3.2508 \(\int \frac{(2+3 x)^4}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=113 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}-\frac{602 \sqrt{1-2 x} (3 x+2)^2}{9075 \sqrt{5 x+3}}-\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (1020 x+12199)}{242000}+\frac{8127 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2000 \sqrt{10}} \]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(165*(3 + 5*x)^(3/2)) - (602*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(9075*Sqrt[3 + 5*x]) -
(7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(12199 + 1020*x))/242000 + (8127*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2000*Sqrt[1
0])

________________________________________________________________________________________

Rubi [A]  time = 0.0324747, antiderivative size = 113, 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 = {98, 150, 147, 54, 216} \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^3}{165 (5 x+3)^{3/2}}-\frac{602 \sqrt{1-2 x} (3 x+2)^2}{9075 \sqrt{5 x+3}}-\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (1020 x+12199)}{242000}+\frac{8127 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^4/(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)),x]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(165*(3 + 5*x)^(3/2)) - (602*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(9075*Sqrt[3 + 5*x]) -
(7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(12199 + 1020*x))/242000 + (8127*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2000*Sqrt[1
0])

Rule 98

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

Rule 150

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]

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 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)^4}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx &=-\frac{2 \sqrt{1-2 x} (2+3 x)^3}{165 (3+5 x)^{3/2}}-\frac{2}{165} \int \frac{\left (-112-\frac{273 x}{2}\right ) (2+3 x)^2}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^3}{165 (3+5 x)^{3/2}}-\frac{602 \sqrt{1-2 x} (2+3 x)^2}{9075 \sqrt{3+5 x}}-\frac{4 \int \frac{\left (-\frac{4809}{2}-\frac{1785 x}{4}\right ) (2+3 x)}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{9075}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^3}{165 (3+5 x)^{3/2}}-\frac{602 \sqrt{1-2 x} (2+3 x)^2}{9075 \sqrt{3+5 x}}-\frac{7 \sqrt{1-2 x} \sqrt{3+5 x} (12199+1020 x)}{242000}+\frac{8127 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{4000}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^3}{165 (3+5 x)^{3/2}}-\frac{602 \sqrt{1-2 x} (2+3 x)^2}{9075 \sqrt{3+5 x}}-\frac{7 \sqrt{1-2 x} \sqrt{3+5 x} (12199+1020 x)}{242000}+\frac{8127 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{2000 \sqrt{5}}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^3}{165 (3+5 x)^{3/2}}-\frac{602 \sqrt{1-2 x} (2+3 x)^2}{9075 \sqrt{3+5 x}}-\frac{7 \sqrt{1-2 x} \sqrt{3+5 x} (12199+1020 x)}{242000}+\frac{8127 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{2000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0899712, size = 65, normalized size = 0.58 \[ -\frac{\sqrt{1-2 x} \left (2940300 x^3+11712195 x^2+10891910 x+2953931\right )}{726000 (5 x+3)^{3/2}}-\frac{8127 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{2000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^4/(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)),x]

[Out]

-(Sqrt[1 - 2*x]*(2953931 + 10891910*x + 11712195*x^2 + 2940300*x^3))/(726000*(3 + 5*x)^(3/2)) - (8127*ArcSin[S
qrt[5/11]*Sqrt[1 - 2*x]])/(2000*Sqrt[10])

________________________________________________________________________________________

Maple [A]  time = 0.012, size = 130, normalized size = 1.2 \begin{align*}{\frac{1}{14520000} \left ( 73752525\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-58806000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-234243900\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -217838200\,x\sqrt{-10\,{x}^{2}-x+3}-59078620\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^4/(3+5*x)^(5/2)/(1-2*x)^(1/2),x)

[Out]

1/14520000*(73752525*10^(1/2)*arcsin(20/11*x+1/11)*x^2-58806000*x^3*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcs
in(20/11*x+1/11)*x-234243900*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)-217838200*x*(-10*x
^2-x+3)^(1/2)-59078620*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

________________________________________________________________________________________

Maxima [A]  time = 2.6137, size = 123, normalized size = 1.09 \begin{align*} \frac{8127}{40000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{81}{500} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{4509}{10000} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{20625 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{32 \, \sqrt{-10 \, x^{2} - x + 3}}{9075 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

8127/40000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 81/500*sqrt(-10*x^2 - x + 3)*x - 4509/10000*sqrt(-10*x^2 -
x + 3) - 2/20625*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) - 32/9075*sqrt(-10*x^2 - x + 3)/(5*x + 3)

________________________________________________________________________________________

Fricas [A]  time = 1.55965, size = 320, normalized size = 2.83 \begin{align*} -\frac{2950101 \, \sqrt{10}{\left (25 \, x^{2} + 30 \, x + 9\right )} \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 ) + 20 \,{\left (2940300 \, x^{3} + 11712195 \, x^{2} + 10891910 \, x + 2953931\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{14520000 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/14520000*(2950101*sqrt(10)*(25*x^2 + 30*x + 9)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)
/(10*x^2 + x - 3)) + 20*(2940300*x^3 + 11712195*x^2 + 10891910*x + 2953931)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(25*
x^2 + 30*x + 9)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (3 x + 2\right )^{4}}{\sqrt{1 - 2 x} \left (5 x + 3\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**4/(3+5*x)**(5/2)/(1-2*x)**(1/2),x)

[Out]

Integral((3*x + 2)**4/(sqrt(1 - 2*x)*(5*x + 3)**(5/2)), x)

________________________________________________________________________________________

Giac [B]  time = 1.8559, size = 238, normalized size = 2.11 \begin{align*} -\frac{27}{50000} \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} + 131 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{18150000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{8127}{20000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{267 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{1512500 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{801 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{1134375 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

-27/50000*(12*sqrt(5)*(5*x + 3) + 131*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) - 1/18150000*sqrt(10)*(sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 8127/20000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 267/15
12500*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 1/1134375*(801*sqrt(10)*(sqrt(2)*sqrt(-10*
x + 5) - sqrt(22))^2/(5*x + 3) + 4*sqrt(10))*(5*x + 3)^(3/2)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3