3.1774 \(\int \sqrt{1-2 x} (2+3 x)^5 (3+5 x) \, dx\)

Optimal. Leaf size=92 \[ -\frac{81}{64} (1-2 x)^{15/2}+\frac{81}{4} (1-2 x)^{13/2}-\frac{97335}{704} (1-2 x)^{11/2}+\frac{4165}{8} (1-2 x)^{9/2}-\frac{74235}{64} (1-2 x)^{7/2}+\frac{12005}{8} (1-2 x)^{5/2}-\frac{184877}{192} (1-2 x)^{3/2} \]

[Out]

(-184877*(1 - 2*x)^(3/2))/192 + (12005*(1 - 2*x)^(5/2))/8 - (74235*(1 - 2*x)^(7/
2))/64 + (4165*(1 - 2*x)^(9/2))/8 - (97335*(1 - 2*x)^(11/2))/704 + (81*(1 - 2*x)
^(13/2))/4 - (81*(1 - 2*x)^(15/2))/64

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Rubi [A]  time = 0.0616982, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{81}{64} (1-2 x)^{15/2}+\frac{81}{4} (1-2 x)^{13/2}-\frac{97335}{704} (1-2 x)^{11/2}+\frac{4165}{8} (1-2 x)^{9/2}-\frac{74235}{64} (1-2 x)^{7/2}+\frac{12005}{8} (1-2 x)^{5/2}-\frac{184877}{192} (1-2 x)^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - 2*x]*(2 + 3*x)^5*(3 + 5*x),x]

[Out]

(-184877*(1 - 2*x)^(3/2))/192 + (12005*(1 - 2*x)^(5/2))/8 - (74235*(1 - 2*x)^(7/
2))/64 + (4165*(1 - 2*x)^(9/2))/8 - (97335*(1 - 2*x)^(11/2))/704 + (81*(1 - 2*x)
^(13/2))/4 - (81*(1 - 2*x)^(15/2))/64

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Rubi in Sympy [A]  time = 9.96896, size = 82, normalized size = 0.89 \[ - \frac{81 \left (- 2 x + 1\right )^{\frac{15}{2}}}{64} + \frac{81 \left (- 2 x + 1\right )^{\frac{13}{2}}}{4} - \frac{97335 \left (- 2 x + 1\right )^{\frac{11}{2}}}{704} + \frac{4165 \left (- 2 x + 1\right )^{\frac{9}{2}}}{8} - \frac{74235 \left (- 2 x + 1\right )^{\frac{7}{2}}}{64} + \frac{12005 \left (- 2 x + 1\right )^{\frac{5}{2}}}{8} - \frac{184877 \left (- 2 x + 1\right )^{\frac{3}{2}}}{192} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**5*(3+5*x)*(1-2*x)**(1/2),x)

[Out]

-81*(-2*x + 1)**(15/2)/64 + 81*(-2*x + 1)**(13/2)/4 - 97335*(-2*x + 1)**(11/2)/7
04 + 4165*(-2*x + 1)**(9/2)/8 - 74235*(-2*x + 1)**(7/2)/64 + 12005*(-2*x + 1)**(
5/2)/8 - 184877*(-2*x + 1)**(3/2)/192

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Mathematica [A]  time = 0.0350631, size = 48, normalized size = 0.52 \[ \frac{1}{33} \sqrt{1-2 x} \left (5346 x^7+24057 x^6+45765 x^5+46875 x^4+26220 x^3+5172 x^2-4120 x-7288\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 - 2*x]*(2 + 3*x)^5*(3 + 5*x),x]

[Out]

(Sqrt[1 - 2*x]*(-7288 - 4120*x + 5172*x^2 + 26220*x^3 + 46875*x^4 + 45765*x^5 +
24057*x^6 + 5346*x^7))/33

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Maple [A]  time = 0.004, size = 40, normalized size = 0.4 \[ -{\frac{2673\,{x}^{6}+13365\,{x}^{5}+29565\,{x}^{4}+38220\,{x}^{3}+32220\,{x}^{2}+18696\,x+7288}{33} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^5*(3+5*x)*(1-2*x)^(1/2),x)

