Optimal. Leaf size=154 \[ -\frac {a^2 (A b-a B)}{2 b^4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a (2 A b-3 a B)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (A b-3 a B) \log (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B x (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.11, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {770, 77} \[ -\frac {a^2 (A b-a B)}{2 b^4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a (2 A b-3 a B)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (A b-3 a B) \log (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B x (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 77
Rule 770
Rubi steps
\begin {align*} \int \frac {x^2 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {x^2 (A+B x)}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \left (\frac {B}{b^6}-\frac {a^2 (-A b+a B)}{b^6 (a+b x)^3}+\frac {a (-2 A b+3 a B)}{b^6 (a+b x)^2}+\frac {A b-3 a B}{b^6 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {a (2 A b-3 a B)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a^2 (A b-a B)}{2 b^4 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B x (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-3 a B) (a+b x) \log (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 89, normalized size = 0.58 \[ \frac {-5 a^3 B+a^2 b (3 A-4 B x)+4 a b^2 x (A+B x)+2 (a+b x)^2 (A b-3 a B) \log (a+b x)+2 b^3 B x^3}{2 b^4 (a+b x) \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 134, normalized size = 0.87 \[ \frac {2 \, B b^{3} x^{3} + 4 \, B a b^{2} x^{2} - 5 \, B a^{3} + 3 \, A a^{2} b - 4 \, {\left (B a^{2} b - A a b^{2}\right )} x - 2 \, {\left (3 \, B a^{3} - A a^{2} b + {\left (3 \, B a b^{2} - A b^{3}\right )} x^{2} + 2 \, {\left (3 \, B a^{2} b - A a b^{2}\right )} x\right )} \log \left (b x + a\right )}{2 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 153, normalized size = 0.99 \[ \frac {\left (2 A \,b^{3} x^{2} \ln \left (b x +a \right )-6 B a \,b^{2} x^{2} \ln \left (b x +a \right )+2 B \,b^{3} x^{3}+4 A a \,b^{2} x \ln \left (b x +a \right )-12 B \,a^{2} b x \ln \left (b x +a \right )+4 B a \,b^{2} x^{2}+2 A \,a^{2} b \ln \left (b x +a \right )+4 A a \,b^{2} x -6 B \,a^{3} \ln \left (b x +a \right )-4 B \,a^{2} b x +3 A \,a^{2} b -5 B \,a^{3}\right ) \left (b x +a \right )}{2 \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 154, normalized size = 1.00 \[ \frac {B x^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac {3 \, B a \log \left (x + \frac {a}{b}\right )}{b^{4}} + \frac {A \log \left (x + \frac {a}{b}\right )}{b^{3}} + \frac {2 \, B a^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4}} - \frac {6 \, B a^{2} x}{b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {2 \, A a x}{b^{4} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {11 \, B a^{3}}{2 \, b^{6} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {3 \, A a^{2}}{2 \, b^{5} {\left (x + \frac {a}{b}\right )}^{2}} \]
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 \[ \int \frac {x^2\,\left (A+B\,x\right )}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{2} \left (A + B x\right )}{\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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