Optimal. Leaf size=125 \[ \frac {a^4 A x^{1+m}}{1+m}+\frac {a^3 (4 A b+a B) x^{2+m}}{2+m}+\frac {2 a^2 b (3 A b+2 a B) x^{3+m}}{3+m}+\frac {2 a b^2 (2 A b+3 a B) x^{4+m}}{4+m}+\frac {b^3 (A b+4 a B) x^{5+m}}{5+m}+\frac {b^4 B x^{6+m}}{6+m} \]
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Rubi [A]
time = 0.05, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77}
\begin {gather*} \frac {a^4 A x^{m+1}}{m+1}+\frac {a^3 x^{m+2} (a B+4 A b)}{m+2}+\frac {2 a^2 b x^{m+3} (2 a B+3 A b)}{m+3}+\frac {b^3 x^{m+5} (4 a B+A b)}{m+5}+\frac {2 a b^2 x^{m+4} (3 a B+2 A b)}{m+4}+\frac {b^4 B x^{m+6}}{m+6} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int x^m (a+b x)^4 (A+B x) \, dx &=\int \left (a^4 A x^m+a^3 (4 A b+a B) x^{1+m}+2 a^2 b (3 A b+2 a B) x^{2+m}+2 a b^2 (2 A b+3 a B) x^{3+m}+b^3 (A b+4 a B) x^{4+m}+b^4 B x^{5+m}\right ) \, dx\\ &=\frac {a^4 A x^{1+m}}{1+m}+\frac {a^3 (4 A b+a B) x^{2+m}}{2+m}+\frac {2 a^2 b (3 A b+2 a B) x^{3+m}}{3+m}+\frac {2 a b^2 (2 A b+3 a B) x^{4+m}}{4+m}+\frac {b^3 (A b+4 a B) x^{5+m}}{5+m}+\frac {b^4 B x^{6+m}}{6+m}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 103, normalized size = 0.82 \begin {gather*} \frac {x^{1+m} \left (B (a+b x)^5+(-a B (1+m)+A b (6+m)) \left (\frac {a^4}{1+m}+\frac {4 a^3 b x}{2+m}+\frac {6 a^2 b^2 x^2}{3+m}+\frac {4 a b^3 x^3}{4+m}+\frac {b^4 x^4}{5+m}\right )\right )}{b (6+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 142, normalized size = 1.14
method | result | size |
norman | \(\frac {A \,a^{4} x \,{\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {B \,b^{4} x^{6} {\mathrm e}^{m \ln \left (x \right )}}{6+m}+\frac {a^{3} \left (4 A b +B a \right ) x^{2} {\mathrm e}^{m \ln \left (x \right )}}{2+m}+\frac {b^{3} \left (A b +4 B a \right ) x^{5} {\mathrm e}^{m \ln \left (x \right )}}{5+m}+\frac {2 a \,b^{2} \left (2 A b +3 B a \right ) x^{4} {\mathrm e}^{m \ln \left (x \right )}}{4+m}+\frac {2 a^{2} b \left (3 A b +2 B a \right ) x^{3} {\mathrm e}^{m \ln \left (x \right )}}{3+m}\) | \(142\) |
risch | \(\frac {x \left (B \,b^{4} m^{5} x^{5}+A \,b^{4} m^{5} x^{4}+4 B a \,b^{3} m^{5} x^{4}+15 B \,b^{4} m^{4} x^{5}+4 A a \,b^{3} m^{5} x^{3}+16 A \,b^{4} m^{4} x^{4}+6 B \,a^{2} b^{2} m^{5} x^{3}+64 B a \,b^{3} m^{4} x^{4}+85 B \,b^{4} m^{3} x^{5}+6 A \,a^{2} b^{2} m^{5} x^{2}+68 A a \,b^{3} m^{4} x^{3}+95 A \,b^{4} m^{3} x^{4}+4 B \,a^{3} b \,m^{5} x^{2}+102 B \,a^{2} b^{2} m^{4} x^{3}+380 B a \,b^{3} m^{3} x^{4}+225 B \,b^{4} m^{2} x^{5}+4 A \,a^{3} b \,m^{5} x +108 A \,a^{2} b^{2} m^{4} x^{2}+428 A a \,b^{3} m^{3} x^{3}+260 A \,b^{4} m^{2} x^{4}+B \,a^{4} m^{5} x +72 B \,a^{3} b \,m^{4} x^{2}+642 B \,a^{2} b^{2} m^{3} x^{3}+1040 B a \,b^{3} m^{2} x^{4}+274 m \,x^{5} B \,b^{4}+A \,a^{4} m^{5}+76 A \,a^{3} b \,m^{4} x +726 A \,a^{2} b^{2} m^{3} x^{2}+1228 A a \,b^{3} m^{2} x^{3}+324 A \,b^{4} x^{4} m +19 B \,a^{4} m^{4} x +484 B \,a^{3} b \,m^{3} x^{2}+1842 B \,a^{2} b^{2} m^{2} x^{3}+1296 B a \,b^{3} x^{4} m +120 B \,b^{4} x^{5}+20 A \,a^{4} m^{4}+548 A \,a^{3} b \,m^{3} x +2232 A \,a^{2} b^{2} m^{2} x^{2}+1584 A a \,b^{3} x^{3} m +144 A \,b^{4} x^{4}+137 B \,a^{4} m^{3} x +1488 B \,a^{3} b \,m^{2} x^{2}+2376 B \,a^{2} b^{2} x^{3} m +576 B a \,b^{3} x^{4}+155 A \,a^{4} m^{3}+1844 A \,a^{3} b \,m^{2} x +3048 A \,a^{2} b^{2} x^{2} m +720 A a \,b^{3} x^{3}+461 B \,a^{4} m^{2} x +2032 B \,a^{3} b \,x^{2} m +1080 B \,a^{2} b^{2} x^{3}+580 A \,a^{4} m^{2}+2808 A \,a^{3} b x m +1440 A \,a^{2} b^{2} x^{2}+702 B \,a^{4} x m +960 B \,a^{3} b \,x^{2}+1044 A \,a^{4} m +1440 A \,a^{3} b x +360 B \,a^{4} x +720 A \,a^{4}\right ) x^{m}}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(721\) |
gosper | \(\frac {x^{1+m} \left (B \,b^{4} m^{5} x^{5}+A \,b^{4} m^{5} x^{4}+4 B a \,b^{3} m^{5} x^{4}+15 B \,b^{4} m^{4} x^{5}+4 A a \,b^{3} m^{5} x^{3}+16 A \,b^{4} m^{4} x^{4}+6 B \,a^{2} b^{2} m^{5} x^{3}+64 B a \,b^{3} m^{4} x^{4}+85 B \,b^{4} m^{3} x^{5}+6 A \,a^{2} b^{2} m^{5} x^{2}+68 A a \,b^{3} m^{4} x^{3}+95 A \,b^{4} m^{3} x^{4}+4 B \,a^{3} b \,m^{5} x^{2}+102 B \,a^{2} b^{2} m^{4} x^{3}+380 B a \,b^{3} m^{3} x^{4}+225 B \,b^{4} m^{2} x^{5}+4 A \,a^{3} b \,m^{5} x +108 A \,a^{2} b^{2} m^{4} x^{2}+428 A a \,b^{3} m^{3} x^{3}+260 A \,b^{4} m^{2} x^{4}+B \,a^{4} m^{5} x +72 B \,a^{3} b \,m^{4} x^{2}+642 B \,a^{2} b^{2} m^{3} x^{3}+1040 B a \,b^{3} m^{2} x^{4}+274 m \,x^{5} B \,b^{4}+A \,a^{4} m^{5}+76 A \,a^{3} b \,m^{4} x +726 A \,a^{2} b^{2} m^{3} x^{2}+1228 A a \,b^{3} m^{2} x^{3}+324 A \,b^{4} x^{4} m +19 B \,a^{4} m^{4} x +484 B \,a^{3} b \,m^{3} x^{2}+1842 B \,a^{2} b^{2} m^{2} x^{3}+1296 B a \,b^{3} x^{4} m +120 B \,b^{4} x^{5}+20 A \,a^{4} m^{4}+548 A \,a^{3} b \,m^{3} x +2232 A \,a^{2} b^{2} m^{2} x^{2}+1584 A a \,b^{3} x^{3} m +144 A \,b^{4} x^{4}+137 B \,a^{4} m^{3} x +1488 B \,a^{3} b \,m^{2} x^{2}+2376 B \,a^{2} b^{2} x^{3} m +576 B a \,b^{3} x^{4}+155 A \,a^{4} m^{3}+1844 A \,a^{3} b \,m^{2} x +3048 A \,a^{2} b^{2} x^{2} m +720 A a \,b^{3} x^{3}+461 B \,a^{4} m^{2} x +2032 B \,a^{3} b \,x^{2} m +1080 B \,a^{2} b^{2} x^{3}+580 A \,a^{4} m^{2}+2808 A \,a^{3} b x m +1440 A \,a^{2} b^{2} x^{2}+702 B \,a^{4} x m +960 B \,a^{3} b \,x^{2}+1044 A \,a^{4} m +1440 A \,a^{3} b x +360 B \,a^{4} x +720 A \,a^{4}\right )}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(722\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 167, normalized size = 1.34 \begin {gather*} \frac {B b^{4} x^{m + 6}}{m + 6} + \frac {4 \, B a b^{3} x^{m + 5}}{m + 5} + \frac {A b^{4} x^{m + 5}}{m + 5} + \frac {6 \, B a^{2} b^{2} x^{m + 4}}{m + 4} + \frac {4 \, A a b^{3} x^{m + 4}}{m + 4} + \frac {4 \, B a^{3} b x^{m + 3}}{m + 3} + \frac {6 \, A a^{2} b^{2} x^{m + 3}}{m + 3} + \frac {B a^{4} x^{m + 2}}{m + 2} + \frac {4 \, A a^{3} b x^{m + 2}}{m + 2} + \frac {A a^{4} x^{m + 1}}{m + 1} \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. 607 vs.
\(2 (125) = 250\).
time = 1.15, size = 607, normalized size = 4.86 \begin {gather*} \frac {{\left ({\left (B b^{4} m^{5} + 15 \, B b^{4} m^{4} + 85 \, B b^{4} m^{3} + 225 \, B b^{4} m^{2} + 274 \, B b^{4} m + 120 \, B b^{4}\right )} x^{6} + {\left ({\left (4 \, B a b^{3} + A b^{4}\right )} m^{5} + 576 \, B a b^{3} + 144 \, A b^{4} + 16 \, {\left (4 \, B a b^{3} + A b^{4}\right )} m^{4} + 95 \, {\left (4 \, B a b^{3} + A b^{4}\right )} m^{3} + 260 \, {\left (4 \, B a b^{3} + A b^{4}\right )} m^{2} + 324 \, {\left (4 \, B a b^{3} + A b^{4}\right )} m\right )} x^{5} + 2 \, {\left ({\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} m^{5} + 540 \, B a^{2} b^{2} + 360 \, A a b^{3} + 17 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} m^{4} + 107 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} m^{3} + 307 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} m^{2} + 396 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} m\right )} x^{4} + 2 \, {\left ({\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} m^{5} + 480 \, B a^{3} b + 720 \, A a^{2} b^{2} + 18 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} m^{4} + 121 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} m^{3} + 372 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} m^{2} + 508 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} m\right )} x^{3} + {\left ({\left (B a^{4} + 4 \, A a^{3} b\right )} m^{5} + 360 \, B a^{4} + 1440 \, A a^{3} b + 19 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} m^{4} + 137 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} m^{3} + 461 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} m^{2} + 702 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} m\right )} x^{2} + {\left (A a^{4} m^{5} + 20 \, A a^{4} m^{4} + 155 \, A a^{4} m^{3} + 580 \, A a^{4} m^{2} + 1044 \, A a^{4} m + 720 \, A a^{4}\right )} x\right )} x^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \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. 3417 vs.
\(2 (117) = 234\).
time = 0.55, size = 3417, normalized size = 27.34 \begin {gather*} \text {Too large to display} \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. 926 vs.
\(2 (125) = 250\).
