Optimal. Leaf size=177 \[ \frac {\left (24 c^2 e-18 b c f+15 b^2 g-16 a c g\right ) \sqrt {a+b x+c x^2}}{24 c^3}+\frac {(6 c f-5 b g) x \sqrt {a+b x+c x^2}}{12 c^2}+\frac {g x^2 \sqrt {a+b x+c x^2}}{3 c}+\frac {\left (16 c^3 d-8 c^2 (b e+a f)-5 b^3 g+6 b c (b f+2 a g)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{7/2}} \]
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Rubi [A]
time = 0.14, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1675, 654, 635,
212} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-8 c^2 (a f+b e)+6 b c (2 a g+b f)-5 b^3 g+16 c^3 d\right )}{16 c^{7/2}}+\frac {\sqrt {a+b x+c x^2} \left (-16 a c g+15 b^2 g-18 b c f+24 c^2 e\right )}{24 c^3}+\frac {x \sqrt {a+b x+c x^2} (6 c f-5 b g)}{12 c^2}+\frac {g x^2 \sqrt {a+b x+c x^2}}{3 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 654
Rule 1675
Rubi steps
\begin {align*} \int \frac {d+e x+f x^2+g x^3}{\sqrt {a+b x+c x^2}} \, dx &=\frac {g x^2 \sqrt {a+b x+c x^2}}{3 c}+\frac {\int \frac {3 c d+(3 c e-2 a g) x+\frac {1}{2} (6 c f-5 b g) x^2}{\sqrt {a+b x+c x^2}} \, dx}{3 c}\\ &=\frac {(6 c f-5 b g) x \sqrt {a+b x+c x^2}}{12 c^2}+\frac {g x^2 \sqrt {a+b x+c x^2}}{3 c}+\frac {\int \frac {\frac {1}{2} \left (12 c^2 d-6 a c f+5 a b g\right )+\frac {1}{4} \left (24 c^2 e-18 b c f+15 b^2 g-16 a c g\right ) x}{\sqrt {a+b x+c x^2}} \, dx}{6 c^2}\\ &=\frac {\left (24 c^2 e-18 b c f+15 b^2 g-16 a c g\right ) \sqrt {a+b x+c x^2}}{24 c^3}+\frac {(6 c f-5 b g) x \sqrt {a+b x+c x^2}}{12 c^2}+\frac {g x^2 \sqrt {a+b x+c x^2}}{3 c}+\frac {\left (16 c^3 d-8 c^2 (b e+a f)-5 b^3 g+6 b c (b f+2 a g)\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16 c^3}\\ &=\frac {\left (24 c^2 e-18 b c f+15 b^2 g-16 a c g\right ) \sqrt {a+b x+c x^2}}{24 c^3}+\frac {(6 c f-5 b g) x \sqrt {a+b x+c x^2}}{12 c^2}+\frac {g x^2 \sqrt {a+b x+c x^2}}{3 c}+\frac {\left (16 c^3 d-8 c^2 (b e+a f)-5 b^3 g+6 b c (b f+2 a g)\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{8 c^3}\\ &=\frac {\left (24 c^2 e-18 b c f+15 b^2 g-16 a c g\right ) \sqrt {a+b x+c x^2}}{24 c^3}+\frac {(6 c f-5 b g) x \sqrt {a+b x+c x^2}}{12 c^2}+\frac {g x^2 \sqrt {a+b x+c x^2}}{3 c}+\frac {\left (16 c^3 d-8 c^2 (b e+a f)-5 b^3 g+6 b c (b f+2 a g)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.55, size = 139, normalized size = 0.79 \begin {gather*} \frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (15 b^2 g-2 c (9 b f+8 a g+5 b g x)+4 c^2 (6 e+x (3 f+2 g x))\right )+3 \left (-16 c^3 d+8 c^2 (b e+a f)+5 b^3 g-6 b c (b f+2 a g)\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{48 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(374\) vs.
\(2(155)=310\).
time = 0.15, size = 375, normalized size = 2.12
method | result | size |
risch | \(-\frac {\left (-8 g \,c^{2} x^{2}+10 b c g x -12 c^{2} f x +16 a c g -15 b^{2} g +18 b c f -24 c^{2} e \right ) \sqrt {c \,x^{2}+b x +a}}{24 c^{3}}+\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a b g}{4 c^{\frac {5}{2}}}-\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a f}{2 c^{\frac {3}{2}}}-\frac {5 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{3} g}{16 c^{\frac {7}{2}}}+\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{2} f}{8 c^{\frac {5}{2}}}-\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b e}{2 c^{\frac {3}{2}}}+\frac {d \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}\) | \(259\) |
default | \(g \left (\frac {x^{2} \sqrt {c \,x^{2}+b x +a}}{3 c}-\frac {5 b \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{6 c}-\frac {2 a \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{3 c}\right )+f \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )+e \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )+\frac {d \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}\) | \(375\) |
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 [A]
time = 0.44, size = 345, normalized size = 1.95 \begin {gather*} \left [-\frac {3 \, {\left (16 \, c^{3} d - 8 \, b c^{2} e + 2 \, {\left (3 \, b^{2} c - 4 \, a c^{2}\right )} f - {\left (5 \, b^{3} - 12 \, a b c\right )} g\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (8 \, c^{3} g x^{2} - 18 \, b c^{2} f + 24 \, c^{3} e + {\left (15 \, b^{2} c - 16 \, a c^{2}\right )} g + 2 \, {\left (6 \, c^{3} f - 5 \, b c^{2} g\right )} x\right )} \sqrt {c x^{2} + b x + a}}{96 \, c^{4}}, -\frac {3 \, {\left (16 \, c^{3} d - 8 \, b c^{2} e + 2 \, {\left (3 \, b^{2} c - 4 \, a c^{2}\right )} f - {\left (5 \, b^{3} - 12 \, a b c\right )} g\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (8 \, c^{3} g x^{2} - 18 \, b c^{2} f + 24 \, c^{3} e + {\left (15 \, b^{2} c - 16 \, a c^{2}\right )} g + 2 \, {\left (6 \, c^{3} f - 5 \, b c^{2} g\right )} x\right )} \sqrt {c x^{2} + b x + a}}{48 \, c^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d + e x + f x^{2} + g x^{3}}{\sqrt {a + b x + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.85, size = 149, normalized size = 0.84 \begin {gather*} \frac {1}{24} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (\frac {4 \, g x}{c} + \frac {6 \, c^{2} f - 5 \, b c g}{c^{3}}\right )} x - \frac {18 \, b c f - 15 \, b^{2} g + 16 \, a c g - 24 \, c^{2} e}{c^{3}}\right )} - \frac {{\left (16 \, c^{3} d + 6 \, b^{2} c f - 8 \, a c^{2} f - 5 \, b^{3} g + 12 \, a b c g - 8 \, b c^{2} e\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{16 \, c^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {g\,x^3+f\,x^2+e\,x+d}{\sqrt {c\,x^2+b\,x+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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