Optimal. Leaf size=84 \[ \frac {8 a^3 (a+b x)^{4+m}}{b (4+m)}-\frac {12 a^2 (a+b x)^{5+m}}{b (5+m)}+\frac {6 a (a+b x)^{6+m}}{b (6+m)}-\frac {(a+b x)^{7+m}}{b (7+m)} \]
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Rubi [A]
time = 0.03, antiderivative size = 84, 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 = {641, 45}
\begin {gather*} \frac {8 a^3 (a+b x)^{m+4}}{b (m+4)}-\frac {12 a^2 (a+b x)^{m+5}}{b (m+5)}+\frac {6 a (a+b x)^{m+6}}{b (m+6)}-\frac {(a+b x)^{m+7}}{b (m+7)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 641
Rubi steps
\begin {align*} \int (a+b x)^m \left (a^2-b^2 x^2\right )^3 \, dx &=\int (a-b x)^3 (a+b x)^{3+m} \, dx\\ &=\int \left (8 a^3 (a+b x)^{3+m}-12 a^2 (a+b x)^{4+m}+6 a (a+b x)^{5+m}-(a+b x)^{6+m}\right ) \, dx\\ &=\frac {8 a^3 (a+b x)^{4+m}}{b (4+m)}-\frac {12 a^2 (a+b x)^{5+m}}{b (5+m)}+\frac {6 a (a+b x)^{6+m}}{b (6+m)}-\frac {(a+b x)^{7+m}}{b (7+m)}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 114, normalized size = 1.36 \begin {gather*} \frac {(a+b x)^{4+m} \left (a^3 \left (384+152 m+21 m^2+m^3\right )-3 a^2 b \left (232+118 m+19 m^2+m^3\right ) x+3 a b^2 \left (160+92 m+17 m^2+m^3\right ) x^2-b^3 \left (120+74 m+15 m^2+m^3\right ) x^3\right )}{b (4+m) (5+m) (6+m) (7+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(177\) vs.
\(2(84)=168\).
time = 0.50, size = 178, normalized size = 2.12
method | result | size |
gosper | \(\frac {\left (b x +a \right )^{4+m} \left (-b^{3} m^{3} x^{3}+3 a \,b^{2} m^{3} x^{2}-15 b^{3} m^{2} x^{3}-3 a^{2} b \,m^{3} x +51 a \,b^{2} m^{2} x^{2}-74 b^{3} m \,x^{3}+a^{3} m^{3}-57 a^{2} b \,m^{2} x +276 a \,b^{2} m \,x^{2}-120 b^{3} x^{3}+21 a^{3} m^{2}-354 a^{2} b m x +480 a \,b^{2} x^{2}+152 a^{3} m -696 a^{2} b x +384 a^{3}\right )}{b \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}\) | \(178\) |
norman | \(\frac {a^{6} \left (m^{3}+27 m^{2}+254 m +840\right ) x \,{\mathrm e}^{m \ln \left (b x +a \right )}}{m^{4}+22 m^{3}+179 m^{2}+638 m +840}+\frac {a^{7} \left (m^{3}+21 m^{2}+152 m +384\right ) {\mathrm e}^{m \ln \left (b x +a \right )}}{b \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}-\frac {b^{6} x^{7} {\mathrm e}^{m \ln \left (b x +a \right )}}{7+m}-\frac {a m \,b^{5} x^{6} {\mathrm e}^{m \ln \left (b x +a \right )}}{m^{2}+13 m +42}-\frac {3 a^{4} b^{2} \left (m^{3}+23 m^{2}+162 m +280\right ) x^{3} {\mathrm e}^{m \ln \left (b x +a \right )}}{m^{4}+22 m^{3}+179 m^{2}+638 m +840}+\frac {3 \left (m^{2}+15 m +42\right ) a^{2} b^{4} x^{5} {\mathrm e}^{m \ln \left (b x +a \right )}}{m^{3}+18 m^{2}+107 m +210}+\frac {3 a^{3} b^{3} m \left (m^{2}+13 m +32\right ) x^{4} {\mathrm e}^{m \ln \left (b x +a \right )}}{m^{4}+22 m^{3}+179 m^{2}+638 m +840}-\frac {3 b m \,a^{5} \left (m^{2}+17 m +76\right ) x^{2} {\mathrm e}^{m \ln \left (b x +a \right )}}{m^{4}+22 m^{3}+179 m^{2}+638 m +840}\) | \(341\) |
