Optimal. Leaf size=178 \[ -\frac {\left (a+\frac {c d^2}{e^2}\right ) \sqrt {f+g x}}{2 (e f-d g) (d+e x)^2}+\frac {\left (3 a e^2 g+c d (8 e f-5 d g)\right ) \sqrt {f+g x}}{4 e^2 (e f-d g)^2 (d+e x)}-\frac {\left (3 a e^2 g^2+c \left (8 e^2 f^2-8 d e f g+3 d^2 g^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{4 e^{5/2} (e f-d g)^{5/2}} \]
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Rubi [A]
time = 0.19, antiderivative size = 182, normalized size of antiderivative = 1.02, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {912, 1171, 393,
214} \begin {gather*} -\frac {\sqrt {f+g x} \left (a e^2+c d^2\right )}{2 e^2 (d+e x)^2 (e f-d g)}-\frac {\left (3 a e^2 g^2+c \left (3 d^2 g^2-8 d e f g+8 e^2 f^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}}+\frac {\sqrt {f+g x} \left (3 a e^2 g+c d (8 e f-5 d g)\right )}{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 912
Rule 1171
Rubi steps
\begin {align*} \int \frac {a+c x^2}{(d+e x)^3 \sqrt {f+g x}} \, dx &=\frac {2 \text {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}}{\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 {c d^2}{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 {4 c f^2}{g^2}+\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 {c d^2}{e^2}\right ) \sqrt {f+g x}}{2 (e f-d g) (d+e x)^2}+\frac {\left (3 a e^2 g+c d (8 e f-5 d g)\right ) \sqrt {f+g x}}{4 e^2 (e f-d g)^2 (d+e x)}+\frac {\left (3 a e^2 g^2+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 {c d^2}{e^2}\right ) \sqrt {f+g x}}{2 (e f-d g) (d+e x)^2}+\frac {\left (3 a e^2 g+c d (8 e f-5 d g)\right ) \sqrt {f+g x}}{4 e^2 (e f-d g)^2 (d+e x)}-\frac {\left (3 a e^2 g^2+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 = 0.72, size = 166, normalized size = 0.93 \begin {gather*} \frac {\frac {\sqrt {e} \sqrt {f+g x} \left (a e^2 (-2 e f+5 d g+3 e g x)+c d \left (-3 d^2 g+8 e^2 f x+d e (6 f-5 g x)\right )\right )}{(e f-d g)^2 (d+e x)^2}+\frac {\left (3 a e^2 g^2+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 = 0.10, size = 220, normalized size = 1.24
method | result | size |
derivativedivides | \(\frac {\frac {g \left (3 a \,e^{2} g -5 c \,d^{2} g +8 c d e f \right ) \left (g x +f \right )^{\frac {3}{2}}}{4 e \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right )}+\frac {\left (5 a \,e^{2} g -3 c \,d^{2} g +8 c d e f \right ) g \sqrt {g x +f}}{4 e^{2} \left (d g -e f \right )}}{\left (e \left (g x +f \right )+d g -e f \right )^{2}}+\frac {\left (3 a \,e^{2} g^{2}+3 c \,d^{2} g^{2}-8 c d e f g +8 c \,e^{2} f^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{4 \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) e^{2} \sqrt {\left (d g -e f \right ) e}}\) | \(220\) |
default | \(\frac {\frac {g \left (3 a \,e^{2} g -5 c \,d^{2} g +8 c d e f \right ) \left (g x +f \right )^{\frac {3}{2}}}{4 e \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right )}+\frac {\left (5 a \,e^{2} g -3 c \,d^{2} g +8 c d e f \right ) g \sqrt {g x +f}}{4 e^{2} \left (d g -e f \right )}}{\left (e \left (g x +f \right )+d g -e f \right )^{2}}+\frac {\left (3 a \,e^{2} g^{2}+3 c \,d^{2} g^{2}-8 c d e f g +8 c \,e^{2} f^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{4 \left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) e^{2} \sqrt {\left (d g -e f \right ) e}}\) | \(220\) |
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. 427 vs.
\(2 (159) = 318\).
time = 3.34, size = 870, normalized size = 4.89 \begin {gather*} \left [-\frac {{\left (3 \, c d^{4} g^{2} + {\left (8 \, c f^{2} + 3 \, a g^{2}\right )} x^{2} e^{4} - 2 \, {\left (4 \, c d f g x^{2} - {\left (8 \, c d f^{2} + 3 \, a d g^{2}\right )} x\right )} e^{3} + {\left (3 \, c d^{2} g^{2} x^{2} - 16 \, c d^{2} f g x + 8 \, c d^{2} f^{2} + 3 \, a d^{2} g^{2}\right )} e^{2} + 2 \, {\left (3 \, c d^{3} g^{2} x - 4 \, c d^{3} f g\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 (3 \, a f g x - 2 \, a f^{2}\right )} e^{5} + {\left (7 \, a d f g + {\left (8 \, c d f^{2} - 3 \, a d g^{2}\right )} x\right )} e^{4} - {\left (13 \, c d^{2} f g x - 6 \, c d^{2} f^{2} + 5 \, a d^{2} g^{2}\right )} e^{3} + {\left (5 \, c d^{3} g^{2} x - 9 \, c d^{3} f g\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} + 3 \, a g^{2}\right )} x^{2} e^{4} - 2 \, {\left (4 \, c d f g x^{2} - {\left (8 \, c d f^{2} + 3 \, a d g^{2}\right )} x\right )} e^{3} + {\left (3 \, c d^{2} g^{2} x^{2} - 16 \, c d^{2} f g x + 8 \, c d^{2} f^{2} + 3 \, a d^{2} g^{2}\right )} e^{2} + 2 \, {\left (3 \, c d^{3} g^{2} x - 4 \, c d^{3} f g\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 (3 \, a f g x - 2 \, a f^{2}\right )} e^{5} + {\left (7 \, a d f g + {\left (8 \, c d f^{2} - 3 \, a d g^{2}\right )} x\right )} e^{4} - {\left (13 \, c d^{2} f g x - 6 \, c d^{2} f^{2} + 5 \, a d^{2} g^{2}\right )} e^{3} + {\left (5 \, c d^{3} g^{2} x - 9 \, c d^{3} f g\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 = 1.36, size = 278, normalized size = 1.56 \begin {gather*} \frac {{\left (3 \, c d^{2} g^{2} - 8 \, c d f g e + 8 \, c f^{2} 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 - 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} - 5 \, \sqrt {g x + f} a d g^{3} e^{2} - 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 = 2.91, size = 224, normalized size = 1.26 \begin {gather*} \frac {\frac {\sqrt {f+g\,x}\,\left (-3\,c\,d^2\,g^2+8\,c\,f\,d\,e\,g+5\,a\,e^2\,g^2\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+8\,c\,f\,d\,e\,g+3\,a\,e^2\,g^2\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}+\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+8\,c\,e^2\,f^2+3\,a\,e^2\,g^2\right )}{4\,e^{5/2}\,{\left (d\,g-e\,f\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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