Optimal. Leaf size=436 \[ \frac {\left (128 c^4 d^2+21 b^4 f^2-56 b^2 c f (b e+a f)-32 c^3 \left (4 b d e+a \left (e^2+2 d f\right )\right )+8 c^2 \left (12 a b e f+2 a^2 f^2+5 b^2 \left (e^2+2 d f\right )\right )\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^5}+\frac {\left (640 c^3 d e-105 b^3 f^2+28 b c f (10 b e+7 a f)-8 c^2 \left (32 a e f+25 b \left (e^2+2 d f\right )\right )\right ) \left (a+b x+c x^2\right )^{3/2}}{960 c^4}+\frac {\left (21 b^2 f^2-4 c f (14 b e+5 a f)+40 c^2 \left (e^2+2 d f\right )\right ) x \left (a+b x+c x^2\right )^{3/2}}{160 c^3}+\frac {f (8 c e-3 b f) x^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}-\frac {\left (b^2-4 a c\right ) \left (128 c^4 d^2+21 b^4 f^2-56 b^2 c f (b e+a f)-32 c^3 \left (4 b d e+a \left (e^2+2 d f\right )\right )+8 c^2 \left (12 a b e f+2 a^2 f^2+5 b^2 \left (e^2+2 d f\right )\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{11/2}} \]
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Rubi [A]
time = 0.50, antiderivative size = 436, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1675, 654, 626,
635, 212} \begin {gather*} -\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 (8 c^2 \left (2 a^2 f^2+12 a b e f+5 b^2 \left (2 d f+e^2\right )\right )-56 b^2 c f (a f+b e)-32 c^3 \left (a \left (2 d f+e^2\right )+4 b d e\right )+21 b^4 f^2+128 c^4 d^2\right )}{1024 c^{11/2}}+\frac {(b+2 c x) \sqrt {a+b x+c x^2} \left (8 c^2 \left (2 a^2 f^2+12 a b e f+5 b^2 \left (2 d f+e^2\right )\right )-56 b^2 c f (a f+b e)-32 c^3 \left (a \left (2 d f+e^2\right )+4 b d e\right )+21 b^4 f^2+128 c^4 d^2\right )}{512 c^5}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-8 c^2 \left (32 a e f+25 b \left (2 d f+e^2\right )\right )+28 b c f (7 a f+10 b e)-105 b^3 f^2+640 c^3 d e\right )}{960 c^4}+\frac {x \left (a+b x+c x^2\right )^{3/2} \left (-4 c f (5 a f+14 b e)+21 b^2 f^2+40 c^2 \left (2 d f+e^2\right )\right )}{160 c^3}+\frac {f x^2 \left (a+b x+c x^2\right )^{3/2} (8 c e-3 b f)}{20 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 626
Rule 635
Rule 654
Rule 1675
Rubi steps
\begin {align*} \int \sqrt {a+b x+c x^2} \left (d+e x+f x^2\right )^2 \, dx &=\frac {f^2 x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}+\frac {\int \sqrt {a+b x+c x^2} \left (6 c d^2+12 c d e x-3 \left (a f^2-2 c \left (e^2+2 d f\right )\right ) x^2+\frac {3}{2} f (8 c e-3 b f) x^3\right ) \, dx}{6 c}\\ &=\frac {f (8 c e-3 b f) x^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}+\frac {\int \sqrt {a+b x+c x^2} \left (30 c^2 d^2+3 \left (20 c^2 d e-8 a c e f+3 a b f^2\right ) x+\frac {3}{4} \left (21 b^2 f^2-4 c f (14 b