Optimal. Leaf size=158 \[ \frac {(3 A b-2 a B) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 a^{5/2}}-\frac {\sqrt {a+b x+c x^2} \left (-8 a A c-2 a b B+3 A b^2\right )}{a^2 x \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \]
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Rubi [A] time = 0.12, antiderivative size = 158, 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 = {822, 806, 724, 206} \begin {gather*} -\frac {\sqrt {a+b x+c x^2} \left (-8 a A c-2 a b B+3 A b^2\right )}{a^2 x \left (b^2-4 a c\right )}+\frac {(3 A b-2 a B) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 a^{5/2}}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a x \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 806
Rule 822
Rubi steps
\begin {align*} \int \frac {A+B x}{x^2 \left (a+b x+c x^2\right )^{3/2}} \, dx &=\frac {2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{a \left (b^2-4 a c\right ) x \sqrt {a+b x+c x^2}}-\frac {2 \int \frac {\frac {1}{2} \left (-3 A b^2+2 a b B+8 a A c\right )-(A b-2 a B) c x}{x^2 \sqrt {a+b x+c x^2}} \, dx}{a \left (b^2-4 a c\right )}\\ &=\frac {2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{a \left (b^2-4 a c\right ) x \sqrt {a+b x+c x^2}}-\frac {\left (3 A b^2-2 a b B-8 a A c\right ) \sqrt {a+b x+c x^2}}{a^2 \left (b^2-4 a c\right ) x}-\frac {(3 A b-2 a B) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{2 a^2}\\ &=\frac {2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{a \left (b^2-4 a c\right ) x \sqrt {a+b x+c x^2}}-\frac {\left (3 A b^2-2 a b B-8 a A c\right ) \sqrt {a+b x+c x^2}}{a^2 \left (b^2-4 a c\right ) x}+\frac {(3 A b-2 a B) \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 )}{a^2}\\ &=\frac {2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{a \left (b^2-4 a c\right ) x \sqrt {a+b x+c x^2}}-\frac {\left (3 A b^2-2 a b B-8 a A c\right ) \sqrt {a+b x+c x^2}}{a^2 \left (b^2-4 a c\right ) x}+\frac {(3 A b-2 a B) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 a^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 150, normalized size = 0.95 \begin {gather*} \frac {\frac {2 \sqrt {a} \left (-4 a^2 c (A-B x)+a A \left (b^2-10 b c x-8 c^2 x^2\right )-2 a b B x (b+c x)+3 A b^2 x (b+c x)\right )}{x \sqrt {a+x (b+c x)}}-\left (b^2-4 a c\right ) (3 A b-2 a B) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )}{2 a^{5/2} \left (4 a c-b^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.77, size = 153, normalized size = 0.97 \begin {gather*} \frac {(2 a B-3 A b) \tanh ^{-1}\left (\frac {\sqrt {c} x-\sqrt {a+b x+c x^2}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {-4 a^2 A c+4 a^2 B c x+a A b^2-10 a A b c x-8 a A c^2 x^2-2 a b^2 B x-2 a b B c x^2+3 A b^3 x+3 A b^2 c x^2}{a^2 x \left (4 a c-b^2\right ) \sqrt {a+b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.86, size = 657, normalized size = 4.16 \begin {gather*} \left [\frac {{\left ({\left (4 \, {\left (2 \, B a^{2} - 3 \, A a b\right )} c^{2} - {\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} c\right )} x^{3} - {\left (2 \, B a b^{3} - 3 \, A b^{4} - 4 \, {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} c\right )} x^{2} - {\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3} - 4 \, {\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} c\right )} x\right )} \sqrt {a} \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 (A a^{2} b^{2} - 4 \, A a^{3} c - {\left (8 \, A a^{2} c^{2} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} c\right )} x^{2} - {\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3} - 2 \, {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} c\right )} x\right )} \sqrt {c x^{2} + b x + a}}{4 \, {\left ({\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} x^{3} + {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{2} + {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x\right )}}, -\frac {{\left ({\left (4 \, {\left (2 \, B a^{2} - 3 \, A a b\right )} c^{2} - {\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} c\right )} x^{3} - {\left (2 \, B a b^{3} - 3 \, A b^{4} - 4 \, {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} c\right )} x^{2} - {\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3} - 4 \, {\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} c\right )} x\right )} \sqrt {-a} \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 (A a^{2} b^{2} - 4 \, A a^{3} c - {\left (8 \, A a^{2} c^{2} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} c\right )} x^{2} - {\left (2 \, B a^{2} b^{2} - 3 \, A a b^{3} - 2 \, {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} c\right )} x\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left ({\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} x^{3} + {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{2} + {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 220, normalized size = 1.39 \begin {gather*} \frac {2 \, {\left (\frac {{\left (B a^{3} b c - A a^{2} b^{2} c + 2 \, A a^{3} c^{2}\right )} x}{a^{4} b^{2} - 4 \, a^{5} c} + \frac {B a^{3} b^{2} - A a^{2} b^{3} - 2 \, B a^{4} c + 3 \, A a^{3} b c}{a^{4} b^{2} - 4 \, a^{5} c}\right )}}{\sqrt {c x^{2} + b x + a}} + \frac {{\left (2 \, B a - 3 \, A b\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A b + 2 \, A a \sqrt {c}}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 330, normalized size = 2.09 \begin {gather*} -\frac {8 A \,c^{2} x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a}+\frac {3 A \,b^{2} c x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{2}}-\frac {2 B b c x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a}-\frac {4 A b c}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a}+\frac {3 A \,b^{3}}{2 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{2}}-\frac {B \,b^{2}}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a}+\frac {3 A b \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{2 a^{\frac {5}{2}}}-\frac {B \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{a^{\frac {3}{2}}}-\frac {3 A b}{2 \sqrt {c \,x^{2}+b x +a}\, a^{2}}+\frac {B}{\sqrt {c \,x^{2}+b x +a}\, a}-\frac {A}{\sqrt {c \,x^{2}+b x +a}\, a x} \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^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,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^{2} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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