3.2600 \(\int \frac{(2+3 x)^2 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=140 \[ -\frac{1183 (5 x+3)^{7/2}}{363 \sqrt{1-2 x}}+\frac{49 (5 x+3)^{7/2}}{66 (1-2 x)^{3/2}}-\frac{24749 \sqrt{1-2 x} (5 x+3)^{5/2}}{2904}-\frac{123745 \sqrt{1-2 x} (5 x+3)^{3/2}}{2112}-\frac{123745}{256} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{272239}{256} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

(-123745*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/256 - (123745*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/2112 - (24749*Sqrt[1 - 2*x]
*(3 + 5*x)^(5/2))/2904 + (49*(3 + 5*x)^(7/2))/(66*(1 - 2*x)^(3/2)) - (1183*(3 + 5*x)^(7/2))/(363*Sqrt[1 - 2*x]
) + (272239*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/256

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Rubi [A]  time = 0.0403028, antiderivative size = 140, 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 = {89, 78, 50, 54, 216} \[ -\frac{1183 (5 x+3)^{7/2}}{363 \sqrt{1-2 x}}+\frac{49 (5 x+3)^{7/2}}{66 (1-2 x)^{3/2}}-\frac{24749 \sqrt{1-2 x} (5 x+3)^{5/2}}{2904}-\frac{123745 \sqrt{1-2 x} (5 x+3)^{3/2}}{2112}-\frac{123745}{256} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{272239}{256} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-123745*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/256 - (123745*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/2112 - (24749*Sqrt[1 - 2*x]
*(3 + 5*x)^(5/2))/2904 + (49*(3 + 5*x)^(7/2))/(66*(1 - 2*x)^(3/2)) - (1183*(3 + 5*x)^(7/2))/(363*Sqrt[1 - 2*x]
) + (272239*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/256

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

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)^2 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx &=\frac{49 (3+5 x)^{7/2}}{66 (1-2 x)^{3/2}}-\frac{1}{66} \int \frac{(3+5 x)^{5/2} \left (\frac{2069}{2}+297 x\right )}{(1-2 x)^{3/2}} \, dx\\ &=\frac{49 (3+5 x)^{7/2}}{66 (1-2 x)^{3/2}}-\frac{1183 (3+5 x)^{7/2}}{363 \sqrt{1-2 x}}+\frac{24749}{484} \int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{24749 \sqrt{1-2 x} (3+5 x)^{5/2}}{2904}+\frac{49 (3+5 x)^{7/2}}{66 (1-2 x)^{3/2}}-\frac{1183 (3+5 x)^{7/2}}{363 \sqrt{1-2 x}}+\frac{123745}{528} \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{123745 \sqrt{1-2 x} (3+5 x)^{3/2}}{2112}-\frac{24749 \sqrt{1-2 x} (3+5 x)^{5/2}}{2904}+\frac{49 (3+5 x)^{7/2}}{66 (1-2 x)^{3/2}}-\frac{1183 (3+5 x)^{7/2}}{363 \sqrt{1-2 x}}+\frac{123745}{128} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{123745}{256} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{123745 \sqrt{1-2 x} (3+5 x)^{3/2}}{2112}-\frac{24749 \sqrt{1-2 x} (3+5 x)^{5/2}}{2904}+\frac{49 (3+5 x)^{7/2}}{66 (1-2 x)^{3/2}}-\frac{1183 (3+5 x)^{7/2}}{363 \sqrt{1-2 x}}+\frac{1361195}{512} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{123745}{256} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{123745 \sqrt{1-2 x} (3+5 x)^{3/2}}{2112}-\frac{24749 \sqrt{1-2 x} (3+5 x)^{5/2}}{2904}+\frac{49 (3+5 x)^{7/2}}{66 (1-2 x)^{3/2}}-\frac{1183 (3+5 x)^{7/2}}{363 \sqrt{1-2 x}}+\frac{1}{256} \left (272239 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )\\ &=-\frac{123745}{256} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{123745 \sqrt{1-2 x} (3+5 x)^{3/2}}{2112}-\frac{24749 \sqrt{1-2 x} (3+5 x)^{5/2}}{2904}+\frac{49 (3+5 x)^{7/2}}{66 (1-2 x)^{3/2}}-\frac{1183 (3+5 x)^{7/2}}{363 \sqrt{1-2 x}}+\frac{272239}{256} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0703303, size = 79, normalized size = 0.56 \[ \frac{816717 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-2 \sqrt{5 x+3} \left (28800 x^4+146160 x^3+497868 x^2-1713440 x+617319\right )}{1536 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[3 + 5*x]*(617319 - 1713440*x + 497868*x^2 + 146160*x^3 + 28800*x^4) + 816717*Sqrt[10 - 20*x]*(-1 + 2*
x)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1536*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.012, size = 154, normalized size = 1.1 \begin{align*}{\frac{1}{3072\, \left ( 2\,x-1 \right ) ^{2}} \left ( -115200\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+3266868\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-584640\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-3266868\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-1991472\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+816717\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +6853760\,x\sqrt{-10\,{x}^{2}-x+3}-2469276\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\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)^2*(3+5*x)^(5/2)/(1-2*x)^(5/2),x)

