3.2468 \(\int \frac{(2+3 x)^3 (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=128 \[ -\frac{3}{50} \sqrt{1-2 x} (3 x+2)^2 (5 x+3)^{5/2}-\frac{3 \sqrt{1-2 x} (3900 x+7889) (5 x+3)^{5/2}}{16000}-\frac{917953 \sqrt{1-2 x} (5 x+3)^{3/2}}{128000}-\frac{30292449 \sqrt{1-2 x} \sqrt{5 x+3}}{512000}+\frac{333216939 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{512000 \sqrt{10}} \]

[Out]

(-30292449*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/512000 - (917953*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/128000 - (3*Sqrt[1 - 2
*x]*(2 + 3*x)^2*(3 + 5*x)^(5/2))/50 - (3*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)*(7889 + 3900*x))/16000 + (333216939*Arc
Sin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(512000*Sqrt[10])

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Rubi [A]  time = 0.0333134, 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} \[ -\frac{3}{50} \sqrt{1-2 x} (3 x+2)^2 (5 x+3)^{5/2}-\frac{3 \sqrt{1-2 x} (3900 x+7889) (5 x+3)^{5/2}}{16000}-\frac{917953 \sqrt{1-2 x} (5 x+3)^{3/2}}{128000}-\frac{30292449 \sqrt{1-2 x} \sqrt{5 x+3}}{512000}+\frac{333216939 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{512000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-30292449*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/512000 - (917953*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/128000 - (3*Sqrt[1 - 2
*x]*(2 + 3*x)^2*(3 + 5*x)^(5/2))/50 - (3*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)*(7889 + 3900*x))/16000 + (333216939*Arc
Sin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(512000*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{(2+3 x)^3 (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx &=-\frac{3}{50} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac{1}{50} \int \frac{\left (-311-\frac{975 x}{2}\right ) (2+3 x) (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{3}{50} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac{3 \sqrt{1-2 x} (3+5 x)^{5/2} (7889+3900 x)}{16000}+\frac{917953 \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx}{32000}\\ &=-\frac{917953 \sqrt{1-2 x} (3+5 x)^{3/2}}{128000}-\frac{3}{50} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac{3 \sqrt{1-2 x} (3+5 x)^{5/2} (7889+3900 x)}{16000}+\frac{30292449 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx}{256000}\\ &=-\frac{30292449 \sqrt{1-2 x} \sqrt{3+5 x}}{512000}-\frac{917953 \sqrt{1-2 x} (3+5 x)^{3/2}}{128000}-\frac{3}{50} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac{3 \sqrt{1-2 x} (3+5 x)^{5/2} (7889+3900 x)}{16000}+\frac{333216939 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{1024000}\\ &=-\frac{30292449 \sqrt{1-2 x} \sqrt{3+5 x}}{512000}-\frac{917953 \sqrt{1-2 x} (3+5 x)^{3/2}}{128000}-\frac{3}{50} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac{3 \sqrt{1-2 x} (3+5 x)^{5/2} (7889+3900 x)}{16000}+\frac{333216939 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{512000 \sqrt{5}}\\ &=-\frac{30292449 \sqrt{1-2 x} \sqrt{3+5 x}}{512000}-\frac{917953 \sqrt{1-2 x} (3+5 x)^{3/2}}{128000}-\frac{3}{50} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac{3 \sqrt{1-2 x} (3+5 x)^{5/2} (7889+3900 x)}{16000}+\frac{333216939 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{512000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.123878, size = 70, normalized size = 0.55 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (6912000 x^4+26870400 x^3+46785120 x^2+51453140 x+49229901\right )-333216939 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{5120000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(49229901 + 51453140*x + 46785120*x^2 + 26870400*x^3 + 6912000*x^4) - 3332169
39*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/5120000

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Maple [A]  time = 0.009, size = 121, normalized size = 1. \begin{align*}{\frac{1}{10240000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -138240000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-537408000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-935702400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -1029062800\,x\sqrt{-10\,{x}^{2}-x+3}-984598020\,\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*(3+5*x)^(3/2)/(1-2*x)^(1/2),x)

[Out]

1/10240000*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(-138240000*x^4*(-10*x^2-x+3)^(1/2)-537408000*x^3*(-10*x^2-x+3)^(1/2)-9
35702400*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)-1029062800*x*(-10*x^2-x+3)^(1/2)-9845
98020*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.75476, size = 124, normalized size = 0.97 \begin{align*} -\frac{27}{2} \, \sqrt{-10 \, x^{2} - x + 3} x^{4} - \frac{8397}{160} \, \sqrt{-10 \, x^{2} - x + 3} x^{3} - \frac{292407}{3200} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} - \frac{2572657}{25600} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{333216939}{10240000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{49229901}{512000} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/2*sqrt(-10*x^2 - x + 3)*x^4 - 8397/160*sqrt(-10*x^2 - x + 3)*x^3 - 292407/3200*sqrt(-10*x^2 - x + 3)*x^2 -
2572657/25600*sqrt(-10*x^2 - x + 3)*x - 333216939/10240000*sqrt(10)*arcsin(-20/11*x - 1/11) - 49229901/512000
*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.87384, size = 293, normalized size = 2.29 \begin{align*} -\frac{1}{512000} \,{\left (6912000 \, x^{4} + 26870400 \, x^{3} + 46785120 \, x^{2} + 51453140 \, x + 49229901\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{333216939}{10240000} \, \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*(3+5*x)^(3/2)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/512000*(6912000*x^4 + 26870400*x^3 + 46785120*x^2 + 51453140*x + 49229901)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 3
33216939/10240000*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*(3+5*x)**(3/2)/(1-2*x)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.7743, size = 97, normalized size = 0.76 \begin{align*} -\frac{1}{25600000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (24 \,{\left (36 \,{\left (80 \, x + 167\right )}{\left (5 \, x + 3\right )} + 27809\right )}{\left (5 \, x + 3\right )} + 4589765\right )}{\left (5 \, x + 3\right )} + 151462245\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 1666084695 \, \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*(3+5*x)^(3/2)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

-1/25600000*sqrt(5)*(2*(4*(24*(36*(80*x + 167)*(5*x + 3) + 27809)*(5*x + 3) + 4589765)*(5*x + 3) + 151462245)*
sqrt(5*x + 3)*sqrt(-10*x + 5) - 1666084695*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))