = 1.26 \begin{align*} \begin{cases} - \frac{2048 a^{5} \sqrt [4]{a - b x^{4}}}{69615 b^{5}} - \frac{512 a^{4} x^{4} \sqrt [4]{a - b x^{4}}}{69615 b^{4}} - \frac{64 a^{3} x^{8} \sqrt [4]{a - b x^{4}}}{13923 b^{3}} - \frac{16 a^{2} x^{12} \sqrt [4]{a - b x^{4}}}{4641 b^{2}} - \frac{a x^{16} \sqrt [4]{a - b x^{4}}}{357 b} + \frac{x^{20} \sqrt [4]{a - b x^{4}}}{21} & \text{for}\: b \neq 0 \\\frac{\sqrt [4]{a} x^{20}}{20} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11257, size = 162, normalized size = 1.53 \begin{align*} \frac{3315 \,{\left (b x^{4} - a\right )}^{5}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} + 16380 \,{\left (b x^{4} - a\right )}^{4}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a + 32130 \,{\left (b x^{4} - a\right )}^{3}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{2} + 30940 \,{\left (b x^{4} - a\right )}^{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{3} - 13923 \,{\left (-b x^{4} + a\right )}^{\frac{5}{4}} a^{4}}{69615 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]