Optimal. Leaf size=38 \[ \frac {(a+b x)^5 (A b-a B)}{5 b^2}+\frac {B (a+b x)^6}{6 b^2} \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 43} \[ \frac {(a+b x)^5 (A b-a B)}{5 b^2}+\frac {B (a+b x)^6}{6 b^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int (a+b x)^4 (A+B x) \, dx\\ &=\int \left (\frac {(A b-a B) (a+b x)^4}{b}+\frac {B (a+b x)^5}{b}\right ) \, dx\\ &=\frac {(A b-a B) (a+b x)^5}{5 b^2}+\frac {B (a+b x)^6}{6 b^2}\\ \end {align*}
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Mathematica [B] time = 0.02, size = 84, normalized size = 2.21 \[ \frac {1}{30} x \left (15 a^4 (2 A+B x)+20 a^3 b x (3 A+2 B x)+15 a^2 b^2 x^2 (4 A+3 B x)+6 a b^3 x^3 (5 A+4 B x)+b^4 x^4 (6 A+5 B x)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.79, size = 97, normalized size = 2.55 \[ \frac {1}{6} x^{6} b^{4} B + \frac {4}{5} x^{5} b^{3} a B + \frac {1}{5} x^{5} b^{4} A + \frac {3}{2} x^{4} b^{2} a^{2} B + x^{4} b^{3} a A + \frac {4}{3} x^{3} b a^{3} B + 2 x^{3} b^{2} a^{2} A + \frac {1}{2} x^{2} a^{4} B + 2 x^{2} b a^{3} A + x a^{4} A \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 97, normalized size = 2.55 \[ \frac {1}{6} \, B b^{4} x^{6} + \frac {4}{5} \, B a b^{3} x^{5} + \frac {1}{5} \, A b^{4} x^{5} + \frac {3}{2} \, B a^{2} b^{2} x^{4} + A a b^{3} x^{4} + \frac {4}{3} \, B a^{3} b x^{3} + 2 \, A a^{2} b^{2} x^{3} + \frac {1}{2} \, B a^{4} x^{2} + 2 \, A a^{3} b x^{2} + A a^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 97, normalized size = 2.55 \[ \frac {B \,b^{4} x^{6}}{6}+A \,a^{4} x +\frac {\left (A \,b^{4}+4 B a \,b^{3}\right ) x^{5}}{5}+\frac {\left (4 A a \,b^{3}+6 b^{2} B \,a^{2}\right ) x^{4}}{4}+\frac {\left (6 A \,b^{2} a^{2}+4 a^{3} b B \right ) x^{3}}{3}+\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.62, size = 96, normalized size = 2.53 \[ \frac {1}{6} \, B b^{4} x^{6} + A a^{4} x + \frac {1}{5} \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{5} + \frac {1}{2} \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{4} + \frac {2}{3} \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.00, size = 88, normalized size = 2.32 \[ x^2\,\left (\frac {B\,a^4}{2}+2\,A\,b\,a^3\right )+x^5\,\left (\frac {A\,b^4}{5}+\frac {4\,B\,a\,b^3}{5}\right )+\frac {B\,b^4\,x^6}{6}+A\,a^4\,x+\frac {2\,a^2\,b\,x^3\,\left (3\,A\,b+2\,B\,a\right )}{3}+\frac {a\,b^2\,x^4\,\left (2\,A\,b+3\,B\,a\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.08, size = 100, normalized size = 2.63 \[ A a^{4} x + \frac {B b^{4} x^{6}}{6} + x^{5} \left (\frac {A b^{4}}{5} + \frac {4 B a b^{3}}{5}\right ) + x^{4} \left (A a b^{3} + \frac {3 B a^{2} b^{2}}{2}\right ) + x^{3} \left (2 A a^{2} b^{2} + \frac {4 B a^{3} b}{3}\right ) + x^{2} \left (2 A a^{3} b + \frac {B a^{4}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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