3.1791 \(\int \sqrt{1-2 x} (2+3 x)^5 (3+5 x) \, dx\)

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 = 0.0180173, 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, Rules used = {77} \[ -\frac{81}{64} (1-2 x)^{15/2}+\frac{81}{4} (1-2 x)^{13/2}-\frac{97335}{704} (1-2 x)^{11/2}+\frac{4165}{8} (1-2 x)^{9/2}-\frac{74235}{64} (1-2 x)^{7/2}+\frac{12005}{8} (1-2 x)^{5/2}-\frac{184877}{192} (1-2 x)^{3/2} \]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \sqrt{1-2 x} (2+3 x)^5 (3+5 x) \, dx &=\int \left (\frac{184877}{64} \sqrt{1-2 x}-\frac{60025}{8} (1-2 x)^{3/2}+\frac{519645}{64} (1-2 x)^{5/2}-\frac{37485}{8} (1-2 x)^{7/2}+\frac{97335}{64} (1-2 x)^{9/2}-\frac{1053}{4} (1-2 x)^{11/2}+\frac{1215}{64} (1-2 x)^{13/2}\right ) \, dx\\ &=-\frac{184877}{192} (1-2 x)^{3/2}+\frac{12005}{8} (1-2 x)^{5/2}-\frac{74235}{64} (1-2 x)^{7/2}+\frac{4165}{8} (1-2 x)^{9/2}-\frac{97335}{704} (1-2 x)^{11/2}+\frac{81}{4} (1-2 x)^{13/2}-\frac{81}{64} (1-2 x)^{15/2}\\ \end{align*}

Mathematica [A]  time = 0.0182522, size = 43, normalized size = 0.47 \[ -\frac{1}{33} (1-2 x)^{3/2} \left (2673 x^6+13365 x^5+29565 x^4+38220 x^3+32220 x^2+18696 x+7288\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 40, normalized size = 0.4 \begin{align*} -{\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}}}} \end{align*}

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 = 1.07487, size = 86, normalized size = 0.93 \begin{align*} -\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}} \end{align*}

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, 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) - 74
235/64*(-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 = 1.34584, size = 144, normalized size = 1.57 \begin{align*} \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} \end{align*}

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, 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 = 2.30543, size = 82, normalized size = 0.89 \begin{align*} - \frac{81 \left (1 - 2 x\right )^{\frac{15}{2}}}{64} + \frac{81 \left (1 - 2 x\right )^{\frac{13}{2}}}{4} - \frac{97335 \left (1 - 2 x\right )^{\frac{11}{2}}}{704} + \frac{4165 \left (1 - 2 x\right )^{\frac{9}{2}}}{8} - \frac{74235 \left (1 - 2 x\right )^{\frac{7}{2}}}{64} + \frac{12005 \left (1 - 2 x\right )^{\frac{5}{2}}}{8} - \frac{184877 \left (1 - 2 x\right )^{\frac{3}{2}}}{192} \end{align*}

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*(1 - 2*x)**(15/2)/64 + 81*(1 - 2*x)**(13/2)/4 - 97335*(1 - 2*x)**(11/2)/704 + 4165*(1 - 2*x)**(9/2)/8 - 74
235*(1 - 2*x)**(7/2)/64 + 12005*(1 - 2*x)**(5/2)/8 - 184877*(1 - 2*x)**(3/2)/192

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Giac [A]  time = 1.78675, size = 143, normalized size = 1.55 \begin{align*} \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}} \end{align*}

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, 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) + 41
65/8*(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) - 1
84877/192*(-2*x + 1)^(3/2)