3.2459 \(\int \frac{(2+3 x)^3 \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=106 \[ -\frac{3}{40} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^2-\frac{3 \sqrt{1-2 x} (5 x+3)^{3/2} (408 x+865)}{1280}-\frac{61547 \sqrt{1-2 x} \sqrt{5 x+3}}{5120}+\frac{677017 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5120 \sqrt{10}} \]

[Out]

(-61547*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/5120 - (3*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(3/2))/40 - (3*Sqrt[1 - 2*x
]*(3 + 5*x)^(3/2)*(865 + 408*x))/1280 + (677017*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5120*Sqrt[10])

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Rubi [A]  time = 0.0267679, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ -\frac{3}{40} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^2-\frac{3 \sqrt{1-2 x} (5 x+3)^{3/2} (408 x+865)}{1280}-\frac{61547 \sqrt{1-2 x} \sqrt{5 x+3}}{5120}+\frac{677017 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5120 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-61547*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/5120 - (3*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(3/2))/40 - (3*Sqrt[1 - 2*x
]*(3 + 5*x)^(3/2)*(865 + 408*x))/1280 + (677017*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5120*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 \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx &=-\frac{3}{40} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac{1}{40} \int \frac{\left (-241-\frac{765 x}{2}\right ) (2+3 x) \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{3}{40} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac{3 \sqrt{1-2 x} (3+5 x)^{3/2} (865+408 x)}{1280}+\frac{61547 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx}{2560}\\ &=-\frac{61547 \sqrt{1-2 x} \sqrt{3+5 x}}{5120}-\frac{3}{40} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac{3 \sqrt{1-2 x} (3+5 x)^{3/2} (865+408 x)}{1280}+\frac{677017 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{10240}\\ &=-\frac{61547 \sqrt{1-2 x} \sqrt{3+5 x}}{5120}-\frac{3}{40} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac{3 \sqrt{1-2 x} (3+5 x)^{3/2} (865+408 x)}{1280}+\frac{677017 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{5120 \sqrt{5}}\\ &=-\frac{61547 \sqrt{1-2 x} \sqrt{3+5 x}}{5120}-\frac{3}{40} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac{3 \sqrt{1-2 x} (3+5 x)^{3/2} (865+408 x)}{1280}+\frac{677017 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{5120 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.106186, size = 65, normalized size = 0.61 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (17280 x^3+57888 x^2+88092 x+97295\right )-677017 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{51200} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(97295 + 88092*x + 57888*x^2 + 17280*x^3) - 677017*Sqrt[10]*ArcSin[Sqrt[5/11]
*Sqrt[1 - 2*x]])/51200

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Maple [A]  time = 0.009, size = 104, normalized size = 1. \begin{align*}{\frac{1}{102400}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -345600\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-1157760\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+677017\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -1761840\,x\sqrt{-10\,{x}^{2}-x+3}-1945900\,\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)^(1/2)/(1-2*x)^(1/2),x)

[Out]

1/102400*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(-345600*x^3*(-10*x^2-x+3)^(1/2)-1157760*x^2*(-10*x^2-x+3)^(1/2)+677017*1
0^(1/2)*arcsin(20/11*x+1/11)-1761840*x*(-10*x^2-x+3)^(1/2)-1945900*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 3.93001, size = 99, normalized size = 0.93 \begin{align*} \frac{27}{80} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{677017}{102400} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{351}{320} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{4383}{256} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{114143}{5120} \, \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)^(1/2)/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