e+5 a f)+40 c^2 \left (e^2+2 d f\right )\right ) x^2\right ) \, dx}{30 c^2}\\ &=\frac {\left (21 b^2 f^2-4 c f (14 b e+5 a f)+40 c^2 \left (e^2+2 d f\right )\right ) x \left (a+b x+c x^2\right )^{3/2}}{160 c^3}+\frac {f (8 c e-3 b f) x^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}+\frac {\int \left (\frac {3}{4} \left (160 c^3 d^2-21 a b^2 f^2+4 a c f (14 b e+5 a f)-40 a c^2 \left (e^2+2 d f\right )\right )+\frac {3}{8} \left (640 c^3 d e-105 b^3 f^2+28 b c f (10 b e+7 a f)-8 c^2 \left (32 a e f+25 b \left (e^2+2 d f\right )\right )\right ) x\right ) \sqrt {a+b x+c x^2} \, dx}{120 c^3}\\ &=\frac {\left (640 c^3 d e-105 b^3 f^2+28 b c f (10 b e+7 a f)-8 c^2 \left (32 a e f+25 b \left (e^2+2 d f\right )\right )\right ) \left (a+b x+c x^2\right )^{3/2}}{960 c^4}+\frac {\left (21 b^2 f^2-4 c f (14 b e+5 a f)+40 c^2 \left (e^2+2 d f\right )\right ) x \left (a+b x+c x^2\right )^{3/2}}{160 c^3}+\frac {f (8 c e-3 b f) x^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}+\frac {\left (128 c^4 d^2+21 b^4 f^2-56 b^2 c f (b e+a f)-32 c^3 \left (4 b d e+a \left (e^2+2 d f\right )\right )+8 c^2 \left (12 a b e f+2 a^2 f^2+5 b^2 \left (e^2+2 d f\right )\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{128 c^4}\\ &=\frac {\left (128 c^4 d^2+21 b^4 f^2-56 b^2 c f (b e+a f)-32 c^3 \left (4 b d e+a \left (e^2+2 d f\right )\right )+8 c^2 \left (12 a b e f+2 a^2 f^2+5 b^2 \left (e^2+2 d f\right )\right )\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^5}+\frac {\left (640 c^3 d e-105 b^3 f^2+28 b c f (10 b e+7 a f)-8 c^2 \left (32 a e f+25 b \left (e^2+2 d f\right )\right )\right ) \left (a+b x+c x^2\right )^{3/2}}{960 c^4}+\frac {\left (21 b^2 f^2-4 c f (14 b e+5 a f)+40 c^2 \left (e^2+2 d f\right )\right ) x \left (a+b x+c x^2\right )^{3/2}}{160 c^3}+\frac {f (8 c e-3 b f) x^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}-\frac {\left (\left (b^2-4 a c\right ) \left (128 c^4 d^2+21 b^4 f^2-56 b^2 c f (b e+a f)-32 c^3 \left (4 b d e+a \left (e^2+2 d f\right )\right )+8 c^2 \left (12 a b e f+2 a^2 f^2+5 b^2 \left (e^2+2 d f\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{1024 c^5}\\ &=\frac {\left (128 c^4 d^2+21 b^4 f^2-56 b^2 c f (b e+a f)-32 c^3 \left (4 b d e+a \left (e^2+2 d f\right )\right )+8 c^2 \left (12 a b e f+2 a^2 f^2+5 b^2 \left (e^2+2 d f\right )\right )\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^5}+\frac {\left (640 c^3 d e-105 b^3 f^2+28 b c f (10 b e+7 a f)-8 c^2 \left (32 a e f+25 b \left (e^2+2 d f\right )\right )\right ) \left (a+b x+c x^2\right )^{3/2}}{960 c^4}+\frac {\left (21 b^2 f^2-4 c f (14 b e+5 a f)+40 c^2 \left (e^2+2 d