Optimal. Leaf size=337 \[ -\frac {(b d-a e)^5 (d+e x)^{1+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (1+m) (a+b x)}+\frac {5 b (b d-a e)^4 (d+e x)^{2+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (2+m) (a+b x)}-\frac {10 b^2 (b d-a e)^3 (d+e x)^{3+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (3+m) (a+b x)}+\frac {10 b^3 (b d-a e)^2 (d+e x)^{4+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (4+m) (a+b x)}-\frac {5 b^4 (b d-a e) (d+e x)^{5+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (5+m) (a+b x)}+\frac {b^5 (d+e x)^{6+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (6+m) (a+b x)} \]
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Rubi [A]
time = 0.10, antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 45}
\begin {gather*} -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (d+e x)^{m+1}}{e^6 (m+1) (a+b x)}+\frac {5 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (d+e x)^{m+2}}{e^6 (m+2) (a+b x)}-\frac {10 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (d+e x)^{m+3}}{e^6 (m+3) (a+b x)}+\frac {b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{m+6}}{e^6 (m+6) (a+b x)}-\frac {5 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (d+e x)^{m+5}}{e^6 (m+5) (a+b x)}+\frac {10 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (d+e x)^{m+4}}{e^6 (m+4) (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 660
Rubi steps
\begin {align*} \int (d+e x)^m \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^5 (d+e x)^m \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^5 (b d-a e)^5 (d+e x)^m}{e^5}+\frac {5 b^6 (b d-a e)^4 (d+e x)^{1+m}}{e^5}-\frac {10 b^7 (b d-a e)^3 (d+e x)^{2+m}}{e^5}+\frac {10 b^8 (b d-a e)^2 (d+e x)^{3+m}}{e^5}-\frac {5 b^9 (b d-a e) (d+e x)^{4+m}}{e^5}+\frac {b^{10} (d+e x)^{5+m}}{e^5}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {(b d-a e)^5 (d+e x)^{1+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (1+m) (a+b x)}+\frac {5 b (b d-a e)^4 (d+e x)^{2+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (2+m) (a+b x)}-\frac {10 b^2 (b d-a e)^3 (d+e x)^{3+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (3+m) (a+b x)}+\frac {10 b^3 (b d-a e)^2 (d+e x)^{4+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (4+m) (a+b x)}-\frac {5 b^4 (b d-a e) (d+e x)^{5+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (5+m) (a+b x)}+\frac {b^5 (d+e x)^{6+m} \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (6+m) (a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 167, normalized size = 0.50 \begin {gather*} \frac {\left ((a+b x)^2\right )^{5/2} (d+e x)^{1+m} \left (-\frac {(b d-a e)^5}{1+m}+\frac {5 b (b d-a e)^4 (d+e x)}{2+m}-\frac {10 b^2 (b d-a e)^3 (d+e x)^2}{3+m}+\frac {10 b^3 (b d-a e)^2 (d+e x)^3}{4+m}-\frac {5 b^4 (b d-a e) (d+e x)^4}{5+m}+\frac {b^5 (d+e x)^5}{6+m}\right )}{e^6 (a+b x)^5} \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. \(1360\) vs.
\(2(271)=542\).
