Optimal. Leaf size=112 \[ -\frac {2 \left (a e^2+c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{3/2} (e f-d g)^{3/2}}+\frac {2 \left (a g^2+c f^2\right )}{g^2 \sqrt {f+g x} (e f-d g)}+\frac {2 c \sqrt {f+g x}}{e g^2} \]
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Rubi [A] time = 0.17, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {898, 1261, 208} \[ -\frac {2 \left (a e^2+c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{3/2} (e f-d g)^{3/2}}+\frac {2 \left (a g^2+c f^2\right )}{g^2 \sqrt {f+g x} (e f-d g)}+\frac {2 c \sqrt {f+g x}}{e g^2} \]
Antiderivative was successfully verified.
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Rule 208
Rule 898
Rule 1261
Rubi steps
\begin {align*} \int \frac {a+c x^2}{(d+e x) (f+g x)^{3/2}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {\frac {c f^2+a g^2}{g^2}-\frac {2 c f x^2}{g^2}+\frac {c x^4}{g^2}}{x^2 \left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )} \, dx,x,\sqrt {f+g x}\right )}{g}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (\frac {c}{e g}+\frac {c f^2+a g^2}{g (-e f+d g) x^2}-\frac {\left (c d^2+a e^2\right ) g}{e (e f-d g) \left (e f-d g-e x^2\right )}\right ) \, dx,x,\sqrt {f+g x}\right )}{g}\\ &=\frac {2 \left (c f^2+a g^2\right )}{g^2 (e f-d g) \sqrt {f+g x}}+\frac {2 c \sqrt {f+g x}}{e g^2}-\frac {\left (2 \left (c d^2+a e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{e f-d g-e x^2} \, dx,x,\sqrt {f+g x}\right )}{e (e f-d g)}\\ &=\frac {2 \left (c f^2+a g^2\right )}{g^2 (e f-d g) \sqrt {f+g x}}+\frac {2 c \sqrt {f+g x}}{e g^2}-\frac {2 \left (c d^2+a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{3/2} (e f-d g)^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 90, normalized size = 0.80 \[ -\frac {2 \left (g^2 \left (a e^2+c d^2\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {e (f+g x)}{e f-d g}\right )+c (e f-d g) (d g+2 e f+e g x)\right )}{e^2 g^2 \sqrt {f+g x} (d g-e f)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.68, size = 492, normalized size = 4.39 \[ \left [-\frac {{\left ({\left (c d^{2} + a e^{2}\right )} g^{3} x + {\left (c d^{2} + a e^{2}\right )} f g^{2}\right )} \sqrt {e^{2} f - d e g} \log \left (\frac {e g x + 2 \, e f - d g + 2 \, \sqrt {e^{2} f - d e g} \sqrt {g x + f}}{e x + d}\right ) - 2 \, {\left (2 \, c e^{3} f^{3} - 3 \, c d e^{2} f^{2} g - a d e^{2} g^{3} + {\left (c d^{2} e + a e^{3}\right )} f g^{2} + {\left (c e^{3} f^{2} g - 2 \, c d e^{2} f g^{2} + c d^{2} e g^{3}\right )} x\right )} \sqrt {g x + f}}{e^{4} f^{3} g^{2} - 2 \, d e^{3} f^{2} g^{3} + d^{2} e^{2} f g^{4} + {\left (e^{4} f^{2} g^{3} - 2 \, d e^{3} f g^{4} + d^{2} e^{2} g^{5}\right )} x}, \frac {2 \, {\left ({\left ({\left (c d^{2} + a e^{2}\right )} g^{3} x + {\left (c d^{2} + a e^{2}\right )} f g^{2}\right )} \sqrt {-e^{2} f + d e g} \arctan \left (\frac {\sqrt {-e^{2} f + d e g} \sqrt {g x + f}}{e g x + e f}\right ) + {\left (2 \, c e^{3} f^{3} - 3 \, c d e^{2} f^{2} g - a d e^{2} g^{3} + {\left (c d^{2} e + a e^{3}\right )} f g^{2} + {\left (c e^{3} f^{2} g - 2 \, c d e^{2} f g^{2} + c d^{2} e g^{3}\right )} x\right )} \sqrt {g x + f}\right )}}{e^{4} f^{3} g^{2} - 2 \, d e^{3} f^{2} g^{3} + d^{2} e^{2} f g^{4} + {\left (e^{4} f^{2} g^{3} - 2 \, d e^{3} f g^{4} + d^{2} e^{2} g^{5}\right )} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 101, normalized size = 0.90 \[ -\frac {2 \, {\left (c d^{2} + a e^{2}\right )} \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {d g e - f e^{2}}}\right )}{{\left (d g e - f e^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, \sqrt {g x + f} c e^{\left (-1\right )}}{g^{2}} - \frac {2 \, {\left (c f^{2} + a g^{2}\right )}}{{\left (d g^{3} - f g^{2} e\right )} \sqrt {g x + f}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 165, normalized size = 1.47 \[ -\frac {2 a e \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}}-\frac {2 c \,d^{2} \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}\, e}-\frac {2 a}{\left (d g -e f \right ) \sqrt {g x +f}}-\frac {2 c \,f^{2}}{\left (d g -e f \right ) \sqrt {g x +f}\, g^{2}}+\frac {2 \sqrt {g x +f}\, c}{e \,g^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 141, normalized size = 1.26 \[ \frac {2\,\mathrm {atan}\left (\frac {2\,\sqrt {f+g\,x}\,\left (c\,d^2+a\,e^2\right )\,\left (e^2\,f-d\,e\,g\right )}{\sqrt {e}\,\left (2\,c\,d^2+2\,a\,e^2\right )\,{\left (d\,g-e\,f\right )}^{3/2}}\right )\,\left (c\,d^2+a\,e^2\right )}{e^{3/2}\,{\left (d\,g-e\,f\right )}^{3/2}}+\frac {2\,c\,\sqrt {f+g\,x}}{e\,g^2}-\frac {2\,\left (c\,e\,f^2+a\,e\,g^2\right )}{e\,g^2\,\sqrt {f+g\,x}\,\left (d\,g-e\,f\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 41.28, size = 104, normalized size = 0.93 \[ \frac {2 c \sqrt {f + g x}}{e g^{2}} - \frac {2 \left (a g^{2} + c f^{2}\right )}{g^{2} \sqrt {f + g x} \left (d g - e f\right )} - \frac {2 \left (a e^{2} + c d^{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {f + g x}}{\sqrt {\frac {d g - e f}{e}}} \right )}}{e^{2} \sqrt {\frac {d g - e f}{e}} \left (d g - e f\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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