Optimal. Leaf size=51 \[ \frac {-a e+c d x}{3 a c \left (a+c x^2\right )^{3/2}}+\frac {2 d x}{3 a^2 \sqrt {a+c x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {653, 197}
\begin {gather*} \frac {2 d x}{3 a^2 \sqrt {a+c x^2}}-\frac {a e-c d x}{3 a c \left (a+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 653
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (a+c x^2\right )^{5/2}} \, dx &=-\frac {a e-c d x}{3 a c \left (a+c x^2\right )^{3/2}}+\frac {(2 d) \int \frac {1}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a}\\ &=-\frac {a e-c d x}{3 a c \left (a+c x^2\right )^{3/2}}+\frac {2 d x}{3 a^2 \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 43, normalized size = 0.84 \begin {gather*} \frac {-a^2 e+3 a c d x+2 c^2 d x^3}{3 a^2 c \left (a+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 50, normalized size = 0.98
method | result | size |
gosper | \(-\frac {-2 c^{2} d \,x^{3}-3 d x a c +a^{2} e}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a^{2} c}\) | \(39\) |
trager | \(-\frac {-2 c^{2} d \,x^{3}-3 d x a c +a^{2} e}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a^{2} c}\) | \(39\) |
default | \(-\frac {e}{3 c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+d \left (\frac {x}{3 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {c \,x^{2}+a}}\right )\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 49, normalized size = 0.96 \begin {gather*} \frac {2 \, d x}{3 \, \sqrt {c x^{2} + a} a^{2}} + \frac {d x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {e}{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 = 4.04, size = 63, normalized size = 1.24 \begin {gather*} \frac {{\left (2 \, c^{2} d x^{3} + 3 \, a c d x - a^{2} e\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 97 vs.
\(2 (44) = 88\).
time = 4.33, size = 146, normalized size = 2.86 \begin {gather*} d \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 ) + e \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.90, size = 38, normalized size = 0.75 \begin {gather*} \frac {{\left (\frac {2 \, c d x^{2}}{a^{2}} + \frac {3 \, d}{a}\right )} x - \frac {e}{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 = 0.39, size = 41, normalized size = 0.80 \begin {gather*} \frac {2\,c\,d\,x\,\left (c\,x^2+a\right )-a^2\,e+a\,c\,d\,x}{3\,a^2\,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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