[Out]

1/3072*(-115200*x^4*(-10*x^2-x+3)^(1/2)+3266868*10^(1/2)*arcsin(20/11*x+1/11)*x^2-584640*x^3*(-10*x^2-x+3)^(1/
2)-3266868*10^(1/2)*arcsin(20/11*x+1/11)*x-1991472*x^2*(-10*x^2-x+3)^(1/2)+816717*10^(1/2)*arcsin(20/11*x+1/11
)+6853760*x*(-10*x^2-x+3)^(1/2)-2469276*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2*x-1)^2/(-10*x^2-x+
3)^(1/2)

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Maxima [B]  time = 1.92991, size = 333, normalized size = 2.38 \begin{align*} \frac{272239}{1024} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{49 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{8 \,{\left (16 \, x^{4} - 32 \, x^{3} + 24 \, x^{2} - 8 \, x + 1\right )}} - \frac{21 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{8 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} - \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{8 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac{5445}{256} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2695 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{96 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac{1155 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{32 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{165 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{64 \,{\left (2 \, x - 1\right )}} + \frac{29645 \, \sqrt{-10 \, x^{2} - x + 3}}{192 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{104335 \, \sqrt{-10 \, x^{2} - x + 3}}{96 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

272239/1024*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 49/8*(-10*x^2 - x + 3)^(5/2)/(16*x^4 - 32*x^3 + 24*x^2 -
8*x + 1) - 21/8*(-10*x^2 - x + 3)^(5/2)/(8*x^3 - 12*x^2 + 6*x - 1) - 3/8*(-10*x^2 - x + 3)^(5/2)/(4*x^2 - 4*x
+ 1) - 5445/256*sqrt(-10*x^2 - x + 3) - 2695/96*(-10*x^2 - x + 3)^(3/2)/(8*x^3 - 12*x^2 + 6*x - 1) + 1155/32*(
-10*x^2 - x + 3)^(3/2)/(4*x^2 - 4*x + 1) + 165/64*(-10*x^2 - x + 3)^(3/2)/(2*x - 1) + 29645/192*sqrt(-10*x^2 -
x + 3)/(4*x^2 - 4*x + 1) + 104335/96*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 1.55376, size = 335, normalized size = 2.39 \begin{align*} -\frac{816717 \, \sqrt{5} \sqrt{2}{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) + 4 \,{\left (28800 \, x^{4} + 146160 \, x^{3} + 497868 \, x^{2} - 1713440 \, x + 617319\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{3072 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3072*(816717*sqrt(5)*sqrt(2)*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2
*x + 1)/(10*x^2 + x - 3)) + 4*(28800*x^4 + 146160*x^3 + 497868*x^2 - 1713440*x + 617319)*sqrt(5*x + 3)*sqrt(-2
*x + 1))/(4*x^2 - 4*x + 1)

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

[Out]

Timed out

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Giac [A]  time = 2.05595, size = 131, normalized size = 0.94 \begin{align*} \frac{272239}{512} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (3 \,{\left (12 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + 107 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 24749 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 2722390 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 44919435 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{96000 \,{\left (2 \, x - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

272239/512*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/96000*(4*(3*(12*(8*sqrt(5)*(5*x + 3) + 107*sqrt(5)
)*(5*x + 3) + 24749*sqrt(5))*(5*x + 3) - 2722390*sqrt(5))*(5*x + 3) + 44919435*sqrt(5))*sqrt(5*x + 3)*sqrt(-10
*x + 5)/(2*x - 1)^2