Optimal. Leaf size=206 \[ -\frac {\left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{2 (e f-d g) (d+e x)^2}+\frac {(c d (8 e f-5 d g)-e (4 b e f-b d g-3 a e g)) \sqrt {f+g x}}{4 e^2 (e f-d g)^2 (d+e x)}+\frac {\left (e g (4 b e f-b d g-3 a e g)-c \left (8 e^2 f^2-8 d e f g+3 d^2 g^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{4 e^{5/2} (e f-d g)^{5/2}} \]
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Rubi [A]
time = 0.24, antiderivative size = 210, normalized size of antiderivative = 1.02, 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, 1171, 393,
214} \begin {gather*} -\frac {\sqrt {f+g x} \left (a e^2-b d e+c d^2\right )}{2 e^2 (d+e x)^2 (e f-d g)}+\frac {\tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (e g (-3 a e g-b d g+4 b e f)-c \left (3 d^2 g^2-8 d e f g+8 e^2 f^2\right )\right )}{4 e^{5/2} (e f-d g)^{5/2}}+\frac {\sqrt {f+g x} (c d (8 e f-5 d g)-e (-3 a e g-b d g+4 b e f))}{4 e^2 (d+e x) (e f-d g)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 393
Rule 911
Rule 1171
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{(d+e x)^3 \sqrt {f+g x}} \, 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}}{\left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )^3} \, dx,x,\sqrt {f+g x}\right )}{g}\\ &=-\frac {\left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{2 (e f-d g) (d+e x)^2}+\frac {\text {Subst}\left (\int \frac {-3 a+\frac {c d^2}{e^2}-\frac {b d}{e}-\frac {4 c f^2}{g^2}+\frac {4 b f}{g}+\frac {4 c (e f-d g) x^2}{e g^2}}{\left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )^2} \, dx,x,\sqrt {f+g x}\right )}{2 (e f-d g)}\\ &=-\frac {\left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{2 (e f-d g) (d+e x)^2}+\frac {(c d (8 e f-5 d g)-e (4 b e f-b d g-3 a e g)) \sqrt {f+g x}}{4 e^2 (e f-d g)^2 (d+e x)}-\frac {\left (e g (4 b e f-b d g-3 a e g)-c \left (8 e^2 f^2-8 d e f g+3 d^2 g^2\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {-e f+d g}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{4 e^2 g (e f-d g)^2}\\ &=-\frac {\left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{2 (e f-d g) (d+e x)^2}+\frac {(c d (8 e f-5 d g)-e (4 b e f-b d g-3 a e g)) \sqrt {f+g x}}{4 e^2 (e f-d g)^2 (d+e x)}+\frac {\left (e g (4 b e f-b d g-3 a e g)-c \left (8 e^2 f^2-8 d e f g+3 d^2 g^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{4 e^{5/2} (e f-d g)^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.83, size = 203, normalized size = 0.99 \begin {gather*} \frac {\frac {\sqrt {e} \sqrt {f+g x} \left (c d \left (-3 d^2 g+8 e^2 f x+d e (6 f-5 g x)\right )+e \left (a e (-2 e f+5 d g+3 e g x)-b \left (2 d e f+d^2 g+4 e^2 f x-d e g x\right )\right )\right )}{(e f-d g)^2 (d+e x)^2}+\frac {\left (e g (-4 b e f+b d g+3 a e g)+c \left (8 e^2 f^2-8 d e f g+3 d^2 g^2\right )\right ) \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {-e f+d g}}\right )}{(-e f+d g)^{5/2}}}{4 e^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 260, normalized size = 1.26
method | result | size |
derivativedivides | \(\frac {\frac {g \left (3 a \,e^{2} g +b d e g -4 b \,e^{2} f -5 c \,d^{2} g +8 c d e f \right ) \left (g x +f \right )^{\frac {3}{2}}}{4 e \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right )}+\frac {\left (5 a \,e^{2} g -b d e g -4 b \,e^{2} f -3 c \,d^{2} g +8 c d e f \right ) g \sqrt {g x +f}}{4 e^{2} \left (d g -e f \right )}}{\left (e \left (g x +f \right )+d g -e f \right )^{2}}+\frac {\left (3 a \,e^{2} g^{2}+b d e \,g^{2}-4 b \,e^{2} f g +3 c \,d^{2} g^{2}-8 c d e f g +8 c \,e^{2} f^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{4 \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) e^{2} \sqrt {\left (d g -e f \right ) e}}\) | \(260\) |
default | \(\frac {\frac {g \left (3 a \,e^{2} g +b d e g -4 b \,e^{2} f -5 c \,d^{2} g +8 c d e f \right ) \left (g x +f \right )^{\frac {3}{2}}}{4 e \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right )}+\frac {\left (5 a \,e^{2} g -b d e g -4 b \,e^{2} f -3 c \,d^{2} g +8 c d e f \right ) g \sqrt {g x +f}}{4 e^{2} \left (d g -e f \right )}}{\left (e \left (g x +f \right )+d g -e f \right )^{2}}+\frac {\left (3 a \,e^{2} g^{2}+b d e \,g^{2}-4 b \,e^{2} f g +3 c \,d^{2} g^{2}-8 c d e f g +8 c \,e^{2} f^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{4 \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) e^{2} \sqrt {\left (d g -e f \right ) e}}\) | \(260\) |
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. 523 vs.
