Optimal. Leaf size=96 \[ \frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {A \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{a^{3/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 96, 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, 12, 724, 206} \begin {gather*} \frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {A \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 724
Rule 822
Rubi steps
\begin {align*} \int \frac {A+B x}{x \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 ) \sqrt {a+b x+c x^2}}-\frac {2 \int -\frac {A \left (b^2-4 a c\right )}{2 x \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 ) \sqrt {a+b x+c x^2}}+\frac {A \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{a}\\ &=\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 ) \sqrt {a+b x+c x^2}}-\frac {(2 A) \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}\\ &=\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 ) \sqrt {a+b x+c x^2}}-\frac {A \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{a^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 104, normalized size = 1.08 \begin {gather*} \frac {\frac {2 \sqrt {a} \left (a B (b+2 c x)-A \left (-2 a c+b^2+b c x\right )\right )}{\sqrt {a+x (b+c x)}}+A \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 )}{a^{3/2} \left (4 a c-b^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.53, size = 103, normalized size = 1.07 \begin {gather*} \frac {2 A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+b x+c x^2}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {2 \left (2 a A c+a b B+2 a B c x-A b^2-A b c x\right )}{a \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 = 0.66, size = 412, normalized size = 4.29 \begin {gather*} \left [\frac {{\left (A a b^{2} - 4 \, A a^{2} c + {\left (A b^{2} c - 4 \, A a c^{2}\right )} x^{2} + {\left (A b^{3} - 4 \, A a b 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 (B a^{2} b - A a b^{2} + 2 \, A a^{2} c + {\left (2 \, B a^{2} - A a b\right )} c x\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left (a^{3} b^{2} - 4 \, a^{4} c + {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{2} + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x\right )}}, \frac {{\left (A a b^{2} - 4 \, A a^{2} c + {\left (A b^{2} c - 4 \, A a c^{2}\right )} x^{2} + {\left (A b^{3} - 4 \, A a b 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 (B a^{2} b - A a b^{2} + 2 \, A a^{2} c + {\left (2 \, B a^{2} - A a b\right )} c x\right )} \sqrt {c x^{2} + b x + a}}{a^{3} b^{2} - 4 \, a^{4} c + {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{2} + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 125, normalized size = 1.30 \begin {gather*} -\frac {2 \, {\left (\frac {{\left (2 \, B a^{2} c - A a b c\right )} x}{a^{2} b^{2} - 4 \, a^{3} c} + \frac {B a^{2} b - A a b^{2} + 2 \, A a^{2} c}{a^{2} b^{2} - 4 \, a^{3} c}\right )}}{\sqrt {c x^{2} + b x + a}} + \frac {2 \, A \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 153, normalized size = 1.59 \begin {gather*} -\frac {2 A b c x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a}-\frac {A \,b^{2}}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a}+\frac {2 \left (2 c x +b \right ) B}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {A \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{a^{\frac {3}{2}}}+\frac {A}{\sqrt {c \,x^{2}+b x +a}\, a} \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\,{\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 \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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