f\right )\right ) x \left (a+b x+c x^2\right )^{3/2}}{160 c^3}+\frac {f (8 c e-3 b f) x^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}-\frac {\left (\left (b^2-4 a c\right ) \left (128 c^4 d^2+21 b^4 f^2-56 b^2 c f (b e+a f)-32 c^3 \left (4 b d e+a \left (e^2+2 d f\right )\right )+8 c^2 \left (12 a b e f+2 a^2 f^2+5 b^2 \left (e^2+2 d f\right )\right )\right )\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 )}{512 c^5}\\ &=\frac {\left (128 c^4 d^2+21 b^4 f^2-56 b^2 c f (b e+a f)-32 c^3 \left (4 b d e+a \left (e^2+2 d f\right )\right )+8 c^2 \left (12 a b e f+2 a^2 f^2+5 b^2 \left (e^2+2 d f\right )\right )\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^5}+\frac {\left (640 c^3 d e-105 b^3 f^2+28 b c f (10 b e+7 a f)-8 c^2 \left (32 a e f+25 b \left (e^2+2 d f\right )\right )\right ) \left (a+b x+c x^2\right )^{3/2}}{960 c^4}+\frac {\left (21 b^2 f^2-4 c f (14 b e+5 a f)+40 c^2 \left (e^2+2 d f\right )\right ) x \left (a+b x+c x^2\right )^{3/2}}{160 c^3}+\frac {f (8 c e-3 b f) x^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2}+\frac {f^2 x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}-\frac {\left (b^2-4 a c\right ) \left (128 c^4 d^2+21 b^4 f^2-56 b^2 c f (b e+a f)-32 c^3 \left (4 b d e+a \left (e^2+2 d f\right )\right )+8 c^2 \left (12 a b e f+2 a^2 f^2+5 b^2 \left (e^2+2 d f\right )\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{11/2}}\\ \end {align*}
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Mathematica [A]
time = 1.98, size = 456, normalized size = 1.05 \begin {gather*} \frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (315 b^5 f^2-210 b^4 c f (4 e+f x)-16 b^2 c^2 \left (-2 a f (115 e+28 f x)+c \left (120 d e+25 e^2 x+50 d f x+28 e f x^2+9 f^2 x^3\right )\right )+8 b^3 c \left (-210 a f^2+c \left (75 e^2+70 e f x+3 f \left (50 d+7 f x^2\right )\right )\right )+16 b c^2 \left (113 a^2 f^2-2 a c \left (65 e^2+58 e f x+f \left (130 d+17 f x^2\right )\right )+4 c^2 \left (30 d^2+10 d x (2 e+f x)+x^2 \left (5 e^2+6 e f x+2 f^2 x^2\right )\right )\right )-32 c^3 \left (a^2 f (64 e+15 f x)-2 a c \left (80 d e+15 e^2 x+30 d f x+16 e f x^2+5 f^2 x^3\right )-4 c^2 x \left (30 d^2+10 d x (4 e+3 f x)+x^2 \left (15 e^2+24 e f x+10 f^2 x^2\right )\right )\right )\right )+15 \left (b^2-4 a c\right ) \left (128 c^4 d^2+21 b^4 f^2-56 b^2 c f (b e+a f)-32 c^3 \left (4 b d e+a \left (e^2+2 d f\right )\right )+8 c^2 \left (12 a b e f+2 a^2 f^2+5 b^2 \left (e^2+2 d f\right )\right )\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{15360 c^{11/2}} \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. \(1185\) vs.
\(2(406)=812\).
