3.2288 \(\int \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=165 \[ -\frac{13}{80} (1-2 x)^{3/2} (5 x+3)^{7/2}-\frac{1}{20} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{7/2}-\frac{1069 (1-2 x)^{3/2} (5 x+3)^{5/2}}{1280}-\frac{11759 (1-2 x)^{3/2} (5 x+3)^{3/2}}{3072}-\frac{129349 (1-2 x)^{3/2} \sqrt{5 x+3}}{8192}+\frac{1422839 \sqrt{1-2 x} \sqrt{5 x+3}}{81920}+\frac{15651229 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{81920 \sqrt{10}} \]

[Out]

(1422839*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/81920 - (129349*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/8192 - (11759*(1 - 2*x)^(
3/2)*(3 + 5*x)^(3/2))/3072 - (1069*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/1280 - (13*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2)
)/80 - ((1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^(7/2))/20 + (15651229*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(81920*Sqr
t[10])

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Rubi [A]  time = 0.0488844, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {90, 80, 50, 54, 216} \[ -\frac{13}{80} (1-2 x)^{3/2} (5 x+3)^{7/2}-\frac{1}{20} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{7/2}-\frac{1069 (1-2 x)^{3/2} (5 x+3)^{5/2}}{1280}-\frac{11759 (1-2 x)^{3/2} (5 x+3)^{3/2}}{3072}-\frac{129349 (1-2 x)^{3/2} \sqrt{5 x+3}}{8192}+\frac{1422839 \sqrt{1-2 x} \sqrt{5 x+3}}{81920}+\frac{15651229 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{81920 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1422839*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/81920 - (129349*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/8192 - (11759*(1 - 2*x)^(
3/2)*(3 + 5*x)^(3/2))/3072 - (1069*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/1280 - (13*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2)
)/80 - ((1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^(7/2))/20 + (15651229*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(81920*Sqr
t[10])

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2} \, dx &=-\frac{1}{20} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{7/2}-\frac{1}{60} \int \left (-318-\frac{975 x}{2}\right ) \sqrt{1-2 x} (3+5 x)^{5/2} \, dx\\ &=-\frac{13}{80} (1-2 x)^{3/2} (3+5 x)^{7/2}-\frac{1}{20} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{7/2}+\frac{1069}{160} \int \sqrt{1-2 x} (3+5 x)^{5/2} \, dx\\ &=-\frac{1069 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1280}-\frac{13}{80} (1-2 x)^{3/2} (3+5 x)^{7/2}-\frac{1}{20} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{7/2}+\frac{11759}{512} \int \sqrt{1-2 x} (3+5 x)^{3/2} \, dx\\ &=-\frac{11759 (1-2 x)^{3/2} (3+5 x)^{3/2}}{3072}-\frac{1069 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1280}-\frac{13}{80} (1-2 x)^{3/2} (3+5 x)^{7/2}-\frac{1}{20} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{7/2}+\frac{129349 \int \sqrt{1-2 x} \sqrt{3+5 x} \, dx}{2048}\\ &=-\frac{129349 (1-2 x)^{3/2} \sqrt{3+5 x}}{8192}-\frac{11759 (1-2 x)^{3/2} (3+5 x)^{3/2}}{3072}-\frac{1069 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1280}-\frac{13}{80} (1-2 x)^{3/2} (3+5 x)^{7/2}-\frac{1}{20} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{7/2}+\frac{1422839 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{16384}\\ &=\frac{1422839 \sqrt{1-2 x} \sqrt{3+5 x}}{81920}-\frac{129349 (1-2 x)^{3/2} \sqrt{3+5 x}}{8192}-\frac{11759 (1-2 x)^{3/2} (3+5 x)^{3/2}}{3072}-\frac{1069 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1280}-\frac{13}{80} (1-2 x)^{3/2} (3+5 x)^{7/2}-\frac{1}{20} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{7/2}+\frac{15651229 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{163840}\\ &=\frac{1422839 \sqrt{1-2 x} \sqrt{3+5 x}}{81920}-\frac{129349 (1-2 x)^{3/2} \sqrt{3+5 x}}{8192}-\frac{11759 (1-2 x)^{3/2} (3+5 x)^{3/2}}{3072}-\frac{1069 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1280}-\frac{13}{80} (1-2 x)^{3/2} (3+5 x)^{7/2}-\frac{1}{20} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{7/2}+\frac{15651229 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{81920 \sqrt{5}}\\ &=\frac{1422839 \sqrt{1-2 x} \sqrt{3+5 x}}{81920}-\frac{129349 (1-2 x)^{3/2} \sqrt{3+5 x}}{8192}-\frac{11759 (1-2 x)^{3/2} (3+5 x)^{3/2}}{3072}-\frac{1069 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1280}-\frac{13}{80} (1-2 x)^{3/2} (3+5 x)^{7/2}-\frac{1}{20} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{7/2}+\frac{15651229 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{81920 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0495547, size = 75, normalized size = 0.45 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (9216000 x^5+28108800 x^4+32887680 x^3+16507936 x^2+17884 x-6023169\right )-46953687 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{2457600} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-6023169 + 17884*x + 16507936*x^2 + 32887680*x^3 + 28108800*x^4 + 9216000*x^5
) - 46953687*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/2457600

