Optimal. Leaf size=167 \[ -\frac {\sqrt {a+b x+c x^2} \left (-16 a A c-18 a b B+15 A b^2\right )}{24 a^3 x}+\frac {(5 A b-6 a B) \sqrt {a+b x+c x^2}}{12 a^2 x^2}+\frac {\left (8 a^2 B c-12 a A b c-6 a b^2 B+5 A b^3\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{16 a^{7/2}}-\frac {A \sqrt {a+b x+c x^2}}{3 a x^3} \]
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Rubi [A] time = 0.16, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 5, 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 (-16 a A c-18 a b B+15 A b^2\right )}{24 a^3 x}+\frac {\left (8 a^2 B c-12 a A b c-6 a b^2 B+5 A b^3\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{16 a^{7/2}}+\frac {(5 A b-6 a B) \sqrt {a+b x+c x^2}}{12 a^2 x^2}-\frac {A \sqrt {a+b x+c x^2}}{3 a x^3} \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^4 \sqrt {a+b x+c x^2}} \, dx &=-\frac {A \sqrt {a+b x+c x^2}}{3 a x^3}-\frac {\int \frac {\frac {1}{2} (5 A b-6 a B)+2 A c x}{x^3 \sqrt {a+b x+c x^2}} \, dx}{3 a}\\ &=-\frac {A \sqrt {a+b x+c x^2}}{3 a x^3}+\frac {(5 A b-6 a B) \sqrt {a+b x+c x^2}}{12 a^2 x^2}+\frac {\int \frac {\frac {1}{4} \left (15 A b^2-18 a b B-16 a A c\right )+\frac {1}{2} (5 A b-6 a B) c x}{x^2 \sqrt {a+b x+c x^2}} \, dx}{6 a^2}\\ &=-\frac {A \sqrt {a+b x+c x^2}}{3 a x^3}+\frac {(5 A b-6 a B) \sqrt {a+b x+c x^2}}{12 a^2 x^2}-\frac {\left (15 A b^2-18 a b B-16 a A c\right ) \sqrt {a+b x+c x^2}}{24 a^3 x}-\frac {\left (5 A b^3-6 a b^2 B-12 a A b c+8 a^2 B c\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{16 a^3}\\ &=-\frac {A \sqrt {a+b x+c x^2}}{3 a x^3}+\frac {(5 A b-6 a B) \sqrt {a+b x+c x^2}}{12 a^2 x^2}-\frac {\left (15 A b^2-18 a b B-16 a A c\right ) \sqrt {a+b x+c x^2}}{24 a^3 x}+\frac {\left (5 A b^3-6 a b^2 B-12 a A b c+8 a^2 B c\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 )}{8 a^3}\\ &=-\frac {A \sqrt {a+b x+c x^2}}{3 a x^3}+\frac {(5 A b-6 a B) \sqrt {a+b x+c x^2}}{12 a^2 x^2}-\frac {\left (15 A b^2-18 a b B-16 a A c\right ) \sqrt {a+b x+c x^2}}{24 a^3 x}+\frac {\left (5 A b^3-6 a b^2 B-12 a A b c+8 a^2 B c\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{16 a^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 132, normalized size = 0.79 \begin {gather*} \frac {\left (8 a^2 B c-12 a A b c-6 a b^2 B+5 A b^3\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )}{16 a^{7/2}}-\frac {\sqrt {a+x (b+c x)} \left (4 a^2 (2 A+3 B x)-2 a x (5 A b+8 A c x+9 b B x)+15 A b^2 x^2\right )}{24 a^3 x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.85, size = 174, normalized size = 1.04 \begin {gather*} -\frac {3 \left (2 A b c+b^2 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x+c x^2}-\sqrt {c} x}{\sqrt {a}}\right )}{4 a^{5/2}}+\frac {\left (-8 a^2 B c-5 A b^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+b x+c x^2}}{\sqrt {a}}\right )}{8 a^{7/2}}+\frac {\sqrt {a+b x+c x^2} \left (-8 a^2 A-12 a^2 B x+10 a A b x+16 a A c x^2+18 a b B x^2-15 A b^2 x^2\right )}{24 a^3 x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 321, normalized size = 1.92 \begin {gather*} \left [\frac {3 \, {\left (6 \, B a b^{2} - 5 \, A b^{3} - 4 \, {\left (2 \, B a^{2} - 3 \, A a b\right )} c\right )} \sqrt {a} x^{3} \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 (8 \, A a^{3} - {\left (18 \, B a^{2} b - 15 \, A a b^{2} + 16 \, A a^{2} c\right )} x^{2} + 2 \, {\left (6 \, B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{96 \, a^{4} x^{3}}, \frac {3 \, {\left (6 \, B a b^{2} - 5 \, A b^{3} - 4 \, {\left (2 \, B a^{2} - 3 \, A a b\right )} c\right )} \sqrt {-a} x^{3} \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 (8 \, A a^{3} - {\left (18 \, B a^{2} b - 15 \, A a b^{2} + 16 \, A a^{2} c\right )} x^{2} + 2 \, {\left (6 \, B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{48 \, a^{4} x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 511, normalized size = 3.06 \begin {gather*} \frac {{\left (6 \, B a b^{2} - 5 \, A b^{3} - 8 \, B a^{2} c + 12 \, A a b c\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a^{3}} - \frac {18 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a b^{2} - 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A b^{3} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a^{2} c + 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a b c - 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a^{2} b^{2} + 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a b^{3} - 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a^{2} b c - 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{3} b \sqrt {c} - 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a^{3} c^{\frac {3}{2}} + 30 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{3} b^{2} - 33 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{2} b^{3} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{4} c - 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{3} b c + 48 \, B a^{4} b \sqrt {c} - 48 \, A a^{3} b^{2} \sqrt {c} + 32 \, A a^{4} c^{\frac {3}{2}}}{24 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{3} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 283, normalized size = 1.69 \begin {gather*} -\frac {3 A b c \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{4 a^{\frac {5}{2}}}+\frac {5 A \,b^{3} \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 {B c \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{2 a^{\frac {3}{2}}}-\frac {3 B \,b^{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 {2 \sqrt {c \,x^{2}+b x +a}\, A c}{3 a^{2} x}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, A \,b^{2}}{8 a^{3} x}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, B b}{4 a^{2} x}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, A b}{12 a^{2} x^{2}}-\frac {\sqrt {c \,x^{2}+b x +a}\, B}{2 a \,x^{2}}-\frac {\sqrt {c \,x^{2}+b x +a}\, A}{3 a \,x^{3}} \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 = -0.01 \begin {gather*} \int \frac {A+B\,x}{x^4\,\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^{4} \sqrt {a + b x + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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