\(2 (193) = 386\).
time = 1.76, size = 1062, normalized size = 5.16 \begin {gather*} \left [-\frac {{\left (3 \, c d^{4} g^{2} + {\left (8 \, c f^{2} - 4 \, b f g + 3 \, a g^{2}\right )} x^{2} e^{4} - {\left ({\left (8 \, c d f g - b d g^{2}\right )} x^{2} - 2 \, {\left (8 \, c d f^{2} - 4 \, b d f g + 3 \, a d g^{2}\right )} x\right )} e^{3} + {\left (3 \, c d^{2} g^{2} x^{2} + 8 \, c d^{2} f^{2} - 4 \, b d^{2} f g + 3 \, a d^{2} g^{2} - 2 \, {\left (8 \, c d^{2} f g - b d^{2} g^{2}\right )} x\right )} e^{2} + {\left (6 \, c d^{3} g^{2} x - 8 \, c d^{3} f g + b d^{3} g^{2}\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 \, {\left (3 \, c d^{4} g^{2} e - {\left (2 \, a f^{2} + {\left (4 \, b f^{2} - 3 \, a f g\right )} x\right )} e^{5} - {\left (2 \, b d f^{2} - 7 \, a d f g - {\left (8 \, c d f^{2} + 5 \, b d f g - 3 \, a d g^{2}\right )} x\right )} e^{4} + {\left (6 \, c d^{2} f^{2} + b d^{2} f g - 5 \, a d^{2} g^{2} - {\left (13 \, c d^{2} f g + b d^{2} g^{2}\right )} x\right )} e^{3} + {\left (5 \, c d^{3} g^{2} x - 9 \, c d^{3} f g + b d^{3} g^{2}\right )} e^{2}\right )} \sqrt {g x + f}}{8 \, {\left (d^{5} g^{3} e^{3} - f^{3} x^{2} e^{8} + {\left (3 \, d f^{2} g x^{2} - 2 \, d f^{3} x\right )} e^{7} - {\left (3 \, d^{2} f g^{2} x^{2} - 6 \, d^{2} f^{2} g x + d^{2} f^{3}\right )} e^{6} + {\left (d^{3} g^{3} x^{2} - 6 \, d^{3} f g^{2} x + 3 \, d^{3} f^{2} g\right )} e^{5} + {\left (2 \, d^{4} g^{3} x - 3 \, d^{4} f g^{2}\right )} e^{4}\right )}}, -\frac {{\left (3 \, c d^{4} g^{2} + {\left (8 \, c f^{2} - 4 \, b f g + 3 \, a g^{2}\right )} x^{2} e^{4} - {\left ({\left (8 \, c d f g - b d g^{2}\right )} x^{2} - 2 \, {\left (8 \, c d f^{2} - 4 \, b d f g + 3 \, a d g^{2}\right )} x\right )} e^{3} + {\left (3 \, c d^{2} g^{2} x^{2} + 8 \, c d^{2} f^{2} - 4 \, b d^{2} f g + 3 \, a d^{2} g^{2} - 2 \, {\left (8 \, c d^{2} f g - b d^{2} g^{2}\right )} x\right )} e^{2} + {\left (6 \, c d^{3} g^{2} x - 8 \, c d^{3} f g + b d^{3} g^{2}\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 ) + {\left (3 \, c d^{4} g^{2} e - {\left (2 \, a f^{2} + {\left (4 \, b f^{2} - 3 \, a f g\right )} x\right )} e^{5} - {\left (2 \, b d f^{2} - 7 \, a d f g - {\left (8 \, c d f^{2} + 5 \, b d f g - 3 \, a d g^{2}\right )} x\right )} e^{4} + {\left (6 \, c d^{2} f^{2} + b d^{2} f g - 5 \, a d^{2} g^{2} - {\left (13 \, c d^{2} f g + b d^{2} g^{2}\right )} x\right )} e^{3} + {\left (5 \, c d^{3} g^{2} x - 9 \, c d^{3} f g + b