Optimal. Leaf size=101 \[ \frac {a^4 \left (a+b x^4\right )^{9/4}}{9 b^5}-\frac {4 a^3 \left (a+b x^4\right )^{13/4}}{13 b^5}+\frac {6 a^2 \left (a+b x^4\right )^{17/4}}{17 b^5}-\frac {4 a \left (a+b x^4\right )^{21/4}}{21 b^5}+\frac {\left (a+b x^4\right )^{25/4}}{25 b^5} \]
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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 )^{9/4}}{9 b^5}-\frac {4 a^3 \left (a+b x^4\right )^{13/4}}{13 b^5}+\frac {6 a^2 \left (a+b x^4\right )^{17/4}}{17 b^5}+\frac {\left (a+b x^4\right )^{25/4}}{25 b^5}-\frac {4 a \left (a+b x^4\right )^{21/4}}{21 b^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^{19} \left (a+b x^4\right )^{5/4} \, dx &=\frac {1}{4} \text {Subst}\left (\int x^4 (a+b x)^{5/4} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (\frac {a^4 (a+b x)^{5/4}}{b^4}-\frac {4 a^3 (a+b x)^{9/4}}{b^4}+\frac {6 a^2 (a+b x)^{13/4}}{b^4}-\frac {4 a (a+b x)^{17/4}}{b^4}+\frac {(a+b x)^{21/4}}{b^4}\right ) \, dx,x,x^4\right )\\ &=\frac {a^4 \left (a+b x^4\right )^{9/4}}{9 b^5}-\frac {4 a^3 \left (a+b x^4\right )^{13/4}}{13 b^5}+\frac {6 a^2 \left (a+b x^4\right )^{17/4}}{17 b^5}-\frac {4 a \left (a+b x^4\right )^{21/4}}{21 b^5}+\frac {\left (a+b x^4\right )^{25/4}}{25 b^5}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 61, normalized size = 0.60 \begin {gather*} \frac {\left (a+b x^4\right )^{9/4} \left (2048 a^4-4608 a^3 b x^4+7488 a^2 b^2 x^8-10608 a b^3 x^{12}+13923 b^4 x^{16}\right )}{348075 b^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 58, normalized size = 0.57
method | result | size |
gosper | \(\frac {\left (b \,x^{4}+a \right )^{\frac {9}{4}} \left (13923 x^{16} b^{4}-10608 a \,b^{3} x^{12}+7488 a^{2} b^{2} x^{8}-4608 a^{3} b \,x^{4}+2048 a^{4}\right )}{348075 b^{5}}\) | \(58\) |
trager | \(\frac {\left (13923 b^{6} x^{24}+17238 a \,b^{5} x^{20}+195 a^{2} b^{4} x^{16}-240 a^{3} b^{3} x^{12}+320 a^{4} b^{2} x^{8}-512 a^{5} b \,x^{4}+2048 a^{6}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{348075 b^{5}}\) | \(80\) |
risch | \(\frac {\left (13923 b^{6} x^{24}+17238 a \,b^{5} x^{20}+195 a^{2} b^{4} x^{16}-240 a^{3} b^{3} x^{12}+320 a^{4} b^{2} x^{8}-512 a^{5} b \,x^{4}+2048 a^{6}\right ) \left (b \,x^{4}+a \right )^{\frac {1}{4}}}{348075 b^{5}}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 81, normalized size = 0.80 \begin {gather*} \frac {{\left (b x^{4} + a\right )}^{\frac {25}{4}}}{25 \, b^{5}} - \frac {4 \, {\left (b x^{4} + a\right )}^{\frac {21}{4}} a}{21 \, b^{5}} + \frac {6 \, {\left (b x^{4} + a\right )}^{\frac {17}{4}} a^{2}}{17 \, b^{5}} - \frac {4 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}} a^{3}}{13 \, b^{5}} + \frac {{\left (b x^{4} + a\right )}^{\frac {9}{4}} a^{4}}{9 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 79, normalized size = 0.78 \begin {gather*} \frac {{\left (13923 \, b^{6} x^{24} + 17238 \, a b^{5} x^{20} + 195 \, a^{2} b^{4} x^{16} - 240 \, a^{3} b^{3} x^{12} + 320 \, a^{4} b^{2} x^{8} - 512 \, a^{5} b x^{4} + 2048 \, a^{6}\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{348075 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.30, size = 156, normalized size = 1.54 \begin {gather*} \begin {cases} \frac {2048 a^{6} \sqrt [4]{a + b x^{4}}}{348075 b^{5}} - \frac {512 a^{5} x^{4} \sqrt [4]{a + b x^{4}}}{348075 b^{4}} + \frac {64 a^{4} x^{8} \sqrt [4]{a + b x^{4}}}{69615 b^{3}} - \frac {16 a^{3} x^{12} \sqrt [4]{a + b x^{4}}}{23205 b^{2}} + \frac {a^{2} x^{16} \sqrt [4]{a + b x^{4}}}{1785 b} + \frac {26 a x^{20} \sqrt [4]{a + b x^{4}}}{525} + \frac {b x^{24} \sqrt [4]{a + b x^{4}}}{25} & \text {for}\: b \neq 0 \\\frac {a^{\frac {5}{4}} x^{20}}{20} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.53, size = 71, normalized size = 0.70 \begin {gather*} \frac {13923 \, {\left (b x^{4} + a\right )}^{\frac {25}{4}} - 66300 \, {\left (b x^{4} + a\right )}^{\frac {21}{4}} a + 122850 \, {\left (b x^{4} + a\right )}^{\frac {17}{4}} a^{2} - 107100 \, {\left (b x^{4} + a\right )}^{\frac {13}{4}} a^{3} + 38675 \, {\left (b x^{4} + a\right )}^{\frac {9}{4}} a^{4}}{348075 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.18, size = 75, normalized size = 0.74 \begin {gather*} {\left (b\,x^4+a\right )}^{1/4}\,\left (\frac {26\,a\,x^{20}}{525}+\frac {b\,x^{24}}{25}+\frac {2048\,a^6}{348075\,b^5}-\frac {512\,a^5\,x^4}{348075\,b^4}+\frac {64\,a^4\,x^8}{69615\,b^3}-\frac {16\,a^3\,x^{12}}{23205\,b^2}+\frac {a^2\,x^{16}}{1785\,b}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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