Optimal. Leaf size=67 \[ \frac {2 \sqrt [4]{a+b x^2} (4 b c-5 a d)}{5 a^2 e^3 \sqrt {e x}}-\frac {2 c \sqrt [4]{a+b x^2}}{5 a e (e x)^{5/2}} \]
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Rubi [A] time = 0.04, 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} \[ \frac {2 \sqrt [4]{a+b x^2} (4 b c-5 a d)}{5 a^2 e^3 \sqrt {e x}}-\frac {2 c \sqrt [4]{a+b x^2}}{5 a e (e x)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 453
Rubi steps
\begin {align*} \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{3/4}} \, dx &=-\frac {2 c \sqrt [4]{a+b x^2}}{5 a e (e x)^{5/2}}-\frac {(4 b c-5 a d) \int \frac {1}{(e x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx}{5 a e^2}\\ &=-\frac {2 c \sqrt [4]{a+b x^2}}{5 a e (e x)^{5/2}}+\frac {2 (4 b c-5 a d) \sqrt [4]{a+b x^2}}{5 a^2 e^3 \sqrt {e x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.66 \[ -\frac {2 x \sqrt [4]{a+b x^2} \left (a \left (c+5 d x^2\right )-4 b c x^2\right )}{5 a^2 (e x)^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 43, normalized size = 0.64 \[ \frac {2 \, {\left ({\left (4 \, b c - 5 \, a d\right )} x^{2} - a c\right )} {\left (b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {e x}}{5 \, a^{2} e^{4} x^{3}} \]
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 \[ \int \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {7}{2}}}\,{d x} \]
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 \[ -\frac {2 \left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (5 a d \,x^{2}-4 b c \,x^{2}+a c \right ) x}{5 \left (e x \right )^{\frac {7}{2}} a^{2}} \]
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 \[ \int \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 49, normalized size = 0.73 \[ -\frac {\left (\frac {2\,c}{5\,a\,e^3}+\frac {x^2\,\left (10\,a\,d-8\,b\,c\right )}{5\,a^2\,e^3}\right )\,{\left (b\,x^2+a\right )}^{1/4}}{x^2\,\sqrt {e\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 59.81, size = 121, normalized size = 1.81 \[ - \frac {\sqrt [4]{b} c \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {5}{4}\right )}{8 a e^{\frac {7}{2}} x^{2} \Gamma \left (\frac {3}{4}\right )} + \frac {\sqrt [4]{b} d \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {1}{4}\right )}{2 a e^{\frac {7}{2}} \Gamma \left (\frac {3}{4}\right )} + \frac {b^{\frac {5}{4}} c \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {5}{4}\right )}{2 a^{2} e^{\frac {7}{2}} \Gamma \left (\frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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