Optimal. Leaf size=306 \[ -\frac {\sqrt {a+b x+c x^2} \left (-64 a A c-70 a b B+63 A b^2\right )}{240 a^3 x^3}+\frac {(9 A b-10 a B) \sqrt {a+b x+c x^2}}{40 a^2 x^4}-\frac {\left (2 a B \left (48 a^2 c^2-120 a b^2 c+35 b^4\right )-A \left (240 a^2 b c^2-280 a b^3 c+63 b^5\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{256 a^{11/2}}+\frac {\sqrt {a+b x+c x^2} \left (50 a b B \left (21 b^2-44 a c\right )-A \left (1024 a^2 c^2-2940 a b^2 c+945 b^4\right )\right )}{1920 a^5 x}+\frac {\sqrt {a+b x+c x^2} \left (360 a^2 B c-644 a A b c-350 a b^2 B+315 A b^3\right )}{960 a^4 x^2}-\frac {A \sqrt {a+b x+c x^2}}{5 a x^5} \]
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Rubi [A] time = 0.40, antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {834, 806, 724, 206} \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (50 a b B \left (21 b^2-44 a c\right )-A \left (1024 a^2 c^2-2940 a b^2 c+945 b^4\right )\right )}{1920 a^5 x}-\frac {\left (2 a B \left (48 a^2 c^2-120 a b^2 c+35 b^4\right )-A \left (240 a^2 b c^2-280 a b^3 c+63 b^5\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{256 a^{11/2}}+\frac {\sqrt {a+b x+c x^2} \left (360 a^2 B c-644 a A b c-350 a b^2 B+315 A b^3\right )}{960 a^4 x^2}-\frac {\sqrt {a+b x+c x^2} \left (-64 a A c-70 a b B+63 A b^2\right )}{240 a^3 x^3}+\frac {(9 A b-10 a B) \sqrt {a+b x+c x^2}}{40 a^2 x^4}-\frac {A \sqrt {a+b x+c x^2}}{5 a x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 806
Rule 834
Rubi steps
\begin {align*} \int \frac {A+B x}{x^6 \sqrt {a+b x+c x^2}} \, dx &=-\frac {A \sqrt {a+b x+c x^2}}{5 a x^5}-\frac {\int \frac {\frac {1}{2} (9 A b-10 a B)+4 A c x}{x^5 \sqrt {a+b x+c x^2}} \, dx}{5 a}\\ &=-\frac {A \sqrt {a+b x+c x^2}}{5 a x^5}+\frac {(9 A b-10 a B) \sqrt {a+b x+c x^2}}{40 a^2 x^4}+\frac {\int \frac {\frac {1}{4} \left (63 A b^2-70 a b B-64 a A c\right )+\frac {3}{2} (9 A b-10 a B) c x}{x^4 \sqrt {a+b x+c x^2}} \, dx}{20 a^2}\\ &=-\frac {A \sqrt {a+b x+c x^2}}{5 a x^5}+\frac {(9 A b-10 a B) \sqrt {a+b x+c x^2}}{40 a^2 x^4}-\frac {\left (63 A b^2-70 a b B-64 a A c\right ) \sqrt {a+b x+c x^2}}{240 a^3 x^3}-\frac {\int \frac {\frac {1}{8} \left (315 A b^3-350 a b^2 B-644 a A b c+360 a^2 B c\right )+\frac {1}{2} c \left (63 A b^2-70 a b B-64 a A c\right ) x}{x^3 \sqrt {a+b x+c x^2}} \, dx}{60 a^3}\\ &=-\frac {A \sqrt {a+b x+c x^2}}{5 a x^5}+\frac {(9 A b-10 a B) \sqrt {a+b x+c x^2}}{40 a^2 x^4}-\frac {\left (63 A b^2-70 a b B-64 a A c\right ) \sqrt {a+b x+c x^2}}{240 a^3 x^3}+\frac {\left (315 A b^3-350 a b^2 B-644 a A b c+360 a^2 B c\right ) \sqrt {a+b x+c x^2}}{960 a^4 x^2}+\frac {\int \frac {\frac {1}{16} \left (945 A b^4-1050 a b^3 B-2940 a A b^2 c+2200 a^2 b B c+1024 a^2 A c^2\right )+\frac {1}{8} c \left (315 A b^3-350 a b^2 B-644 a A b c+360 a^2 B c\right ) x}{x^2 \sqrt {a+b x+c x^2}} \, dx}{120 a^4}\\ &=-\frac {A \sqrt {a+b x+c x^2}}{5 a x^5}+\frac {(9 A b-10 a B) \sqrt {a+b x+c x^2}}{40 a^2 x^4}-\frac {\left (63 A b^2-70 a b B-64 a A c\right ) \sqrt {a+b x+c x^2}}{240 a^3 x^3}+\frac {\left (315 A b^3-350 a b^2 B-644 a A b c+360 a^2 B c\right ) \sqrt {a+b x+c x^2}}{960 a^4 x^2}+\frac {\left (50 a b B \left (21 b^2-44 a c\right )-A \left (945 b^4-2940 a b^2 c+1024 a^2 