Optimal. Leaf size=50 \[ \frac {-A-B x}{3 c \left (a+c x^2\right )^{3/2}}+\frac {B x}{3 a c \sqrt {a+c x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 47, normalized size of antiderivative = 0.94, number of steps
used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {792, 197}
\begin {gather*} \frac {B x}{3 a c \sqrt {a+c x^2}}-\frac {A+B x}{3 c \left (a+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 792
Rubi steps
\begin {align*} \int \frac {x (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx &=-\frac {A+B x}{3 c \left (a+c x^2\right )^{3/2}}+\frac {B \int \frac {1}{\left (a+c x^2\right )^{3/2}} \, dx}{3 c}\\ &=-\frac {A+B x}{3 c \left (a+c x^2\right )^{3/2}}+\frac {B x}{3 a c \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 32, normalized size = 0.64 \begin {gather*} \frac {-a A+B c x^3}{3 a c \left (a+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.59, size = 72, normalized size = 1.44
method | result | size |
gosper | \(-\frac {-B c \,x^{3}+A a}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a c}\) | \(29\) |
trager | \(-\frac {-B c \,x^{3}+A a}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a c}\) | \(29\) |
default | \(B \left (-\frac {x}{2 c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {c \,x^{2}+a}}\right )}{2 c}\right )-\frac {A}{3 c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 51, normalized size = 1.02 \begin {gather*} -\frac {B x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {B x}{3 \, \sqrt {c x^{2} + a} a c} - \frac {A}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.70, size = 49, normalized size = 0.98 \begin {gather*} \frac {{\left (B c x^{3} - A a\right )} \sqrt {c x^{2} + a}}{3 \, {\left (a c^{3} x^{4} + 2 \, a^{2} c^{2} x^{2} + a^{3} c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 5.82, size = 95, normalized size = 1.90 \begin {gather*} A \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 {B 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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.57, size = 26, normalized size = 0.52 \begin {gather*} \frac {\frac {B x^{3}}{a} - \frac {A}{c}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.09, size = 34, normalized size = 0.68 \begin {gather*} \frac {B\,x^3}{3\,a\,{\left (c\,x^2+a\right )}^{3/2}}-\frac {A}{3\,c\,{\left (c\,x^2+a\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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