Optimal. Leaf size=116 \[ -\frac {\left (-4 a A c-4 a b B+3 A b^2\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{5/2}}+\frac {(3 A b-4 a B) \sqrt {a+b x+c x^2}}{4 a^2 x}-\frac {A \sqrt {a+b x+c x^2}}{2 a x^2} \]
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Rubi [A] time = 0.09, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {\left (-4 a A c-4 a b B+3 A b^2\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{5/2}}+\frac {(3 A b-4 a B) \sqrt {a+b x+c x^2}}{4 a^2 x}-\frac {A \sqrt {a+b x+c x^2}}{2 a x^2} \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^3 \sqrt {a+b x+c x^2}} \, dx &=-\frac {A \sqrt {a+b x+c x^2}}{2 a x^2}-\frac {\int \frac {\frac {1}{2} (3 A b-4 a B)+A c x}{x^2 \sqrt {a+b x+c x^2}} \, dx}{2 a}\\ &=-\frac {A \sqrt {a+b x+c x^2}}{2 a x^2}+\frac {(3 A b-4 a B) \sqrt {a+b x+c x^2}}{4 a^2 x}+\frac {\left (3 A b^2-4 a b B-4 a A c\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{8 a^2}\\ &=-\frac {A \sqrt {a+b x+c x^2}}{2 a x^2}+\frac {(3 A b-4 a B) \sqrt {a+b x+c x^2}}{4 a^2 x}-\frac {\left (3 A b^2-4 a b B-4 a A 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 )}{4 a^2}\\ &=-\frac {A \sqrt {a+b x+c x^2}}{2 a x^2}+\frac {(3 A b-4 a B) \sqrt {a+b x+c x^2}}{4 a^2 x}-\frac {\left (3 A b^2-4 a b B-4 a A c\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 95, normalized size = 0.82 \begin {gather*} \frac {\left (4 a A c+4 a b B-3 A b^2\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )}{8 a^{5/2}}+\frac {\sqrt {a+x (b+c x)} (3 A b x-2 a (A+2 B x))}{4 a^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.59, size = 131, normalized size = 1.13 \begin {gather*} \frac {3 A b^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+b x+c x^2}}{\sqrt {a}}\right )}{4 a^{5/2}}+\frac {(A c+b B) \tanh ^{-1}\left (\frac {\sqrt {a+b x+c x^2}-\sqrt {c} x}{\sqrt {a}}\right )}{a^{3/2}}+\frac {\sqrt {a+b x+c x^2} (-2 a A-4 a B x+3 A b x)}{4 a^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 235, normalized size = 2.03 \begin {gather*} \left [\frac {{\left (4 \, B a b - 3 \, A b^{2} + 4 \, A a c\right )} \sqrt {a} x^{2} \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 (2 \, A a^{2} + {\left (4 \, B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{16 \, a^{3} x^{2}}, -\frac {{\left (4 \, B a b - 3 \, A b^{2} + 4 \, A a c\right )} \sqrt {-a} x^{2} \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 (2 \, A a^{2} + {\left (4 \, B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{8 \, a^{3} x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.29, size = 303, normalized size = 2.61 \begin {gather*} -\frac {{\left (4 \, B a b - 3 \, A b^{2} + 4 \, A a c\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a^{2}} + \frac {4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a b - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A b^{2} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a c + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{2} \sqrt {c} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{2} b + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a b^{2} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{2} c - 8 \, B a^{3} \sqrt {c} + 8 \, A a^{2} b \sqrt {c}}{4 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{2} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 176, normalized size = 1.52 \begin {gather*} \frac {A 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 A \,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 {B b \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 \sqrt {c \,x^{2}+b x +a}\, A b}{4 a^{2} x}-\frac {\sqrt {c \,x^{2}+b x +a}\, B}{a x}-\frac {\sqrt {c \,x^{2}+b x +a}\, A}{2 a \,x^{2}} \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^3\,\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^{3} \sqrt {a + b x + c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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