3.2529 \(\int \frac{(2+3 x)^4 (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=161 \[ \frac{(5 x+3)^{3/2} (3 x+2)^4}{\sqrt{1-2 x}}+\frac{33}{20} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^3+\frac{10377 \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^2}{1600}+\frac{9 \sqrt{1-2 x} (5 x+3)^{3/2} (2253560 x+4772357)}{256000}+\frac{1018114917 \sqrt{1-2 x} \sqrt{5 x+3}}{1024000}-\frac{11199264087 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1024000 \sqrt{10}} \]

[Out]

(1018114917*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1024000 + (10377*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(3/2))/1600 + (3
3*Sqrt[1 - 2*x]*(2 + 3*x)^3*(3 + 5*x)^(3/2))/20 + ((2 + 3*x)^4*(3 + 5*x)^(3/2))/Sqrt[1 - 2*x] + (9*Sqrt[1 - 2*
x]*(3 + 5*x)^(3/2)*(4772357 + 2253560*x))/256000 - (11199264087*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1024000*Sqr
t[10])

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Rubi [A]  time = 0.0444022, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {97, 153, 147, 50, 54, 216} \[ \frac{(5 x+3)^{3/2} (3 x+2)^4}{\sqrt{1-2 x}}+\frac{33}{20} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^3+\frac{10377 \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^2}{1600}+\frac{9 \sqrt{1-2 x} (5 x+3)^{3/2} (2253560 x+4772357)}{256000}+\frac{1018114917 \sqrt{1-2 x} \sqrt{5 x+3}}{1024000}-\frac{11199264087 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1024000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1018114917*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1024000 + (10377*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(3/2))/1600 + (3
3*Sqrt[1 - 2*x]*(2 + 3*x)^3*(3 + 5*x)^(3/2))/20 + ((2 + 3*x)^4*(3 + 5*x)^(3/2))/Sqrt[1 - 2*x] + (9*Sqrt[1 - 2*
x]*(3 + 5*x)^(3/2)*(4772357 + 2253560*x))/256000 - (11199264087*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1024000*Sqr
t[10])

Rule 97

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

Rule 153

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 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)^4 (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx &=\frac{(2+3 x)^4 (3+5 x)^{3/2}}{\sqrt{1-2 x}}-\int \frac{(2+3 x)^3 \sqrt{3+5 x} \left (51+\frac{165 x}{2}\right )}{\sqrt{1-2 x}} \, dx\\ &=\frac{33}{20} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{3/2}+\frac{(2+3 x)^4 (3+5 x)^{3/2}}{\sqrt{1-2 x}}+\frac{1}{50} \int \frac{\left (-8070-\frac{51885 x}{4}\right ) (2+3 x)^2 \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=\frac{10377 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}{1600}+\frac{33}{20} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{3/2}+\frac{(2+3 x)^4 (3+5 x)^{3/2}}{\sqrt{1-2 x}}-\frac{\int \frac{(2+3 x) \sqrt{3+5 x} \left (\frac{3983295}{4}+\frac{12676275 x}{8}\right )}{\sqrt{1-2 x}} \, dx}{2000}\\ &=\frac{10377 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}{1600}+\frac{33}{20} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{3/2}+\frac{(2+3 x)^4 (3+5 x)^{3/2}}{\sqrt{1-2 x}}+\frac{9 \sqrt{1-2 x} (3+5 x)^{3/2} (4772357+2253560 x)}{256000}-\frac{1018114917 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx}{512000}\\ &=\frac{1018114917 \sqrt{1-2 x} \sqrt{3+5 x}}{1024000}+\frac{10377 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}{1600}+\frac{33}{20} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{3/2}+\frac{(2+3 x)^4 (3+5 x)^{3/2}}{\sqrt{1-2 x}}+\frac{9 \sqrt{1-2 x} (3+5 x)^{3/2} (4772357+2253560 x)}{256000}-\frac{11199264087 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{2048000}\\ &=\frac{1018114917 \sqrt{1-2 x} \sqrt{3+5 x}}{1024000}+\frac{10377 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}{1600}+\frac{33}{20} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{3/2}+\frac{(2+3 x)^4 (3+5 x)^{3/2}}{\sqrt{1-2 x}}+\frac{9 \sqrt{1-2 x} (3+5 x)^{3/2} (4772357+2253560 x)}{256000}-\frac{11199264087 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{1024000 \sqrt{5}}\\ &=\frac{1018114917 \sqrt{1-2 x} \sqrt{3+5 x}}{1024000}+\frac{10377 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}{1600}+\frac{33}{20} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{3/2}+\frac{(2+3 x)^4 (3+5 x)^{3/2}}{\sqrt{1-2 x}}+\frac{9 \sqrt{1-2 x} (3+5 x)^{3/2} (4772357+2253560 x)}{256000}-\frac{11199264087 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{1024000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0485931, size = 79, normalized size = 0.49 \[ \frac{11199264087 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (41472000 x^5+200966400 x^4+461171520 x^3+732415080 x^2+1206337246 x-1702927233\right )}{10240000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(-1702927233 + 1206337246*x + 732415080*x^2 + 461171520*x^3 + 200966400*x^4 + 41472000*x^5)
+ 11199264087*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(10240000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.012, size = 157, normalized size = 1. \begin{align*} -{\frac{1}{40960000\,x-20480000} \left ( -829440000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-4019328000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-9223430400\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-14648301600\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-11199264087\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -24126744920\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\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)^4*(3+5*x)^(3/2)/(1-2*x)^(3/2),x)

