Optimal. Leaf size=102 \[ \frac {d \sqrt {e x} \sqrt [4]{a+b x^2}}{b e}-\frac {(2 b c-a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} \sqrt {b} e^2 \left (a+b x^2\right )^{3/4}} \]
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Rubi [A]
time = 0.06, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {470, 335, 243,
342, 281, 237} \begin {gather*} \frac {d \sqrt {e x} \sqrt [4]{a+b x^2}}{b e}-\frac {(e x)^{3/2} \left (\frac {a}{b x^2}+1\right )^{3/4} (2 b c-a d) F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} \sqrt {b} e^2 \left (a+b x^2\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 237
Rule 243
Rule 281
Rule 335
Rule 342
Rule 470
Rubi steps
\begin {align*} \int \frac {c+d x^2}{\sqrt {e x} \left (a+b x^2\right )^{3/4}} \, dx &=\frac {d \sqrt {e x} \sqrt [4]{a+b x^2}}{b e}-\frac {\left (-b c+\frac {a d}{2}\right ) \int \frac {1}{\sqrt {e x} \left (a+b x^2\right )^{3/4}} \, dx}{b}\\ &=\frac {d \sqrt {e x} \sqrt [4]{a+b x^2}}{b e}+\frac {(2 b c-a d) \text {Subst}\left (\int \frac {1}{\left (a+\frac {b x^4}{e^2}\right )^{3/4}} \, dx,x,\sqrt {e x}\right )}{b e}\\ &=\frac {d \sqrt {e x} \sqrt [4]{a+b x^2}}{b e}+\frac {\left ((2 b c-a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a e^2}{b x^4}\right )^{3/4} x^3} \, dx,x,\sqrt {e x}\right )}{b e \left (a+b x^2\right )^{3/4}}\\ &=\frac {d \sqrt {e x} \sqrt [4]{a+b x^2}}{b e}-\frac {\left ((2 b c-a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2}\right ) \text {Subst}\left (\int \frac {x}{\left (1+\frac {a e^2 x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{\sqrt {e x}}\right )}{b e \left (a+b x^2\right )^{3/4}}\\ &=\frac {d \sqrt {e x} \sqrt [4]{a+b x^2}}{b e}-\frac {\left ((2 b c-a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a e^2 x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{e x}\right )}{2 b e \left (a+b x^2\right )^{3/4}}\\ &=\frac {d \sqrt {e x} \sqrt [4]{a+b x^2}}{b e}-\frac {(2 b c-a d) \left (1+\frac {a}{b x^2}\right )^{3/4} (e x)^{3/2} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} \sqrt {b} e^2 \left (a+b x^2\right )^{3/4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.05, size = 77, normalized size = 0.75 \begin {gather*} \frac {d x \left (a+b x^2\right )+(2 b c-a d) x \left (1+\frac {b x^2}{a}\right )^{3/4} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {5}{4};-\frac {b x^2}{a}\right )}{b \sqrt {e x} \left (a+b x^2\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {d \,x^{2}+c}{\sqrt {e x}\, \left (b \,x^{2}+a \right )^{\frac {3}{4}}}\, dx\]
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 = 0.00, 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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Sympy [C] Result contains complex when optimal does not.
time = 2.20, size = 78, normalized size = 0.76 \begin {gather*} - \frac {c {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {3}{2} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{b^{\frac {3}{4}} \sqrt {e} x} + \frac {d x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{4}} \sqrt {e} \Gamma \left (\frac {9}{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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {d\,x^2+c}{\sqrt {e\,x}\,{\left (b\,x^2+a\right )}^{3/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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