Optimal. Leaf size=383 \[ \frac {e^4 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{10} (-6 a B e+A b e+5 b B d)}{11 b^7}+\frac {e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e) (-3 a B e+A b e+2 b B d)}{2 b^7}+\frac {10 e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^2 (-2 a B e+A b e+b B d)}{9 b^7}+\frac {5 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{8 b^7}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^4 (-6 a B e+5 A b e+b B d)}{7 b^7}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B) (b d-a e)^5}{6 b^7}+\frac {B e^5 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{11}}{12 b^7} \]
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Rubi [A] time = 0.79, antiderivative size = 383, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {770, 77} \begin {gather*} \frac {e^4 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{10} (-6 a B e+A b e+5 b B d)}{11 b^7}+\frac {e^3 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^9 (b d-a e) (-3 a B e+A b e+2 b B d)}{2 b^7}+\frac {10 e^2 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^8 (b d-a e)^2 (-2 a B e+A b e+b B d)}{9 b^7}+\frac {5 e \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^7 (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{8 b^7}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^6 (b d-a e)^4 (-6 a B e+5 A b e+b B d)}{7 b^7}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (A b-a B) (b d-a e)^5}{6 b^7}+\frac {B e^5 \sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^{11}}{12 b^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
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 \left (a b+b^2 x\right )^5 (A+B x) (d+e x)^5 \, 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 {(A b-a B) (b d-a e)^5 \left (a b+b^2 x\right )^5}{b^6}+\frac {(b d-a e)^4 (b B d+5 A b e-6 a B e) \left (a b+b^2 x\right )^6}{b^7}+\frac {5 e (b d-a e)^3 (b B d+2 A b e-3 a B e) \left (a b+b^2 x\right )^7}{b^8}+\frac {10 e^2 (b d-a e)^2 (b B d+A b e-2 a B e) \left (a b+b^2 x\right )^8}{b^9}+\frac {5 e^3 (b d-a e) (2 b B d+A b e-3 a B e) \left (a b+b^2 x\right )^9}{b^{10}}+\frac {e^4 (5 b B d+A b e-6 a B e) \left (a b+b^2 x\right )^{10}}{b^{11}}+\frac {B e^5 \left (a b+b^2 x\right )^{11}}{b^{12}}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {(A b-a B) (b d-a e)^5 (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^7}+\frac {(b d-a e)^4 (b B d+5 A b e-6 a B e) (a+b x)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 b^7}+\frac {5 e (b d-a e)^3 (b B d+2 A b e-3 a B e) (a+b x)^7 \sqrt {a^2+2 a b x+b^2 x^2}}{8 b^7}+\frac {10 e^2 (b d-a e)^2 (b B d+A b e-2 a B e) (a+b x)^8 \sqrt {a^2+2 a b x+b^2 x^2}}{9 b^7}+\frac {e^3 (b d-a e) (2 b B d+A b e-3 a B e) (a+b x)^9 \sqrt {a^2+2 a b x+b^2 x^2}}{2 b^7}+\frac {e^4 (5 b B d+A b e-6 a B e) (a+b x)^{10} \sqrt {a^2+2 a b x+b^2 x^2}}{11 b^7}+\frac {B e^5 (a+b x)^{11} \sqrt {a^2+2 a b x+b^2 x^2}}{12 b^7}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 740, normalized size = 1.93 \begin {gather*} \frac {x \sqrt {(a+b x)^2} \left (132 a^5 \left (7 A \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 )+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 )\right )+165 a^4 b x \left (4 A \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 )+B x \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 )\right )+110 a^3 b^2 x^2 \left (3 A \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 )+B x \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 )\right )+22 a^2 b^3 x^3 \left (5 A \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 )+2 B x \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 )\right )+2 a b^4 x^4 \left (11 A \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 )+5 B x \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 )\right )+b^5 x^5 \left (A \left (924 d^5+3960 d^4 e x+6930 d^3 e^2 x^2+6160 d^2 e^3 x^3+2772 d e^4 x^4+504 e^5 x^5\right )+B x \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 )\right )}{5544 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 10.37, 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 = 0.42, size = 813, normalized size = 2.12 \begin {gather*} \frac {1}{12} \, B b^{5} e^{5} x^{12} + A a^{5} d^{5} x + \frac {1}{11} \, {\left (5 \, B b^{5} d e^{4} + {\left (5 \, B a b^{4} + A b^{5}\right )} e^{5}\right )} x^{11} + \frac {1}{2} \, {\left (2 \, B b^{5} d^{2} e^{3} + {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{4} + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{5}\right )} x^{10} + \frac {5}{9} \, {\left (2 \, B b^{5} d^{3} e^{2} + 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{3} + 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{4} + 