Optimal. Leaf size=257 \[ \frac {\left (b^2-4 a c\right )^2 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{9/2}}-\frac {\left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{512 c^4}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{192 c^3}+\frac {7 e \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{60 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c} \]
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Rubi [A] time = 0.33, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {742, 640, 612, 621, 206} \[ \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{192 c^3}-\frac {\left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right )}{512 c^4}+\frac {\left (b^2-4 a c\right )^2 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{9/2}}+\frac {7 e \left (a+b x+c x^2\right )^{5/2} (2 c d-b e)}{60 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 640
Rule 742
Rubi steps
\begin {align*} \int (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\int \left (\frac {1}{2} \left (12 c d^2-2 e \left (\frac {5 b d}{2}+a e\right )\right )+\frac {7}{2} e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{6 c}\\ &=\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{5/2}}{6 c}+\frac {\left (-\frac {7}{2} b e (2 c d-b e)+c \left (12 c d^2-2 e \left (\frac {5 b d}{2}+a e\right )\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{12 c^2}\\ &=\frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e 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^2+7 b^2 e^2-4 c e (6 b d+a e)\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^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e 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^2+7 b^2 e^2-4 c e (6 b d+a e)\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^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e 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^2+7 b^2 e^2-4 c e (6 b d+a e)\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^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}+\frac {\left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac {7 e (2 c d-b e) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}+\frac {e (d+e 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^2+7 b^2 e^2-4 c e (6 b d+a e)\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 = 0.30, size = 188, normalized size = 0.73 \[ \frac {\frac {\left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{512 c^{7/2}}+\frac {7 e (a+x (b+c x))^{5/2} (2 c d-b e)}{10 c}+e (d+e x) (a+x (b+c x))^{5/2}}{6 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 899, normalized size = 3.50 \[ \left [-\frac {15 \, {\left (24 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{2} - 24 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d e + {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} e^{2}\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} e^{2} x^{5} + 128 \, {\left (24 \, c^{6} d e + 13 \, b c^{5} e^{2}\right )} x^{4} + 16 \, {\left (120 \, c^{6} d^{2} + 264 \, b c^{5} d e + {\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} e^{2}\right )} x^{3} - 120 \, {\left (3 \, b^{3} c^{3} - 20 \, a b c^{4}\right )} d^{2} + 24 \, {\left (15 \, b^{4} c^{2} - 100 \, a b^{2} c^{3} + 128 \, a^{2} c^{4}\right )} d e - {\left (105 \, b^{5} c - 760 \, a b^{3} c^{2} + 1296 \, a^{2} b c^{3}\right )} e^{2} + 8 \, {\left (360 \, b c^{5} d^{2} + 24 \, {\left (b^{2} c^{4} + 32 \, a c^{5}\right )} d e - {\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} e^{2}\right )} x^{2} + 2 \, {\left (120 \, {\left (b^{2} c^{4} + 20 \, a c^{5}\right )} d^{2} - 24 \, {\left (5 \, b^{3} c^{3} - 28 \, a b c^{4}\right )} d e + {\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} e^{2}\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^{2} - 24 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d e + {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} e^{2}\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} e^{2} x^{5} + 128 \, {\left (24 \, c^{6} d e + 13 \, b c^{5} e^{2}\right )} x^{4} + 16 \, {\left (120 \, c^{6} d^{2} + 264 \, b c^{5} d e + {\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} e^{2}\right )} x^{3} - 120 \, {\left (3 \, b^{3} c^{3} - 20 \, a b c^{4}\right )} d^{2} + 24 \, {\left (15 \, b^{4} c^{2} - 100 \, a b^{2} c^{3} + 128 \, a^{2} c^{4}\right )} d e - {\left (105 \, b^{5} c - 760 \, a b^{3} c^{2} + 1296 \, a^{2} b c^{3}\right )} e^{2} + 8 \, {\left (360 \, b c^{5} d^{2} + 24 \, {\left (b^{2} c^{4} + 32 \, a c^{5}\right )} d e - {\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} e^{2}\right )} x^{2} + 2 \, {\left (120 \, {\left (b^{2} c^{4} + 20 \, a c^{5}\right )} d^{2} - 24 \, {\left (5 \, b^{3} c^{3} - 28 \, a b c^{4}\right )} d e + {\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} e^{2}\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 [B] time = 0.32, size = 463, normalized size = 1.80 \[ \frac {1}{7680} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c x e^{2} + \frac {24 \, c^{6} d e + 13 \, b c^{5} e^{2}}{c^{5}}\right )} x + \frac {120 \, c^{6} d^{2} + 264 \, b c^{5} d e + 3 \, b^{2} c^{4} e^{2} + 140 \, a c^{5} e^{2}}{c^{5}}\right )} x + \frac {360 \, b c^{5} d^{2} + 24 \, b^{2} c^{4} d e + 768 \, a c^{5} d e - 7 \, b^{3} c^{3} e^{2} + 36 \, a b c^{4} e^{2}}{c^{5}}\right )} x + \frac {120 \, b^{2} c^{4} d^{2} + 2400 \, a c^{5} d^{2} - 120 \, b^{3} c^{3} d e + 672 \, a b c^{4} d e + 35 \, b^{4} c^{2} e^{2} - 216 \, a b^{2} c^{3} e^{2} + 240 \, a^{2} c^{4} e^{2}}{c^{5}}\right )} x - \frac {360 \, b^{3} c^{3} d^{2} - 2400 \, a b c^{4} d^{2} - 360 \, b^{4} c^{2} d e + 2400 \, a b^{2} c^{3} d e - 3072 \, a^{2} c^{4} d e + 105 \, b^{5} c e^{2} - 760 \, a b^{3} c^{2} e^{2} + 1296 \, a^{2} b c^{3} e^{2}}{c^{5}}\right )} - \frac {{\left (24 \, b^{4} c^{2} d^{2} - 192 \, a b^{2} c^{3} d^{2} + 384 \, a^{2} c^{4} d^{2} - 24 \, b^{5} c d e + 192 \, a b^{3} c^{2} d e - 384 \, a^{2} b c^{3} d e + 7 \, b^{6} e^{2} - 60 \, a b^{4} c e^{2} + 144 \, a^{2} b^{2} c^{2} e^{2} - 64 \, a^{3} c^{3} e^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{1024 \, c^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 922, normalized size = 3.59 \[ -\frac {a^{3} e^{2} \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} e^{2} \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 d e \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^{2} d^{2} \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} e^{2} \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} d 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 {3 a \,b^{2} d^{2} \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} e^{2} \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} d e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {7}{2}}}+\frac {3 b^{4} d^{2} \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} e^{2} x}{16 c}+\frac {\sqrt {c \,x^{2}+b x +a}\, a \,b^{2} e^{2} x}{8 c^{2}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, a b d e x}{8 c}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, a \,d^{2} x}{8}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, b^{4} e^{2} x}{256 c^{3}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, b^{3} d e x}{32 c^{2}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, b^{2} d^{2} x}{32 c}-\frac {\sqrt {c \,x^{2}+b x +a}\, a^{2} b \,e^{2}}{32 c^{2}}+\frac {\sqrt {c \,x^{2}+b x +a}\, a \,b^{3} e^{2}}{16 c^{3}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} d e}{16 c^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, a b \,d^{2}}{16 c}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,e^{2} x}{24 c}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, b^{5} e^{2}}{512 c^{4}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, b^{4} d e}{64 c^{3}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, b^{3} d^{2}}{64 c^{2}}+\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{2} e^{2} x}{96 c^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b d e x}{4 c}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} d^{2} x}{4}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a b \,e^{2}}{48 c^{2}}+\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{3} e^{2}}{192 c^{3}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{2} d e}{8 c^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b \,d^{2}}{8 c}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} e^{2} x}{6 c}-\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b \,e^{2}}{60 c^{2}}+\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} d 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 \[ \int {\left (d+e\,x\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2} \,d x \]
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 \[ \int \left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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