Optimal. Leaf size=61 \[ \frac {4 a^2 (a+b x)^{m+3}}{b (m+3)}-\frac {4 a (a+b x)^{m+4}}{b (m+4)}+\frac {(a+b x)^{m+5}}{b (m+5)} \]
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Rubi [A] time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {627, 43} \begin {gather*} \frac {4 a^2 (a+b x)^{m+3}}{b (m+3)}-\frac {4 a (a+b x)^{m+4}}{b (m+4)}+\frac {(a+b x)^{m+5}}{b (m+5)} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 627
Rubi steps
\begin {align*} \int (a+b x)^m \left (a^2-b^2 x^2\right )^2 \, dx &=\int (a-b x)^2 (a+b x)^{2+m} \, dx\\ &=\int \left (4 a^2 (a+b x)^{2+m}-4 a (a+b x)^{3+m}+(a+b x)^{4+m}\right ) \, dx\\ &=\frac {4 a^2 (a+b x)^{3+m}}{b (3+m)}-\frac {4 a (a+b x)^{4+m}}{b (4+m)}+\frac {(a+b x)^{5+m}}{b (5+m)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 50, normalized size = 0.82 \begin {gather*} \frac {(a+b x)^{m+3} \left (\frac {4 a^2}{m+3}-\frac {4 a (a+b x)}{m+4}+\frac {(a+b x)^2}{m+5}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.16, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^m \left (a^2-b^2 x^2\right )^2 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.42, size = 173, normalized size = 2.84 \begin {gather*} \frac {{\left (a^{5} m^{2} + 11 \, a^{5} m + {\left (b^{5} m^{2} + 7 \, b^{5} m + 12 \, b^{5}\right )} x^{5} + 32 \, a^{5} + {\left (a b^{4} m^{2} + 3 \, a b^{4} m\right )} x^{4} - 2 \, {\left (a^{2} b^{3} m^{2} + 11 \, a^{2} b^{3} m + 20 \, a^{2} b^{3}\right )} x^{3} - 2 \, {\left (a^{3} b^{2} m^{2} + 7 \, a^{3} b^{2} m\right )} x^{2} + {\left (a^{4} b m^{2} + 15 \, a^{4} b m + 60 \, a^{4} b\right )} x\right )} {\left (b x + a\right )}^{m}}{b m^{3} + 12 \, b m^{2} + 47 \, b m + 60 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 288, normalized size = 4.72 \begin {gather*} \frac {{\left (b x + a\right )}^{m} b^{5} m^{2} x^{5} + {\left (b x + a\right )}^{m} a b^{4} m^{2} x^{4} + 7 \, {\left (b x + a\right )}^{m} b^{5} m x^{5} - 2 \, {\left (b x + a\right )}^{m} a^{2} b^{3} m^{2} x^{3} + 3 \, {\left (b x + a\right )}^{m} a b^{4} m x^{4} + 12 \, {\left (b x + a\right )}^{m} b^{5} x^{5} - 2 \, {\left (b x + a\right )}^{m} a^{3} b^{2} m^{2} x^{2} - 22 \, {\left (b x + a\right )}^{m} a^{2} b^{3} m x^{3} + {\left (b x + a\right )}^{m} a^{4} b m^{2} x - 14 \, {\left (b x + a\right )}^{m} a^{3} b^{2} m x^{2} - 40 \, {\left (b x + a\right )}^{m} a^{2} b^{3} x^{3} + {\left (b x + a\right )}^{m} a^{5} m^{2} + 15 \, {\left (b x + a\right )}^{m} a^{4} b m x + 11 \, {\left (b x + a\right )}^{m} a^{5} m + 60 \, {\left (b x + a\right )}^{m} a^{4} b x + 32 \, {\left (b x + a\right )}^{m} a^{5}}{b m^{3} + 12 \, b m^{2} + 47 \, b m + 60 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 94, normalized size = 1.54 \begin {gather*} \frac {\left (b^{2} m^{2} x^{2}-2 a b \,m^{2} x +7 b^{2} m \,x^{2}+a^{2} m^{2}-18 a b m x +12 b^{2} x^{2}+11 a^{2} m -36 a b x +32 a^{2}\right ) \left (b x +a \right )^{m +3}}{\left (m^{3}+12 m^{2}+47 m +60\right ) b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.50, size = 233, normalized size = 3.82 \begin {gather*} \frac {{\left (b x + a\right )}^{m + 1} a^{4}}{b {\left (m + 1\right )}} - \frac {2 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} b^{3} x^{3} + {\left (m^{2} + m\right )} a b^{2} x^{2} - 2 \, a^{2} b m