Optimal. Leaf size=65 \[ -\frac {2 (e x)^{3/2} (4 b c-a d)}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}-\frac {2 c}{a e \sqrt {e x} \left (a+b x^2\right )^{3/4}} \]
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Rubi [A] time = 0.03, antiderivative size = 65, 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} \[ -\frac {2 (e x)^{3/2} (4 b c-a d)}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}-\frac {2 c}{a e \sqrt {e x} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 453
Rubi steps
\begin {align*} \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{7/4}} \, dx &=-\frac {2 c}{a e \sqrt {e x} \left (a+b x^2\right )^{3/4}}-\frac {(4 b c-a d) \int \frac {\sqrt {e x}}{\left (a+b x^2\right )^{7/4}} \, dx}{a e^2}\\ &=-\frac {2 c}{a e \sqrt {e x} \left (a+b x^2\right )^{3/4}}-\frac {2 (4 b c-a d) (e x)^{3/2}}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.68 \[ \frac {2 x \left (-3 a c+a d x^2-4 b c x^2\right )}{3 a^2 (e x)^{3/2} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.37, size = 56, normalized size = 0.86 \[ -\frac {2 \, {\left ({\left (4 \, b c - a d\right )} x^{2} + 3 \, a c\right )} {\left (b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {e x}}{3 \, {\left (a^{2} b e^{2} x^{3} + a^{3} e^{2} x\right )}} \]
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 \[ \int \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {7}{4}} \left (e x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 40, normalized size = 0.62 \[ -\frac {2 \left (-a d \,x^{2}+4 b c \,x^{2}+3 a c \right ) x}{3 \left (b \,x^{2}+a \right )^{\frac {3}{4}} \left (e x \right )^{\frac {3}{2}} a^{2}} \]
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 \[ \int \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {7}{4}} \left (e x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 69, normalized size = 1.06 \[ -\frac {{\left (b\,x^2+a\right )}^{1/4}\,\left (\frac {2\,c}{a\,b\,e}-\frac {x^2\,\left (2\,a\,d-8\,b\,c\right )}{3\,a^2\,b\,e}\right )}{x^2\,\sqrt {e\,x}+\frac {a\,\sqrt {e\,x}}{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 79.59, size = 119, normalized size = 1.83 \[ c \left (\frac {3 \Gamma \left (- \frac {1}{4}\right )}{8 a b^{\frac {3}{4}} e^{\frac {3}{2}} x^{2} \left (\frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (\frac {7}{4}\right )} + \frac {\sqrt [4]{b} \Gamma \left (- \frac {1}{4}\right )}{2 a^{2} e^{\frac {3}{2}} \left (\frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (\frac {7}{4}\right )}\right ) + \frac {d \Gamma \left (\frac {3}{4}\right )}{2 a b^{\frac {3}{4}} e^{\frac {3}{2}} \left (\frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (\frac {7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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