Optimal. Leaf size=202 \[ -\frac {a^2 (3 A b-4 a B)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 a (a+b x) (A b-2 a B) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x (a+b x) (A b-3 a B)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B x^2 (a+b x)}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^3 (A b-a B)}{2 b^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.14, antiderivative size = 202, 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} \[ \frac {a^3 (A b-a B)}{2 b^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {a^2 (3 A b-4 a B)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x (a+b x) (A b-3 a B)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 a (a+b x) (A b-2 a B) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B x^2 (a+b x)}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 77
Rule 770
Rubi steps
\begin {align*} \int \frac {x^3 (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^3 (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 {A b-3 a B}{b^7}+\frac {B x}{b^6}+\frac {a^3 (-A b+a B)}{b^7 (a+b x)^3}-\frac {a^2 (-3 A b+4 a B)}{b^7 (a+b x)^2}+\frac {3 a (-A b+2 a B)}{b^7 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {a^2 (3 A b-4 a B)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^3 (A b-a B)}{2 b^5 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(A b-3 a B) x (a+b x)}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B x^2 (a+b x)}{2 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 a (A b-2 a B) (a+b x) \log (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 117, normalized size = 0.58 \[ \frac {7 a^4 B+a^3 (2 b B x-5 A b)-a^2 b^2 x (4 A+11 B x)+4 a b^3 x^2 (A-B x)+6 a (a+b x)^2 (2 a B-A b) \log (a+b x)+b^4 x^3 (2 A+B x)}{2 b^5 (a+b x) \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 171, normalized size = 0.85 \[ \frac {B b^{4} x^{4} + 7 \, B a^{4} - 5 \, A a^{3} b - 2 \, {\left (2 \, B a b^{3} - A b^{4}\right )} x^{3} - {\left (11 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{2} + 2 \, {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x + 6 \, {\left (2 \, B a^{4} - A a^{3} b + {\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} x^{2} + 2 \, {\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} x\right )} \log \left (b x + a\right )}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \]
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 = 0.10, size = 191, normalized size = 0.95 \[ -\frac {\left (-B \,b^{4} x^{4}+6 A a \,b^{3} x^{2} \ln \left (b x +a \right )-2 A \,b^{4} x^{3}-12 B \,a^{2} b^{2} x^{2} \ln \left (b x +a \right )+4 B a \,b^{3} x^{3}+12 A \,a^{2} b^{2} x \ln \left (b x +a \right )-4 A a \,b^{3} x^{2}-24 B \,a^{3} b x \ln \left (b x +a \right )+11 B \,a^{2} b^{2} x^{2}+6 A \,a^{3} b \ln \left (b x +a \right )+4 A \,a^{2} b^{2} x -12 B \,a^{4} \ln \left (b x +a \right )-2 B \,a^{3} b x +5 A \,a^{3} b -7 B \,a^{4}\right ) \left (b x +a \right )}{2 \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 242, normalized size = 1.20 \[ \frac {B x^{3}}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac {5 \, B a x^{2}}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{3}} + \frac {A x^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac {6 \, B a^{2} \log \left (x + \frac {a}{b}\right )}{b^{5}} - \frac {3 \, A a \log \left (x + \frac {a}{b}\right )}{b^{4}} - \frac {5 \, B a^{3}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{5}} + \frac {2 \, A a^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4}} + \frac {12 \, B a^{3} x}{b^{6} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {6 \, A a^{2} x}{b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {23 \, B a^{4}}{2 \, b^{7} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {11 \, A a^{3}}{2 \, b^{6} {\left (x + \frac {a}{b}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,\left (A+B\,x\right )}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{3} \left (A + B x\right )}{\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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