Optimal. Leaf size=28 \[ -\frac {a e-c d x}{a c \sqrt {a+c x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {637} \begin {gather*} -\frac {a e-c d x}{a c \sqrt {a+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 637
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (a+c x^2\right )^{3/2}} \, dx &=-\frac {a e-c d x}{a c \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.96 \begin {gather*} \frac {c d x-a e}{a c \sqrt {a+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.31, size = 27, normalized size = 0.96 \begin {gather*} \frac {c d x-a e}{a c \sqrt {a+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 35, normalized size = 1.25 \begin {gather*} \frac {{\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c^{2} x^{2} + a^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 24, normalized size = 0.86 \begin {gather*} \frac {\frac {d x}{a} - \frac {e}{c}}{\sqrt {c x^{2} + a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 27, normalized size = 0.96 \begin {gather*} -\frac {-c d x +a e}{\sqrt {c \,x^{2}+a}\, a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 31, normalized size = 1.11 \begin {gather*} \frac {d x}{\sqrt {c x^{2} + a} a} - \frac {e}{\sqrt {c x^{2} + a} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 24, normalized size = 0.86 \begin {gather*} -\frac {\frac {e}{c}-\frac {d\,x}{a}}{\sqrt {c\,x^2+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.66, size = 46, normalized size = 1.64 \begin {gather*} e \left (\begin {cases} - \frac {1}{c \sqrt {a + c x^{2}}} & \text {for}\: c \neq 0 \\\frac {x^{2}}{2 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) + \frac {d x}{a^{\frac {3}{2}} \sqrt {1 + \frac {c x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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