Optimal. Leaf size=133 \[ \frac {\left (-4 a A c-4 a b B+A b^2\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{3/2}}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}+B \sqrt {c} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \]
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Rubi [A] time = 0.10, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {810, 843, 621, 206, 724} \begin {gather*} \frac {\left (-4 a A c-4 a b B+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^{3/2}}-\frac {\sqrt {a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}+B \sqrt {c} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 810
Rule 843
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^3} \, dx &=-\frac {(2 a A+(A b+4 a B) x) \sqrt {a+b x+c x^2}}{4 a x^2}-\frac {\int \frac {\frac {1}{2} \left (-4 a b B+A \left (b^2-4 a c\right )\right )-4 a B c x}{x \sqrt {a+b x+c x^2}} \, dx}{4 a}\\ &=-\frac {(2 a A+(A b+4 a B) x) \sqrt {a+b x+c x^2}}{4 a x^2}+(B c) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx-\frac {\left (-4 a b B+A \left (b^2-4 a c\right )\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{8 a}\\ &=-\frac {(2 a A+(A b+4 a B) x) \sqrt {a+b x+c x^2}}{4 a x^2}+(2 B c) \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 )+\frac {\left (-4 a b B+A \left (b^2-4 a c\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 )}{4 a}\\ &=-\frac {(2 a A+(A b+4 a B) x) \sqrt {a+b x+c x^2}}{4 a x^2}-\frac {\left (4 a b B-A \left (b^2-4 a c\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{3/2}}+B \sqrt {c} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.35, size = 129, normalized size = 0.97 \begin {gather*} \frac {\left (A \left (b^2-4 a c\right )-4 a b B\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )}{8 a^{3/2}}-\frac {\sqrt {a+x (b+c x)} (2 a (A+2 B x)+A b x)}{4 a x^2}+B \sqrt {c} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.70, size = 131, normalized size = 0.98 \begin {gather*} \frac {\left (-4 a A c-4 a b B+A b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x+c x^2}-\sqrt {c} x}{\sqrt {a}}\right )}{4 a^{3/2}}+\frac {\sqrt {a+b x+c x^2} (-2 a A-4 a B x-A b x)}{4 a x^2}-B \sqrt {c} \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 699, normalized size = 5.26 \begin {gather*} \left [\frac {8 \, B a^{2} \sqrt {c} x^{2} \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 ) + {\left (4 \, B a b - 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} + A a b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{16 \, a^{2} x^{2}}, -\frac {16 \, B a^{2} \sqrt {-c} x^{2} \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 ) - {\left (4 \, B a b - 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} + A a b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{16 \, a^{2} x^{2}}, \frac {4 \, B a^{2} \sqrt {c} x^{2} \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 ) + {\left (4 \, B a b - 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} + A a b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{8 \, a^{2} x^{2}}, -\frac {8 \, B a^{2} \sqrt {-c} x^{2} \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 ) - {\left (4 \, B a b - 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} + A a b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{8 \, a^{2} x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 359, normalized size = 2.70 \begin {gather*} -B \sqrt {c} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} c - b \sqrt {c} \right |}\right ) + \frac {{\left (4 \, B a b - 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} + \frac {4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a b + {\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} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a b \sqrt {c} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{2} b + {\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}}{4 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{2} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 304, normalized size = 2.29 \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 \sqrt {a}}+\frac {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 {3}{2}}}-\frac {B b \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{2 \sqrt {a}}+B \sqrt {c}\, \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )-\frac {\sqrt {c \,x^{2}+b x +a}\, A b c x}{4 a^{2}}+\frac {\sqrt {c \,x^{2}+b x +a}\, B c x}{a}+\frac {\sqrt {c \,x^{2}+b x +a}\, A c}{2 a}-\frac {\sqrt {c \,x^{2}+b x +a}\, A \,b^{2}}{4 a^{2}}+\frac {\sqrt {c \,x^{2}+b x +a}\, B b}{a}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A b}{4 a^{2} x}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B}{a x}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} 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 {\left (A+B\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{x^3} \,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 {\left (A + B x\right ) \sqrt {a + b x + c x^{2}}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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