Optimal. Leaf size=210 \[ \frac {b^2 x^6 \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{6 (a+b x)}+\frac {3 a b x^5 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{5 (a+b x)}+\frac {a^2 x^4 \sqrt {a^2+2 a b x+b^2 x^2} (a B+3 A b)}{4 (a+b x)}+\frac {b^3 B x^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {a^3 A x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)} \]
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Rubi [A] time = 0.09, antiderivative size = 210, 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 {b^2 x^6 \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{6 (a+b x)}+\frac {3 a b x^5 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{5 (a+b x)}+\frac {a^2 x^4 \sqrt {a^2+2 a b x+b^2 x^2} (a B+3 A b)}{4 (a+b x)}+\frac {a^3 A x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {b^3 B x^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 (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) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x^2 \left (a b+b^2 x\right )^3 (A+B x) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a^3 A b^3 x^2+a^2 b^3 (3 A b+a B) x^3+3 a b^4 (A b+a B) x^4+b^5 (A b+3 a B) x^5+b^6 B x^6\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {a^3 A x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {a^2 (3 A b+a B) x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac {3 a b (A b+a B) x^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac {b^2 (A b+3 a B) x^6 \sqrt {a^2+2 a b x+b^2 x^2}}{6 (a+b x)}+\frac {b^3 B x^7 \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 87, normalized size = 0.41 \begin {gather*} \frac {x^3 \sqrt {(a+b x)^2} \left (35 a^3 (4 A+3 B x)+63 a^2 b x (5 A+4 B x)+42 a b^2 x^2 (6 A+5 B x)+10 b^3 x^3 (7 A+6 B x)\right )}{420 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.94, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 73, normalized size = 0.35 \begin {gather*} \frac {1}{7} \, B b^{3} x^{7} + \frac {1}{3} \, A a^{3} x^{3} + \frac {1}{6} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + \frac {3}{5} \, {\left (B a^{2} b + A a b^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 150, normalized size = 0.71 \begin {gather*} \frac {1}{7} \, B b^{3} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, B a b^{2} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, A b^{3} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{5} \, B a^{2} b x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{5} \, A a b^{2} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, B a^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{4} \, A a^{2} b x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, A a^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) - \frac {{\left (3 \, B a^{7} - 7 \, A a^{6} b\right )} \mathrm {sgn}\left (b x + a\right )}{420 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 92, normalized size = 0.44 \begin {gather*} \frac {\left (60 b^{3} B \,x^{4}+70 A \,b^{3} x^{3}+210 x^{3} B a \,b^{2}+252 x^{2} A a \,b^{2}+252 B \,a^{2} b \,x^{2}+315 x A \,a^{2} b +105 B \,a^{3} x +140 A \,a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} x^{3}}{420 \left (b x +a \right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 241, normalized size = 1.15 \begin {gather*} -\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a^{3} x}{4 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A a^{2} x}{4 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B x^{2}}{7 \, b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B a^{4}}{4 \, b^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A a^{3}}{4 \, b^{3}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a x}{14 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A x}{6 \, b^{2}} + \frac {17 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B a^{2}}{70 \, b^{4}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A a}{30 \, b^{3}} \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 = -0.00 \begin {gather*} \int x^2\,\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\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 x^{2} \left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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