time = 0.69, size = 1361, normalized size = 4.04
method | result | size |
gosper | \(\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (e x +d \right )^{1+m} \left (b^{5} e^{5} m^{5} x^{5}+5 a \,b^{4} e^{5} m^{5} x^{4}+15 b^{5} e^{5} m^{4} x^{5}+10 a^{2} b^{3} e^{5} m^{5} x^{3}+80 a \,b^{4} e^{5} m^{4} x^{4}-5 b^{5} d \,e^{4} m^{4} x^{4}+85 b^{5} e^{5} m^{3} x^{5}+10 a^{3} b^{2} e^{5} m^{5} x^{2}+170 a^{2} b^{3} e^{5} m^{4} x^{3}-20 a \,b^{4} d \,e^{4} m^{4} x^{3}+475 a \,b^{4} e^{5} m^{3} x^{4}-50 b^{5} d \,e^{4} m^{3} x^{4}+225 b^{5} e^{5} m^{2} x^{5}+5 a^{4} b \,e^{5} m^{5} x +180 a^{3} b^{2} e^{5} m^{4} x^{2}-30 a^{2} b^{3} d \,e^{4} m^{4} x^{2}+1070 a^{2} b^{3} e^{5} m^{3} x^{3}-240 a \,b^{4} d \,e^{4} m^{3} x^{3}+1300 a \,b^{4} e^{5} m^{2} x^{4}+20 b^{5} d^{2} e^{3} m^{3} x^{3}-175 b^{5} d \,e^{4} m^{2} x^{4}+274 b^{5} e^{5} m \,x^{5}+a^{5} e^{5} m^{5}+95 a^{4} b \,e^{5} m^{4} x -20 a^{3} b^{2} d \,e^{4} m^{4} x +1210 a^{3} b^{2} e^{5} m^{3} x^{2}-420 a^{2} b^{3} d \,e^{4} m^{3} x^{2}+3070 a^{2} b^{3} e^{5} m^{2} x^{3}+60 a \,b^{4} d^{2} e^{3} m^{3} x^{2}-940 a \,b^{4} d \,e^{4} m^{2} x^{3}+1620 a \,b^{4} e^{5} m \,x^{4}+120 b^{5} d^{2} e^{3} m^{2} x^{3}-250 b^{5} d \,e^{4} m \,x^{4}+120 b^{5} e^{5} x^{5}+20 a^{5} e^{5} m^{4}-5 a^{4} b d \,e^{4} m^{4}+685 a^{4} b \,e^{5} m^{3} x -320 a^{3} b^{2} d \,e^{4} m^{3} x +3720 a^{3} b^{2} e^{5} m^{2} x^{2}+60 a^{2} b^{3} d^{2} e^{3} m^{3} x -1950 a^{2} b^{3} d \,e^{4} m^{2} x^{2}+3960 a^{2} b^{3} e^{5} m \,x^{3}+540 a \,b^{4} d^{2} e^{3} m^{2} x^{2}-1440 a \,b^{4} d \,e^{4} m \,x^{3}+720 a \,b^{4} e^{5} x^{4}-60 b^{5} d^{3} e^{2} m^{2} x^{2}+220 b^{5} d^{2} e^{3} m \,x^{3}-120 b^{5} d \,e^{4} x^{4}+155 a^{5} e^{5} m^{3}-90 a^{4} b d \,e^{4} m^{3}+2305 a^{4} b \,e^{5} m^{2} x +20 a^{3} b^{2} d^{2} e^{3} m^{3}-1780 a^{3} b^{2} d \,e^{4} m^{2} x +5080 a^{3} b^{2} e^{5} m \,x^{2}+720 a^{2} b^{3} d^{2} e^{3} m^{2} x -3360 a^{2} b^{3} d \,e^{4} m \,x^{2}+1800 a^{2} b^{3} e^{5} x^{3}-120 a \,b^{4} d^{3} e^{2} m^{2} x +1200 a \,b^{4} d^{2} e^{3} m \,x^{2}-720 a \,b^{4} d \,e^{4} x^{3}-180 b^{5} d^{3} e^{2} m \,x^{2}+120 b^{5} d^{2} e^{3} x^{3}+580 a^{5} e^{5} m^{2}-595 a^{4} b d \,e^{4} m^{2}+3510 a^{4} b \,e^{5} m x +300 a^{3} b^{2} d^{2} e^{3} m^{2}-3880 a^{3} b^{2} d \,e^{4} m x +2400 a^{3} b^{2} e^{5} x^{2}-60 a^{2} b^{3} d^{3} e^{2} m^{2}+2460 a^{2} b^{3} d^{2} e^{3} m x -1800 a^{2} b^{3} d \,e^{4} x^{2}-840 a \,b^{4} d^{3} e^{2} m x +720 a \,b^{4} d^{2} e^{3} x^{2}+120 b^{5} d^{4} e m x -120 b^{5} d^{3} e^{2} x^{2}+1044 a^{5} e^{5} m -1710 a^{4} b d \,e^{4} m +1800 a^{4} b \,e^{5} x +1480 a^{3} b^{2} d^{2} e^{3} m -2400 a^{3} b^{2} d \,e^{4} x -660 a^{2} b^{3} d^{3} e^{2} m +1800 a^{2} b^{3} d^{2} e^{3} x +120 a \,b^{4} d^{4} e m -720 a \,b^{4} d^{3} e^{2} x +120 b^{5} d^{4} e x +720 a^{5} e^{5}-1800 a^{4} b d \,e^{4}+2400 a^{3} b^{2} d^{2} e^{3}-1800 a^{2} b^{3} d^{3} e^{2}+720 a \,b^{4} d^{4} e -120 b^{5} d^{5}\right )}{\left (b x +a \right )^{5} e^{6} \left (m^{6}+21 m^{5}+175 m^{4}+735 m^{3}+1624 m^{2}+1764 m +720\right )}\) | \(1361\) |
risch | \(\text {Expression too large to display}\) | \(1745\) |
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. 770 vs.
\(2 (276) = 552\).
time = 0.30, size = 770, normalized size = 2.28 \begin {gather*} \frac {{\left ({\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} b^{5} x^{6} e^{6} + 120 \, a b^{4} d^{5} {\left (m + 6\right )} e - 120 \, b^{5} d^{6} - 60 \, {\left (m^{2} + 11 \, m + 30\right )} a^{2} b^{3} d^{4} e^{2} + 20 \, {\left (m^{3} + 15 \, m^{2} + 74 \, m + 120\right )} a^{3} b^{2} d^{3} e^{3} - 5 \, {\left (m^{4} + 18 \, m^{3} + 119 \, m^{2} + 342 \, m + 360\right )} a^{4} b d^{2} e^{4} + {\left (m^{5} + 20 \, m^{4} + 155 \, m^{3} + 580 \, m^{2} + 1044 \, m + 720\right )} a^{5} d e^{5} + {\left ({\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} b^{5} d e^{5} + 5 \, {\left (m^{5} + 16 \, m^{4} + 95 \, m^{3} + 260 \, m^{2} + 324 \, m + 144\right )} a b^{4} e^{6}\right )} x^{5} - 5 \, {\left ({\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} b^{5} d^{2} e^{4} - {\left (m^{5} + 12 \, m^{4} + 47 \, m^{3} + 72 \, m^{2} + 36 \, m\right )} a b^{4} d e^{5} - 2 \, {\left (m^{5} + 17 \, m^{4} + 107 \, m^{3} + 307 \, m^{2} + 396 \, m + 180\right )} a^{2} b^{3} e^{6}\right )} x^{4} + 10 \, {\left (2 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} b^{5} d^{3} e^{3} - 2 \, {\left (m^{4} + 9 \, m^{3} + 20 \, m^{2} + 12 \, m\right )} a b^{4} d^{2} e^{4} + {\left (m^{5} + 14 \, m^{4} + 65 \, m^{3} + 112 \, m^{2} + 60 \, m\right )} a^{2} b^{3} d e^{5} + {\left (m^{5} + 18 \, m^{4} + 121 \, m^{3} + 372 \, m^{2} + 508 \, m + 240\right )} a^{3} b^{2} e^{6}\right )} x^{3} - 5 \, {\left (12 \, {\left (m^{2} + m\right )} b^{5} d^{4} e^{2} - 12 \, {\left (m^{3} + 7 \, m^{2} + 6 \, m\right )} a b^{4} d^{3} e^{3} + 6 \, {\left (m^{4} + 12 \, m^{3} + 41 \, m^{2} + 30 \, m\right )} a^{2} b^{3} d^{2} e^{4} - 2 \, {\left (m^{5} + 16 \, m^{4} + 89 \, m^{3} + 194 \, m^{2} + 120 \, m\right )} a^{3} b^{2} d e^{5} - {\left (m^{5} + 19 \, m^{4} + 137 \, m^{3} + 461 \, m^{2} + 702 \, m + 360\right )} a^{4} b e^{6}\right )} x^{2} + {\left (120 \, b^{5} d^{5} m e - 120 \, {\left (m^{2} + 6 \, m\right )} a b^{4} d^{4} e^{2} + 60 \, {\left (m^{3} + 11 \, m^{2} + 30 \, m\right )} a^{2} b^{3} d^{3} e^{3} - 20 \, {\left (m^{4} + 15 \, m^{3} + 74 \, m^{2} + 120 \, m\right )} a^{3} b^{2} d^{2} e^{4} + 5 \, {\left (m^{5} + 18 \, m^{4} + 119 \, m^{3} + 342 \, m^{2} + 360 \, m\right )} a^{4} b d e^{5} + {\left (m^{5} + 20 \, m^{4} + 155 \, m^{3} + 580 \, m^{2} + 1044 \, m + 720\right )} a^{5} e^{6}\right )} x\right )} e^{\left (m \log \left (x e + d\right ) - 6\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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1283 vs.
\(2 (276) = 552\).
time = 2.54, size = 1283, normalized size = 3.81 \begin {gather*} -\frac {{\left (120 \, b^{5} d^{6} - {\left ({\left (b^{5} m^{5} + 15 \, b^{5} m^{4} + 85 \, b^{5} m^{3} + 225 \, b^{5} m^{2} + 274 \, b^{5} m + 120 \, b^{5}\right )} x^{6} + 5 \, {\left (a b^{4} m^{5} + 16 \, a b^{4} m^{4} + 95 \, a b^{4} m^{3} + 260 \, a b^{4} m^{2} + 324 \, a b^{4} m + 144 \, a b^{4}\right )} x^{5} + 10 \, {\left (a^{2} b^{3} m^{5} + 17 \, a^{2} b^{3} m^{4} + 107 \, a^{2} b^{3} m^{3} + 307 \, a^{2} b^{3} m^{2} + 396 \, a^{2} b^{3} m + 180 \, a^{2} b^{3}\right )} x^{4} + 10 \, {\left (a^{3} b^{2} m^{5} + 18 \, a^{3} b^{2} m^{4} + 121 \, a^{3} b^{2} m^{3} + 372 \, a^{3} b^{2} m^{2} + 508 \, a^{3} b^{2} m + 240 \, a^{3} b^{2}\right )} x^{3} + 5 \, {\left (a^{4} b m^{5} + 19 \, a^{4} b m^{4} + 137 \, a^{4} b m^{3} + 461 \, a^{4} b m^{2} + 702 \, a^{4} b m + 360 \, a^{4} b\right )} x^{2} + {\left (a^{5} m^{5} + 20 \, a^{5} m^{4} + 155 \, a^{5} m^{3} + 580 \, a^{5} m^{2} + 1044 \, a^{5} m + 720 \, a^{5}\right )} x\right )} e^{6} - {\left (a^{5} d m^{5} + 20 \, a^{5} d m^{4} + 155 \, a^{5} d m^{3} + 580 \, a^{5} d m^{2} + 1044 \, a^{5} d m + 720 \, a^{5} d + {\left (b^{5} d m^{5} + 10 \, b^{5} d m^{4} + 35 \, b^{5} d m^{3} + 50 \, b^{5} d m^{2} + 24 \, b^{5} d m\right )} x^{5} + 5 \, {\left (a b^{4} d m^{5} + 12 \, a b^{4} d m^{4} + 47 \, a b^{4} d m^{3} + 72 \, a b^{4} d m^{2} + 36 \, a b^{4} d m\right )} x^{4} + 10 \, {\left (a^{2} b^{3} d