3.1804 \(\int \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^2 \, dx\)

Optimal. Leaf size=79 \[ \frac{675}{416} (1-2 x)^{13/2}-\frac{7695}{352} (1-2 x)^{11/2}+\frac{1949}{16} (1-2 x)^{9/2}-\frac{5711}{16} (1-2 x)^{7/2}+\frac{91091}{160} (1-2 x)^{5/2}-\frac{41503}{96} (1-2 x)^{3/2} \]

[Out]

(-41503*(1 - 2*x)^(3/2))/96 + (91091*(1 - 2*x)^(5/2))/160 - (5711*(1 - 2*x)^(7/2))/16 + (1949*(1 - 2*x)^(9/2))
/16 - (7695*(1 - 2*x)^(11/2))/352 + (675*(1 - 2*x)^(13/2))/416

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Rubi [A]  time = 0.0151205, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {88} \[ \frac{675}{416} (1-2 x)^{13/2}-\frac{7695}{352} (1-2 x)^{11/2}+\frac{1949}{16} (1-2 x)^{9/2}-\frac{5711}{16} (1-2 x)^{7/2}+\frac{91091}{160} (1-2 x)^{5/2}-\frac{41503}{96} (1-2 x)^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-41503*(1 - 2*x)^(3/2))/96 + (91091*(1 - 2*x)^(5/2))/160 - (5711*(1 - 2*x)^(7/2))/16 + (1949*(1 - 2*x)^(9/2))
/16 - (7695*(1 - 2*x)^(11/2))/352 + (675*(1 - 2*x)^(13/2))/416

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^2 \, dx &=\int \left (\frac{41503}{32} \sqrt{1-2 x}-\frac{91091}{32} (1-2 x)^{3/2}+\frac{39977}{16} (1-2 x)^{5/2}-\frac{17541}{16} (1-2 x)^{7/2}+\frac{7695}{32} (1-2 x)^{9/2}-\frac{675}{32} (1-2 x)^{11/2}\right ) \, dx\\ &=-\frac{41503}{96} (1-2 x)^{3/2}+\frac{91091}{160} (1-2 x)^{5/2}-\frac{5711}{16} (1-2 x)^{7/2}+\frac{1949}{16} (1-2 x)^{9/2}-\frac{7695}{352} (1-2 x)^{11/2}+\frac{675}{416} (1-2 x)^{13/2}\\ \end{align*}

Mathematica [A]  time = 0.0170215, size = 38, normalized size = 0.48 \[ -\frac{(1-2 x)^{3/2} \left (111375 x^5+471825 x^4+868215 x^3+913245 x^2+607254 x+253898\right )}{2145} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(3/2)*(253898 + 607254*x + 913245*x^2 + 868215*x^3 + 471825*x^4 + 111375*x^5))/2145

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Maple [A]  time = 0.005, size = 35, normalized size = 0.4 \begin{align*} -{\frac{111375\,{x}^{5}+471825\,{x}^{4}+868215\,{x}^{3}+913245\,{x}^{2}+607254\,x+253898}{2145} \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)^3*(3+5*x)^2*(1-2*x)^(1/2),x)

[Out]

-1/2145*(111375*x^5+471825*x^4+868215*x^3+913245*x^2+607254*x+253898)*(1-2*x)^(3/2)

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Maxima [A]  time = 1.16972, size = 74, normalized size = 0.94 \begin{align*} \frac{675}{416} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - \frac{7695}{352} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{1949}{16} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{5711}{16} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{91091}{160} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{41503}{96} \,{\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)^3*(3+5*x)^2*(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

675/416*(-2*x + 1)^(13/2) - 7695/352*(-2*x + 1)^(11/2) + 1949/16*(-2*x + 1)^(9/2) - 5711/16*(-2*x + 1)^(7/2) +
91091/160*(-2*x + 1)^(5/2) - 41503/96*(-2*x + 1)^(3/2)

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Fricas [A]  time = 1.47567, size = 146, normalized size = 1.85 \begin{align*} \frac{1}{2145} \,{\left (222750 \, x^{6} + 832275 \, x^{5} + 1264605 \, x^{4} + 958275 \, x^{3} + 301263 \, x^{2} - 99458 \, x - 253898\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^3*(3+5*x)^2*(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/2145*(222750*x^6 + 832275*x^5 + 1264605*x^4 + 958275*x^3 + 301263*x^2 - 99458*x - 253898)*sqrt(-2*x + 1)

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Sympy [A]  time = 2.1094, size = 70, normalized size = 0.89 \begin{align*} \frac{675 \left (1 - 2 x\right )^{\frac{13}{2}}}{416} - \frac{7695 \left (1 - 2 x\right )^{\frac{11}{2}}}{352} + \frac{1949 \left (1 - 2 x\right )^{\frac{9}{2}}}{16} - \frac{5711 \left (1 - 2 x\right )^{\frac{7}{2}}}{16} + \frac{91091 \left (1 - 2 x\right )^{\frac{5}{2}}}{160} - \frac{41503 \left (1 - 2 x\right )^{\frac{3}{2}}}{96} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

675*(1 - 2*x)**(13/2)/416 - 7695*(1 - 2*x)**(11/2)/352 + 1949*(1 - 2*x)**(9/2)/16 - 5711*(1 - 2*x)**(7/2)/16 +
91091*(1 - 2*x)**(5/2)/160 - 41503*(1 - 2*x)**(3/2)/96

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Giac [A]  time = 1.91022, size = 122, normalized size = 1.54 \begin{align*} \frac{675}{416} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + \frac{7695}{352} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{1949}{16} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{5711}{16} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{91091}{160} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{41503}{96} \,{\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)^3*(3+5*x)^2*(1-2*x)^(1/2),x, algorithm="giac")

[Out]

675/416*(2*x - 1)^6*sqrt(-2*x + 1) + 7695/352*(2*x - 1)^5*sqrt(-2*x + 1) + 1949/16*(2*x - 1)^4*sqrt(-2*x + 1)
+ 5711/16*(2*x - 1)^3*sqrt(-2*x + 1) + 91091/160*(2*x - 1)^2*sqrt(-2*x + 1) - 41503/96*(-2*x + 1)^(3/2)