d^{3} g^{2}\right )} e^{2}\right )} \sqrt {g x + f}}{4 \, {\left (d^{5} g^{3} e^{3} - f^{3} x^{2} e^{8} + {\left (3 \, d f^{2} g x^{2} - 2 \, d f^{3} x\right )} e^{7} - {\left (3 \, d^{2} f g^{2} x^{2} - 6 \, d^{2} f^{2} g x + d^{2} f^{3}\right )} e^{6} + {\left (d^{3} g^{3} x^{2} - 6 \, d^{3} f g^{2} x + 3 \, d^{3} f^{2} g\right )} e^{5} + {\left (2 \, d^{4} g^{3} x - 3 \, d^{4} f g^{2}\right )} e^{4}\right )}}\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 = 4.01, size = 373, normalized size = 1.81 \begin {gather*} \frac {{\left (3 \, c d^{2} g^{2} - 8 \, c d f g e + b d g^{2} e + 8 \, c f^{2} e^{2} - 4 \, b f g e^{2} + 3 \, a g^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {d g e - f e^{2}}}\right )}{4 \, {\left (d^{2} g^{2} e^{2} - 2 \, d f g e^{3} + f^{2} e^{4}\right )} \sqrt {d g e - f e^{2}}} - \frac {3 \, \sqrt {g x + f} c d^{3} g^{3} + 5 \, {\left (g x + f\right )}^{\frac {3}{2}} c d^{2} g^{2} e - 11 \, \sqrt {g x + f} c d^{2} f g^{2} e + \sqrt {g x + f} b d^{2} g^{3} e - 8 \, {\left (g x + f\right )}^{\frac {3}{2}} c d f g e^{2} + 8 \, \sqrt {g x + f} c d f^{2} g e^{2} - {\left (g x + f\right )}^{\frac {3}{2}} b d g^{2} e^{2} + 3 \, \sqrt {g x + f} b d f g^{2} e^{2} - 5 \, \sqrt {g x + f} a d g^{3} e^{2} + 4 \, {\left (g x + f\right )}^{\frac {3}{2}} b f g e^{3} - 4 \, \sqrt {g x + f} b f^{2} g e^{3} - 3 \, {\left (g x + f\right )}^{\frac {3}{2}} a g^{2} e^{3} + 5 \, \sqrt {g x + f} a f g^{2} e^{3}}{4 \, {\left (d^{2} g^{2} e^{2} - 2 \, d f g e^{3} + f^{2} e^{4}\right )} {\left (d g + {\left (g x + f\right )} e - f e\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.28, size = 270, normalized size = 1.31 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {e}\,\sqrt {f+g\,x}}{\sqrt {d\,g-e\,f}}\right )\,\left (3\,c\,d^2\,g^2-8\,c\,d\,e\,f\,g+b\,d\,e\,g^2+8\,c\,e^2\,f^2-4\,b\,e^2\,f\,g+3\,a\,e^2\,g^2\right )}{4\,e^{5/2}\,{\left (d\,g-e\,f\right )}^{5/2}}-\frac {\frac {\sqrt {f+g\,x}\,\left (3\,c\,d^2\,g^2+b\,d\,e\,g^2-8\,c\,f\,d\,e\,g-5\,a\,e^2\,g^2+4\,b\,f\,e^2\,g\right )}{4\,e^2\,\left (d\,g-e\,f\right )}-\frac {{\left (f+g\,x\right )}^{3/2}\,\left (-5\,c\,d^2\,g^2+b\,d\,e\,g^2+8\,c\,f\,d\,e\,g+3\,a\,e^2\,g^2-4\,b\,f\,e^2\,g\right )}{4\,e\,{\left (d\,g-e\,f\right )}^2}}{e^2\,{\left (f+g\,x\right )}^2-\left (f+g\,x\right )\,\left (2\,e^2\,f-2\,d\,e\,g\right )+d^2\,g^2+e^2\,f^2-2\,d\,e\,f\,g} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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