Optimal. Leaf size=236 \[ \frac {\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c (a f+3 b e)+7 b^2 f+24 c^2 d\right )}{1024 c^{9/2}}-\frac {\left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} \left (-4 c (a f+3 b e)+7 b^2 f+24 c^2 d\right )}{512 c^4}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a c f+7 b^2 f-12 b c e+24 c^2 d\right )}{192 c^3}+\frac {\left (a+b x+c x^2\right )^{5/2} (12 c e-7 b f)}{60 c^2}+\frac {f x \left (a+b x+c x^2\right )^{5/2}}{6 c} \]
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Rubi [A] time = 0.24, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a c f+7 b^2 f-12 b c e+24 c^2 d\right )}{192 c^3}-\frac {\left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} \left (-4 c (a f+3 b e)+7 b^2 f+24 c^2 d\right )}{512 c^4}+\frac {\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c (a f+3 b e)+7 b^2 f+24 c^2 d\right )}{1024 c^{9/2}}+\frac {\left (a+b x+c x^2\right )^{5/2} (12 c e-7 b f)}{60 c^2}+\frac {f x \left (a+b x+c x^2\right )^{5/2}}{6 c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 640
Rule 1661
Rubi steps
\begin {align*} \int \left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx &=\frac {f x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\int \left (6 c d-a f+\frac {1}{2} (12 c e-7 b f) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{6 c}\\ &=\frac {(12 c e-7 b f) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {f x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (2 c (6 c d-a f)-\frac {1}{2} b (12 c e-7 b f)\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{12 c^2}\\ &=\frac {\left (24 c^2 d-12 b c e+7 b^2 f-4 a c f\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac {(12 c e-7 b f) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {f x \left (a+b x+c x^2\right )^{5/2}}{6 c}-\frac {\left (\left (b^2-4 a c\right ) \left (24 c^2 d+7 b^2 f-4 c (3 b e+a f)\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{128 c^3}\\ &=-\frac {\left (b^2-4 a c\right ) \left (24 c^2 d+7 b^2 f-4 c (3 b e+a f)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (24 c^2 d-12 b c e+7 b^2 f-4 a c f\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac {(12 c e-7 b f) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {f x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (\left (b^2-4 a c\right )^2 \left (24 c^2 d+7 b^2 f-4 c (3 b e+a f)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{1024 c^4}\\ &=-\frac {\left (b^2-4 a c\right ) \left (24 c^2 d+7 b^2 f-4 c (3 b e+a f)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (24 c^2 d-12 b c e+7 b^2 f-4 a c f\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac {(12 c e-7 b f) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {f x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (\left (b^2-4 a c\right )^2 \left (24 c^2 d+7 b^2 f-4 c (3 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 )}{512 c^4}\\ &=-\frac {\left (b^2-4 a c\right ) \left (24 c^2 d+7 b^2 f-4 c (3 b e+a f)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (24 c^2 d-12 b c e+7 b^2 f-4 a c f\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac {(12 c e-7 b f) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {f x \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (b^2-4 a c\right )^2 \left (24 c^2 d+7 b^2 f-4 c (3 b e+a f)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.69, size = 392, normalized size = 1.66 \[ \frac {\frac {360 d \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}\right )}{c^{3/2}}-60 b e \left (\frac {3 \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}\right )}{c^{5/2}}+\frac {16 (b+2 c x) (a+x (b+c x))^{3/2}}{c}\right )+\frac {f \left (5 \left (7 b^2-4 a c\right ) \left (\frac {3 \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )-2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}\right )}{c^{5/2}}+\frac {16 (b+2 c x) (a+x (b+c x))^{3/2}}{c}\right )-1792 b (a+x (b+c x))^{5/2}\right )}{c}+1920 d (b+2 c x) (a+x (b+c x))^{3/2}+3072 e (a+x (b+c x))^{5/2}+2560 f x (a+x (b+c x))^{5/2}}{15360 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 839, normalized size = 3.56 \[ \left [-\frac {15 \, {\left (24 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d - 12 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} e + {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\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 (1280 \, c^{6} f x^{5} + 128 \, {\left (12 \, c^{6} e + 13 \, b c^{5} f\right )} x^{4} + 16 \, {\left (120 \, c^{6} d + 132 \, b c^{5} e + {\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} f\right )} x^{3} + 8 \, {\left (360 \, b c^{5} d + 12 \, {\left (b^{2} c^{4} + 32 \, a c^{5}\right )} e - {\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} f\right )} x^{2} - 120 \, {\left (3 \, b^{3} c^{3} - 20 \, a b c^{4}\right )} d + 12 \, {\left (15 \, b^{4} c^{2} - 100 \, a b^{2} c^{3} + 128 \, a^{2} c^{4}\right )} e - {\left (105 \, b^{5} c - 760 \, a b^{3} c^{2} + 1296 \, a^{2} b c^{3}\right )} f + 2 \, {\left (120 \, {\left (b^{2} c^{4} + 20 \, a c^{5}\right )} d - 12 \, {\left (5 \, b^{3} c^{3} - 28 \, a b c^{4}\right )} e + {\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} f\right )} x\right )} \sqrt {c x^{2} + b x + a}}{30720 \, c^{5}}, -\frac {15 \, {\left (24 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d - 12 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} e + {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\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 (1280 \, c^{6} f x^{5} + 128 \, {\left (12 \, c^{6} e + 13 \, b c^{5} f\right )} x^{4} + 16 \, {\left (120 \, c^{6} d + 132 \, b c^{5} e + {\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} f\right )} x^{3} + 8 \, {\left (360 \, b c^{5} d + 12 \, {\left (b^{2} c^{4} + 32 \, a c^{5}\right )} e - {\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} f\right )} x^{2} - 120 \, {\left (3 \, b^{3} c^{3} - 20 \, a b c^{4}\right )} d + 12 \, {\left (15 \, b^{4} c^{2} - 100 \, a b^{2} c^{3} + 128 \, a^{2} c^{4}\right )} e - {\left (105 \, b^{5} c - 760 \, a b^{3} c^{2} + 1296 \, a^{2} b c^{3}\right )} f + 2 \, {\left (120 \, {\left (b^{2} c^{4} + 20 \, a c^{5}\right )} d - 12 \, {\left (5 \, b^{3} c^{3} - 28 \, a b c^{4}\right )} e + {\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} f\right )} x\right )} \sqrt {c x^{2} + b x + a}}{15360 \, c^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 417, normalized size = 1.77 \[ \frac {1}{7680} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c f x + \frac {13 \, b c^{5} f + 12 \, c^{6} e}{c^{5}}\right )} x + \frac {120 \, c^{6} d + 3 \, b^{2} c^{4} f + 140 \, a c^{5} f + 132 \, b c^{5} e}{c^{5}}\right )} x + \frac {360 \, b c^{5} d - 7 \, b^{3} c^{3} f + 36 \, a b c^{4} f + 12 \, b^{2} c^{4} e + 384 \, a c^{5} e}{c^{5}}\right )} x + \frac {120 \, b^{2} c^{4} d + 2400 \, a c^{5} d + 35 \, b^{4} c^{2} f - 216 \, a b^{2} c^{3} f + 240 \, a^{2} c^{4} f - 60 \, b^{3} c^{3} e + 336 \, a b c^{4} e}{c^{5}}\right )} x - \frac {360 \, b^{3} c^{3} d - 2400 \, a b c^{4} d + 105 \, b^{5} c f - 760 \, a b^{3} c^{2} f + 1296 \, a^{2} b c^{3} f - 180 \, b^{4} c^{2} e + 1200 \, a b^{2} c^{3} e - 1536 \, a^{2} c^{4} e}{c^{5}}\right )} - \frac {{\left (24 \, b^{4} c^{2} d - 192 \, a b^{2} c^{3} d + 384 \, a^{2} c^{4} d + 7 \, b^{6} f - 60 \, a b^{4} c f + 144 \, a^{2} b^{2} c^{2} f - 64 \, a^{3} c^{3} f - 12 \, b^{5} c e + 96 \, a b^{3} c^{2} e - 192 \, a^{2} b c^{3} e\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{1024 \, c^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 862, normalized size = 3.65 \[ -\frac {a^{3} f \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {3}{2}}}+\frac {9 a^{2} b^{2} f \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{64 c^{\frac {5}{2}}}-\frac {3 a^{2} b e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {3}{2}}}+\frac {3 a^{2} d \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 \sqrt {c}}-\frac {15 a \,b^{4} f \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {7}{2}}}+\frac {3 a \,b^{3} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{32 c^{\frac {5}{2}}}-\frac {3 a \,b^{2} d \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {3}{2}}}+\frac {7 b^{6} f \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{1024 c^{\frac {9}{2}}}-\frac {3 b^{5} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {7}{2}}}+\frac {3 b^{4} d \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {5}{2}}}-\frac {\sqrt {c \,x^{2}+b x +a}\, a^{2} f x}{16 c}+\frac {\sqrt {c \,x^{2}+b x +a}\, a \,b^{2} f x}{8 c^{2}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, a b e x}{16 c}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, a d x}{8}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, b^{4} f x}{256 c^{3}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, b^{3} e x}{64 c^{2}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, b^{2} d x}{32 c}-\frac {\sqrt {c \,x^{2}+b x +a}\, a^{2} b f}{32 c^{2}}+\frac {\sqrt {c \,x^{2}+b x +a}\, a \,b^{3} f}{16 c^{3}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} e}{32 c^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, a b d}{16 c}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a f x}{24 c}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, b^{5} f}{512 c^{4}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, b^{4} e}{128 c^{3}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, b^{3} d}{64 c^{2}}+\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{2} f x}{96 c^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b e x}{8 c}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} d x}{4}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a b f}{48 c^{2}}+\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{3} f}{192 c^{3}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{2} e}{16 c^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b d}{8 c}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} f x}{6 c}-\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b f}{60 c^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} e}{5 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 [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (c\,x^2+b\,x+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right ) \,d x \]
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 \[ \int \left (a + b x + c x^{2}\right )^{\frac {3}{2}} \left (d + e x + f x^{2}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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