[Out]

-1/20480000*(-829440000*x^5*(-10*x^2-x+3)^(1/2)-4019328000*x^4*(-10*x^2-x+3)^(1/2)-9223430400*x^3*(-10*x^2-x+3
)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x-14648301600*x^2*(-10*x^2-x+3)^(1/2)-11199264087*10^(1/2)*a
rcsin(20/11*x+1/11)-24126744920*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(
1/2)/(2*x-1)/(-10*x^2-x+3)^(1/2)

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Maxima [C]  time = 3.5268, size = 267, normalized size = 1.66 \begin{align*} \frac{81}{400} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} - \frac{6669}{640} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{12607994487}{20480000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{1760913}{25600} i \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x - \frac{21}{11}\right ) - \frac{359469}{12800} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{14553}{64} \, \sqrt{10 \, x^{2} - 21 \, x + 8} x - \frac{2420847}{51200} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{305613}{1280} \, \sqrt{10 \, x^{2} - 21 \, x + 8} + \frac{540891153}{1024000} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2401 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{32 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac{1029 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{16 \,{\left (2 \, x - 1\right )}} - \frac{79233 \, \sqrt{-10 \, x^{2} - x + 3}}{64 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

81/400*(-10*x^2 - x + 3)^(5/2) - 6669/640*(-10*x^2 - x + 3)^(3/2)*x - 12607994487/20480000*sqrt(5)*sqrt(2)*arc
sin(20/11*x + 1/11) - 1760913/25600*I*sqrt(5)*sqrt(2)*arcsin(20/11*x - 21/11) - 359469/12800*(-10*x^2 - x + 3)
^(3/2) + 14553/64*sqrt(10*x^2 - 21*x + 8)*x - 2420847/51200*sqrt(-10*x^2 - x + 3)*x - 305613/1280*sqrt(10*x^2
- 21*x + 8) + 540891153/1024000*sqrt(-10*x^2 - x + 3) - 2401/32*(-10*x^2 - x + 3)^(3/2)/(4*x^2 - 4*x + 1) - 10
29/16*(-10*x^2 - x + 3)^(3/2)/(2*x - 1) - 79233/64*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 1.858, size = 350, normalized size = 2.17 \begin{align*} \frac{11199264087 \, \sqrt{10}{\left (2 \, x - 1\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 (41472000 \, x^{5} + 200966400 \, x^{4} + 461171520 \, x^{3} + 732415080 \, x^{2} + 1206337246 \, x - 1702927233\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20480000 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20480000*(11199264087*sqrt(10)*(2*x - 1)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^
2 + x - 3)) + 20*(41472000*x^5 + 200966400*x^4 + 461171520*x^3 + 732415080*x^2 + 1206337246*x - 1702927233)*sq
rt(5*x + 3)*sqrt(-2*x + 1))/(2*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)**4*(3+5*x)**(3/2)/(1-2*x)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.23926, size = 149, normalized size = 0.93 \begin{align*} -\frac{11199264087}{10240000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (2 \,{\left (12 \,{\left (24 \,{\left (12 \,{\left (48 \, \sqrt{5}{\left (5 \, x + 3\right )} + 443 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 44497 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 10283927 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 1696858195 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 55996320435 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{128000000 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-11199264087/10240000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/128000000*(2*(12*(24*(12*(48*sqrt(5)*(5
*x + 3) + 443*sqrt(5))*(5*x + 3) + 44497*sqrt(5))*(5*x + 3) + 10283927*sqrt(5))*(5*x + 3) + 1696858195*sqrt(5)
)*(5*x + 3) - 55996320435*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)