Optimal. Leaf size=263 \[ \frac {5 e^4 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{10} (b d-a e)}{11 b^6}+\frac {e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e)^2}{b^6}+\frac {10 e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^3}{9 b^6}+\frac {5 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^4}{8 b^6}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^5}{7 b^6}+\frac {e^5 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{11}}{12 b^6} \]
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Rubi [A] time = 0.38, antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {770, 21, 43} \begin {gather*} \frac {5 e^4 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{10} (b d-a e)}{11 b^6}+\frac {e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e)^2}{b^6}+\frac {10 e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^3}{9 b^6}+\frac {5 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^4}{8 b^6}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^5}{7 b^6}+\frac {e^5 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{11}}{12 b^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 43
Rule 770
Rubi steps
\begin {align*} \int (a+b x) (d+e x)^5 \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 (a+b x) \left (a b+b^2 x\right )^5 (d+e x)^5 \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^6 (d+e x)^5 \, dx}{a b+b^2 x}\\ &=\frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(b d-a e)^5 (a+b x)^6}{b^5}+\frac {5 e (b d-a e)^4 (a+b x)^7}{b^5}+\frac {10 e^2 (b d-a e)^3 (a+b x)^8}{b^5}+\frac {10 e^3 (b d-a e)^2 (a+b x)^9}{b^5}+\frac {5 e^4 (b d-a e) (a+b x)^{10}}{b^5}+\frac {e^5 (a+b x)^{11}}{b^5}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e)^5 (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^6}+\frac {5 e (b d-a e)^4 (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^6}+\frac {10 e^2 (b d-a e)^3 (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^6}+\frac {e^3 (b d-a e)^2 (a+b x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{b^6}+\frac {5 e^4 (b d-a e) (a+b x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{11 b^6}+\frac {e^5 (a+b x)^{11} \sqrt {a^2+2 a b x+b^2 x^2}}{12 b^6}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 448, normalized size = 1.70 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (924 a^6 \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )+792 a^5 b x \left (21 d^5+70 d^4 e x+105 d^3 e^2 x^2+84 d^2 e^3 x^3+35 d e^4 x^4+6 e^5 x^5\right )+495 a^4 b^2 x^2 \left (56 d^5+210 d^4 e x+336 d^3 e^2 x^2+280 d^2 e^3 x^3+120 d e^4 x^4+21 e^5 x^5\right )+220 a^3 b^3 x^3 \left (126 d^5+504 d^4 e x+840 d^3 e^2 x^2+720 d^2 e^3 x^3+315 d e^4 x^4+56 e^5 x^5\right )+66 a^2 b^4 x^4 \left (252 d^5+1050 d^4 e x+1800 d^3 e^2 x^2+1575 d^2 e^3 x^3+700 d e^4 x^4+126 e^5 x^5\right )+12 a b^5 x^5 \left (462 d^5+1980 d^4 e x+3465 d^3 e^2 x^2+3080 d^2 e^3 x^3+1386 d e^4 x^4+252 e^5 x^5\right )+b^6 x^6 \left (792 d^5+3465 d^4 e x+6160 d^3 e^2 x^2+5544 d^2 e^3 x^3+2520 d e^4 x^4+462 e^5 x^5\right )\right )}{5544 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 4.48, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.43, size = 517, normalized size = 1.97 \begin {gather*} \frac {1}{12} \, b^{6} e^{5} x^{12} + a^{6} d^{5} x + \frac {1}{11} \, {\left (5 \, b^{6} d e^{4} + 6 \, a b^{5} e^{5}\right )} x^{11} + \frac {1}{2} \, {\left (2 \, b^{6} d^{2} e^{3} + 6 \, a b^{5} d e^{4} + 3 \, a^{2} b^{4} e^{5}\right )} x^{10} + \frac {5}{9} \, {\left (2 \, b^{6} d^{3} e^{2} + 12 \, a b^{5} d^{2} e^{3} + 15 \, a^{2} b^{4} d e^{4} + 4 \, a^{3} b^{3} e^{5}\right )} x^{9} + \frac {5}{8} \, {\left (b^{6} d^{4} e + 12 \, a b^{5} d^{3} e^{2} + 30 \, a^{2} b^{4} d^{2} e^{3} + 20 \, a^{3} b^{3} d e^{4} + 3 \, a^{4} b^{2} e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (b^{6} d^{5} + 30 \, a b^{5} d^{4} e + 150 \, a^{2} b^{4} d^{3} e^{2} + 200 \, a^{3} b^{3} d^{2} e^{3} + 75 \, a^{4} b^{2} d e^{4} + 6 \, a^{5} b e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (6 \, a b^{5} d^{5} + 75 \, a^{2} b^{4} d^{4} e + 200 \, a^{3} b^{3} d^{3} e^{2} + 150 \, a^{4} b^{2} d^{2} e^{3} + 30 \, a^{5} b d e^{4} + a^{6} e^{5}\right )} x^{6} + {\left (3 \, a^{2} b^{4} d^{5} + 20 \, a^{3} b^{3} d^{4} e + 30 \, a^{4} b^{2} d^{3} e^{2} + 12 \, a^{5} b d^{2} e^{3} + a^{6} d e^{4}\right )} x^{5} + \frac {5}{4} \, {\left (4 \, a^{3} b^{3} d^{5} + 15 \, a^{4} b^{2} d^{4} e + 12 \, a^{5} b d^{3} e^{2} + 2 \, a^{6} d^{2} e^{3}\right )} x^{4} + \frac {5}{3} \, {\left (3 \, a^{4} b^{2} d^{5} + 6 \, a^{5} b d^{4} e + 2 \, a^{6} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (6 \, a^{5} b d^{5} + 5 \, a^{6} d^{4} e\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 810, normalized size = 3.08 \begin {gather*} \frac {1}{12} \, b^{6} x^{12} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{11} \, b^{6} d