Optimal. Leaf size=33 \[ -\frac {2 (x (2 c d-b e)+b d)}{b^2 \sqrt {b x+c x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {636} \[ -\frac {2 (x (2 c d-b e)+b d)}{b^2 \sqrt {b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 636
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (b d+(2 c d-b e) x)}{b^2 \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 0.91 \[ \frac {2 (-b d+b e x-2 c d x)}{b^2 \sqrt {x (b+c x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 44, normalized size = 1.33 \[ -\frac {2 \, \sqrt {c x^{2} + b x} {\left (b d + {\left (2 \, c d - b e\right )} x\right )}}{b^{2} c x^{2} + b^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 34, normalized size = 1.03 \[ -\frac {2 \, {\left (\frac {d}{b} + \frac {{\left (2 \, c d - b e\right )} x}{b^{2}}\right )}}{\sqrt {c x^{2} + b x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 37, normalized size = 1.12 \[ -\frac {2 \left (c x +b \right ) \left (-b e x +2 c d x +b d \right ) x}{\left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 55, normalized size = 1.67 \[ -\frac {4 \, c d x}{\sqrt {c x^{2} + b x} b^{2}} + \frac {2 \, e x}{\sqrt {c x^{2} + b x} b} - \frac {2 \, d}{\sqrt {c x^{2} + b x} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 31, normalized size = 0.94 \[ -\frac {2\,b\,d-2\,b\,e\,x+4\,c\,d\,x}{b^2\,\sqrt {c\,x^2+b\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {d + e x}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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