Optimal. Leaf size=128 \[ -\frac {77 b^{5/2} \sqrt [4]{\frac {b x^4}{a}+1} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{40 a^{7/2} \sqrt [4]{a+b x^4}}-\frac {77 b^2}{120 a^3 x^2 \sqrt [4]{a+b x^4}}+\frac {11 b}{60 a^2 x^6 \sqrt [4]{a+b x^4}}-\frac {1}{10 a x^{10} \sqrt [4]{a+b x^4}} \]
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Rubi [A] time = 0.08, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {275, 286, 197, 196} \[ -\frac {77 b^2}{120 a^3 x^2 \sqrt [4]{a+b x^4}}-\frac {77 b^{5/2} \sqrt [4]{\frac {b x^4}{a}+1} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{40 a^{7/2} \sqrt [4]{a+b x^4}}+\frac {11 b}{60 a^2 x^6 \sqrt [4]{a+b x^4}}-\frac {1}{10 a x^{10} \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
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Rule 196
Rule 197
Rule 275
Rule 286
Rubi steps
\begin {align*} \int \frac {1}{x^{11} \left (a+b x^4\right )^{5/4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^6 \left (a+b x^2\right )^{5/4}} \, dx,x,x^2\right )\\ &=-\frac {1}{10 a x^{10} \sqrt [4]{a+b x^4}}-\frac {(11 b) \operatorname {Subst}\left (\int \frac {1}{x^4 \left (a+b x^2\right )^{5/4}} \, dx,x,x^2\right )}{20 a}\\ &=-\frac {1}{10 a x^{10} \sqrt [4]{a+b x^4}}+\frac {11 b}{60 a^2 x^6 \sqrt [4]{a+b x^4}}+\frac {\left (77 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2\right )^{5/4}} \, dx,x,x^2\right )}{120 a^2}\\ &=-\frac {1}{10 a x^{10} \sqrt [4]{a+b x^4}}+\frac {11 b}{60 a^2 x^6 \sqrt [4]{a+b x^4}}-\frac {77 b^2}{120 a^3 x^2 \sqrt [4]{a+b x^4}}-\frac {\left (77 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^{5/4}} \, dx,x,x^2\right )}{80 a^3}\\ &=-\frac {1}{10 a x^{10} \sqrt [4]{a+b x^4}}+\frac {11 b}{60 a^2 x^6 \sqrt [4]{a+b x^4}}-\frac {77 b^2}{120 a^3 x^2 \sqrt [4]{a+b x^4}}-\frac {\left (77 b^3 \sqrt [4]{1+\frac {b x^4}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/4}} \, dx,x,x^2\right )}{80 a^4 \sqrt [4]{a+b x^4}}\\ &=-\frac {1}{10 a x^{10} \sqrt [4]{a+b x^4}}+\frac {11 b}{60 a^2 x^6 \sqrt [4]{a+b x^4}}-\frac {77 b^2}{120 a^3 x^2 \sqrt [4]{a+b x^4}}-\frac {77 b^{5/2} \sqrt [4]{1+\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{40 a^{7/2} \sqrt [4]{a+b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 54, normalized size = 0.42 \[ -\frac {\sqrt [4]{\frac {b x^4}{a}+1} \, _2F_1\left (-\frac {5}{2},\frac {5}{4};-\frac {3}{2};-\frac {b x^4}{a}\right )}{10 a x^{10} \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{b^{2} x^{19} + 2 \, a b x^{15} + a^{2} x^{11}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x^{4} + a\right )}^{\frac {5}{4}} x^{11}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {5}{4}} x^{11}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{4} + a\right )}^{\frac {5}{4}} x^{11}}\,{d x} \]
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 \[ \int \frac {1}{x^{11}\,{\left (b\,x^4+a\right )}^{5/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.15, size = 32, normalized size = 0.25 \[ - \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, \frac {5}{4} \\ - \frac {3}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{10 a^{\frac {5}{4}} x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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