Optimal. Leaf size=67 \[ \frac{x (a C+2 A c)}{3 a^2 c \sqrt{a+c x^2}}-\frac{a B-x (A c-a C)}{3 a c \left (a+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0422141, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1814, 12, 191} \[ \frac{x (a C+2 A c)}{3 a^2 c \sqrt{a+c x^2}}-\frac{a B-x (A c-a C)}{3 a c \left (a+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 1814
Rule 12
Rule 191
Rubi steps
\begin{align*} \int \frac{A+B x+C x^2}{\left (a+c x^2\right )^{5/2}} \, dx &=-\frac{a B-(A c-a C) x}{3 a c \left (a+c x^2\right )^{3/2}}-\frac{\int \frac{-2 A-\frac{a C}{c}}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a}\\ &=-\frac{a B-(A c-a C) x}{3 a c \left (a+c x^2\right )^{3/2}}+\frac{(2 A c+a C) \int \frac{1}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a c}\\ &=-\frac{a B-(A c-a C) x}{3 a c \left (a+c x^2\right )^{3/2}}+\frac{(2 A c+a C) x}{3 a^2 c \sqrt{a+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.033789, size = 50, normalized size = 0.75 \[ \frac{-a^2 B+a c x \left (3 A+C x^2\right )+2 A c^2 x^3}{3 a^2 c \left (a+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 47, normalized size = 0.7 \begin{align*}{\frac{2\,A{x}^{3}{c}^{2}+Cac{x}^{3}+3\,Axac-B{a}^{2}}{3\,{a}^{2}c} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.979341, size = 112, normalized size = 1.67 \begin{align*} \frac{2 \, A x}{3 \, \sqrt{c x^{2} + a} a^{2}} + \frac{A x}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} a} - \frac{C x}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} c} + \frac{C x}{3 \, \sqrt{c x^{2} + a} a c} - \frac{B}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59628, size = 139, normalized size = 2.07 \begin{align*} \frac{{\left (3 \, A a c x +{\left (C a c + 2 \, A c^{2}\right )} x^{3} - B a^{2}\right )} \sqrt{c x^{2} + a}}{3 \,{\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.8247, size = 194, normalized size = 2.9 \begin{align*} A \left (\frac{3 a x}{3 a^{\frac{7}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 3 a^{\frac{5}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{2 c x^{3}}{3 a^{\frac{7}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 3 a^{\frac{5}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}}}\right ) + B \left (\begin{cases} - \frac{1}{3 a c \sqrt{a + c x^{2}} + 3 c^{2} x^{2} \sqrt{a + c x^{2}}} & \text{for}\: c \neq 0 \\\frac{x^{2}}{2 a^{\frac{5}{2}}} & \text{otherwise} \end{cases}\right ) + \frac{C x^{3}}{3 a^{\frac{5}{2}} \sqrt{1 + \frac{c x^{2}}{a}} + 3 a^{\frac{3}{2}} c x^{2} \sqrt{1 + \frac{c x^{2}}{a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16968, size = 65, normalized size = 0.97 \begin{align*} \frac{x{\left (\frac{3 \, A}{a} + \frac{{\left (C a c + 2 \, A c^{2}\right )} x^{2}}{a^{2} c}\right )} - \frac{B}{c}}{3 \,{\left (c x^{2} + a\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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