Optimal. Leaf size=101 \[ \frac {a^4 \left (a+b x^4\right )^{5/4}}{5 b^5}-\frac {4 a^3 \left (a+b x^4\right )^{9/4}}{9 b^5}+\frac {6 a^2 \left (a+b x^4\right )^{13/4}}{13 b^5}-\frac {4 a \left (a+b x^4\right )^{17/4}}{17 b^5}+\frac {\left (a+b x^4\right )^{21/4}}{21 b^5} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 101, normalized size of antiderivative = 1.00, 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 = {272, 45}
\begin {gather*} \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 {\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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^{19} \sqrt [4]{a+b x^4} \, dx &=\frac {1}{4} \text {Subst}\left (\int x^4 \sqrt [4]{a+b x} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {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 = 0.04, size = 61, normalized size = 0.60 \begin {gather*} \frac {\left (a+b x^4\right )^{5/4} \left (2048 a^4-2560 a^3 b x^4+2880 a^2 b^2 x^8-3120 a b^3 x^{12}+3315 b^4 x^{16}\right )}{69615 b^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.15, size = 58, normalized size = 0.57
method | result | size |
gosper | \(\frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}} \left (3315 x^{16} b^{4}-3120 a \,b^{3} x^{12}+2880 a^{2} b^{2} x^{8}-2560 a^{3} b \,x^{4}+2048 a^{4}\right )}{69615 b^{5}}\) | \(58\) |
trager | \(\frac {\left (3315 b^{5} x^{20}+195 a \,b^{4} x^{16}-240 a^{2} b^{3} x^{12}+320 a^{3} b^{2} x^{8}-512 a^{4} b \,x^{4}+2048 a^{5}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{69615 b^{5}}\) | \(69\) |
risch | \(\frac {\left (3315 b^{5} x^{20}+195 a \,b^{4} x^{16}-240 a^{2} b^{3} x^{12}+320 a^{3} b^{2} x^{8}-512 a^{4} b \,x^{4}+2048 a^{5}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{69615 b^{5}}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.31, size = 81, normalized size = 0.80 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 68, normalized size = 0.67 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 1.07, size = 134, normalized size = 1.33 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.28, size = 71, normalized size = 0.70 \begin {gather*} \frac {3315 \, {\left (b x^{4} + a\right )}^{\frac {21}{4}} - 16380 \, {\left (b x^{4} + a\right )}^{\frac {17}{4}} a + 32130 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}} a^{2} - 30940 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} a^{3} + 13923 \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} a^{4}}{69615 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.15, size = 66, normalized size = 0.65 \begin {gather*} {\left (b\,x^4+a\right )}^{1/4}\,\left (\frac {x^{20}}{21}+\frac {2048\,a^5}{69615\,b^5}+\frac {a\,x^{16}}{357\,b}-\frac {512\,a^4\,x^4}{69615\,b^4}+\frac {64\,a^3\,x^8}{13923\,b^3}-\frac {16\,a^2\,x^{12}}{4641\,b^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________