Optimal. Leaf size=165 \[ -\frac {2 \left (c f^2-b f g+a g^2\right )}{g (e f-d g)^2 \sqrt {f+g x}}-\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {f+g x}}{e (e f-d g)^2 (d+e x)}+\frac {(c d (4 e f-d g)-e (2 b e f+b d g-3 a e g)) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{3/2} (e f-d g)^{5/2}} \]
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Rubi [A]
time = 0.23, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {911, 1273, 464,
214} \begin {gather*} -\frac {\sqrt {f+g x} \left (a e^2-b d e+c d^2\right )}{e (d+e x) (e f-d g)^2}+\frac {\tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) (c d (4 e f-d g)-e (-3 a e g+b d g+2 b e f))}{e^{3/2} (e f-d g)^{5/2}}-\frac {2 \left (a g^2-b f g+c f^2\right )}{g \sqrt {f+g x} (e f-d g)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 464
Rule 911
Rule 1273
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{(d+e x)^2 (f+g x)^{3/2}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {\frac {c f^2-b f g+a g^2}{g^2}-\frac {(2 c f-b g) 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 )^2} \, dx,x,\sqrt {f+g x}\right )}{g}\\ &=-\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {f+g x}}{e (e f-d g)^2 (d+e x)}-\frac {g^3 \text {Subst}\left (\int \frac {\frac {2 e^2 (e f-d g) \left (c f^2-b f g+a g^2\right )}{g^5}-\frac {e \left (e (b d-a e) g^2+c \left (2 e^2 f^2-4 d e f g+d^2 g^2\right )\right ) x^2}{g^5}}{x^2 \left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )} \, dx,x,\sqrt {f+g x}\right )}{e^2 (e f-d g)^2}\\ &=-\frac {2 \left (c f^2-b f g+a g^2\right )}{g (e f-d g)^2 \sqrt {f+g x}}-\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {f+g x}}{e (e f-d g)^2 (d+e x)}-\frac {(c d (4 e f-d g)-e (2 b e f+b d g-3 a e g)) \text {Subst}\left (\int \frac {1}{\frac {-e f+d g}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{e g (e f-d g)^2}\\ &=-\frac {2 \left (c f^2-b f g+a g^2\right )}{g (e f-d g)^2 \sqrt {f+g x}}-\frac {\left (c d^2-b d e+a e^2\right ) \sqrt {f+g x}}{e (e f-d g)^2 (d+e x)}+\frac {(c d (4 e f-d g)-e (2 b e f+b d g-3 a e g)) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{3/2} (e f-d g)^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.65, size = 176, normalized size = 1.07 \begin {gather*} \frac {-c \left (2 d e f^2+2 e^2 f^2 x+d^2 g (f+g x)\right )+e g (b (3 d f+2 e f x+d g x)-a (e f+2 d g+3 e g x))}{e g (e f-d g)^2 (d+e x) \sqrt {f+g x}}+\frac {(c d (-4 e f+d g)+e (2 b e f+b d g-3 a e g)) \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {-e f+d g}}\right )}{e^{3/2} (-e f+d g)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 175, normalized size = 1.06
method | result | size |
derivativedivides | \(\frac {-\frac {2 g \left (\frac {g \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {g x +f}}{2 e \left (e \left (g x +f \right )+d g -e f \right )}+\frac {\left (3 a \,e^{2} g -b d e g -2 b \,e^{2} f -c \,d^{2} g +4 c d e f \right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{2 e \sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right )^{2}}-\frac {2 \left (a \,g^{2}-b f g +c \,f^{2}\right )}{\left (d g -e f \right )^{2} \sqrt {g x +f}}}{g}\) | \(175\) |
default | \(\frac {-\frac {2 g \left (\frac {g \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {g x +f}}{2 e \left (e \left (g x +f \right )+d g -e f \right )}+\frac {\left (3 a \,e^{2} g -b d e g -2 b \,e^{2} f -c \,d^{2} g +4 c d e f \right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{2 e \sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right )^{2}}-\frac {2 \left (a \,g^{2}-b f g +c \,f^{2}\right )}{\left (d g -e f \right )^{2} \sqrt {g x +f}}}{g}\) | \(175\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 547 vs.
