3.2263 \(\int \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=128 \[ -\frac{3}{50} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^2-\frac{21 (1-2 x)^{3/2} (5 x+3)^{3/2} (444 x+731)}{16000}-\frac{323491 (1-2 x)^{3/2} \sqrt{5 x+3}}{128000}+\frac{3558401 \sqrt{1-2 x} \sqrt{5 x+3}}{1280000}+\frac{39142411 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1280000 \sqrt{10}} \]

[Out]

(3558401*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1280000 - (323491*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/128000 - (3*(1 - 2*x)^(
3/2)*(2 + 3*x)^2*(3 + 5*x)^(3/2))/50 - (21*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)*(731 + 444*x))/16000 + (39142411*Ar
cSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1280000*Sqrt[10])

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Rubi [A]  time = 0.036617, antiderivative size = 128, 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 = {100, 147, 50, 54, 216} \[ -\frac{3}{50} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^2-\frac{21 (1-2 x)^{3/2} (5 x+3)^{3/2} (444 x+731)}{16000}-\frac{323491 (1-2 x)^{3/2} \sqrt{5 x+3}}{128000}+\frac{3558401 \sqrt{1-2 x} \sqrt{5 x+3}}{1280000}+\frac{39142411 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1280000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3558401*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1280000 - (323491*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/128000 - (3*(1 - 2*x)^(
3/2)*(2 + 3*x)^2*(3 + 5*x)^(3/2))/50 - (21*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)*(731 + 444*x))/16000 + (39142411*Ar
cSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1280000*Sqrt[10])

Rule 100

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

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 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)^3 \sqrt{3+5 x} \, dx &=-\frac{3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{1}{50} \int \left (-245-\frac{777 x}{2}\right ) \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x} \, dx\\ &=-\frac{3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{21 (1-2 x)^{3/2} (3+5 x)^{3/2} (731+444 x)}{16000}+\frac{323491 \int \sqrt{1-2 x} \sqrt{3+5 x} \, dx}{32000}\\ &=-\frac{323491 (1-2 x)^{3/2} \sqrt{3+5 x}}{128000}-\frac{3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{21 (1-2 x)^{3/2} (3+5 x)^{3/2} (731+444 x)}{16000}+\frac{3558401 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{256000}\\ &=\frac{3558401 \sqrt{1-2 x} \sqrt{3+5 x}}{1280000}-\frac{323491 (1-2 x)^{3/2} \sqrt{3+5 x}}{128000}-\frac{3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{21 (1-2 x)^{3/2} (3+5 x)^{3/2} (731+444 x)}{16000}+\frac{39142411 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{2560000}\\ &=\frac{3558401 \sqrt{1-2 x} \sqrt{3+5 x}}{1280000}-\frac{323491 (1-2 x)^{3/2} \sqrt{3+5 x}}{128000}-\frac{3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{21 (1-2 x)^{3/2} (3+5 x)^{3/2} (731+444 x)}{16000}+\frac{39142411 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{1280000 \sqrt{5}}\\ &=\frac{3558401 \sqrt{1-2 x} \sqrt{3+5 x}}{1280000}-\frac{323491 (1-2 x)^{3/2} \sqrt{3+5 x}}{128000}-\frac{3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{21 (1-2 x)^{3/2} (3+5 x)^{3/2} (731+444 x)}{16000}+\frac{39142411 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{1280000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.152811, size = 79, normalized size = 0.62 \[ -\frac{10 \sqrt{5 x+3} \left (13824000 x^5+27820800 x^4+12527040 x^3-8941640 x^2-11567238 x+4282349\right )[phone] \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{12800000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(10*Sqrt[3 + 5*x]*(4282349 - 11567238*x - 8941640*x^2 + 12527040*x^3 + 27820800*x^4 + 13824000*x^5) + 3914241
1*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(12800000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.01, size = 121, normalized size = 1. \begin{align*}{\frac{1}{25600000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 138240000\,{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 ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-85646980\,\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)^3*(1-2*x)^(1/2)*(3+5*x)^(1/2),x)

[Out]

1/25600000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(138240000*x^4*(-10*x^2-x+3)^(1/2)[phone]*x^3*(-10*x^2-x+3)^(1/2)+29
8934400*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)-85646980
*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.93748, size = 117, normalized size = 0.91 \begin{align*} -\frac{27}{50} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} - \frac{5211}{4000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{19191}{16000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{323491}{64000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{39142411}{25600000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{323491}{1280000} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/50*(-10*x^2 - x + 3)^(3/2)*x^2 - 5211/4000*(-10*x^2 - x + 3)^(3/2)*x - 19191/16000*(-10*x^2 - x + 3)^(3/2)
+ 323491/64000*sqrt(-10*x^2 - x + 3)*x - 39142411/25600000*sqrt(10)*arcsin(-20/11*x - 1/11) + 323491/1280000*
sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.93203, size = 289, normalized size = 2.26 \begin{align*} \frac{1}{1280000} \,{\left (6912000 \, x^{4} + 17366400 \, x^{3} + 14946720 \, x^{2} + 3002540 \, x - 4282349\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{39142411}{25600000} \, \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)^3*(1-2*x)^(1/2)*(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

1/1280000*(6912000*x^4 + 17366400*x^3 + 14946720*x^2 + 3002540*x - 4282349)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 391
42411/25600000*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)**3*(1-2*x)**(1/2)*(3+5*x)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 2.3003, size = 317, normalized size = 2.48 \begin{align*} \frac{9}{64000000} \, \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{9}{320000} \, \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{3}{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}{50} \, \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)^3*(1-2*x)^(1/2)*(3+5*x)^(1/2),x, algorithm="giac")

[Out]

9/64000000*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))) + 9/320000*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)*s
qrt(5*x + 3))) + 3/2000*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))) + 1/50*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)))