3.2426 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^2}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=143 \[ -\frac{3}{50} (3 x+2) \sqrt{5 x+3} (1-2 x)^{7/2}-\frac{369 \sqrt{5 x+3} (1-2 x)^{7/2}}{4000}+\frac{4907 \sqrt{5 x+3} (1-2 x)^{5/2}}{120000}+\frac{53977 \sqrt{5 x+3} (1-2 x)^{3/2}}{480000}+\frac{593747 \sqrt{5 x+3} \sqrt{1-2 x}}{1600000}+\frac{6531217 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1600000 \sqrt{10}} \]

[Out]

(593747*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1600000 + (53977*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/480000 + (4907*(1 - 2*x)^
(5/2)*Sqrt[3 + 5*x])/120000 - (369*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/4000 - (3*(1 - 2*x)^(7/2)*(2 + 3*x)*Sqrt[3 +
5*x])/50 + (6531217*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1600000*Sqrt[10])

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Rubi [A]  time = 0.040883, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{3}{50} (3 x+2) \sqrt{5 x+3} (1-2 x)^{7/2}-\frac{369 \sqrt{5 x+3} (1-2 x)^{7/2}}{4000}+\frac{4907 \sqrt{5 x+3} (1-2 x)^{5/2}}{120000}+\frac{53977 \sqrt{5 x+3} (1-2 x)^{3/2}}{480000}+\frac{593747 \sqrt{5 x+3} \sqrt{1-2 x}}{1600000}+\frac{6531217 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1600000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(593747*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1600000 + (53977*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/480000 + (4907*(1 - 2*x)^
(5/2)*Sqrt[3 + 5*x])/120000 - (369*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/4000 - (3*(1 - 2*x)^(7/2)*(2 + 3*x)*Sqrt[3 +
5*x])/50 + (6531217*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1600000*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 \frac{(1-2 x)^{5/2} (2+3 x)^2}{\sqrt{3+5 x}} \, dx &=-\frac{3}{50} (1-2 x)^{7/2} (2+3 x) \sqrt{3+5 x}-\frac{1}{50} \int \frac{\left (-116-\frac{369 x}{2}\right ) (1-2 x)^{5/2}}{\sqrt{3+5 x}} \, dx\\ &=-\frac{369 (1-2 x)^{7/2} \sqrt{3+5 x}}{4000}-\frac{3}{50} (1-2 x)^{7/2} (2+3 x) \sqrt{3+5 x}+\frac{4907 \int \frac{(1-2 x)^{5/2}}{\sqrt{3+5 x}} \, dx}{8000}\\ &=\frac{4907 (1-2 x)^{5/2} \sqrt{3+5 x}}{120000}-\frac{369 (1-2 x)^{7/2} \sqrt{3+5 x}}{4000}-\frac{3}{50} (1-2 x)^{7/2} (2+3 x) \sqrt{3+5 x}+\frac{53977 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{48000}\\ &=\frac{53977 (1-2 x)^{3/2} \sqrt{3+5 x}}{480000}+\frac{4907 (1-2 x)^{5/2} \sqrt{3+5 x}}{120000}-\frac{369 (1-2 x)^{7/2} \sqrt{3+5 x}}{4000}-\frac{3}{50} (1-2 x)^{7/2} (2+3 x) \sqrt{3+5 x}+\frac{593747 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{320000}\\ &=\frac{593747 \sqrt{1-2 x} \sqrt{3+5 x}}{1600000}+\frac{53977 (1-2 x)^{3/2} \sqrt{3+5 x}}{480000}+\frac{4907 (1-2 x)^{5/2} \sqrt{3+5 x}}{120000}-\frac{369 (1-2 x)^{7/2} \sqrt{3+5 x}}{4000}-\frac{3}{50} (1-2 x)^{7/2} (2+3 x) \sqrt{3+5 x}+\frac{6531217 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{3200000}\\ &=\frac{593747 \sqrt{1-2 x} \sqrt{3+5 x}}{1600000}+\frac{53977 (1-2 x)^{3/2} \sqrt{3+5 x}}{480000}+\frac{4907 (1-2 x)^{5/2} \sqrt{3+5 x}}{120000}-\frac{369 (1-2 x)^{7/2} \sqrt{3+5 x}}{4000}-\frac{3}{50} (1-2 x)^{7/2} (2+3 x) \sqrt{3+5 x}+\frac{6531217 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{1600000 \sqrt{5}}\\ &=\frac{593747 \sqrt{1-2 x} \sqrt{3+5 x}}{1600000}+\frac{53977 (1-2 x)^{3/2} \sqrt{3+5 x}}{480000}+\frac{4907 (1-2 x)^{5/2} \sqrt{3+5 x}}{120000}-\frac{369 (1-2 x)^{7/2} \sqrt{3+5 x}}{4000}-\frac{3}{50} (1-2 x)^{7/2} (2+3 x) \sqrt{3+5 x}+\frac{6531217 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{1600000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0416867, size = 79, normalized size = 0.55 \[ \frac{10 \sqrt{5 x+3} \left (-13824000 x^5+11347200 x^4+10295360 x^3-13024760 x^2+387158 x+1498491\right )-19593651 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{48000000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[3 + 5*x]*(1498491 + 387158*x - 13024760*x^2 + 10295360*x^3 + 11347200*x^4 - 13824000*x^5) - 19593651*
Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(48000000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.01, size = 121, normalized size = 0.9 \begin{align*}{\frac{1}{96000000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 138240000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-44352000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-125129600\,{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}[phone]\,\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((1-2*x)^(5/2)*(2+3*x)^2/(3+5*x)^(1/2),x)