time = 0.20, size = 1186, normalized size = 2.72
method | result | size |
risch | \(\frac {\left (1280 f^{2} c^{5} x^{5}+128 b \,c^{4} f^{2} x^{4}+3072 c^{5} e f \,x^{4}+320 a \,c^{4} f^{2} x^{3}-144 b^{2} c^{3} f^{2} x^{3}+384 b \,c^{4} e f \,x^{3}+3840 c^{5} d f \,x^{3}+1920 c^{5} e^{2} x^{3}-544 a b \,c^{3} f^{2} x^{2}+1024 a \,c^{4} e f \,x^{2}+168 b^{3} c^{2} f^{2} x^{2}-448 b^{2} c^{3} e f \,x^{2}+640 b \,c^{4} d f \,x^{2}+320 b \,c^{4} e^{2} x^{2}+5120 c^{5} d e \,x^{2}-480 a^{2} c^{3} f^{2} x +896 a \,b^{2} c^{2} f^{2} x -1856 a b \,c^{3} e f x +1920 a \,c^{4} d f x +960 a \,c^{4} e^{2} x -210 b^{4} c \,f^{2} x +560 b^{3} c^{2} e f x -800 b^{2} c^{3} d f x -400 b^{2} c^{3} e^{2} x +1280 b \,c^{4} d e x +3840 c^{5} d^{2} x +1808 a^{2} b \,c^{2} f^{2}-2048 a^{2} c^{3} e f -1680 a \,b^{3} c \,f^{2}+3680 a \,b^{2} c^{2} e f -4160 a b \,c^{3} d f -2080 a b \,c^{3} e^{2}+5120 a \,c^{4} d e +315 b^{5} f^{2}-840 b^{4} c e f +1200 b^{3} c^{2} d f +600 b^{3} c^{2} e^{2}-1920 b^{2} c^{3} d e +1920 b \,c^{4} d^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{7680 c^{5}}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{3} f^{2}}{16 c^{\frac {5}{2}}}-\frac {15 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{2} b^{2} f^{2}}{64 c^{\frac {7}{2}}}+\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{2} b e 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 ) a^{2} d f}{4 c^{\frac {3}{2}}}-\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a^{2} e^{2}}{8 c^{\frac {3}{2}}}+\frac {35 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a \,b^{4} f^{2}}{256 c^{\frac {9}{2}}}-\frac {5 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a \,b^{3} e f}{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 ) a \,b^{2} d f}{8 c^{\frac {5}{2}}}+\frac {3 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a \,b^{2} e^{2}}{16 c^{\frac {5}{2}}}-\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a b d e}{2 c^{\frac {3}{2}}}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) a \,d^{2}}{2 \sqrt {c}}-\frac {21 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{6} f^{2}}{1024 c^{\frac {11}{2}}}+\frac {7 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{5} e f}{128 c^{\frac {9}{2}}}-\frac {5 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{4} d f}{64 c^{\frac {7}{2}}}-\frac {5 \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{4} e^{2}}{128 c^{\frac {7}{2}}}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) b^{3} d e}{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^{2} d^{2}}{8 c^{\frac {3}{2}}}\) | \(1046\) |
default | \(\text {Expression too large to display}\) | \(1186\) |
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 = 3.81, size = 1267, normalized size = 2.91 \begin {gather*} \left [\frac {15 \, {\left (128 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{2} + 16 \, {\left (5 \, b^{4} c^{2} - 24 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d f + {\left (21 \, b^{6} - 140 \, a b^{4} c + 240 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} f^{2} + 8 \, {\left (5 \, b^{4} c^{2} - 24 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} e^{2} - 8 \, {\left (16 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d + {\left (7 \, b^{5} c - 40 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} f\right )} e\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 (1280 \, c^{6} f^{2} x^{5} + 128 \, b c^{5} f^{2} x^{4} + 1920 \, b