[Out]

-1/33*(2673*x^6+13365*x^5+29565*x^4+38220*x^3+32220*x^2+18696*x+7288)*(1-2*x)^(3
/2)

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Maxima [A]  time = 1.34956, size = 86, normalized size = 0.93 \[ -\frac{81}{64} \,{\left (-2 \, x + 1\right )}^{\frac{15}{2}} + \frac{81}{4} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - \frac{97335}{704} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{4165}{8} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{74235}{64} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{12005}{8} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{184877}{192} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*(3*x + 2)^5*sqrt(-2*x + 1),x, algorithm="maxima")

[Out]

-81/64*(-2*x + 1)^(15/2) + 81/4*(-2*x + 1)^(13/2) - 97335/704*(-2*x + 1)^(11/2)
+ 4165/8*(-2*x + 1)^(9/2) - 74235/64*(-2*x + 1)^(7/2) + 12005/8*(-2*x + 1)^(5/2)
- 184877/192*(-2*x + 1)^(3/2)

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Fricas [A]  time = 0.212056, size = 59, normalized size = 0.64 \[ \frac{1}{33} \,{\left (5346 \, x^{7} + 24057 \, x^{6} + 45765 \, x^{5} + 46875 \, x^{4} + 26220 \, x^{3} + 5172 \, x^{2} - 4120 \, x - 7288\right )} \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*(3*x + 2)^5*sqrt(-2*x + 1),x, algorithm="fricas")

[Out]

1/33*(5346*x^7 + 24057*x^6 + 45765*x^5 + 46875*x^4 + 26220*x^3 + 5172*x^2 - 4120
*x - 7288)*sqrt(-2*x + 1)

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Sympy [A]  time = 3.59212, size = 82, normalized size = 0.89 \[ - \frac{81 \left (- 2 x + 1\right )^{\frac{15}{2}}}{64} + \frac{81 \left (- 2 x + 1\right )^{\frac{13}{2}}}{4} - \frac{97335 \left (- 2 x + 1\right )^{\frac{11}{2}}}{704} + \frac{4165 \left (- 2 x + 1\right )^{\frac{9}{2}}}{8} - \frac{74235 \left (- 2 x + 1\right )^{\frac{7}{2}}}{64} + \frac{12005 \left (- 2 x + 1\right )^{\frac{5}{2}}}{8} - \frac{184877 \left (- 2 x + 1\right )^{\frac{3}{2}}}{192} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+3*x)**5*(3+5*x)*(1-2*x)**(1/2),x)

[Out]

-81*(-2*x + 1)**(15/2)/64 + 81*(-2*x + 1)**(13/2)/4 - 97335*(-2*x + 1)**(11/2)/7
04 + 4165*(-2*x + 1)**(9/2)/8 - 74235*(-2*x + 1)**(7/2)/64 + 12005*(-2*x + 1)**(
5/2)/8 - 184877*(-2*x + 1)**(3/2)/192

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GIAC/XCAS [A]  time = 0.216387, size = 143, normalized size = 1.55 \[ \frac{81}{64} \,{\left (2 \, x - 1\right )}^{7} \sqrt{-2 \, x + 1} + \frac{81}{4} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + \frac{97335}{704} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{4165}{8} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{74235}{64} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{12005}{8} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{184877}{192} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)*(3*x + 2)^5*sqrt(-2*x + 1),x, algorithm="giac")

[Out]

81/64*(2*x - 1)^7*sqrt(-2*x + 1) + 81/4*(2*x - 1)^6*sqrt(-2*x + 1) + 97335/704*(
2*x - 1)^5*sqrt(-2*x + 1) + 4165/8*(2*x - 1)^4*sqrt(-2*x + 1) + 74235/64*(2*x -
1)^3*sqrt(-2*x + 1) + 12005/8*(2*x - 1)^2*sqrt(-2*x + 1) - 184877/192*(-2*x + 1)
^(3/2)