m^{5} + 14 \, a^{2} b^{3} d m^{4} + 65 \, a^{2} b^{3} d m^{3} + 112 \, a^{2} b^{3} d m^{2} + 60 \, a^{2} b^{3} d m\right )} x^{3} + 10 \, {\left (a^{3} b^{2} d m^{5} + 16 \, a^{3} b^{2} d m^{4} + 89 \, a^{3} b^{2} d m^{3} + 194 \, a^{3} b^{2} d m^{2} + 120 \, a^{3} b^{2} d m\right )} x^{2} + 5 \, {\left (a^{4} b d m^{5} + 18 \, a^{4} b d m^{4} + 119 \, a^{4} b d m^{3} + 342 \, a^{4} b d m^{2} + 360 \, a^{4} b d m\right )} x\right )} e^{5} + 5 \, {\left (a^{4} b d^{2} m^{4} + 18 \, a^{4} b d^{2} m^{3} + 119 \, a^{4} b d^{2} m^{2} + 342 \, a^{4} b d^{2} m + 360 \, a^{4} b d^{2} + {\left (b^{5} d^{2} m^{4} + 6 \, b^{5} d^{2} m^{3} + 11 \, b^{5} d^{2} m^{2} + 6 \, b^{5} d^{2} m\right )} x^{4} + 4 \, {\left (a b^{4} d^{2} m^{4} + 9 \, a b^{4} d^{2} m^{3} + 20 \, a b^{4} d^{2} m^{2} + 12 \, a b^{4} d^{2} m\right )} x^{3} + 6 \, {\left (a^{2} b^{3} d^{2} m^{4} + 12 \, a^{2} b^{3} d^{2} m^{3} + 41 \, a^{2} b^{3} d^{2} m^{2} + 30 \, a^{2} b^{3} d^{2} m\right )} x^{2} + 4 \, {\left (a^{3} b^{2} d^{2} m^{4} + 15 \, a^{3} b^{2} d^{2} m^{3} + 74 \, a^{3} b^{2} d^{2} m^{2} + 120 \, a^{3} b^{2} d^{2} m\right )} x\right )} e^{4} - 20 \, {\left (a^{3} b^{2} d^{3} m^{3} + 15 \, a^{3} b^{2} d^{3} m^{2} + 74 \, a^{3} b^{2} d^{3} m + 120 \, a^{3} b^{2} d^{3} + {\left (b^{5} d^{3} m^{3} + 3 \, b^{5} d^{3} m^{2} + 2 \, b^{5} d^{3} m\right )} x^{3} + 3 \, {\left (a b^{4} d^{3} m^{3} + 7 \, a b^{4} d^{3} m^{2} + 6 \, a b^{4} d^{3} m\right )} x^{2} + 3 \, {\left (a^{2} b^{3} d^{3} m^{3} + 11 \, a^{2} b^{3} d^{3} m^{2} + 30 \, a^{2} b^{3} d^{3} m\right )} x\right )} e^{3} + 60 \, {\left (a^{2} b^{3} d^{4} m^{2} + 11 \, a^{2} b^{3} d^{4} m + 30 \, a^{2} b^{3} d^{4} + {\left (b^{5} d^{4} m^{2} + b^{5} d^{4} m\right )} x^{2} + 2 \, {\left (a b^{4} d^{4} m^{2} + 6 \, a b^{4} d^{4} m\right )} x\right )} e^{2} - 120 \, {\left (b^{5} d^{5} m x + a b^{4} d^{5} m + 6 \, a b^{4} d^{5}\right )} e\right )} {\left (x e + d\right )}^{m} e^{\left (-6\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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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \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. 3197 vs.
\(2 (276) = 552\).
time = 0.90, size = 3197, normalized size = 9.49 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (d+e\,x\right )}^m\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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