Optimal. Leaf size=121 \[ -\frac {\left (b^2-4 a c\right ) (A b-2 a B) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{16 a^{5/2}}+\frac {(2 a+b x) (A b-2 a B) \sqrt {a+b x+c x^2}}{8 a^2 x^2}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{3 a x^3} \]
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Rubi [A] time = 0.07, antiderivative size = 121, 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 = {806, 720, 724, 206} \begin {gather*} -\frac {\left (b^2-4 a c\right ) (A b-2 a B) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{16 a^{5/2}}+\frac {(2 a+b x) (A b-2 a B) \sqrt {a+b x+c x^2}}{8 a^2 x^2}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{3 a x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 720
Rule 724
Rule 806
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {a+b x+c x^2}}{x^4} \, dx &=-\frac {A \left (a+b x+c x^2\right )^{3/2}}{3 a x^3}-\frac {(A b-2 a B) \int \frac {\sqrt {a+b x+c x^2}}{x^3} \, dx}{2 a}\\ &=\frac {(A b-2 a B) (2 a+b x) \sqrt {a+b x+c x^2}}{8 a^2 x^2}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{3 a x^3}+\frac {\left ((A b-2 a B) \left (b^2-4 a c\right )\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{16 a^2}\\ &=\frac {(A b-2 a B) (2 a+b x) \sqrt {a+b x+c x^2}}{8 a^2 x^2}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{3 a x^3}-\frac {\left ((A b-2 a B) \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 )}{8 a^2}\\ &=\frac {(A b-2 a B) (2 a+b x) \sqrt {a+b x+c x^2}}{8 a^2 x^2}-\frac {A \left (a+b x+c x^2\right )^{3/2}}{3 a x^3}-\frac {(A b-2 a B) \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{16 a^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 116, normalized size = 0.96 \begin {gather*} \frac {3 x (A b-2 a B) \left (2 \sqrt {a} (2 a+b x) \sqrt {a+x (b+c x)}-x^2 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )\right )-16 a^{3/2} A (a+x (b+c x))^{3/2}}{48 a^{5/2} x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.98, size = 173, normalized size = 1.43 \begin {gather*} \frac {\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^{3/2}}+\frac {\sqrt {a+b x+c x^2} \left (-8 a^2 A-12 a^2 B x-2 a A b x-8 a A c x^2-6 a b B x^2+3 A b^2 x^2\right )}{24 a^2 x^3}+\frac {\left (8 a^2 B c+A b^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+b x+c x^2}}{\sqrt {a}}\right )}{8 a^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 317, normalized size = 2.62 \begin {gather*} \left [\frac {3 \, {\left (2 \, B a b^{2} - A b^{3} - 4 \, {\left (2 \, B a^{2} - 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 (6 \, B a^{2} b - 3 \, A a b^{2} + 8 \, A a^{2} c\right )} x^{2} + 2 \, {\left (6 \, B a^{3} + A a^{2} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{96 \, a^{3} x^{3}}, -\frac {3 \, {\left (2 \, B a b^{2} - A b^{3} - 4 \, {\left (2 \, B a^{2} - 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 (6 \, B a^{2} b - 3 \, A a b^{2} + 8 \, A a^{2} c\right )} x^{2} + 2 \, {\left (6 \, B a^{3} + A a^{2} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{48 \, a^{3} x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 524, normalized size = 4.33 \begin {gather*} -\frac {{\left (2 \, B a b^{2} - A b^{3} - 8 \, B a^{2} c + 4 \, 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^{2}} + \frac {6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a b^{2} - 3 \, {\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 + 12 \, {\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 )}^{4} B a^{2} b \sqrt {c} + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} A a^{2} c^{\frac {3}{2}} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a b^{3} + 48 \, {\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} + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a^{2} b^{2} \sqrt {c} - 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{3} b^{2} + 3 \, {\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 + 16 \, 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^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 386, normalized size = 3.19 \begin {gather*} \frac {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 {3}{2}}}-\frac {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 {5}{2}}}-\frac {B c \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{2 \sqrt {a}}+\frac {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 {3}{2}}}+\frac {\sqrt {c \,x^{2}+b x +a}\, A \,b^{2} c x}{8 a^{3}}-\frac {\sqrt {c \,x^{2}+b x +a}\, B b c x}{4 a^{2}}-\frac {\sqrt {c \,x^{2}+b x +a}\, A b c}{4 a^{2}}+\frac {\sqrt {c \,x^{2}+b x +a}\, A \,b^{3}}{8 a^{3}}+\frac {\sqrt {c \,x^{2}+b x +a}\, B c}{2 a}-\frac {\sqrt {c \,x^{2}+b x +a}\, B \,b^{2}}{4 a^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{2}}{8 a^{3} x}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B b}{4 a^{2} x}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A b}{4 a^{2} x^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B}{2 a \,x^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} 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 {\left (A+B\,x\right )\,\sqrt {c\,x^2+b\,x+a}}{x^4} \,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^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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