3.2518 \(\int \frac{(2+3 x)^5 \sqrt{3+5 x}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=168 \[ \frac{\sqrt{5 x+3} (3 x+2)^5}{\sqrt{1-2 x}}+\frac{33}{20} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^4+\frac{10389 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^3}{1600}+\frac{847637 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{32000}+\frac{49 \sqrt{1-2 x} \sqrt{5 x+3} (36265980 x+87394471)}{5120000}-\frac{35439958001 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5120000 \sqrt{10}} \]

[Out]

(847637*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/32000 + (10389*Sqrt[1 - 2*x]*(2 + 3*x)^3*Sqrt[3 + 5*x])/1600
+ (33*Sqrt[1 - 2*x]*(2 + 3*x)^4*Sqrt[3 + 5*x])/20 + ((2 + 3*x)^5*Sqrt[3 + 5*x])/Sqrt[1 - 2*x] + (49*Sqrt[1 - 2
*x]*Sqrt[3 + 5*x]*(87394471 + 36265980*x))/5120000 - (35439958001*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5120000*S
qrt[10])

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Rubi [A]  time = 0.0512781, antiderivative size = 168, 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 = {97, 153, 147, 54, 216} \[ \frac{\sqrt{5 x+3} (3 x+2)^5}{\sqrt{1-2 x}}+\frac{33}{20} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^4+\frac{10389 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^3}{1600}+\frac{847637 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{32000}+\frac{49 \sqrt{1-2 x} \sqrt{5 x+3} (36265980 x+87394471)}{5120000}-\frac{35439958001 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{5120000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(847637*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/32000 + (10389*Sqrt[1 - 2*x]*(2 + 3*x)^3*Sqrt[3 + 5*x])/1600
+ (33*Sqrt[1 - 2*x]*(2 + 3*x)^4*Sqrt[3 + 5*x])/20 + ((2 + 3*x)^5*Sqrt[3 + 5*x])/Sqrt[1 - 2*x] + (49*Sqrt[1 - 2
*x]*Sqrt[3 + 5*x]*(87394471 + 36265980*x))/5120000 - (35439958001*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5120000*S
qrt[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 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)^5 \sqrt{3+5 x}}{(1-2 x)^{3/2}} \, dx &=\frac{(2+3 x)^5 \sqrt{3+5 x}}{\sqrt{1-2 x}}-\int \frac{(2+3 x)^4 \left (50+\frac{165 x}{2}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{33}{20} \sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}+\frac{(2+3 x)^5 \sqrt{3+5 x}}{\sqrt{1-2 x}}+\frac{1}{50} \int \frac{\left (-\frac{15775}{2}-\frac{51945 x}{4}\right ) (2+3 x)^3}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{10389 \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}}{1600}+\frac{33}{20} \sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}+\frac{(2+3 x)^5 \sqrt{3+5 x}}{\sqrt{1-2 x}}-\frac{\int \frac{(2+3 x)^2 \left (\frac{1937285}{2}+\frac{12714555 x}{8}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{2000}\\ &=\frac{847637 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{32000}+\frac{10389 \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}}{1600}+\frac{33}{20} \sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}+\frac{(2+3 x)^5 \sqrt{3+5 x}}{\sqrt{1-2 x}}+\frac{\int \frac{\left (-\frac{681095835}{8}-\frac{2221291275 x}{16}\right ) (2+3 x)}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{60000}\\ &=\frac{847637 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{32000}+\frac{10389 \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}}{1600}+\frac{33}{20} \sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}+\frac{(2+3 x)^5 \sqrt{3+5 x}}{\sqrt{1-2 x}}+\frac{49 \sqrt{1-2 x} \sqrt{3+5 x} (87394471+36265980 x)}{5120000}-\frac{35439958001 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{10240000}\\ &=\frac{847637 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{32000}+\frac{10389 \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}}{1600}+\frac{33}{20} \sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}+\frac{(2+3 x)^5 \sqrt{3+5 x}}{\sqrt{1-2 x}}+\frac{49 \sqrt{1-2 x} \sqrt{3+5 x} (87394471+36265980 x)}{5120000}-\frac{35439958001 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{5120000 \sqrt{5}}\\ &=\frac{847637 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{32000}+\frac{10389 \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}}{1600}+\frac{33}{20} \sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}+\frac{(2+3 x)^5 \sqrt{3+5 x}}{\sqrt{1-2 x}}+\frac{49 \sqrt{1-2 x} \sqrt{3+5 x} (87394471+36265980 x)}{5120000}-\frac{35439958001 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{5120000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.114899, size = 79, normalized size = 0.47 \[ \frac{35439958001 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (124416000 x^5+613267200 x^4+1429191360 x^3+2297649240 x^2+3810769458 x-5389783159\right )}{51200000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(-5389783159 + 3810769458*x + 2297649240*x^2 + 1429191360*x^3 + 613267200*x^4 + 124416000*x
^5) + 35439958001*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(51200000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.014, size = 157, normalized size = 0.9 \begin{align*} -{\frac{1}{204800000\,x-102400000} \left ( -2488320000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-12265344000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-28583827200\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-45952984800\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-35439958001\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -76215389160\,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)^5*(3+5*x)^(1/2)/(1-2*x)^(3/2),x)