[Out]

1/96000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(138240000*x^4*(-10*x^2-x+3)^(1/2)-44352000*x^3*(-10*x^2-x+3)^(1/2)-125
129600*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)[phone]*
(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 2.71411, size = 124, normalized size = 0.87 \begin{align*} \frac{36}{25} \, \sqrt{-10 \, x^{2} - x + 3} x^{4} - \frac{231}{500} \, \sqrt{-10 \, x^{2} - x + 3} x^{3} - \frac{39103}{30000} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + \frac{169207}{240000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{6531217}{32000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{499497}{1600000} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

36/25*sqrt(-10*x^2 - x + 3)*x^4 - 231/500*sqrt(-10*x^2 - x + 3)*x^3 - 39103/30000*sqrt(-10*x^2 - x + 3)*x^2 +
169207/240000*sqrt(-10*x^2 - x + 3)*x - 6531217/32000000*sqrt(10)*arcsin(-20/11*x - 1/11) + 499497/1600000*sqr
t(-10*x^2 - x + 3)

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Fricas [A]  time = 1.68028, size = 285, normalized size = 1.99 \begin{align*} \frac{1}{4800000} \,{\left (6912000 \, x^{4} - 2217600 \, x^{3} - 6256480 \, x^{2} + 3384140 \, x + 1498491\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{6531217}{32000000} \, \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((1-2*x)^(5/2)*(2+3*x)^2/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

1/4800000*(6912000*x^4 - 2217600*x^3 - 6256480*x^2 + 3384140*x + 1498491)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 65312
17/32000000*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((1-2*x)**(5/2)*(2+3*x)**2/(3+5*x)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 2.16279, size = 371, normalized size = 2.59 \begin{align*} \frac{3}{80000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 203\right )}{\left (5 \, x + 3\right )} + 19073\right )}{\left (5 \, x + 3\right )} - 506185\right )}{\left (5 \, x + 3\right )} + 4031895\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 10392195 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{800000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 119\right )}{\left (5 \, x + 3\right )} + 6163\right )}{\left (5 \, x + 3\right )} - 66189\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 184305 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{23}{120000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 59\right )}{\left (5 \, x + 3\right )} + 1293\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 4785 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{1}{500} \, \sqrt{5}{\left (2 \,{\left (20 \, x - 23\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 143 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{2}{25} \, \sqrt{5}{\left (11 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + 2 \, \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/80000000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) - 506185)*(5*x + 3) + 4031895)*sqrt(
5*x + 3)*sqrt(-10*x + 5) + 10392195*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1/800000*sqrt(5)*(2*(4*(8*(
60*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 184305*sqrt(2)*arcsin(1/11*sq
rt(22)*sqrt(5*x + 3))) - 23/120000*sqrt(5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqrt(-10*x + 5) +
4785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 1/500*sqrt(5)*(2*(20*x - 23)*sqrt(5*x + 3)*sqrt(-10*x + 5
) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 2/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x
+ 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))