Optimal. Leaf size=175 \[ -\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c (a f+2 b e)+5 b^2 f+16 c^2 d\right )}{128 c^{7/2}}+\frac {(b+2 c x) \sqrt {a+b x+c x^2} \left (-4 a c f+5 b^2 f-8 b c e+16 c^2 d\right )}{64 c^3}+\frac {\left (a+b x+c x^2\right )^{3/2} (8 c e-5 b f)}{24 c^2}+\frac {f x \left (a+b x+c x^2\right )^{3/2}}{4 c} \]
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Rubi [A] time = 0.16, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1661, 640, 612, 621, 206} \[ \frac {(b+2 c x) \sqrt {a+b x+c x^2} \left (-4 a c f+5 b^2 f-8 b c e+16 c^2 d\right )}{64 c^3}-\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c (a f+2 b e)+5 b^2 f+16 c^2 d\right )}{128 c^{7/2}}+\frac {\left (a+b x+c x^2\right )^{3/2} (8 c e-5 b f)}{24 c^2}+\frac {f x \left (a+b x+c x^2\right )^{3/2}}{4 c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 640
Rule 1661
Rubi steps
\begin {align*} \int \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right ) \, dx &=\frac {f x \left (a+b x+c x^2\right )^{3/2}}{4 c}+\frac {\int \left (4 c d-a f+\frac {1}{2} (8 c e-5 b f) x\right ) \sqrt {a+b x+c x^2} \, dx}{4 c}\\ &=\frac {(8 c e-5 b f) \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac {f x \left (a+b x+c x^2\right )^{3/2}}{4 c}+\frac {\left (16 c^2 d-8 b c e+5 b^2 f-4 a c f\right ) \int \sqrt {a+b x+c x^2} \, dx}{16 c^2}\\ &=\frac {\left (16 c^2 d-8 b c e+5 b^2 f-4 a c f\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{64 c^3}+\frac {(8 c e-5 b f) \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac {f x \left (a+b x+c x^2\right )^{3/2}}{4 c}-\frac {\left (\left (b^2-4 a c\right ) \left (16 c^2 d+5 b^2 f-4 c (2 b e+a f)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{128 c^3}\\ &=\frac {\left (16 c^2 d-8 b c e+5 b^2 f-4 a c f\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{64 c^3}+\frac {(8 c e-5 b f) \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac {f x \left (a+b x+c x^2\right )^{3/2}}{4 c}-\frac {\left (\left (b^2-4 a c\right ) \left (16 c^2 d+5 b^2 f-4 c (2 b e+a f)\right )\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 )}{64 c^3}\\ &=\frac {\left (16 c^2 d-8 b c e+5 b^2 f-4 a c f\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{64 c^3}+\frac {(8 c e-5 b f) \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac {f x \left (a+b x+c x^2\right )^{3/2}}{4 c}-\frac {\left (b^2-4 a c\right ) \left (16 c^2 d+5 b^2 f-4 c (2 b e+a f)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 173, normalized size = 0.99 \[ \frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (4 b c \left (2 c \left (6 d+2 e x+f x^2\right )-13 a f\right )+8 c^2 \left (a (8 e+3 f x)+2 c x \left (6 d+4 e x+3 f x^2\right )\right )+15 b^3 f-2 b^2 c (12 e+5 f x)\right )-3 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right ) \left (-4 c (a f+2 b e)+5 b^2 f+16 c^2 d\right )}{384 c^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 465, normalized size = 2.66 \[ \left [\frac {3 \, {\left (16 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d - 8 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} e + {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} f\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (48 \, c^{4} f x^{3} + 48 \, b c^{3} d + 8 \, {\left (8 \, c^{4} e + b c^{3} f\right )} x^{2} - 8 \, {\left (3 \, b^{2} c^{2} - 8 \, a c^{3}\right )} e + {\left (15 \, b^{3} c - 52 \, a b c^{2}\right )} f + 2 \, {\left (48 \, c^{4} d + 8 \, b c^{3} e - {\left (5 \, b^{2} c^{2} - 12 \, a c^{3}\right )} f\right )} x\right )} \sqrt {c x^{2} + b x + a}}{768 \, c^{4}}, \frac {3 \, {\left (16 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d - 8 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} e + {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} f\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (48 \, c^{4} f x^{3} + 48 \, b c^{3} d + 8 \, {\left (8 \, c^{4} e + b c^{3} f\right )} x^{2} - 8 \, {\left (3 \, b^{2} c^{2} - 8 \, a c^{3}\right )} e + {\left (15 \, b^{3} c - 52 \, a b c^{2}\right )} f + 2 \, {\left (48 \, c^{4} d + 8 \, b c^{3} e - {\left (5 \, b^{2} c^{2} - 12 \, a c^{3}\right )} f\right )} x\right )} \sqrt {c x^{2} + b x + a}}{384 \, c^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 212, normalized size = 1.21 \[ \frac {1}{192} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (6 \, f x + \frac {b c^{2} f + 8 \, c^{3} e}{c^{3}}\right )} x + \frac {48 \, c^{3} d - 5 \, b^{2} c f + 12 \, a c^{2} f + 8 \, b c^{2} e}{c^{3}}\right )} x + \frac {48 \, b c^{2} d + 15 \, b^{3} f - 52 \, a b c f - 24 \, b^{2} c e + 64 \, a c^{2} e}{c^{3}}\right )} + \frac {{\left (16 \, b^{2} c^{2} d - 64 \, a c^{3} d + 5 \, b^{4} f - 24 \, a b^{2} c f + 16 \, a^{2} c^{2} f - 8 \, b^{3} c e + 32 \, a b c^{2} e\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{128 \, c^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 453, normalized size = 2.59 \[ -\frac {a^{2} f \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}+\frac {3 a \,b^{2} f \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {5}{2}}}-\frac {a b e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{4 c^{\frac {3}{2}}}+\frac {a d \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}-\frac {5 b^{4} f \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {7}{2}}}+\frac {b^{3} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {5}{2}}}-\frac {b^{2} d \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}-\frac {\sqrt {c \,x^{2}+b x +a}\, a f x}{8 c}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, b^{2} f x}{32 c^{2}}-\frac {\sqrt {c \,x^{2}+b x +a}\, b e x}{4 c}+\frac {\sqrt {c \,x^{2}+b x +a}\, d x}{2}-\frac {\sqrt {c \,x^{2}+b x +a}\, a b f}{16 c^{2}}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, b^{3} f}{64 c^{3}}-\frac {\sqrt {c \,x^{2}+b x +a}\, b^{2} e}{8 c^{2}}+\frac {\sqrt {c \,x^{2}+b x +a}\, b d}{4 c}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} f x}{4 c}-\frac {5 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b f}{24 c^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} e}{3 c} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.91, size = 320, normalized size = 1.83 \[ d\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}-\frac {a\,f\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}+\frac {d\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}+\frac {e\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}-\frac {5\,b\,f\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}+\frac {e\,\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}+\frac {f\,x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c} \]
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 \[ \int \sqrt {a + b x + c x^{2}} \left (d + e x + f x^{2}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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