27/80*(-10*x^2 - x + 3)^(3/2)*x + 677017/102400*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 351/320*(-10*x^2 - x
+ 3)^(3/2) - 4383/256*sqrt(-10*x^2 - x + 3)*x - 114143/5120*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.82309, size = 248, normalized size = 2.34 \begin{align*} -\frac{1}{5120} \,{\left (17280 \, x^{3} + 57888 \, x^{2} + 88092 \, x + 97295\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{677017}{102400} \, \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)^(1/2)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/5120*(17280*x^3 + 57888*x^2 + 88092*x + 97295)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 677017/102400*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 [A]  time = 27.4437, size = 466, normalized size = 4.4 \begin{align*} \frac{2 \sqrt{5} \left (\begin{cases} \frac{11 \sqrt{2} \left (- \frac{\sqrt{2} \sqrt{5 - 10 x} \sqrt{5 x + 3}}{22} + \frac{\operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{2}\right )}{4} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{625} + \frac{18 \sqrt{5} \left (\begin{cases} \frac{121 \sqrt{2} \left (\frac{\sqrt{2} \sqrt{5 - 10 x} \left (- 20 x - 1\right ) \sqrt{5 x + 3}}{968} - \frac{\sqrt{2} \sqrt{5 - 10 x} \sqrt{5 x + 3}}{22} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{8}\right )}{8} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{625} + \frac{54 \sqrt{5} \left (\begin{cases} \frac{1331 \sqrt{2} \left (\frac{\sqrt{2} \left (5 - 10 x\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3993} + \frac{3 \sqrt{2} \sqrt{5 - 10 x} \left (- 20 x - 1\right ) \sqrt{5 x + 3}}{1936} - \frac{\sqrt{2} \sqrt{5 - 10 x} \sqrt{5 x + 3}}{22} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{16}\right )}{16} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{625} + \frac{54 \sqrt{5} \left (\begin{cases} \frac{14641 \sqrt{2} \left (\frac{2 \sqrt{2} \left (5 - 10 x\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3993} + \frac{7 \sqrt{2} \sqrt{5 - 10 x} \left (- 20 x - 1\right ) \sqrt{5 x + 3}}{3872} + \frac{\sqrt{2} \sqrt{5 - 10 x} \sqrt{5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{1874048} - \frac{\sqrt{2} \sqrt{5 - 10 x} \sqrt{5 x + 3}}{22} + \frac{35 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{128}\right )}{32} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{625} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(5)*Piecewise((11*sqrt(2)*(-sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)/22 + asin(sqrt(22)*sqrt(5*x + 3)/11)/2)
/4, (x >= -3/5) & (x < 1/2)))/625 + 18*sqrt(5)*Piecewise((121*sqrt(2)*(sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt
(5*x + 3)/968 - sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)/22 + 3*asin(sqrt(22)*sqrt(5*x + 3)/11)/8)/8, (x >= -3/5)
& (x < 1/2)))/625 + 54*sqrt(5)*Piecewise((1331*sqrt(2)*(sqrt(2)*(5 - 10*x)**(3/2)*(5*x + 3)**(3/2)/3993 + 3*sq
rt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/1936 - sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)/22 + 5*asin(sqrt(22
)*sqrt(5*x + 3)/11)/16)/16, (x >= -3/5) & (x < 1/2)))/625 + 54*sqrt(5)*Piecewise((14641*sqrt(2)*(2*sqrt(2)*(5
- 10*x)**(3/2)*(5*x + 3)**(3/2)/3993 + 7*sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/3872 + sqrt(2)*sqrt(
5 - 10*x)*sqrt(5*x + 3)*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x + 3)**2 - 5929)/1874048 - sqrt(2)*sqrt(5 - 10
*x)*sqrt(5*x + 3)/22 + 35*asin(sqrt(22)*sqrt(5*x + 3)/11)/128)/32, (x >= -3/5) & (x < 1/2)))/625

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Giac [A]  time = 2.14158, size = 85, normalized size = 0.8 \begin{align*} -\frac{1}{1280000} \, \sqrt{5}{\left (2 \,{\left (36 \,{\left (24 \,{\left (20 \, x + 43\right )}{\left (5 \, x + 3\right )} + 5179\right )}{\left (5 \, x + 3\right )} + 1538675\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 16925425 \, \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)^(1/2)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

-1/1280000*sqrt(5)*(2*(36*(24*(20*x + 43)*(5*x + 3) + 5179)*(5*x + 3) + 1538675)*sqrt(5*x + 3)*sqrt(-10*x + 5)
- 16925425*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))