Optimal. Leaf size=98 \[ \frac {3 b x^2}{2 \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{2 x^2}-\frac {3 \sqrt {a} \sqrt {b} \sqrt [4]{1+\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{2 \sqrt [4]{a+b x^4}} \]
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Rubi [A]
time = 0.04, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {281, 283, 235,
233, 202} \begin {gather*} -\frac {3 \sqrt {a} \sqrt {b} \sqrt [4]{\frac {b x^4}{a}+1} E\left (\left .\frac {1}{2} \text {ArcTan}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{2 \sqrt [4]{a+b x^4}}+\frac {3 b x^2}{2 \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 202
Rule 233
Rule 235
Rule 281
Rule 283
Rubi steps
\begin {align*} \int \frac {\left (a+b x^4\right )^{3/4}}{x^3} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\left (a+b x^2\right )^{3/4}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{2 x^2}+\frac {1}{4} (3 b) \text {Subst}\left (\int \frac {1}{\sqrt [4]{a+b x^2}} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{2 x^2}+\frac {\left (3 b \sqrt [4]{1+\frac {b x^4}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+\frac {b x^2}{a}}} \, dx,x,x^2\right )}{4 \sqrt [4]{a+b x^4}}\\ &=\frac {3 b x^2}{2 \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{2 x^2}-\frac {\left (3 b \sqrt [4]{1+\frac {b x^4}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{5/4}} \, dx,x,x^2\right )}{4 \sqrt [4]{a+b x^4}}\\ &=\frac {3 b x^2}{2 \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{2 x^2}-\frac {3 \sqrt {a} \sqrt {b} \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 )}{2 \sqrt [4]{a+b x^4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.02, size = 51, normalized size = 0.52 \begin {gather*} -\frac {\left (a+b x^4\right )^{3/4} \, _2F_1\left (-\frac {3}{4},-\frac {1}{2};\frac {1}{2};-\frac {b x^4}{a}\right )}{2 x^2 \left (1+\frac {b x^4}{a}\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}}}{x^{3}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.09, size = 15, normalized size = 0.15 \begin {gather*} {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{x^{3}}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.48, size = 32, normalized size = 0.33 \begin {gather*} - \frac {a^{\frac {3}{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{2 x^{2}} \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (b\,x^4+a\right )}^{3/4}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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