Optimal. Leaf size=142 \[ -\frac {2 c}{a e \sqrt {e x} \left (a+b x^2\right )^{5/4}}-\frac {2 (6 b c-a d) (e x)^{3/2}}{5 a^2 e^3 \left (a+b x^2\right )^{5/4}}+\frac {4 (6 b c-a d) \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 a^{5/2} \sqrt {b} e^2 \sqrt [4]{a+b x^2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {464, 296, 290,
342, 202} \begin {gather*} \frac {4 \sqrt {e x} \sqrt [4]{\frac {a}{b x^2}+1} (6 b c-a d) E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 a^{5/2} \sqrt {b} e^2 \sqrt [4]{a+b x^2}}-\frac {2 (e x)^{3/2} (6 b c-a d)}{5 a^2 e^3 \left (a+b x^2\right )^{5/4}}-\frac {2 c}{a e \sqrt {e x} \left (a+b x^2\right )^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 202
Rule 290
Rule 296
Rule 342
Rule 464
Rubi steps
\begin {align*} \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{9/4}} \, dx &=-\frac {2 c}{a e \sqrt {e x} \left (a+b x^2\right )^{5/4}}-\frac {(6 b c-a d) \int \frac {\sqrt {e x}}{\left (a+b x^2\right )^{9/4}} \, dx}{a e^2}\\ &=-\frac {2 c}{a e \sqrt {e x} \left (a+b x^2\right )^{5/4}}-\frac {2 (6 b c-a d) (e x)^{3/2}}{5 a^2 e^3 \left (a+b x^2\right )^{5/4}}-\frac {(2 (6 b c-a d)) \int \frac {\sqrt {e x}}{\left (a+b x^2\right )^{5/4}} \, dx}{5 a^2 e^2}\\ &=-\frac {2 c}{a e \sqrt {e x} \left (a+b x^2\right )^{5/4}}-\frac {2 (6 b c-a d) (e x)^{3/2}}{5 a^2 e^3 \left (a+b x^2\right )^{5/4}}-\frac {\left (2 (6 b c-a d) \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {e x}\right ) \int \frac {1}{\left (1+\frac {a}{b x^2}\right )^{5/4} x^2} \, dx}{5 a^2 b e^2 \sqrt [4]{a+b x^2}}\\ &=-\frac {2 c}{a e \sqrt {e x} \left (a+b x^2\right )^{5/4}}-\frac {2 (6 b c-a d) (e x)^{3/2}}{5 a^2 e^3 \left (a+b x^2\right )^{5/4}}+\frac {\left (2 (6 b c-a d) \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {e x}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )}{5 a^2 b e^2 \sqrt [4]{a+b x^2}}\\ &=-\frac {2 c}{a e \sqrt {e x} \left (a+b x^2\right )^{5/4}}-\frac {2 (6 b c-a d) (e x)^{3/2}}{5 a^2 e^3 \left (a+b x^2\right )^{5/4}}+\frac {4 (6 b c-a d) \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {e x} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{5 a^{5/2} \sqrt {b} e^2 \sqrt [4]{a+b x^2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.04, size = 85, normalized size = 0.60 \begin {gather*} \frac {2 x \left (-3 a^2 c+(-6 b c+a d) x^2 \left (a+b x^2\right ) \sqrt [4]{1+\frac {b x^2}{a}} \, _2F_1\left (\frac {3}{4},\frac {9}{4};\frac {7}{4};-\frac {b x^2}{a}\right )\right )}{3 a^3 (e x)^{3/2} \left (a+b x^2\right )^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {d \,x^{2}+c}{\left (e x \right )^{\frac {3}{2}} \left (b \,x^{2}+a \right )^{\frac {9}{4}}}\, dx\]
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 = 128.49, size = 97, normalized size = 0.68 \begin {gather*} \frac {c \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {9}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {9}{4}} e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {d x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {9}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {9}{4}} e^{\frac {3}{2}} \Gamma \left (\frac {7}{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}{{\left (e\,x\right )}^{3/2}\,{\left (b\,x^2+a\right )}^{9/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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