[Out]

-1/102400000*(-2488320000*x^5*(-10*x^2-x+3)^(1/2)-12265344000*x^4*(-10*x^2-x+3)^(1/2)-28583827200*x^3*(-10*x^2
-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x-45952984800*x^2*(-10*x^2-x+3)^(1/2)-35439958001*10^(1/
2)*arcsin(20/11*x+1/11)-76215389160*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 [A]  time = 3.58018, size = 150, normalized size = 0.89 \begin{align*} -\frac{243}{200} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} - \frac{103599}{16000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{35439958001}{102400000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{1086219}{64000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{80155719}{256000} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{2961355719}{5120000} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{16807 \, \sqrt{-10 \, x^{2} - x + 3}}{32 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/200*(-10*x^2 - x + 3)^(3/2)*x^2 - 103599/16000*(-10*x^2 - x + 3)^(3/2)*x - 35439958001/102400000*sqrt(5)*
sqrt(2)*arcsin(20/11*x + 1/11) - 1086219/64000*(-10*x^2 - x + 3)^(3/2) + 80155719/256000*sqrt(-10*x^2 - x + 3)
*x + 2961355719/5120000*sqrt(-10*x^2 - x + 3) - 16807/32*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 1.82087, size = 355, normalized size = 2.11 \begin{align*} \frac{35439958001 \, \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 (124416000 \, x^{5} + 613267200 \, x^{4} + 1429191360 \, x^{3} + 2297649240 \, x^{2} + 3810769458 \, x - 5389783159\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{102400000 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/102400000*(35439958001*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*(124416000*x^5 + 613267200*x^4 + 1429191360*x^3 + 2297649240*x^2 + 3810769458*x - 5389783159
)*sqrt(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)**5*(3+5*x)**(1/2)/(1-2*x)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.76431, size = 149, normalized size = 0.89 \begin{align*} -\frac{35439958001}{51200000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (6 \,{\left (12 \,{\left (8 \,{\left (36 \,{\left (48 \, \sqrt{5}{\left (5 \, x + 3\right )} + 463 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 140711 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 10847547 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 1789896455 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 177199790005 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{640000000 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-35439958001/51200000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/640000000*(6*(12*(8*(36*(48*sqrt(5)*(5*
x + 3) + 463*sqrt(5))*(5*x + 3) + 140711*sqrt(5))*(5*x + 3) + 10847547*sqrt(5))*(5*x + 3) + 1789896455*sqrt(5)
)*(5*x + 3) - 177199790005*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)