Optimal. Leaf size=177 \[ \frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-8 c^2 (a f+b e)+6 b c (2 a g+b f)-5 b^3 g+16 c^3 d\right )}{16 c^{7/2}}+\frac {\sqrt {a+b x+c x^2} \left (-16 a c g+15 b^2 g-18 b c f+24 c^2 e\right )}{24 c^3}+\frac {x \sqrt {a+b x+c x^2} (6 c f-5 b g)}{12 c^2}+\frac {g x^2 \sqrt {a+b x+c x^2}}{3 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.24, 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 = {1661, 640, 621, 206} \[ \frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-8 c^2 (a f+b e)+6 b c (2 a g+b f)-5 b^3 g+16 c^3 d\right )}{16 c^{7/2}}+\frac {\sqrt {a+b x+c x^2} \left (-16 a c g+15 b^2 g-18 b c f+24 c^2 e\right )}{24 c^3}+\frac {x \sqrt {a+b x+c x^2} (6 c f-5 b g)}{12 c^2}+\frac {g x^2 \sqrt {a+b x+c x^2}}{3 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 621
Rule 640
Rule 1661
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 ) \operatorname {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*}
________________________________________________________________________________________
Mathematica [A] time = 0.26, size = 141, normalized size = 0.80 \[ \frac {3 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right ) \left (-8 c^2 (a f+b e)+6 b c (2 a g+b f)-5 b^3 g+16 c^3 d\right )+2 \sqrt {c} \sqrt {a+x (b+c x)} \left (-2 c (8 a g+9 b f+5 b g x)+15 b^2 g+4 c^2 (6 e+x (3 f+2 g x))\right )}{48 c^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.76, size = 341, normalized size = 1.93 \[ \left [\frac {3 \, {\left (16 \, c^{3} d - 8 \, b c^{2} e + 2 \, {\left (3 \, b^{2} c - 4 \, a c^{2}\right )} f - {\left (5 \, b^{3} - 12 \, a b c\right )} g\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (8 \, c^{3} g x^{2} + 24 \, c^{3} e - 18 \, b c^{2} f + {\left (15 \, b^{2} c - 16 \, a c^{2}\right )} g + 2 \, {\left (6 \, c^{3} f - 5 \, b c^{2} g\right )} x\right )} \sqrt {c x^{2} + b x + a}}{96 \, c^{4}}, -\frac {3 \, {\left (16 \, c^{3} d - 8 \, b c^{2} e + 2 \, {\left (3 \, b^{2} c - 4 \, a c^{2}\right )} f - {\left (5 \, b^{3} - 12 \, a b c\right )} g\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (8 \, c^{3} g x^{2} + 24 \, c^{3} e - 18 \, b c^{2} f + {\left (15 \, b^{2} c - 16 \, a c^{2}\right )} g + 2 \, {\left (6 \, c^{3} f - 5 \, b c^{2} g\right )} x\right )} \sqrt {c x^{2} + b x + a}}{48 \, c^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.27, size = 149, normalized size = 0.84 \[ \frac {1}{24} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (\frac {4 \, g x}{c} + \frac {6 \, c^{2} f - 5 \, b c g}{c^{3}}\right )} x - \frac {18 \, b c f - 15 \, b^{2} g + 16 \, a c g - 24 \, c^{2} e}{c^{3}}\right )} - \frac {{\left (16 \, c^{3} d + 6 \, b^{2} c f - 8 \, a c^{2} f - 5 \, b^{3} g + 12 \, a b c g - 8 \, b c^{2} e\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{16 \, c^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.01, size = 333, normalized size = 1.88 \[ \frac {\sqrt {c \,x^{2}+b x +a}\, g \,x^{2}}{3 c}+\frac {3 a b g \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{4 c^{\frac {5}{2}}}-\frac {a f \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}-\frac {5 b^{3} g \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {7}{2}}}+\frac {3 b^{2} f \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {5}{2}}}-\frac {b e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}+\frac {d \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, b g x}{12 c^{2}}+\frac {\sqrt {c \,x^{2}+b x +a}\, f x}{2 c}-\frac {2 \sqrt {c \,x^{2}+b x +a}\, a g}{3 c^{2}}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, b^{2} g}{8 c^{3}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, b f}{4 c^{2}}+\frac {\sqrt {c \,x^{2}+b x +a}\, e}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {g\,x^3+f\,x^2+e\,x+d}{\sqrt {c\,x^2+b\,x+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {d + e x + f x^{2} + g x^{3}}{\sqrt {a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________