risch | \(\frac {\left (-b^{7} m^{3} x^{7}-a \,b^{6} m^{3} x^{6}-15 b^{7} m^{2} x^{7}+3 a^{2} b^{5} m^{3} x^{5}-9 m^{2} b^{6} a \,x^{6}-74 m \,b^{7} x^{7}+3 a^{3} b^{4} m^{3} x^{4}+57 a^{2} b^{5} m^{2} x^{5}-20 a \,b^{6} m \,x^{6}-120 b^{7} x^{7}-3 a^{4} b^{3} m^{3} x^{3}+39 a^{3} b^{4} m^{2} x^{4}+306 a^{2} b^{5} m \,x^{5}-3 a^{5} b^{2} m^{3} x^{2}-69 a^{4} b^{3} m^{2} x^{3}+96 a^{3} b^{4} m \,x^{4}+504 a^{2} b^{5} x^{5}+a^{6} b \,m^{3} x -51 a^{5} b^{2} m^{2} x^{2}-486 a^{4} b^{3} m \,x^{3}+a^{7} m^{3}+27 a^{6} b \,m^{2} x -228 a^{5} b^{2} m \,x^{2}-840 a^{4} b^{3} x^{3}+21 a^{7} m^{2}+254 a^{6} b m x +152 a^{7} m +840 a^{6} b x +384 a^{7}\right ) \left (b x +a \right )^{m}}{\left (6+m \right ) \left (7+m \right ) \left (5+m \right ) b \left (4+m \right )}\) | \(348\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 468 vs.
\(2 (84) = 168\).
time = 0.29, size = 468, normalized size = 5.57 \begin {gather*} \frac {{\left (b x + a\right )}^{m + 1} a^{6}}{b {\left (m + 1\right )}} - \frac {3 \, {\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^{4}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b} + \frac {3 \, {\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} a^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} b} - \frac {{\left ({\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} b^{7} x^{7} + {\left (m^{6} + 15 \, m^{5} + 85 \, m^{4} + 225 \, m^{3} + 274 \, m^{2} + 120 \, m\right )} a b^{6} x^{6} - 6 \, {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} a^{2} b^{5} x^{5} + 30 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a^{3} b^{4} x^{4} - 120 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{4} b^{3} x^{3} + 360 \, {\left (m^{2} + m\right )} a^{5} b^{2} x^{2} - 720 \, a^{6} b m x + 720 \, a^{7}\right )} {\left (b x + a\right )}^{m}}{{\left (m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040\right )} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 317 vs.
\(2 (84) = 168\).
time = 3.02, size = 317, normalized size = 3.77 \begin {gather*} \frac {{\left (a^{7} m^{3} + 21 \, a^{7} m^{2} + 152 \, a^{7} m - {\left (b^{7} m^{3} + 15 \, b^{7} m^{2} + 74 \, b^{7} m + 120 \, b^{7}\right )} x^{7} + 384 \, a^{7} - {\left (a b^{6} m^{3} + 9 \, a b^{6} m^{2} + 20 \, a b^{6} m\right )} x^{6} + 3 \, {\left (a^{2} b^{5} m^{3} + 19 \, a^{2} b^{5} m^{2} + 102 \, a^{2} b^{5} m + 168 \, a^{2} b^{5}\right )} x^{5} + 3 \, {\left (a^{3} b^{4} m^{3} + 13 \, a^{3} b^{4} m^{2} + 32 \, a^{3} b^{4} m\right )} x^{4} - 3 \, {\left (a^{4} b^{3} m^{3} + 23 \, a^{4} b^{3} m^{2} + 162 \, a^{4} b^{3} m + 280 \, a^{4} b^{3}\right )} x^{3} - 3 \, {\left (a^{5} b^{2} m^{3} + 17 \, a^{5} b^{2} m^{2} + 76 \, a^{5} b^{2} m\right )} x^{2} + {\left (a^{6} b m^{3} + 27 \, a^{6} b m^{2} + 254 \, a^{6} b m + 840 \, a^{6} b\right )} x\right )} {\left (b x + a\right )}^{m}}{b m^{4} + 22 \, b m^{3} + 179 \, b m^{2} + 638 \, b m + 840 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2059 vs.
