3.2264 \(\int \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=121 \[ -\frac{3}{40} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac{37}{160} (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac{1313 \sqrt{5 x+3} (1-2 x)^{3/2}}{1280}+\frac{14443 \sqrt{5 x+3} \sqrt{1-2 x}}{12800}+\frac{158873 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{12800 \sqrt{10}} \]

[Out]

(14443*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/12800 - (1313*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/1280 - (37*(1 - 2*x)^(3/2)*(3
+ 5*x)^(3/2))/160 - (3*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^(3/2))/40 + (158873*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x
]])/(12800*Sqrt[10])

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Rubi [A]  time = 0.0337888, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{3}{40} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac{37}{160} (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac{1313 \sqrt{5 x+3} (1-2 x)^{3/2}}{1280}+\frac{14443 \sqrt{5 x+3} \sqrt{1-2 x}}{12800}+\frac{158873 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{12800 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(14443*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/12800 - (1313*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/1280 - (37*(1 - 2*x)^(3/2)*(3
+ 5*x)^(3/2))/160 - (3*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^(3/2))/40 + (158873*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x
]])/(12800*Sqrt[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 \sqrt{3+5 x} \, dx &=-\frac{3}{40} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}-\frac{1}{40} \int \left (-178-\frac{555 x}{2}\right ) \sqrt{1-2 x} \sqrt{3+5 x} \, dx\\ &=-\frac{37}{160} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac{3}{40} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}+\frac{1313}{320} \int \sqrt{1-2 x} \sqrt{3+5 x} \, dx\\ &=-\frac{1313 (1-2 x)^{3/2} \sqrt{3+5 x}}{1280}-\frac{37}{160} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac{3}{40} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}+\frac{14443 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{2560}\\ &=\frac{14443 \sqrt{1-2 x} \sqrt{3+5 x}}{12800}-\frac{1313 (1-2 x)^{3/2} \sqrt{3+5 x}}{1280}-\frac{37}{160} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac{3}{40} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}+\frac{158873 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{25600}\\ &=\frac{14443 \sqrt{1-2 x} \sqrt{3+5 x}}{12800}-\frac{1313 (1-2 x)^{3/2} \sqrt{3+5 x}}{1280}-\frac{37}{160} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac{3}{40} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}+\frac{158873 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{12800 \sqrt{5}}\\ &=\frac{14443 \sqrt{1-2 x} \sqrt{3+5 x}}{12800}-\frac{1313 (1-2 x)^{3/2} \sqrt{3+5 x}}{1280}-\frac{37}{160} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac{3}{40} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}+\frac{158873 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{12800 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0401899, size = 65, normalized size = 0.54 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (28800 x^3+51680 x^2+22500 x-13327\right )-158873 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{128000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-13327 + 22500*x + 51680*x^2 + 28800*x^3) - 158873*Sqrt[10]*ArcSin[Sqrt[5/11]
*Sqrt[1 - 2*x]])/128000

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Maple [A]  time = 0.008, size = 104, normalized size = 0.9 \begin{align*}{\frac{1}{256000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 576000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+1033600\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+158873\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +450000\,x\sqrt{-10\,{x}^{2}-x+3}-266540\,\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*(1-2*x)^(1/2)*(3+5*x)^(1/2),x)

[Out]

1/256000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(576000*x^3*(-10*x^2-x+3)^(1/2)+1033600*x^2*(-10*x^2-x+3)^(1/2)+158873*10
^(1/2)*arcsin(20/11*x+1/11)+450000*x*(-10*x^2-x+3)^(1/2)-266540*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 2.92998, size = 95, normalized size = 0.79 \begin{align*} -\frac{9}{40} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{61}{160} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{1313}{640} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{158873}{256000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{1313}{12800} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/40*(-10*x^2 - x + 3)^(3/2)*x - 61/160*(-10*x^2 - x + 3)^(3/2) + 1313/640*sqrt(-10*x^2 - x + 3)*x - 158873/2
56000*sqrt(10)*arcsin(-20/11*x - 1/11) + 1313/12800*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.76978, size = 248, normalized size = 2.05 \begin{align*} \frac{1}{12800} \,{\left (28800 \, x^{3} + 51680 \, x^{2} + 22500 \, x - 13327\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{158873}{256000} \, \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*(1-2*x)^(1/2)*(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

1/12800*(28800*x^3 + 51680*x^2 + 22500*x - 13327)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 158873/256000*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 [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 2.70399, size = 220, normalized size = 1.82 \begin{align*} \frac{3}{640000} \, \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{1}{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{1}{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*(1-2*x)^(1/2)*(3+5*x)^(1/2),x, algorithm="giac")

[Out]

3/640000*sqrt(5)*(2*(4*(8*(60*x - 71)*(5*x + 3) + 2179)*(5*x + 3) - 4125)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 4537
5*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1/2000*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))) + 1/100*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)))