x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{m} a^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} b^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a b^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{2} b^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b m x + 24 \, a^{5}\right )} {\left (b x + a\right )}^{m}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.55, size = 186, normalized size = 3.05 \begin {gather*} {\left (a+b\,x\right )}^m\,\left (\frac {a^4\,x\,\left (m^2+15\,m+60\right )}{m^3+12\,m^2+47\,m+60}+\frac {a^5\,\left (m^2+11\,m+32\right )}{b\,\left (m^3+12\,m^2+47\,m+60\right )}+\frac {b^4\,x^5\,\left (m^2+7\,m+12\right )}{m^3+12\,m^2+47\,m+60}-\frac {2\,a^2\,b^2\,x^3\,\left (m^2+11\,m+20\right )}{m^3+12\,m^2+47\,m+60}+\frac {a\,b^3\,m\,x^4\,\left (m+3\right )}{m^3+12\,m^2+47\,m+60}-\frac {2\,a^3\,b\,m\,x^2\,\left (m+7\right )}{m^3+12\,m^2+47\,m+60}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.05, size = 876, normalized size = 14.36 \begin {gather*} \begin {cases} a^{4} a^{m} x & \text {for}\: b = 0 \\\frac {a^{2} \log {\left (\frac {a}{b} + x \right )}}{a^{2} b + 2 a b^{2} x + b^{3} x^{2}} + \frac {2 a^{2}}{a^{2} b + 2 a b^{2} x + b^{3} x^{2}} + \frac {2 a b x \log {\left (\frac {a}{b} + x \right )}}{a^{2} b + 2 a b^{2} x + b^{3} x^{2}} + \frac {4 a b x}{a^{2} b + 2 a b^{2} x + b^{3} x^{2}} + \frac {b^{2} x^{2} \log {\left (\frac {a}{b} + x \right )}}{a^{2} b + 2 a b^{2} x + b^{3} x^{2}} & \text {for}\: m = -5 \\- \frac {4 a^{2} \log {\left (\frac {a}{b} + x \right )}}{a b + b^{2} x} - \frac {7 a^{2}}{a b + b^{2} x} - \frac {4 a b x \log {\left (\frac {a}{b} + x \right )}}{a b + b^{2} x} - \frac {2 a b x}{a b + b^{2} x} + \frac {b^{2} x^{2}}{a b + b^{2} x} & \text {for}\: m = -4 \\\frac {4 a^{2} \log {\left (\frac {a}{b} + x \right )}}{b} - 3 a x + \frac {b x^{2}}{2} & \text {for}\: m = -3 \\\frac {a^{5} m^{2} \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} + \frac {11 a^{5} m \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} + \frac {32 a^{5} \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} + \frac {a^{4} b m^{2} x \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} + \frac {15 a^{4} b m x \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} + \frac {60 a^{4} b x \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} - \frac {2 a^{3} b^{2} m^{2} x^{2} \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} - \frac {14 a^{3} b^{2} m x^{2} \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} - \frac {2 a^{2} b^{3} m^{2} x^{3} \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} - \frac {22 a^{2} b^{3} m x^{3} \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} - \frac {40 a^{2} b^{3} x^{3} \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} + \frac {a b^{4} m^{2} x^{4} \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} + \frac {3 a b^{4} m x^{4} \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} + \frac {b^{5} m^{2} x^{5} \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} + \frac {7 b^{5} m x^{5} \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} + \frac {12 b^{5} x^{5} \left (a + b x\right )^{m}}{b m^{3} + 12 b m^{2} + 47 b m + 60 b} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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