\(2 (66) = 132\).
time = 1.27, size = 2059, normalized size = 24.51 \begin {gather*} \begin {cases} a^{6} a^{m} x & \text {for}\: b = 0 \\- \frac {3 a^{3} \log {\left (\frac {a}{b} + x \right )}}{3 a^{3} b + 9 a^{2} b^{2} x + 9 a b^{3} x^{2} + 3 b^{4} x^{3}} - \frac {8 a^{3}}{3 a^{3} b + 9 a^{2} b^{2} x + 9 a b^{3} x^{2} + 3 b^{4} x^{3}} - \frac {9 a^{2} b x \log {\left (\frac {a}{b} + x \right )}}{3 a^{3} b + 9 a^{2} b^{2} x + 9 a b^{3} x^{2} + 3 b^{4} x^{3}} - \frac {18 a^{2} b x}{3 a^{3} b + 9 a^{2} b^{2} x + 9 a b^{3} x^{2} + 3 b^{4} x^{3}} - \frac {9 a b^{2} x^{2} \log {\left (\frac {a}{b} + x \right )}}{3 a^{3} b + 9 a^{2} b^{2} x + 9 a b^{3} x^{2} + 3 b^{4} x^{3}} - \frac {18 a b^{2} x^{2}}{3 a^{3} b + 9 a^{2} b^{2} x + 9 a b^{3} x^{2} + 3 b^{4} x^{3}} - \frac {3 b^{3} x^{3} \log {\left (\frac {a}{b} + x \right )}}{3 a^{3} b + 9 a^{2} b^{2} x + 9 a b^{3} x^{2} + 3 b^{4} x^{3}} & \text {for}\: m = -7 \\\frac {6 a^{3} \log {\left (\frac {a}{b} + x \right )}}{a^{2} b + 2 a b^{2} x + b^{3} x^{2}} + \frac {13 a^{3}}{a^{2} b + 2 a b^{2} x + b^{3} x^{2}} + \frac {12 a^{2} b x \log {\left (\frac {a}{b} + x \right )}}{a^{2} b + 2 a b^{2} x + b^{3} x^{2}} + \frac {21 a^{2} b x}{a^{2} b + 2 a b^{2} x + b^{3} x^{2}} + \frac {6 a b^{2} x^{2} \log {\left (\frac {a}{b} + x \right )}}{a^{2} b + 2 a b^{2} x + b^{3} x^{2}} + \frac {3 a b^{2} x^{2}}{a^{2} b + 2 a b^{2} x + b^{3} x^{2}} - \frac {b^{3} x^{3}}{a^{2} b + 2 a b^{2} x + b^{3} x^{2}} & \text {for}\: m = -6 \\- \frac {24 a^{3} \log {\left (\frac {a}{b} + x \right )}}{2 a b + 2 b^{2} x} - \frac {50 a^{3}}{2 a b + 2 b^{2} x} - \frac {24 a^{2} b x \log {\left (\frac {a}{b} + x \right )}}{2 a b + 2 b^{2} x} - \frac {24 a^{2} b x}{2 a b + 2 b^{2} x} + \frac {9 a b^{2} x^{2}}{2 a b + 2 b^{2} x} - \frac {b^{3} x^{3}}{2 a b + 2 b^{2} x} & \text {for}\: m = -5 \\\frac {8 a^{3} \log {\left (\frac {a}{b} + x \right )}}{b} - 7 a^{2} x + 2 a b x^{2} - \frac {b^{2} x^{3}}{3} & \text {for}\: m = -4 \\\frac {a^{7} m^{3} \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} + \frac {21 a^{7} m^{2} \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} + \frac {152 a^{7} m \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} + \frac {384 a^{7} \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} + \frac {a^{6} b m^{3} x \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} + \frac {27 a^{6} b m^{2} x \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} + \frac {254 a^{6} b m x \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} + \frac {840 a^{6} b x \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} - \frac {3 a^{5} b^{2} m^{3} x^{2} \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} - \frac {51 a^{5} b^{2} m^{2} x^{2} \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} - \frac {228 a^{5} b^{2} m x^{2} \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} - \frac {3 a^{4} b^{3} m^{3} x^{3} \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} - \frac {69 a^{4} b^{3} m^{2} x^{3} \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} - \frac {486 a^{4} b^{3} m x^{3} \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} - \frac {840 a^{4} b^{3} x^{3} \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} + \frac {3 a^{3} b^{4} m^{3} x^{4} \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} + \frac {39 a^{3} b^{4} m^{2} x^{4} \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} + \frac {96 a^{3} b^{4} m x^{4} \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} + \frac {3 a^{2} b^{5} m^{3} x^{5} \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} + \frac {57 a^{2} b^{5} m^{2} x^{5} \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} + \frac {306 a^{2} b^{5} m x^{5} \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} + \frac {504 a^{2} b^{5} x^{5} \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} - \frac {a b^{6} m^{3} x^{6} \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} - \frac {9 a b^{6} m^{2} x^{6} \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} - \frac {20 a b^{6} m x^{6} \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} - \frac {b^{7} m^{3} x^{7} \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} - \frac {15 b^{7} m^{2} x^{7} \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} - \frac {74 b^{7} m x^{7} \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} - \frac {120 b^{7} x^{7} \left (a + b x\right )^{m}}{b m^{4} + 22 b m^{3} + 179 b m^{2} + 638 b m + 840 b} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 548 vs.
