Optimal. Leaf size=66 \[ -\frac {1}{7 a x^7 \sqrt [4]{a+b x^4}}+\frac {8 b}{21 a^2 x^3 \sqrt [4]{a+b x^4}}+\frac {32 b^2 x}{21 a^3 \sqrt [4]{a+b x^4}} \]
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Rubi [A]
time = 0.01, antiderivative size = 66, 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 = {277, 197}
\begin {gather*} \frac {32 b^2 x}{21 a^3 \sqrt [4]{a+b x^4}}+\frac {8 b}{21 a^2 x^3 \sqrt [4]{a+b x^4}}-\frac {1}{7 a x^7 \sqrt [4]{a+b x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 277
Rubi steps
\begin {align*} \int \frac {1}{x^8 \left (a+b x^4\right )^{5/4}} \, dx &=-\frac {1}{7 a x^7 \sqrt [4]{a+b x^4}}-\frac {(8 b) \int \frac {1}{x^4 \left (a+b x^4\right )^{5/4}} \, dx}{7 a}\\ &=-\frac {1}{7 a x^7 \sqrt [4]{a+b x^4}}+\frac {8 b}{21 a^2 x^3 \sqrt [4]{a+b x^4}}+\frac {\left (32 b^2\right ) \int \frac {1}{\left (a+b x^4\right )^{5/4}} \, dx}{21 a^2}\\ &=-\frac {1}{7 a x^7 \sqrt [4]{a+b x^4}}+\frac {8 b}{21 a^2 x^3 \sqrt [4]{a+b x^4}}+\frac {32 b^2 x}{21 a^3 \sqrt [4]{a+b x^4}}\\ \end {align*}
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Mathematica [A]
time = 0.36, size = 42, normalized size = 0.64 \begin {gather*} \frac {-3 a^2+8 a b x^4+32 b^2 x^8}{21 a^3 x^7 \sqrt [4]{a+b x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 39, normalized size = 0.59
method | result | size |
gosper | \(-\frac {-32 b^{2} x^{8}-8 a b \,x^{4}+3 a^{2}}{21 x^{7} \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{3}}\) | \(39\) |
trager | \(-\frac {-32 b^{2} x^{8}-8 a b \,x^{4}+3 a^{2}}{21 x^{7} \left (b \,x^{4}+a \right )^{\frac {1}{4}} a^{3}}\) | \(39\) |
risch | \(-\frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}} \left (-11 b \,x^{4}+3 a \right )}{21 a^{3} x^{7}}+\frac {b^{2} x}{a^{3} \left (b \,x^{4}+a \right )^{\frac {1}{4}}}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 53, normalized size = 0.80 \begin {gather*} \frac {b^{2} x}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{3}} + \frac {\frac {14 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}} b}{x^{3}} - \frac {3 \, {\left (b x^{4} + a\right )}^{\frac {7}{4}}}{x^{7}}}{21 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 50, normalized size = 0.76 \begin {gather*} \frac {{\left (32 \, b^{2} x^{8} + 8 \, a b x^{4} - 3 \, a^{2}\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{21 \, {\left (a^{3} b x^{11} + a^{4} x^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 323 vs.
\(2 (61) = 122\).
time = 0.87, size = 323, normalized size = 4.89 \begin {gather*} - \frac {3 a^{3} b^{\frac {19}{4}} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{64 a^{5} b^{4} x^{4} \Gamma \left (\frac {5}{4}\right ) + 128 a^{4} b^{5} x^{8} \Gamma \left (\frac {5}{4}\right ) + 64 a^{3} b^{6} x^{12} \Gamma \left (\frac {5}{4}\right )} + \frac {5 a^{2} b^{\frac {23}{4}} x^{4} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{64 a^{5} b^{4} x^{4} \Gamma \left (\frac {5}{4}\right ) + 128 a^{4} b^{5} x^{8} \Gamma \left (\frac {5}{4}\right ) + 64 a^{3} b^{6} x^{12} \Gamma \left (\frac {5}{4}\right )} + \frac {40 a b^{\frac {27}{4}} x^{8} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{64 a^{5} b^{4} x^{4} \Gamma \left (\frac {5}{4}\right ) + 128 a^{4} b^{5} x^{8} \Gamma \left (\frac {5}{4}\right ) + 64 a^{3} b^{6} x^{12} \Gamma \left (\frac {5}{4}\right )} + \frac {32 b^{\frac {31}{4}} x^{12} \left (\frac {a}{b x^{4}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{64 a^{5} b^{4} x^{4} \Gamma \left (\frac {5}{4}\right ) + 128 a^{4} b^{5} x^{8} \Gamma \left (\frac {5}{4}\right ) + 64 a^{3} b^{6} x^{12} \Gamma \left (\frac {5}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.31, size = 70, normalized size = 1.06 \begin {gather*} -\frac {32\,{\left (b\,x^4+a\right )}^2-56\,a\,\left (b\,x^4+a\right )+21\,a^2}{\left (\frac {21\,a^4\,x^3}{b}-\frac {21\,a^3\,x^3\,\left (b\,x^4+a\right )}{b}\right )\,{\left (b\,x^4+a\right )}^{1/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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