Optimal. Leaf size=59 \[ -\frac {2 \left (a g^2+c f^2\right )}{g^3 \sqrt {f+g x}}+\frac {2 c (f+g x)^{3/2}}{3 g^3}-\frac {4 c f \sqrt {f+g x}}{g^3} \]
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Rubi [A] time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {697} \[ -\frac {2 \left (a g^2+c f^2\right )}{g^3 \sqrt {f+g x}}+\frac {2 c (f+g x)^{3/2}}{3 g^3}-\frac {4 c f \sqrt {f+g x}}{g^3} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin {align*} \int \frac {a+c x^2}{(f+g x)^{3/2}} \, dx &=\int \left (\frac {c f^2+a g^2}{g^2 (f+g x)^{3/2}}-\frac {2 c f}{g^2 \sqrt {f+g x}}+\frac {c \sqrt {f+g x}}{g^2}\right ) \, dx\\ &=-\frac {2 \left (c f^2+a g^2\right )}{g^3 \sqrt {f+g x}}-\frac {4 c f \sqrt {f+g x}}{g^3}+\frac {2 c (f+g x)^{3/2}}{3 g^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 43, normalized size = 0.73 \[ \frac {2 \left (c \left (-8 f^2-4 f g x+g^2 x^2\right )-3 a g^2\right )}{3 g^3 \sqrt {f+g x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 49, normalized size = 0.83 \[ \frac {2 \, {\left (c g^{2} x^{2} - 4 \, c f g x - 8 \, c f^{2} - 3 \, a g^{2}\right )} \sqrt {g x + f}}{3 \, {\left (g^{4} x + f g^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 56, normalized size = 0.95 \[ -\frac {2 \, {\left (c f^{2} + a g^{2}\right )}}{\sqrt {g x + f} g^{3}} + \frac {2 \, {\left ({\left (g x + f\right )}^{\frac {3}{2}} c g^{6} - 6 \, \sqrt {g x + f} c f g^{6}\right )}}{3 \, g^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 41, normalized size = 0.69 \[ -\frac {2 \left (-c \,x^{2} g^{2}+4 c f x g +3 a \,g^{2}+8 c \,f^{2}\right )}{3 \sqrt {g x +f}\, g^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 54, normalized size = 0.92 \[ \frac {2 \, {\left (\frac {{\left (g x + f\right )}^{\frac {3}{2}} c - 6 \, \sqrt {g x + f} c f}{g^{2}} - \frac {3 \, {\left (c f^{2} + a g^{2}\right )}}{\sqrt {g x + f} g^{2}}\right )}}{3 \, g} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 44, normalized size = 0.75 \[ -\frac {6\,a\,g^2-2\,c\,{\left (f+g\,x\right )}^2+6\,c\,f^2+12\,c\,f\,\left (f+g\,x\right )}{3\,g^3\,\sqrt {f+g\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.14, size = 58, normalized size = 0.98 \[ - \frac {4 c f \sqrt {f + g x}}{g^{3}} + \frac {2 c \left (f + g x\right )^{\frac {3}{2}}}{3 g^{3}} - \frac {2 \left (a g^{2} + c f^{2}\right )}{g^{3} \sqrt {f + g x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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