Optimal. Leaf size=60 \[ \frac {a^2 (a+b x)^{p+1}}{b^3 (p+1)}-\frac {2 a (a+b x)^{p+2}}{b^3 (p+2)}+\frac {(a+b x)^{p+3}}{b^3 (p+3)} \]
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Rubi [A] time = 0.02, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {43} \[ \frac {a^2 (a+b x)^{p+1}}{b^3 (p+1)}-\frac {2 a (a+b x)^{p+2}}{b^3 (p+2)}+\frac {(a+b x)^{p+3}}{b^3 (p+3)} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int x^2 (a+b x)^p \, dx &=\int \left (\frac {a^2 (a+b x)^p}{b^2}-\frac {2 a (a+b x)^{1+p}}{b^2}+\frac {(a+b x)^{2+p}}{b^2}\right ) \, dx\\ &=\frac {a^2 (a+b x)^{1+p}}{b^3 (1+p)}-\frac {2 a (a+b x)^{2+p}}{b^3 (2+p)}+\frac {(a+b x)^{3+p}}{b^3 (3+p)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 57, normalized size = 0.95 \[ \frac {(a+b x)^{p+1} \left (2 a^2-2 a b (p+1) x+b^2 \left (p^2+3 p+2\right ) x^2\right )}{b^3 (p+1) (p+2) (p+3)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 96, normalized size = 1.60 \[ -\frac {{\left (2 \, a^{2} b p x - {\left (b^{3} p^{2} + 3 \, b^{3} p + 2 \, b^{3}\right )} x^{3} - 2 \, a^{3} - {\left (a b^{2} p^{2} + a b^{2} p\right )} x^{2}\right )} {\left (b x + a\right )}^{p}}{b^{3} p^{3} + 6 \, b^{3} p^{2} + 11 \, b^{3} p + 6 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.22, size = 140, normalized size = 2.33 \[ \frac {{\left (b x + a\right )}^{p} b^{3} p^{2} x^{3} + {\left (b x + a\right )}^{p} a b^{2} p^{2} x^{2} + 3 \, {\left (b x + a\right )}^{p} b^{3} p x^{3} + {\left (b x + a\right )}^{p} a b^{2} p x^{2} + 2 \, {\left (b x + a\right )}^{p} b^{3} x^{3} - 2 \, {\left (b x + a\right )}^{p} a^{2} b p x + 2 \, {\left (b x + a\right )}^{p} a^{3}}{b^{3} p^{3} + 6 \, b^{3} p^{2} + 11 \, b^{3} p + 6 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 73, normalized size = 1.22 \[ \frac {\left (b^{2} p^{2} x^{2}+3 b^{2} p \,x^{2}-2 a b p x +2 x^{2} b^{2}-2 a x b +2 a^{2}\right ) \left (b x +a \right )^{p +1}}{\left (p^{3}+6 p^{2}+11 p +6\right ) b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 68, normalized size = 1.13 \[ \frac {{\left ({\left (p^{2} + 3 \, p + 2\right )} b^{3} x^{3} + {\left (p^{2} + p\right )} a b^{2} x^{2} - 2 \, a^{2} b p x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{p}}{{\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.60, size = 192, normalized size = 3.20 \[ \left \{\begin {array}{cl} \frac {2\,a^2\,\ln \left (a+b\,x\right )+b^2\,x^2-2\,a\,b\,x}{2\,b^3} & \text {\ if\ \ }p=-1\\ \frac {x}{b^2}-\frac {a^2}{b^3\,\left (a+b\,x\right )}-\frac {2\,a\,\ln \left (a+b\,x\right )}{b^3} & \text {\ if\ \ }p=-2\\ \frac {\ln \left (a+b\,x\right )+\frac {2\,a}{a+b\,x}-\frac {a^2}{2\,{\left (a+b\,x\right )}^2}}{b^3} & \text {\ if\ \ }p=-3\\ \frac {2\,{\left (a+b\,x\right )}^{p+1}\,\left (8\,a^2-8\,a\,b\,p\,x-8\,a\,b\,x+4\,b^2\,p^2\,x^2+12\,b^2\,p\,x^2+8\,b^2\,x^2\right )}{b^3\,\left (8\,p^3+48\,p^2+88\,p+48\right )} & \text {\ if\ \ }p\neq -1\wedge p\neq -2\wedge p\neq -3 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.24, size = 597, normalized size = 9.95 \[ \begin {cases} \frac {a^{p} x^{3}}{3} & \text {for}\: b = 0 \\\frac {2 a^{2} \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {3 a^{2}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {4 a b x \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {4 a b x}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {2 b^{2} x^{2} \log {\left (\frac {a}{b} + x \right )}}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} & \text {for}\: p = -3 \\- \frac {2 a^{2} \log {\left (\frac {a}{b} + x \right )}}{a b^{3} + b^{4} x} - \frac {2 a^{2}}{a b^{3} + b^{4} x} - \frac {2 a b x \log {\left (\frac {a}{b} + x \right )}}{a b^{3} + b^{4} x} + \frac {b^{2} x^{2}}{a b^{3} + b^{4} x} & \text {for}\: p = -2 \\\frac {a^{2} \log {\left (\frac {a}{b} + x \right )}}{b^{3}} - \frac {a x}{b^{2}} + \frac {x^{2}}{2 b} & \text {for}\: p = -1 \\\frac {2 a^{3} \left (a + b x\right )^{p}}{b^{3} p^{3} + 6 b^{3} p^{2} + 11 b^{3} p + 6 b^{3}} - \frac {2 a^{2} b p x \left (a + b x\right )^{p}}{b^{3} p^{3} + 6 b^{3} p^{2} + 11 b^{3} p + 6 b^{3}} + \frac {a b^{2} p^{2} x^{2} \left (a + b x\right )^{p}}{b^{3} p^{3} + 6 b^{3} p^{2} + 11 b^{3} p + 6 b^{3}} + \frac {a b^{2} p x^{2} \left (a + b x\right )^{p}}{b^{3} p^{3} + 6 b^{3} p^{2} + 11 b^{3} p + 6 b^{3}} + \frac {b^{3} p^{2} x^{3} \left (a + b x\right )^{p}}{b^{3} p^{3} + 6 b^{3} p^{2} + 11 b^{3} p + 6 b^{3}} + \frac {3 b^{3} p x^{3} \left (a + b x\right )^{p}}{b^{3} p^{3} + 6 b^{3} p^{2} + 11 b^{3} p + 6 b^{3}} + \frac {2 b^{3} x^{3} \left (a + b x\right )^{p}}{b^{3} p^{3} + 6 b^{3} p^{2} + 11 b^{3} p + 6 b^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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