3.2507 \(\int \frac{(2+3 x)^5}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=142 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^4}{165 (5 x+3)^{3/2}}-\frac{734 \sqrt{1-2 x} (3 x+2)^3}{9075 \sqrt{5 x+3}}+\frac{511 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{30250}-\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (366420 x+938509)}{4840000}+\frac{462357 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{40000 \sqrt{10}} \]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^4)/(165*(3 + 5*x)^(3/2)) - (734*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(9075*Sqrt[3 + 5*x]) +
(511*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/30250 - (7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(938509 + 366420*x))/4840
000 + (462357*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(40000*Sqrt[10])

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Rubi [A]  time = 0.0432441, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {98, 150, 153, 147, 54, 216} \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^4}{165 (5 x+3)^{3/2}}-\frac{734 \sqrt{1-2 x} (3 x+2)^3}{9075 \sqrt{5 x+3}}+\frac{511 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{30250}-\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (366420 x+938509)}{4840000}+\frac{462357 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{40000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^4)/(165*(3 + 5*x)^(3/2)) - (734*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(9075*Sqrt[3 + 5*x]) +
(511*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/30250 - (7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(938509 + 366420*x))/4840
000 + (462357*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(40000*Sqrt[10])

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 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{1-2 x} (3+5 x)^{5/2}} \, dx &=-\frac{2 \sqrt{1-2 x} (2+3 x)^4}{165 (3+5 x)^{3/2}}-\frac{2}{165} \int \frac{\left (-115-\frac{261 x}{2}\right ) (2+3 x)^3}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^4}{165 (3+5 x)^{3/2}}-\frac{734 \sqrt{1-2 x} (2+3 x)^3}{9075 \sqrt{3+5 x}}-\frac{4 \int \frac{(2+3 x)^2 \left (-3087+\frac{4599 x}{4}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{9075}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^4}{165 (3+5 x)^{3/2}}-\frac{734 \sqrt{1-2 x} (2+3 x)^3}{9075 \sqrt{3+5 x}}+\frac{511 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{30250}+\frac{2 \int \frac{(2+3 x) \left (\frac{662697}{4}+\frac{1923705 x}{8}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{136125}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^4}{165 (3+5 x)^{3/2}}-\frac{734 \sqrt{1-2 x} (2+3 x)^3}{9075 \sqrt{3+5 x}}+\frac{511 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{30250}-\frac{7 \sqrt{1-2 x} \sqrt{3+5 x} (938509+366420 x)}{4840000}+\frac{462357 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{80000}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^4}{165 (3+5 x)^{3/2}}-\frac{734 \sqrt{1-2 x} (2+3 x)^3}{9075 \sqrt{3+5 x}}+\frac{511 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{30250}-\frac{7 \sqrt{1-2 x} \sqrt{3+5 x} (938509+366420 x)}{4840000}+\frac{462357 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{40000 \sqrt{5}}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^4}{165 (3+5 x)^{3/2}}-\frac{734 \sqrt{1-2 x} (2+3 x)^3}{9075 \sqrt{3+5 x}}+\frac{511 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{30250}-\frac{7 \sqrt{1-2 x} \sqrt{3+5 x} (938509+366420 x)}{4840000}+\frac{462357 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{40000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.071923, size = 70, normalized size = 0.49 \[ -\frac{\sqrt{1-2 x} \left (117612000 x^4+502791300 x^3+1030526145 x^2+795297410 x+199549721\right )}{14520000 (5 x+3)^{3/2}}-\frac{462357 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{40000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*(199549721 + 795297410*x + 1030526145*x^2 + 502791300*x^3 + 117612000*x^4))/(14520000*(3 + 5*x
)^(3/2)) - (462357*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(40000*Sqrt[10])

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Maple [A]  time = 0.014, size = 147, normalized size = 1. \begin{align*}{\frac{1}{290400000} \left ( -2352240000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-10055826000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-20610522900\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -15905948200\,x\sqrt{-10\,{x}^{2}-x+3}-3990994420\,\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)^5/(3+5*x)^(5/2)/(1-2*x)^(1/2),x)

[Out]

1/290400000*(-2352240000*x^4*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x^2-10055826000*x^3*
(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x-20610522900*x^2*(-10*x^2-x+3)^(1/2)[phone]*
10^(1/2)*arcsin(20/11*x+1/11)-15905948200*x*(-10*x^2-x+3)^(1/2)-3990994420*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/
(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [A]  time = 4.0801, size = 146, normalized size = 1.03 \begin{align*} -\frac{81}{250} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + \frac{462357}{800000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{9963}{10000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{305343}{200000} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{103125 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{998 \, \sqrt{-10 \, x^{2} - x + 3}}{1134375 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81/250*sqrt(-10*x^2 - x + 3)*x^2 + 462357/800000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 9963/10000*sqrt(-10
*x^2 - x + 3)*x - 305343/200000*sqrt(-10*x^2 - x + 3) - 2/103125*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) - 9
98/1134375*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]  time = 1.62346, size = 355, normalized size = 2.5 \begin{align*} -\frac{167835591 \, \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 (117612000 \, x^{4} + 502791300 \, x^{3} + 1030526145 \, x^{2} + 795297410 \, x + 199549721\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{290400000 \,{\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)^5/(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/290400000*(167835591*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*(117612000*x^4 + 502791300*x^3 + 1030526145*x^2 + 795297410*x + 199549721)*sqrt(5*x
+ 3)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)

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

[Out]

Timed out

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Giac [A]  time = 1.96011, size = 255, normalized size = 1.8 \begin{align*} -\frac{27}{1000000} \,{\left (12 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + 75 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 7745 \, \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}}{90750000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{462357}{400000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{333 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{7562500 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{999 \, \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}}}{5671875 \,{\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)^5/(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

-27/1000000*(12*(8*sqrt(5)*(5*x + 3) + 75*sqrt(5))*(5*x + 3) + 7745*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) - 1
/90750000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 462357/400000*sqrt(10)*arcsin(1/11
*sqrt(22)*sqrt(5*x + 3)) - 333/7562500*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 1/5671875
*(999*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(-1
0*x + 5) - sqrt(22))^3