Optimal. Leaf size=98 \[ \frac {5 \sqrt {a} b^{3/2} \left (\frac {b x^4}{a}+1\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{12 \left (a+b x^4\right )^{3/4}}-\frac {\left (a+b x^4\right )^{5/4}}{6 x^6}-\frac {5 b \sqrt [4]{a+b x^4}}{12 x^2} \]
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Rubi [A] time = 0.06, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {275, 277, 233, 231} \[ \frac {5 \sqrt {a} b^{3/2} \left (\frac {b x^4}{a}+1\right )^{3/4} F\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{12 \left (a+b x^4\right )^{3/4}}-\frac {5 b \sqrt [4]{a+b x^4}}{12 x^2}-\frac {\left (a+b x^4\right )^{5/4}}{6 x^6} \]
Antiderivative was successfully verified.
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Rule 231
Rule 233
Rule 275
Rule 277
Rubi steps
\begin {align*} \int \frac {\left (a+b x^4\right )^{5/4}}{x^7} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^{5/4}}{x^4} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^4\right )^{5/4}}{6 x^6}+\frac {1}{12} (5 b) \operatorname {Subst}\left (\int \frac {\sqrt [4]{a+b x^2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {5 b \sqrt [4]{a+b x^4}}{12 x^2}-\frac {\left (a+b x^4\right )^{5/4}}{6 x^6}+\frac {1}{24} \left (5 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^{3/4}} \, dx,x,x^2\right )\\ &=-\frac {5 b \sqrt [4]{a+b x^4}}{12 x^2}-\frac {\left (a+b x^4\right )^{5/4}}{6 x^6}+\frac {\left (5 b^2 \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 )}{24 \left (a+b x^4\right )^{3/4}}\\ &=-\frac {5 b \sqrt [4]{a+b x^4}}{12 x^2}-\frac {\left (a+b x^4\right )^{5/4}}{6 x^6}+\frac {5 \sqrt {a} b^{3/2} \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 )}{12 \left (a+b x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 52, normalized size = 0.53 \[ -\frac {a \sqrt [4]{a+b x^4} \, _2F_1\left (-\frac {3}{2},-\frac {5}{4};-\frac {1}{2};-\frac {b x^4}{a}\right )}{6 x^6 \sqrt [4]{\frac {b x^4}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}}}{x^{7}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}}}{x^{7}}\,{d x} \]
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 \[ \int \frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}}}{x^{7}}\, dx \]
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 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}}}{x^{7}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^4+a\right )}^{5/4}}{x^7} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.13, size = 34, normalized size = 0.35 \[ - \frac {a^{\frac {5}{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, - \frac {5}{4} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{6 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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