c^2\right )\right ) \sqrt {a+b x+c x^2}}{1920 a^5 x}+\frac {\left (2 a B \left (35 b^4-120 a b^2 c+48 a^2 c^2\right )-A \left (63 b^5-280 a b^3 c+240 a^2 b c^2\right )\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{256 a^5}\\ &=-\frac {A \sqrt {a+b x+c x^2}}{5 a x^5}+\frac {(9 A b-10 a B) \sqrt {a+b x+c x^2}}{40 a^2 x^4}-\frac {\left (63 A b^2-70 a b B-64 a A c\right ) \sqrt {a+b x+c x^2}}{240 a^3 x^3}+\frac {\left (315 A b^3-350 a b^2 B-644 a A b c+360 a^2 B c\right ) \sqrt {a+b x+c x^2}}{960 a^4 x^2}+\frac {\left (50 a b B \left (21 b^2-44 a c\right )-A \left (945 b^4-2940 a b^2 c+1024 a^2 c^2\right )\right ) \sqrt {a+b x+c x^2}}{1920 a^5 x}-\frac {\left (2 a B \left (35 b^4-120 a b^2 c+48 a^2 c^2\right )-A \left (63 b^5-280 a b^3 c+240 a^2 b c^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x}{\sqrt {a+b x+c x^2}}\right )}{128 a^5}\\ &=-\frac {A \sqrt {a+b x+c x^2}}{5 a x^5}+\frac {(9 A b-10 a B) \sqrt {a+b x+c x^2}}{40 a^2 x^4}-\frac {\left (63 A b^2-70 a b B-64 a A c\right ) \sqrt {a+b x+c x^2}}{240 a^3 x^3}+\frac {\left (315 A b^3-350 a b^2 B-644 a A b c+360 a^2 B c\right ) \sqrt {a+b x+c x^2}}{960 a^4 x^2}+\frac {\left (50 a b B \left (21 b^2-44 a c\right )-A \left (945 b^4-2940 a b^2 c+1024 a^2 c^2\right )\right ) \sqrt {a+b x+c x^2}}{1920 a^5 x}-\frac {\left (2 a B \left (35 b^4-120 a b^2 c+48 a^2 c^2\right )-A \left (63 b^5-280 a b^3 c+240 a^2 b c^2\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{256 a^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 233, normalized size = 0.76 \begin {gather*} \frac {\left (A \left (240 a^2 b c^2-280 a b^3 c+63 b^5\right )-2 a B \left (48 a^2 c^2-120 a b^2 c+35 b^4\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )}{256 a^{11/2}}-\frac {\sqrt {a+x (b+c x)} \left (96 a^4 (4 A+5 B x)-16 a^3 x (A (27 b+32 c x)+5 B x (7 b+9 c x))+4 a^2 x^2 \left (2 A \left (63 b^2+161 b c x+128 c^2 x^2\right )+25 b B x (7 b+22 c x)\right )-210 a b^2 x^3 (3 A b+14 A c x+5 b B x)+945 A b^4 x^4\right )}{1920 a^5 x^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.82, size = 300, normalized size = 0.98 \begin {gather*} \frac {5 \left (24 a A b c^2+24 a b^2 B c-28 A b^3 c-7 b^4 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x+c x^2}-\sqrt {c} x}{\sqrt {a}}\right )}{64 a^{9/2}}+\frac {3 \left (32 a^3 B c^2-21 A b^5\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+b x+c x^2}}{\sqrt {a}}\right )}{128 a^{11/2}}+\frac {\sqrt {a+b x+c x^2} \left (-384 a^4 A-480 a^4 B x+432 a^3 A b x+512 a^3 A c x^2+560 a^3 b B x^2+720 a^3 B c x^3-504 a^2 A b^2 x^2-1288 a^2 A b c x^3-1024 a^2 A c^2 x^4-700 a^2 b^2 B x^3-2200 a^2 b B c x^4+630 a A b^3 x^3+2940 a A b^2 c x^4+1050 a b^3 B x^4-945 A b^4 x^4\right )}{1920 a^5 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.94, size = 557, normalized size = 1.82 \begin {gather*} \left [-\frac {15 \, {\left (70 \, B a b^{4} - 63 \, A b^{5} + 48 \, {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} c^{2} - 40 \, {\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} c\right )} \sqrt {a} x^{5} \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) + 4 \, {\left (384 \, A a^{5} - {\left (1050 \, B a^{2} b^{3} - 945 \, A a b^{4} - 1024 \, A