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Maple [A]  time = 0.008, size = 138, normalized size = 0.8 \begin{align*}{\frac{1}{4915200}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 184320000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +357680\,x\sqrt{-10\,{x}^{2}-x+3}-120463380\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4915200*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(184320000*x^5*(-10*x^2-x+3)^(1/2)[phone]*x^4*(-10*x^2-x+3)^(1/2)+657
753600*x^3*(-10*x^2-x+3)^(1/2)[phone]*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)+357680
*x*(-10*x^2-x+3)^(1/2)-120463380*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 2.52187, size = 140, normalized size = 0.85 \begin{align*} -\frac{15}{4} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} - \frac{177}{16} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} - \frac{17153}{1280} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{133567}{15360} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{129349}{4096} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{15651229}{1638400} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{129349}{81920} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-15/4*(-10*x^2 - x + 3)^(3/2)*x^3 - 177/16*(-10*x^2 - x + 3)^(3/2)*x^2 - 17153/1280*(-10*x^2 - x + 3)^(3/2)*x
- 133567/15360*(-10*x^2 - x + 3)^(3/2) + 129349/4096*sqrt(-10*x^2 - x + 3)*x - 15651229/1638400*sqrt(10)*arcsi
n(-20/11*x - 1/11) + 129349/81920*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.79803, size = 304, normalized size = 1.84 \begin{align*} \frac{1}{245760} \,{\left (9216000 \, x^{5} + 28108800 \, x^{4} + 32887680 \, x^{3} + 16507936 \, x^{2} + 17884 \, x - 6023169\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{15651229}{1638400} \, \sqrt{10} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/245760*(9216000*x^5 + 28108800*x^4 + 32887680*x^3 + 16507936*x^2 + 17884*x - 6023169)*sqrt(5*x + 3)*sqrt(-2*
x + 1) - 15651229/1638400*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x -
3))

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Sympy [A]  time = 131.543, size = 694, normalized size = 4.21 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5929*sqrt(2)*Piecewise((121*sqrt(5)*(-sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/121 + asin(sqrt(55)*sqr
t(1 - 2*x)/11))/200, (x <= 1/2) & (x > -3/5)))/32 + 1309*sqrt(2)*Piecewise((1331*sqrt(5)*(-5*sqrt(5)*(1 - 2*x)
**(3/2)*(10*x + 6)**(3/2)/7986 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/1936 + asin(sqrt(55)*sqrt(1 -
2*x)/11)/16)/125, (x <= 1/2) & (x > -3/5)))/4 - 3467*sqrt(2)*Piecewise((14641*sqrt(5)*(-5*sqrt(5)*(1 - 2*x)**
(3/2)*(10*x + 6)**(3/2)/7986 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/3872 - sqrt(5)*sqrt(1 - 2*x)*sq
rt(10*x + 6)*(12100*x - 2000*(1 - 2*x)**3 + 6600*(1 - 2*x)**2 - 4719)/1874048 + 5*asin(sqrt(55)*sqrt(1 - 2*x)/
11)/128)/625, (x <= 1/2) & (x > -3/5)))/16 + 255*sqrt(2)*Piecewise((161051*sqrt(5)*(5*sqrt(5)*(1 - 2*x)**(5/2)
*(10*x + 6)**(5/2)/322102 - 5*sqrt(5)*(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)/7986 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*
x + 6)*(20*x + 1)/7744 - 3*sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(12100*x - 2000*(1 - 2*x)**3 + 6600*(1 - 2*x)*
*2 - 4719)/3748096 + 7*asin(sqrt(55)*sqrt(1 - 2*x)/11)/256)/3125, (x <= 1/2) & (x > -3/5)))/4 - 225*sqrt(2)*Pi
ecewise((1771561*sqrt(5)*(5*sqrt(5)*(1 - 2*x)**(5/2)*(10*x + 6)**(5/2)/161051 + 5*sqrt(5)*(1 - 2*x)**(3/2)*(10
*x + 6)**(3/2)*(20*x + 1)**3/170069856 - 5*sqrt(5)*(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)/7986 - sqrt(5)*sqrt(1 -
2*x)*sqrt(10*x + 6)*(20*x + 1)/15488 - 13*sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(12100*x - 2000*(1 - 2*x)**3 +
6600*(1 - 2*x)**2 - 4719)/14992384 + 21*asin(sqrt(55)*sqrt(1 - 2*x)/11)/1024)/15625, (x <= 1/2) & (x > -3/5)))
/32

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Giac [B]  time = 2.34447, size = 427, normalized size = 2.59 \begin{align*} \frac{3}{102400000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (100 \, x - 239\right )}{\left (5 \, x + 3\right )} + 27999\right )}{\left (5 \, x + 3\right )} - 318159\right )}{\left (5 \, x + 3\right )} + 3237255\right )}{\left (5 \, x + 3\right )} - 2656665\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 29223315 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{19}{6400000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 143\right )}{\left (5 \, x + 3\right )} + 9773\right )}{\left (5 \, x + 3\right )} - 136405\right )}{\left (5 \, x + 3\right )} + 60555\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 666105 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{541}{1920000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{19}{2000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{9}{100} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/102400000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 239)*(5*x + 3) + 27999)*(5*x + 3) - 318159)*(5*x + 3) + 3237255)*
(5*x + 3) - 2656665)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 29223315*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1
9/6400000*sqrt(5)*(2*(4*(8*(12*(80*x - 143)*(5*x + 3) + 9773)*(5*x + 3) - 136405)*(5*x + 3) + 60555)*sqrt(5*x
+ 3)*sqrt(-10*x + 5) - 666105*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 541/1920000*sqrt(5)*(2*(4*(8*(60*
x - 71)*(5*x + 3) + 2179)*(5*x + 3) - 4125)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 45375*sqrt(2)*arcsin(1/11*sqrt(22)
*sqrt(5*x + 3))) + 19/2000*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 363*sqrt(
2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 9/100*sqrt(5)*(2*(20*x + 1)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt
(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))