3.2361 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^3}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=128 \[ -\frac{3 \sqrt{5 x+3} (11580 x+14629) (1-2 x)^{5/2}}{80000}-\frac{3}{50} (3 x+2)^2 \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{51373 \sqrt{5 x+3} (1-2 x)^{3/2}}{320000}+\frac{1695309 \sqrt{5 x+3} \sqrt{1-2 x}}{3200000}+\frac{18648399 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{3200000 \sqrt{10}} \]

[Out]

(1695309*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/3200000 + (51373*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/320000 - (3*(1 - 2*x)^(5
/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/50 - (3*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x]*(14629 + 11580*x))/80000 + (18648399*ArcS
in[Sqrt[2/11]*Sqrt[3 + 5*x]])/(3200000*Sqrt[10])

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Rubi [A]  time = 0.0374545, 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} \[ -\frac{3 \sqrt{5 x+3} (11580 x+14629) (1-2 x)^{5/2}}{80000}-\frac{3}{50} (3 x+2)^2 \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{51373 \sqrt{5 x+3} (1-2 x)^{3/2}}{320000}+\frac{1695309 \sqrt{5 x+3} \sqrt{1-2 x}}{3200000}+\frac{18648399 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{3200000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1695309*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/3200000 + (51373*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/320000 - (3*(1 - 2*x)^(5
/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/50 - (3*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x]*(14629 + 11580*x))/80000 + (18648399*ArcS
in[Sqrt[2/11]*Sqrt[3 + 5*x]])/(3200000*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 \frac{(1-2 x)^{3/2} (2+3 x)^3}{\sqrt{3+5 x}} \, dx &=-\frac{3}{50} (1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x}-\frac{1}{50} \int \frac{\left (-179-\frac{579 x}{2}\right ) (1-2 x)^{3/2} (2+3 x)}{\sqrt{3+5 x}} \, dx\\ &=-\frac{3}{50} (1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x}-\frac{3 (1-2 x)^{5/2} \sqrt{3+5 x} (14629+11580 x)}{80000}+\frac{51373 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{32000}\\ &=\frac{51373 (1-2 x)^{3/2} \sqrt{3+5 x}}{320000}-\frac{3}{50} (1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x}-\frac{3 (1-2 x)^{5/2} \sqrt{3+5 x} (14629+11580 x)}{80000}+\frac{1695309 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{640000}\\ &=\frac{1695309 \sqrt{1-2 x} \sqrt{3+5 x}}{3200000}+\frac{51373 (1-2 x)^{3/2} \sqrt{3+5 x}}{320000}-\frac{3}{50} (1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x}-\frac{3 (1-2 x)^{5/2} \sqrt{3+5 x} (14629+11580 x)}{80000}+\frac{18648399 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{6400000}\\ &=\frac{1695309 \sqrt{1-2 x} \sqrt{3+5 x}}{3200000}+\frac{51373 (1-2 x)^{3/2} \sqrt{3+5 x}}{320000}-\frac{3}{50} (1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x}-\frac{3 (1-2 x)^{5/2} \sqrt{3+5 x} (14629+11580 x)}{80000}+\frac{18648399 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{3200000 \sqrt{5}}\\ &=\frac{1695309 \sqrt{1-2 x} \sqrt{3+5 x}}{3200000}+\frac{51373 (1-2 x)^{3/2} \sqrt{3+5 x}}{320000}-\frac{3}{50} (1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x}-\frac{3 (1-2 x)^{5/2} \sqrt{3+5 x} (14629+11580 x)}{80000}+\frac{18648399 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{3200000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.125669, size = 79, normalized size = 0.62 \[ \frac{10 \sqrt{5 x+3} \left (13824000 x^5+8812800 x^4-13767360 x^3-7793240 x^2+6001742 x-314441\right )-18648399 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{32000000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[3 + 5*x]*(-314441 + 6001742*x - 7793240*x^2 - 13767360*x^3 + 8812800*x^4 + 13824000*x^5) - 18648399*S
qrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(32000000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.009, size = 121, normalized size = 1. \begin{align*}{\frac{1}{64000000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -138240000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-157248000\,{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}-6288820\,\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)^(3/2)*(2+3*x)^3/(3+5*x)^(1/2),x)

[Out]

1/64000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-138240000*x^4*(-10*x^2-x+3)^(1/2)-157248000*x^3*(-10*x^2-x+3)^(1/2)+5
9049600*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)-6288820
*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 2.48825, size = 124, normalized size = 0.97 \begin{align*} -\frac{54}{25} \, \sqrt{-10 \, x^{2} - x + 3} x^{4} - \frac{2457}{1000} \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + \frac{18453}{20000} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + \frac{268643}{160000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{18648399}{64000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{314441}{3200000} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-54/25*sqrt(-10*x^2 - x + 3)*x^4 - 2457/1000*sqrt(-10*x^2 - x + 3)*x^3 + 18453/20000*sqrt(-10*x^2 - x + 3)*x^2
+ 268643/160000*sqrt(-10*x^2 - x + 3)*x - 18648399/64000000*sqrt(10)*arcsin(-20/11*x - 1/11) - 314441/3200000
*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.53287, size = 286, normalized size = 2.23 \begin{align*} -\frac{1}{3200000} \,{\left (6912000 \, x^{4} + 7862400 \, x^{3} - 2952480 \, x^{2} - 5372860 \, x + 314441\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{18648399}{64000000} \, \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)^(3/2)*(2+3*x)^3/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

-1/3200000*(6912000*x^4 + 7862400*x^3 - 2952480*x^2 - 5372860*x + 314441)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 18648
399/64000000*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)**(3/2)*(2+3*x)**3/(3+5*x)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 1.76313, size = 371, normalized size = 2.9 \begin{align*} -\frac{9}{160000000} \, \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{27}{3200000} \, \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{3}{20000} \, \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}{100} \, \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{4}{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)^(3/2)*(2+3*x)^3/(3+5*x)^(1/2),x, algorithm="giac")

[Out]

-9/160000000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) - 506185)*(5*x + 3) + 4031895)*sqr
t(5*x + 3)*sqrt(-10*x + 5) + 10392195*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 27/3200000*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(1/1
1*sqrt(22)*sqrt(5*x + 3))) - 3/20000*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))) + 1/100*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))) + 4/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*
x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))