a^{3} c^{2} - 20 \, {\left (110 \, B a^{3} b - 147 \, A a^{2} b^{2}\right )} c\right )} x^{4} + 2 \, {\left (350 \, B a^{3} b^{2} - 315 \, A a^{2} b^{3} - 4 \, {\left (90 \, B a^{4} - 161 \, A a^{3} b\right )} c\right )} x^{3} - 8 \, {\left (70 \, B a^{4} b - 63 \, A a^{3} b^{2} + 64 \, A a^{4} c\right )} x^{2} + 48 \, {\left (10 \, B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{7680 \, a^{6} x^{5}}, \frac {15 \, {\left (70 \, B a b^{4} - 63 \, A b^{5} + 48 \, {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} c^{2} - 40 \, {\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} c\right )} \sqrt {-a} x^{5} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right ) - 2 \, {\left (384 \, A a^{5} - {\left (1050 \, B a^{2} b^{3} - 945 \, A a b^{4} - 1024 \, A a^{3} c^{2} - 20 \, {\left (110 \, B a^{3} b - 147 \, A a^{2} b^{2}\right )} c\right )} x^{4} + 2 \, {\left (350 \, B a^{3} b^{2} - 315 \, A a^{2} b^{3} - 4 \, {\left (90 \, B a^{4} - 161 \, A a^{3} b\right )} c\right )} x^{3} - 8 \, {\left (70 \, B a^{4} b - 63 \, A a^{3} b^{2} + 64 \, A a^{4} c\right )} x^{2} + 48 \, {\left (10 \, B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3840 \, a^{6} x^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 1266, normalized size = 4.14
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 578, normalized size = 1.89 \begin {gather*} \frac {15 A b \,c^{2} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{16 a^{\frac {7}{2}}}-\frac {35 A \,b^{3} c \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{32 a^{\frac {9}{2}}}+\frac {63 A \,b^{5} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{256 a^{\frac {11}{2}}}-\frac {3 B \,c^{2} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{8 a^{\frac {5}{2}}}+\frac {15 B \,b^{2} c \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{16 a^{\frac {7}{2}}}-\frac {35 B \,b^{4} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{128 a^{\frac {9}{2}}}-\frac {8 \sqrt {c \,x^{2}+b x +a}\, A \,c^{2}}{15 a^{3} x}+\frac {49 \sqrt {c \,x^{2}+b x +a}\, A \,b^{2} c}{32 a^{4} x}-\frac {63 \sqrt {c \,x^{2}+b x +a}\, A \,b^{4}}{128 a^{5} x}-\frac {55 \sqrt {c \,x^{2}+b x +a}\, B b c}{48 a^{3} x}+\frac {35 \sqrt {c \,x^{2}+b x +a}\, B \,b^{3}}{64 a^{4} x}-\frac {161 \sqrt {c \,x^{2}+b x +a}\, A b c}{240 a^{3} x^{2}}+\frac {21 \sqrt {c \,x^{2}+b x +a}\, A \,b^{3}}{64 a^{4} x^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, B c}{8 a^{2} x^{2}}-\frac {35 \sqrt {c \,x^{2}+b x +a}\, B \,b^{2}}{96 a^{3} x^{2}}+\frac {4 \sqrt {c \,x^{2}+b x +a}\, A c}{15 a^{2} x^{3}}-\frac {21 \sqrt {c \,x^{2}+b x +a}\, A \,b^{2}}{80 a^{3} x^{3}}+\frac {7 \sqrt {c \,x^{2}+b x +a}\, B b}{24 a^{2} x^{3}}+\frac {9 \sqrt {c \,x^{2}+b x +a}\, A b}{40 a^{2} x^{4}}-\frac {\sqrt {c \,x^{2}+b x +a}\, B}{4 a \,x^{4}}-\frac {\sqrt {c \,x^{2}+b x +a}\, A}{5 a \,x^{5}} \end {gather*}
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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{x^6\,\sqrt {c\,x^2+b\,x+a}} \,d x \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 \frac {A + B x}{x^{6} \sqrt {a + b x + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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