Optimal. Leaf size=47 \[ \frac {2 c x}{3 a^2 \sqrt {a+b x^2}}+\frac {x \left (c+d x^2\right )}{3 a \left (a+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {378, 191} \begin {gather*} \frac {2 c x}{3 a^2 \sqrt {a+b x^2}}+\frac {x \left (c+d x^2\right )}{3 a \left (a+b x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 191
Rule 378
Rubi steps
\begin {align*} \int \frac {c+d x^2}{\left (a+b x^2\right )^{5/2}} \, dx &=\frac {x \left (c+d x^2\right )}{3 a \left (a+b x^2\right )^{3/2}}+\frac {(2 c) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{3 a}\\ &=\frac {2 c x}{3 a^2 \sqrt {a+b x^2}}+\frac {x \left (c+d x^2\right )}{3 a \left (a+b x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 37, normalized size = 0.79 \begin {gather*} \frac {x \left (3 a c+a d x^2+2 b c x^2\right )}{3 a^2 \left (a+b x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 37, normalized size = 0.79 \begin {gather*} \frac {3 a c x+a d x^3+2 b c x^3}{3 a^2 \left (a+b x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 54, normalized size = 1.15 \begin {gather*} \frac {{\left ({\left (2 \, b c + a d\right )} x^{3} + 3 \, a c x\right )} \sqrt {b x^{2} + a}}{3 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.62, size = 40, normalized size = 0.85 \begin {gather*} \frac {x {\left (\frac {3 \, c}{a} + \frac {{\left (2 \, b^{2} c + a b d\right )} x^{2}}{a^{2} b}\right )}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 34, normalized size = 0.72 \begin {gather*} \frac {\left (a d \,x^{2}+2 b c \,x^{2}+3 a c \right ) x}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 68, normalized size = 1.45 \begin {gather*} \frac {2 \, c x}{3 \, \sqrt {b x^{2} + a} a^{2}} + \frac {c x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {d x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {d x}{3 \, \sqrt {b x^{2} + a} a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.78, size = 33, normalized size = 0.70 \begin {gather*} \frac {3\,a\,c\,x+a\,d\,x^3+2\,b\,c\,x^3}{3\,a^2\,{\left (b\,x^2+a\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 11.03, size = 144, normalized size = 3.06 \begin {gather*} c \left (\frac {3 a x}{3 a^{\frac {7}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {5}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {2 b x^{3}}{3 a^{\frac {7}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {5}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + \frac {d x^{3}}{3 a^{\frac {5}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {3}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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