c^{5} d^{2} + 16 \, {\left (240 \, c^{6} d f - {\left (9 \, b^{2} c^{4} - 20 \, a c^{5}\right )} f^{2}\right )} x^{3} + 80 \, {\left (15 \, b^{3} c^{3} - 52 \, a b c^{4}\right )} d f + {\left (315 \, b^{5} c - 1680 \, a b^{3} c^{2} + 1808 \, a^{2} b c^{3}\right )} f^{2} + 8 \, {\left (80 \, b c^{5} d f + {\left (21 \, b^{3} c^{3} - 68 \, a b c^{4}\right )} f^{2}\right )} x^{2} + 2 \, {\left (1920 \, c^{6} d^{2} - 80 \, {\left (5 \, b^{2} c^{4} - 12 \, a c^{5}\right )} d f - {\left (105 \, b^{4} c^{2} - 448 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} f^{2}\right )} x + 40 \, {\left (48 \, c^{6} x^{3} + 8 \, b c^{5} x^{2} + 15 \, b^{3} c^{3} - 52 \, a b c^{4} - 2 \, {\left (5 \, b^{2} c^{4} - 12 \, a c^{5}\right )} x\right )} e^{2} + 8 \, {\left (384 \, c^{6} f x^{4} + 48 \, b c^{5} f x^{3} + 8 \, {\left (80 \, c^{6} d - {\left (7 \, b^{2} c^{4} - 16 \, a c^{5}\right )} f\right )} x^{2} - 80 \, {\left (3 \, b^{2} c^{4} - 8 \, a c^{5}\right )} d - {\left (105 \, b^{4} c^{2} - 460 \, a b^{2} c^{3} + 256 \, a^{2} c^{4}\right )} f + 2 \, {\left (80 \, b c^{5} d + {\left (35 \, b^{3} c^{3} - 116 \, a b c^{4}\right )} f\right )} x\right )} e\right )} \sqrt {c x^{2} + b x + a}}{30720 \, c^{6}}, \frac {15 \, {\left (128 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{2} + 16 \, {\left (5 \, b^{4} c^{2} - 24 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d f + {\left (21 \, b^{6} - 140 \, a b^{4} c + 240 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} f^{2} + 8 \, {\left (5 \, b^{4} c^{2} - 24 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} e^{2} - 8 \, {\left (16 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d + {\left (7 \, b^{5} c - 40 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} f\right )} e\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 (1280 \, c^{6} f^{2} x^{5} + 128 \, b c^{5} f^{2} x^{4} + 1920 \, b c^{5} d^{2} + 16 \, {\left (240 \, c^{6} d f - {\left (9 \, b^{2} c^{4} - 20 \, a c^{5}\right )} f^{2}\right )} x^{3} + 80 \, {\left (15 \, b^{3} c^{3} - 52 \, a b c^{4}\right )} d f + {\left (315 \, b^{5} c - 1680 \, a b^{3} c^{2} + 1808 \, a^{2} b c^{3}\right )} f^{2} + 8 \, {\left (80 \, b c^{5} d f + {\left (21 \, b^{3} c^{3} - 68 \, a b c^{4}\right )} f^{2}\right )} x^{2} + 2 \, {\left (1920 \, c^{6} d^{2} - 80 \, {\left (5 \, b^{2} c^{4} - 12 \, a c^{5}\right )} d f - {\left (105 \, b^{4} c^{2} - 448 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} f^{2}\right )} x + 40 \, {\left (48 \, c^{6} x^{3} + 8 \, b c^{5} x^{2} + 15 \, b^{3} c^{3} - 52 \, a b c^{4} - 2 \, {\left (5 \, b^{2} c^{4} - 12 \, a c^{5}\right )} x\right )} e^{2} + 8 \, {\left (384 \, c^{6} f x^{4} + 48 \, b c^{5} f x^{3} + 8 \, {\left (80 \, c^{6} d - {\left (7 \, b^{2} c^{4} - 16 \, a c^{5}\right )} f\right )} x^{2} - 80 \, {\left (3 \, b^{2} c^{4} - 8 \, a c^{5}\right )} d - {\left (105 \, b^{4} c^{2} - 460 \, a b^{2} c^{3} + 256 \, a^{2} c^{4}\right )} f + 2 \, {\left (80 \, b c^{5} d + {\left (35 \, b^{3} c^{3} - 116 \, a b c^{4}\right )} f\right )} x\right )} e\right )} \sqrt {c x^{2} + b x + a}}{15360 \, c^{6}}\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 \sqrt {a + b x + c x^{2}} \left (d + e x + f x^{2}\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.19, size = 638, normalized size = 1.46 \begin {gather*} \frac {1}{7680} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, f^{2} x + \frac {b c^{4} f^{2} + 24 \, c^{5} f e}{c^{5}}\right )} x + \frac {240 \, c^{5} d f - 9 \, b^{2} c^{3} f^{2} + 20 \, a c^{4} f^{2} + 24 \, b c^{4} f e + 120 \, c^{5} e^{2}}{c^{5}}\right )} x + \frac {80 \, b c^{4} d f + 21 \, b^{3} c^{2} f^{2} - 68 \, a b c^{3} f^{2} + 640 \, c^{5} d e - 56 \, b^{2} c^{3} f e + 128 \, a c^{4} f e + 40 \, b c^{4} e^{2}}{c^{5}}\right )} x + \frac {1920 \, c^{5} d^{2} - 400 \, b^{2} c^{3} d f + 960 \, a c^{4} d f - 105 \, b^{4} c f^{2} + 448 \, a b^{2} c^{2} f^{2} - 240 \, a^{2} c^{3} f^{2} + 640 \, b c^{4} d e + 280 \, b^{3} c^{2} f e - 928 \, a b c^{3} f e - 200 \, b^{2} c^{3} e^{2} + 480 \, a c^{4} e^{2}}{c^{5}}\right )} x + \frac {1920 \, b c^{4} d^{2} + 1200 \, b^{3} c^{2} d f - 4160 \, a b c^{3} d f + 315 \, b^{5} f^{2} - 1680 \, a b^{3} c f^{2} + 1808 \, a^{2} b c^{2} f^{2} - 1920 \, b^{2} c^{3} d e + 5120 \, a c^{4} d e - 840 \, b^{4} c f e + 3680 \, a b^{2} c^{2} f e - 2048 \, a^{2} c^{3} f e + 600 \, b^{3} c^{2} e^{2} - 2080 \, a b c^{3} e^{2}}{c^{5}}\right )} + \frac {{\left (128 \, b^{2} c^{4} d^{2} - 512 \, a c^{5} d^{2} + 80 \, b^{4} c^{2} d f - 384 \, a b^{2} c^{3} d f + 256 \, a^{2} c^{4} d f + 21 \, b^{6} f^{2} - 140 \, a b^{4} c f^{2} + 240 \, a^{2} b^{2} c^{2} f^{2} - 64 \, a^{3} c^{3} f^{2} - 128 \, b^{3} c^{3} d e + 512 \, a b c^{4} d e - 56 \, b^{5} c f e + 320 \, a b^{3} c^{2} f e - 384 \, a^{2} b c^{3} f e + 40 \, b^{4} c^{2} e^{2} - 192 \, a b^{2} c^{3} e^{2} + 128 \, a^{2} c^{4} e^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{1024 \, c^{\frac {11}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.31, size = 1299, normalized size = 2.98 \begin {gather*} d^2\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {e^2\,x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {a\,f^2\,\left (\frac {5\,b\,\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 {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {a\,\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}\right )}{2\,c}-\frac {3\,b\,f^2\,\left (\frac {7\,b\,\left (\frac {5\,b\,\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 {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {a\,\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}\right )}{10\,c}-\frac {2\,a\,\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 )}{5\,c}+\frac {x^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{5\,c}\right )}{4\,c}+\frac {f^2\,x^3\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{6\,c}-\frac {a\,e^2\,\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^2\,\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 {5\,b\,e^2\,\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 {4\,a\,e\,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 )}{5\,c}-\frac {5\,b\,d\,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 )}{4\,c}+\frac {d\,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}}{12\,c^2}+\frac {d\,f\,x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{2\,c}+\frac {7\,b\,e\,f\,\left (\frac {5\,b\,\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 {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {a\,\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}\right )}{5\,c}+\frac {2\,e\,f\,x^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{5\,c}-\frac {a\,d\,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 )}{2\,c}+\frac {d\,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 )}{8\,c^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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