2 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{5}\right )} x^{9} + \frac {5}{8} \, {\left (B b^{5} d^{4} e + 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{2} + 10 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{3} + 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{4} + {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (B b^{5} d^{5} + 5 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e + 50 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{2} + 100 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{3} + 25 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{4} + {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (A a^{5} e^{5} + {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} + 25 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e + 100 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{2} + 50 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{3} + 5 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{4}\right )} x^{6} + {\left (A a^{5} d e^{4} + {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{5} + 10 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{4} e + 10 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{3} e^{2} + 2 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{2} e^{3}\right )} x^{5} + \frac {5}{4} \, {\left (2 \, A a^{5} d^{2} e^{3} + 2 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{5} + 5 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{4} e + 2 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{3} e^{2}\right )} x^{4} + \frac {5}{3} \, {\left (2 \, A a^{5} d^{3} e^{2} + {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{5} + {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{4} e\right )} x^{3} + \frac {1}{2} \, {\left (5 \, A a^{5} d^{4} e + {\left (B a^{5} + 5 \, A a^{4} b\right )} d^{5}\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 1446, normalized size = 3.78
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 1068, normalized size = 2.79 \begin {gather*} \frac {\left (462 B \,e^{5} b^{5} x^{11}+504 x^{10} A \,b^{5} e^{5}+2520 x^{10} B \,e^{5} a \,b^{4}+2520 x^{10} B \,b^{5} d \,e^{4}+2772 x^{9} A a \,b^{4} e^{5}+2772 x^{9} A \,b^{5} d \,e^{4}+5544 x^{9} B \,e^{5} a^{2} b^{3}+13860 x^{9} B a \,b^{4} d \,e^{4}+5544 x^{9} B \,b^{5} d^{2} e^{3}+6160 x^{8} A \,a^{2} b^{3} e^{5}+15400 x^{8} A a \,b^{4} d \,e^{4}+6160 x^{8} A \,b^{5} d^{2} e^{3}+6160 x^{8} B \,e^{5} a^{3} b^{2}+30800 x^{8} B \,a^{2} b^{3} d \,e^{4}+30800 x^{8} B a \,b^{4} d^{2} e^{3}+6160 x^{8} B \,b^{5} d^{3} e^{2}+6930 x^{7} A \,a^{3} b^{2} e^{5}+34650 x^{7} A \,a^{2} b^{3} d \,e^{4}+34650 x^{7} A a \,b^{4} d^{2} e^{3}+6930 x^{7} A \,b^{5} d^{3} e^{2}+3465 x^{7} B \,e^{5} a^{4} b +34650 x^{7} B \,a^{3} b^{2} d \,e^{4}+69300 x^{7} B \,a^{2} b^{3} d^{2} e^{3}+34650 x^{7} B a \,b^{4} d^{3} e^{2}+3465 x^{7} B \,b^{5} d^{4} e +3960 x^{6} A \,a^{4} b \,e^{5}+39600 x^{6} A \,a^{3} b^{2} d \,e^{4}+79200 x^{6} A \,a^{2} b^{3} d^{2} e^{3}+39600 x^{6} A a \,b^{4} d^{3} e^{2}+3960 x^{6} A \,b^{5} d^{4} e +792 x^{6} B \,e^{5} a^{5}+19800 x^{6} B \,a^{4} b d \,e^{4}+79200 x^{6} B \,a^{3} b^{2} d^{2} e^{3}+79200 x^{6} B \,a^{2} b^{3} d^{3} e^{2}+19800 x^{6} B a \,b^{4} d^{4} e +792 x^{6} B \,b^{5} d^{5}+924 x^{5} A \,a^{5} e^{5}+23100 x^{5} A \,a^{4} b d \,e^{4}+92400 x^{5} A \,a^{3} b^{2} d^{2} e^{3}+92400 x^{5} A \,a^{2} b^{3} d^{3} e^{2}+23100 x^{5} A a \,b^{4} d^{4} e +924 x^{5} A \,d^{5} b^{5}+4620 x^{5} B \,a^{5} d \,e^{4}+46200 x^{5} B \,a^{4} b \,d^{2} e^{3}+92400 x^{5} B \,a^{3} b^{2} d^{3} e^{2}+46200 x^{5} B \,a^{2} b^{3} d^{4} e +4620 x^{5} B a \,b^{4} d^{5}+5544 A \,a^{5} d \,e^{4} x^{4}+55440 A \,a^{4} b \,d^{2} e^{3} x^{4}+110880 A \,a^{3} b^{2} d^{3} e^{2} x^{4}+55440 A \,a^{2} b^{3} d^{4} e \,x^{4}+5544 A a \,b^{4} d^{5} x^{4}+11088 B \,a^{5} d^{2} e^{3} x^{4}+55440 B \,a^{4} b \,d^{3} e^{2} x^{4}+55440 B \,a^{3} b^{2} d^{4} e \,x^{4}+11088 B \,a^{2} b^{3} d^{5} x^{4}+13860 x^{3} A \,a^{5} d^{2} e^{3}+69300 x^{3} A \,a^{4} b \,d^{3} e^{2}+69300 x^{3} A \,a^{3} b^{2} d^{4} e +13860 x^{3} A \,d^{5} a^{2} b^{3}+13860 x^{3} B \,a^{5} d^{3} e^{2}+34650 x^{3} B \,a^{4} b \,d^{4} e +13860 x^{3} B \,a^{3} b^{2} d^{5}+18480 x^{2} A \,a^{5} d^{3} e^{2}+46200 x^{2} A \,a^{4} b \,d^{4} e +18480 x^{2} A \,d^{5} a^{3} b^{2}+9240 x^{2} B \,a^{5} d^{4} e +9240 x^{2} B \,a^{4} b \,d^{5}+13860 x A \,a^{5} d^{4} e +13860 x A \,d^{5} a^{4} b +2772 x B \,a^{5} d^{5}+5544 A \,d^{5} a^{5}\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 = 0.81, size = 1330, normalized size = 3.47
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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