\(2 (84) = 168\).
time = 1.26, size = 548, normalized size = 6.52 \begin {gather*} -\frac {{\left (b x + a\right )}^{m} b^{7} m^{3} x^{7} + {\left (b x + a\right )}^{m} a b^{6} m^{3} x^{6} + 15 \, {\left (b x + a\right )}^{m} b^{7} m^{2} x^{7} - 3 \, {\left (b x + a\right )}^{m} a^{2} b^{5} m^{3} x^{5} + 9 \, {\left (b x + a\right )}^{m} a b^{6} m^{2} x^{6} + 74 \, {\left (b x + a\right )}^{m} b^{7} m x^{7} - 3 \, {\left (b x + a\right )}^{m} a^{3} b^{4} m^{3} x^{4} - 57 \, {\left (b x + a\right )}^{m} a^{2} b^{5} m^{2} x^{5} + 20 \, {\left (b x + a\right )}^{m} a b^{6} m x^{6} + 120 \, {\left (b x + a\right )}^{m} b^{7} x^{7} + 3 \, {\left (b x + a\right )}^{m} a^{4} b^{3} m^{3} x^{3} - 39 \, {\left (b x + a\right )}^{m} a^{3} b^{4} m^{2} x^{4} - 306 \, {\left (b x + a\right )}^{m} a^{2} b^{5} m x^{5} + 3 \, {\left (b x + a\right )}^{m} a^{5} b^{2} m^{3} x^{2} + 69 \, {\left (b x + a\right )}^{m} a^{4} b^{3} m^{2} x^{3} - 96 \, {\left (b x + a\right )}^{m} a^{3} b^{4} m x^{4} - 504 \, {\left (b x + a\right )}^{m} a^{2} b^{5} x^{5} - {\left (b x + a\right )}^{m} a^{6} b m^{3} x + 51 \, {\left (b x + a\right )}^{m} a^{5} b^{2} m^{2} x^{2} + 486 \, {\left (b x + a\right )}^{m} a^{4} b^{3} m x^{3} - {\left (b x + a\right )}^{m} a^{7} m^{3} - 27 \, {\left (b x + a\right )}^{m} a^{6} b m^{2} x + 228 \, {\left (b x + a\right )}^{m} a^{5} b^{2} m x^{2} + 840 \, {\left (b x + a\right )}^{m} a^{4} b^{3} x^{3} - 21 \, {\left (b x + a\right )}^{m} a^{7} m^{2} - 254 \, {\left (b x + a\right )}^{m} a^{6} b m x - 152 \, {\left (b x + a\right )}^{m} a^{7} m - 840 \, {\left (b x + a\right )}^{m} a^{6} b x - 384 \, {\left (b x + a\right )}^{m} a^{7}}{b m^{4} + 22 \, b m^{3} + 179 \, b m^{2} + 638 \, b m + 840 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.75, size = 332, normalized size = 3.95 \begin {gather*} {\left (a+b\,x\right )}^m\,\left (\frac {a^6\,x\,\left (m^3+27\,m^2+254\,m+840\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}+\frac {a^7\,\left (m^3+21\,m^2+152\,m+384\right )}{b\,\left (m^4+22\,m^3+179\,m^2+638\,m+840\right )}-\frac {b^6\,x^7\,\left (m^3+15\,m^2+74\,m+120\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}+\frac {3\,a^2\,b^4\,x^5\,\left (m^3+19\,m^2+102\,m+168\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}-\frac {3\,a^4\,b^2\,x^3\,\left (m^3+23\,m^2+162\,m+280\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}-\frac {a\,b^5\,m\,x^6\,\left (m^2+9\,m+20\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}-\frac {3\,a^5\,b\,m\,x^2\,\left (m^2+17\,m+76\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}+\frac {3\,a^3\,b^3\,m\,x^4\,\left (m^2+13\,m+32\right )}{m^4+22\,m^3+179\,m^2+638\,m+840}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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