time = 2.55, size = 926, normalized size = 7.41 \begin {gather*} \frac {B b^{4} m^{5} x^{6} x^{m} + 4 \, B a b^{3} m^{5} x^{5} x^{m} + A b^{4} m^{5} x^{5} x^{m} + 15 \, B b^{4} m^{4} x^{6} x^{m} + 6 \, B a^{2} b^{2} m^{5} x^{4} x^{m} + 4 \, A a b^{3} m^{5} x^{4} x^{m} + 64 \, B a b^{3} m^{4} x^{5} x^{m} + 16 \, A b^{4} m^{4} x^{5} x^{m} + 85 \, B b^{4} m^{3} x^{6} x^{m} + 4 \, B a^{3} b m^{5} x^{3} x^{m} + 6 \, A a^{2} b^{2} m^{5} x^{3} x^{m} + 102 \, B a^{2} b^{2} m^{4} x^{4} x^{m} + 68 \, A a b^{3} m^{4} x^{4} x^{m} + 380 \, B a b^{3} m^{3} x^{5} x^{m} + 95 \, A b^{4} m^{3} x^{5} x^{m} + 225 \, B b^{4} m^{2} x^{6} x^{m} + B a^{4} m^{5} x^{2} x^{m} + 4 \, A a^{3} b m^{5} x^{2} x^{m} + 72 \, B a^{3} b m^{4} x^{3} x^{m} + 108 \, A a^{2} b^{2} m^{4} x^{3} x^{m} + 642 \, B a^{2} b^{2} m^{3} x^{4} x^{m} + 428 \, A a b^{3} m^{3} x^{4} x^{m} + 1040 \, B a b^{3} m^{2} x^{5} x^{m} + 260 \, A b^{4} m^{2} x^{5} x^{m} + 274 \, B b^{4} m x^{6} x^{m} + A a^{4} m^{5} x x^{m} + 19 \, B a^{4} m^{4} x^{2} x^{m} + 76 \, A a^{3} b m^{4} x^{2} x^{m} + 484 \, B a^{3} b m^{3} x^{3} x^{m} + 726 \, A a^{2} b^{2} m^{3} x^{3} x^{m} + 1842 \, B a^{2} b^{2} m^{2} x^{4} x^{m} + 1228 \, A a b^{3} m^{2} x^{4} x^{m} + 1296 \, B a b^{3} m x^{5} x^{m} + 324 \, A b^{4} m x^{5} x^{m} + 120 \, B b^{4} x^{6} x^{m} + 20 \, A a^{4} m^{4} x x^{m} + 137 \, B a^{4} m^{3} x^{2} x^{m} + 548 \, A a^{3} b m^{3} x^{2} x^{m} + 1488 \, B a^{3} b m^{2} x^{3} x^{m} + 2232 \, A a^{2} b^{2} m^{2} x^{3} x^{m} + 2376 \, B a^{2} b^{2} m x^{4} x^{m} + 1584 \, A a b^{3} m x^{4} x^{m} + 576 \, B a b^{3} x^{5} x^{m} + 144 \, A b^{4} x^{5} x^{m} + 155 \, A a^{4} m^{3} x x^{m} + 461 \, B a^{4} m^{2} x^{2} x^{m} + 1844 \, A a^{3} b m^{2} x^{2} x^{m} + 2032 \, B a^{3} b m x^{3} x^{m} + 3048 \, A a^{2} b^{2} m x^{3} x^{m} + 1080 \, B a^{2} b^{2} x^{4} x^{m} + 720 \, A a b^{3} x^{4} x^{m} + 580 \, A a^{4} m^{2} x x^{m} + 702 \, B a^{4} m x^{2} x^{m} + 2808 \, A a^{3} b m x^{2} x^{m} + 960 \, B a^{3} b x^{3} x^{m} + 1440 \, A a^{2} b^{2} x^{3} x^{m} + 1044 \, A a^{4} m x x^{m} + 360 \, B a^{4} x^{2} x^{m} + 1440 \, A a^{3} b x^{2} x^{m} + 720 \, A a^{4} x x^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.75, size = 417, normalized size = 3.34 \begin {gather*} \frac {B\,b^4\,x^m\,x^6\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {a^3\,x^m\,x^2\,\left (4\,A\,b+B\,a\right )\,\left (m^5+19\,m^4+137\,m^3+461\,m^2+702\,m+360\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {b^3\,x^m\,x^5\,\left (A\,b+4\,B\,a\right )\,\left (m^5+16\,m^4+95\,m^3+260\,m^2+324\,m+144\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {A\,a^4\,x\,x^m\,\left (m^5+20\,m^4+155\,m^3+580\,m^2+1044\,m+720\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {2\,a\,b^2\,x^m\,x^4\,\left (2\,A\,b+3\,B\,a\right )\,\left (m^5+17\,m^4+107\,m^3+307\,m^2+396\,m+180\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {2\,a^2\,b\,x^m\,x^3\,\left (3\,A\,b+2\,B\,a\right )\,\left (m^5+18\,m^4+121\,m^3+372\,m^2+508\,m+240\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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