Optimal. Leaf size=122 \[ -\frac {b^{5/2} \left (\frac {b x^4}{a}+1\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{24 \sqrt {a} \left (a+b x^4\right )^{3/4}}-\frac {b^2 \sqrt [4]{a+b x^4}}{24 a x^2}-\frac {\left (a+b x^4\right )^{5/4}}{10 x^{10}}-\frac {b \sqrt [4]{a+b x^4}}{12 x^6} \]
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Rubi [A] time = 0.08, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {275, 277, 325, 233, 231} \[ -\frac {b^2 \sqrt [4]{a+b x^4}}{24 a x^2}-\frac {b^{5/2} \left (\frac {b x^4}{a}+1\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{24 \sqrt {a} \left (a+b x^4\right )^{3/4}}-\frac {b \sqrt [4]{a+b x^4}}{12 x^6}-\frac {\left (a+b x^4\right )^{5/4}}{10 x^{10}} \]
Antiderivative was successfully verified.
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Rule 231
Rule 233
Rule 275
Rule 277
Rule 325
Rubi steps
\begin {align*} \int \frac {\left (a+b x^4\right )^{5/4}}{x^{11}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^{5/4}}{x^6} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^4\right )^{5/4}}{10 x^{10}}+\frac {1}{4} b \operatorname {Subst}\left (\int \frac {\sqrt [4]{a+b x^2}}{x^4} \, dx,x,x^2\right )\\ &=-\frac {b \sqrt [4]{a+b x^4}}{12 x^6}-\frac {\left (a+b x^4\right )^{5/4}}{10 x^{10}}+\frac {1}{24} b^2 \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2\right )^{3/4}} \, dx,x,x^2\right )\\ &=-\frac {b \sqrt [4]{a+b x^4}}{12 x^6}-\frac {b^2 \sqrt [4]{a+b x^4}}{24 a x^2}-\frac {\left (a+b x^4\right )^{5/4}}{10 x^{10}}-\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^{3/4}} \, dx,x,x^2\right )}{48 a}\\ &=-\frac {b \sqrt [4]{a+b x^4}}{12 x^6}-\frac {b^2 \sqrt [4]{a+b x^4}}{24 a x^2}-\frac {\left (a+b x^4\right )^{5/4}}{10 x^{10}}-\frac {\left (b^3 \left (1+\frac {b x^4}{a}\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{3/4}} \, dx,x,x^2\right )}{48 a \left (a+b x^4\right )^{3/4}}\\ &=-\frac {b \sqrt [4]{a+b x^4}}{12 x^6}-\frac {b^2 \sqrt [4]{a+b x^4}}{24 a x^2}-\frac {\left (a+b x^4\right )^{5/4}}{10 x^{10}}-\frac {b^{5/2} \left (1+\frac {b x^4}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{24 \sqrt {a} \left (a+b x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 52, normalized size = 0.43 \[ -\frac {a \sqrt [4]{a+b x^4} \, _2F_1\left (-\frac {5}{2},-\frac {5}{4};-\frac {3}{2};-\frac {b x^4}{a}\right )}{10 x^{10} \sqrt [4]{\frac {b x^4}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}}}{x^{11}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}}}{x^{11}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}}}{x^{11}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}}}{x^{11}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,x^4+a\right )}^{5/4}}{x^{11}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 6.72, size = 34, normalized size = 0.28 \[ - \frac {a^{\frac {5}{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, - \frac {5}{4} \\ - \frac {3}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{10 x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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