x^{11} e^{4} \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{2} x^{10} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{9} \, b^{6} d^{3} x^{9} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{8} \, b^{6} d^{4} x^{8} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{7} \, b^{6} d^{5} x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{11} \, a b^{5} x^{11} e^{5} \mathrm {sgn}\left (b x + a\right ) + 3 \, a b^{5} d x^{10} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{3} \, a b^{5} d^{2} x^{9} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{2} \, a b^{5} d^{3} x^{8} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {30}{7} \, a b^{5} d^{4} x^{7} e \mathrm {sgn}\left (b x + a\right ) + a b^{5} d^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {3}{2} \, a^{2} b^{4} x^{10} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {25}{3} \, a^{2} b^{4} d x^{9} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {75}{4} \, a^{2} b^{4} d^{2} x^{8} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {150}{7} \, a^{2} b^{4} d^{3} x^{7} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {25}{2} \, a^{2} b^{4} d^{4} x^{6} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{4} d^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{9} \, a^{3} b^{3} x^{9} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {25}{2} \, a^{3} b^{3} d x^{8} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {200}{7} \, a^{3} b^{3} d^{2} x^{7} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {100}{3} \, a^{3} b^{3} d^{3} x^{6} e^{2} \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{3} b^{3} d^{4} x^{5} e \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{3} b^{3} d^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {15}{8} \, a^{4} b^{2} x^{8} e^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {75}{7} \, a^{4} b^{2} d x^{7} e^{4} \mathrm {sgn}\left (b x + a\right ) + 25 \, a^{4} b^{2} d^{2} x^{6} e^{3} \mathrm {sgn}\left (b x + a\right ) + 30 \, a^{4} b^{2} d^{3} x^{5} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {75}{4} \, a^{4} b^{2} d^{4} x^{4} e \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{4} b^{2} d^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{7} \, a^{5} b x^{7} e^{5} \mathrm {sgn}\left (b x + a\right ) + 5 \, a^{5} b d x^{6} e^{4} \mathrm {sgn}\left (b x + a\right ) + 12 \, a^{5} b d^{2} x^{5} e^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{5} b d^{3} x^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{5} b d^{4} x^{3} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{5} b d^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, a^{6} x^{6} e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} d x^{5} e^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{6} d^{2} x^{4} e^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{6} d^{3} x^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{6} d^{4} x^{2} e \mathrm {sgn}\left (b x + a\right ) + a^{6} d^{5} x \mathrm {sgn}\left (b x + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 598, normalized size = 2.27 \begin {gather*} \frac {\left (462 b^{6} e^{5} x^{11}+3024 x^{10} e^{5} a \,b^{5}+2520 x^{10} d \,e^{4} b^{6}+8316 x^{9} e^{5} a^{2} b^{4}+16632 x^{9} d \,e^{4} a \,b^{5}+5544 x^{9} d^{2} e^{3} b^{6}+12320 x^{8} e^{5} a^{3} b^{3}+46200 x^{8} d \,e^{4} a^{2} b^{4}+36960 x^{8} d^{2} e^{3} a \,b^{5}+6160 x^{8} d^{3} e^{2} b^{6}+10395 x^{7} e^{5} a^{4} b^{2}+69300 x^{7} d \,e^{4} a^{3} b^{3}+103950 x^{7} d^{2} e^{3} a^{2} b^{4}+41580 x^{7} d^{3} e^{2} a \,b^{5}+3465 x^{7} d^{4} e \,b^{6}+4752 x^{6} e^{5} a^{5} b +59400 x^{6} d \,e^{4} a^{4} b^{2}+158400 x^{6} d^{2} e^{3} a^{3} b^{3}+118800 x^{6} d^{3} e^{2} a^{2} b^{4}+23760 x^{6} d^{4} e a \,b^{5}+792 x^{6} d^{5} b^{6}+924 x^{5} e^{5} a^{6}+27720 x^{5} d \,e^{4} a^{5} b +138600 x^{5} d^{2} e^{3} a^{4} b^{2}+184800 x^{5} d^{3} e^{2} a^{3} b^{3}+69300 x^{5} d^{4} e \,a^{2} b^{4}+5544 x^{5} d^{5} a \,b^{5}+5544 a^{6} d \,e^{4} x^{4}+66528 a^{5} b \,d^{2} e^{3} x^{4}+166320 a^{4} b^{2} d^{3} e^{2} x^{4}+110880 a^{3} b^{3} d^{4} e \,x^{4}+16632 a^{2} b^{4} d^{5} x^{4}+13860 x^{3} d^{2} e^{3} a^{6}+83160 x^{3} d^{3} e^{2} a^{5} b +103950 x^{3} d^{4} e \,a^{4} b^{2}+27720 x^{3} d^{5} a^{3} b^{3}+18480 x^{2} d^{3} e^{2} a^{6}+55440 x^{2} d^{4} e \,a^{5} b +27720 x^{2} d^{5} a^{4} b^{2}+13860 x \,d^{4} e \,a^{6}+16632 x \,d^{5} a^{5} b +5544 d^{5} a^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x}{5544 \left (b x +a \right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.64, size = 1323, normalized size = 5.03
result too large to display
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 (a+b\,x\right )\,{\left (d+e\,x\right )}^5\,{\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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b x\right ) \left (d + e x\right )^{5} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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