\(2 (156) = 312\).
time = 12.16, size = 1110, normalized size = 6.73 \begin {gather*} \left [-\frac {{\left (c d^{3} g^{3} x + c d^{3} f g^{2} + {\left ({\left (2 \, b f g^{2} - 3 \, a g^{3}\right )} x^{2} + {\left (2 \, b f^{2} g - 3 \, a f g^{2}\right )} x\right )} e^{3} + {\left (2 \, b d f^{2} g - 3 \, a d f g^{2} - {\left (4 \, c d f g^{2} - b d g^{3}\right )} x^{2} - {\left (4 \, c d f^{2} g - 3 \, b d f g^{2} + 3 \, a d g^{3}\right )} x\right )} e^{2} + {\left (c d^{2} g^{3} x^{2} - 4 \, c d^{2} f^{2} g + b d^{2} f g^{2} - {\left (3 \, c d^{2} f g^{2} - b d^{2} g^{3}\right )} x\right )} e\right )} \sqrt {-d g e + f e^{2}} \log \left (-\frac {d g - {\left (g x + 2 \, f\right )} e - 2 \, \sqrt {-d g e + f e^{2}} \sqrt {g x + f}}{x e + d}\right ) + 2 \, \sqrt {g x + f} {\left ({\left (a f^{2} g + {\left (2 \, c f^{3} - 2 \, b f^{2} g + 3 \, a f g^{2}\right )} x\right )} e^{4} + {\left (2 \, c d f^{3} - 3 \, b d f^{2} g + a d f g^{2} - {\left (2 \, c d f^{2} g - b d f g^{2} + 3 \, a d g^{3}\right )} x\right )} e^{3} - {\left (c d^{2} f^{2} g - 3 \, b d^{2} f g^{2} + 2 \, a d^{2} g^{3} - {\left (c d^{2} f g^{2} + b d^{2} g^{3}\right )} x\right )} e^{2} - {\left (c d^{3} g^{3} x + c d^{3} f g^{2}\right )} e\right )}}{2 \, {\left ({\left (f^{3} g^{2} x^{2} + f^{4} g x\right )} e^{6} - {\left (3 \, d f^{2} g^{3} x^{2} + 2 \, d f^{3} g^{2} x - d f^{4} g\right )} e^{5} + 3 \, {\left (d^{2} f g^{4} x^{2} - d^{2} f^{3} g^{2}\right )} e^{4} - {\left (d^{3} g^{5} x^{2} - 2 \, d^{3} f g^{4} x - 3 \, d^{3} f^{2} g^{3}\right )} e^{3} - {\left (d^{4} g^{5} x + d^{4} f g^{4}\right )} e^{2}\right )}}, \frac {{\left (c d^{3} g^{3} x + c d^{3} f g^{2} + {\left ({\left (2 \, b f g^{2} - 3 \, a g^{3}\right )} x^{2} + {\left (2 \, b f^{2} g - 3 \, a f g^{2}\right )} x\right )} e^{3} + {\left (2 \, b d f^{2} g - 3 \, a d f g^{2} - {\left (4 \, c d f g^{2} - b d g^{3}\right )} x^{2} - {\left (4 \, c d f^{2} g - 3 \, b d f g^{2} + 3 \, a d g^{3}\right )} x\right )} e^{2} + {\left (c d^{2} g^{3} x^{2} - 4 \, c d^{2} f^{2} g + b d^{2} f g^{2} - {\left (3 \, c d^{2} f g^{2} - b d^{2} g^{3}\right )} x\right )} e\right )} \sqrt {d g e - f e^{2}} \arctan \left (-\frac {\sqrt {d g e - f e^{2}} \sqrt {g x + f}}{d g - f e}\right ) - \sqrt {g x + f} {\left ({\left (a f^{2} g + {\left (2 \, c f^{3} - 2 \, b f^{2} g + 3 \, a f g^{2}\right )} x\right )} e^{4} + {\left (2 \, c d f^{3} - 3 \, b d f^{2} g + a d f g^{2} - {\left (2 \, c d f^{2} g - b d f g^{2} + 3 \, a d g^{3}\right )} x\right )} e^{3} - {\left (c d^{2} f^{2} g - 3 \, b d^{2} f g^{2} + 2 \, a d^{2} g^{3} - {\left (c d^{2} f g^{2} + b d^{2} g^{3}\right )} x\right )} e^{2} - {\left (c d^{3} g^{3} x + c d^{3} f g^{2}\right )} e\right )}}{{\left (f^{3} g^{2} x^{2} + f^{4} g x\right )} e^{6} - {\left (3 \, d f^{2} g^{3} x^{2} + 2 \, d f^{3} g^{2} x - d f^{4} g\right )} e^{5} + 3 \, {\left (d^{2} f g^{4} x^{2} - d^{2} f^{3} g^{2}\right )} e^{4} - {\left (d^{3} g^{5} x^{2} - 2 \, d^{3} f g^{4} x - 3 \, d^{3} f^{2} g^{3}\right )} e^{3} - {\left (d^{4} g^{5} x + d^{4} f g^{4}\right )} e^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.49, size = 282, normalized size = 1.71 \begin {gather*} \frac {{\left (c d^{2} g - 4 \, c d f e + b d g e + 2 \, b f e^{2} - 3 \, a g e^{2}\right )} \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {d g e - f e^{2}}}\right )}{{\left (d^{2} g^{2} e - 2 \, d f g e^{2} + f^{2} e^{3}\right )} \sqrt {d g e - f e^{2}}} - \frac {{\left (g x + f\right )} c d^{2} g^{2} + 2 \, c d f^{2} g e - {\left (g x + f\right )} b d g^{2} e - 2 \, b d f g^{2} e + 2 \, a d g^{3} e + 2 \, {\left (g x + f\right )} c f^{2} e^{2} - 2 \, c f^{3} e^{2} - 2 \, {\left (g x + f\right )} b f g e^{2} + 2 \, b f^{2} g e^{2} + 3 \, {\left (g x + f\right )} a g^{2} e^{2} - 2 \, a f g^{2} e^{2}}{{\left (d^{2} g^{3} e - 2 \, d f g^{2} e^{2} + f^{2} g e^{3}\right )} {\left (\sqrt {g x + f} d g + {\left (g x + f\right )}^{\frac {3}{2}} e - \sqrt {g x + f} f e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.30, size = 218, normalized size = 1.32 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {f+g\,x}\,\left (d^2\,e\,g^2-2\,d\,e^2\,f\,g+e^3\,f^2\right )}{\sqrt {e}\,{\left (d\,g-e\,f\right )}^{5/2}}\right )\,\left (2\,b\,e^2\,f-3\,a\,e^2\,g+c\,d^2\,g+b\,d\,e\,g-4\,c\,d\,e\,f\right )}{e^{3/2}\,{\left (d\,g-e\,f\right )}^{5/2}}-\frac {\frac {2\,\left (c\,f^2-b\,f\,g+a\,g^2\right )}{d\,g-e\,f}+\frac {\left (f+g\,x\right )\,\left (c\,d^2\,g^2-b\,d\,e\,g^2+2\,c\,e^2\,f^2-2\,b\,e^2\,f\,g+3\,a\,e^2\,g^2\right )}{e\,{\left (d\,g-e\,f\right )}^2}}{\sqrt {f+g\,x}\,\left (d\,g^2-e\,f\,g\right )+e\,g\,{\left (f+g\,x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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