Optimal. Leaf size=67 \[ -\frac {2 \sqrt {e x} (4 b c-3 a d)}{3 a^2 e^3 \sqrt [4]{a+b x^2}}-\frac {2 c}{3 a e (e x)^{3/2} \sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {453, 264} \[ -\frac {2 \sqrt {e x} (4 b c-3 a d)}{3 a^2 e^3 \sqrt [4]{a+b x^2}}-\frac {2 c}{3 a e (e x)^{3/2} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 453
Rubi steps
\begin {align*} \int \frac {c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{5/4}} \, dx &=-\frac {2 c}{3 a e (e x)^{3/2} \sqrt [4]{a+b x^2}}-\frac {(4 b c-3 a d) \int \frac {1}{\sqrt {e x} \left (a+b x^2\right )^{5/4}} \, dx}{3 a e^2}\\ &=-\frac {2 c}{3 a e (e x)^{3/2} \sqrt [4]{a+b x^2}}-\frac {2 (4 b c-3 a d) \sqrt {e x}}{3 a^2 e^3 \sqrt [4]{a+b x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 0.67 \[ \frac {x \left (-2 a c+6 a d x^2-8 b c x^2\right )}{3 a^2 (e x)^{5/2} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 57, normalized size = 0.85 \[ -\frac {2 \, {\left ({\left (4 \, b c - 3 \, a d\right )} x^{2} + a c\right )} {\left (b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {e x}}{3 \, {\left (a^{2} b e^{3} x^{4} + a^{3} e^{3} x^{2}\right )}} \]
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 \[ \int \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} \left (e x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 39, normalized size = 0.58 \[ -\frac {2 \left (-3 a d \,x^{2}+4 b c \,x^{2}+a c \right ) x}{3 \left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (e x \right )^{\frac {5}{2}} a^{2}} \]
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 \[ \int \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} \left (e x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 70, normalized size = 1.04 \[ -\frac {{\left (b\,x^2+a\right )}^{3/4}\,\left (\frac {2\,c}{3\,a\,b\,e^2}-\frac {x^2\,\left (6\,a\,d-8\,b\,c\right )}{3\,a^2\,b\,e^2}\right )}{x^3\,\sqrt {e\,x}+\frac {a\,x\,\sqrt {e\,x}}{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 64.29, size = 117, normalized size = 1.75 \[ c \left (\frac {\Gamma \left (- \frac {3}{4}\right )}{8 a \sqrt [4]{b} e^{\frac {5}{2}} x^{2} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (\frac {5}{4}\right )} + \frac {b^{\frac {3}{4}} \Gamma \left (- \frac {3}{4}\right )}{2 a^{2} e^{\frac {5}{2}} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (\frac {5}{4}\right )}\right ) + \frac {d \Gamma \left (\frac {1}{4}\right )}{2 a \sqrt [4]{b} e^{\frac {5}{2}} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (\frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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