Optimal. Leaf size=114 \[ \frac {x^4 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{4 (a+b x)}+\frac {a A x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {b B x^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)} \]
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Rubi [A] time = 0.06, antiderivative size = 114, 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, 76} \begin {gather*} \frac {x^4 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{4 (a+b x)}+\frac {a A x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {b B x^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rule 770
Rubi steps
\begin {align*} \int x^2 (A+B x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x^2 \left (a b+b^2 x\right ) (A+B x) \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a A b x^2+b (A b+a B) x^3+b^2 B x^4\right ) \, dx}{a b+b^2 x}\\ &=\frac {a A x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {(A b+a B) x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac {b B x^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 49, normalized size = 0.43 \begin {gather*} \frac {x^3 \sqrt {(a+b x)^2} (5 a (4 A+3 B x)+3 b x (5 A+4 B x))}{60 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.68, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 (A+B x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 27, normalized size = 0.24 \begin {gather*} \frac {1}{5} \, B b x^{5} + \frac {1}{3} \, A a x^{3} + \frac {1}{4} \, {\left (B a + A b\right )} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 78, normalized size = 0.68 \begin {gather*} \frac {1}{5} \, B b x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, B a x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, A b x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, A a x^{3} \mathrm {sgn}\left (b x + a\right ) - \frac {{\left (3 \, B a^{5} - 5 \, A a^{4} b\right )} \mathrm {sgn}\left (b x + a\right )}{60 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 44, normalized size = 0.39 \begin {gather*} \frac {\left (12 B b \,x^{2}+15 A b x +15 B a x +20 A a \right ) \sqrt {\left (b x +a \right )^{2}}\, x^{3}}{60 b x +60 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.57, size = 241, normalized size = 2.11 \begin {gather*} -\frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a^{3} x}{2 \, b^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A a^{2} x}{2 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B x^{2}}{5 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a^{4}}{2 \, b^{4}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A a^{3}}{2 \, b^{3}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a x}{20 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A x}{4 \, b^{2}} + \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a^{2}}{20 \, b^{4}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A a}{12 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.30, size = 271, normalized size = 2.38 \begin {gather*} \frac {B\,x^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{5\,b^2}+\frac {A\,x\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{4\,b^2}-\frac {7\,B\,a\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a^3-5\,a\,b^2\,x^2+3\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-4\,a^2\,b\,x\right )}{60\,b^4}-\frac {5\,A\,a\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{96\,b^5}-\frac {B\,a^2\,\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{60\,b^6}-\frac {A\,a^2\,\left (\frac {x}{2}+\frac {a}{2\,b}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 29, normalized size = 0.25 \begin {gather*} \frac {A a x^{3}}{3} + \frac {B b x^{5}}{5} + x^{4} \left (\frac {A b}{4} + \frac {B a}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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