3.2589 \(\int \frac {(2+3 x)^4 (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=164 \[ \frac {(5 x+3)^{3/2} (3 x+2)^4}{3 (1-2 x)^{3/2}}-\frac {123 (5 x+3)^{3/2} (3 x+2)^3}{22 \sqrt {1-2 x}}-\frac {3315}{352} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^2-\frac {3 \sqrt {1-2 x} (5 x+3)^{3/2} (10798680 x+22868329)}{281600}-\frac {1626211523 \sqrt {1-2 x} \sqrt {5 x+3}}{1126400}+\frac {1626211523 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{102400 \sqrt {10}} \]

[Out]

1/3*(2+3*x)^4*(3+5*x)^(3/2)/(1-2*x)^(3/2)[phone]/1024000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-123/
22*(2+3*x)^3*(3+5*x)^(3/2)/(1-2*x)^(1/2)-3315/352*(2+3*x)^2*(3+5*x)^(3/2)*(1-2*x)^(1/2)-3/281600*(3+5*x)^(3/2)
*(22868329+10798680*x)*(1-2*x)^(1/2)-1626211523/1126400*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {97, 150, 153, 147, 50, 54, 216} \[ \frac {(5 x+3)^{3/2} (3 x+2)^4}{3 (1-2 x)^{3/2}}-\frac {123 (5 x+3)^{3/2} (3 x+2)^3}{22 \sqrt {1-2 x}}-\frac {3315}{352} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^2-\frac {3 \sqrt {1-2 x} (5 x+3)^{3/2} (10798680 x+22868329)}{281600}-\frac {1626211523 \sqrt {1-2 x} \sqrt {5 x+3}}{1126400}+\frac {1626211523 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{102400 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1626211523*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1126400 - (3315*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(3/2))/352 - (12
3*(2 + 3*x)^3*(3 + 5*x)^(3/2))/(22*Sqrt[1 - 2*x]) + ((2 + 3*x)^4*(3 + 5*x)^(3/2))/(3*(1 - 2*x)^(3/2)) - (3*Sqr
t[1 - 2*x]*(3 + 5*x)^(3/2)*(22868329 + 10798680*x))/281600 + (1626211523*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(10
2400*Sqrt[10])

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 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 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 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 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)^{5/2}} \, dx &=\frac {(2+3 x)^4 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {(2+3 x)^3 \sqrt {3+5 x} \left (51+\frac {165 x}{2}\right )}{(1-2 x)^{3/2}} \, dx\\ &=-\frac {123 (2+3 x)^3 (3+5 x)^{3/2}}{22 \sqrt {1-2 x}}+\frac {(2+3 x)^4 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {\left (-7734-\frac {49725 x}{4}\right ) (2+3 x)^2 \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {3315}{352} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac {123 (2+3 x)^3 (3+5 x)^{3/2}}{22 \sqrt {1-2 x}}+\frac {(2+3 x)^4 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}+\frac {\int \frac {(2+3 x) \sqrt {3+5 x} \left (\frac {3817455}{4}+\frac {12148515 x}{8}\right )}{\sqrt {1-2 x}} \, dx}{1320}\\ &=-\frac {3315}{352} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac {123 (2+3 x)^3 (3+5 x)^{3/2}}{22 \sqrt {1-2 x}}+\frac {(2+3 x)^4 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2} (22868329+10798680 x)}{281600}+\frac {1626211523 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{563200}\\ &=-\frac {1626211523 \sqrt {1-2 x} \sqrt {3+5 x}}{1126400}-\frac {3315}{352} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac {123 (2+3 x)^3 (3+5 x)^{3/2}}{22 \sqrt {1-2 x}}+\frac {(2+3 x)^4 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2} (22868329+10798680 x)}{281600}+\frac {1626211523 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{204800}\\ &=-\frac {1626211523 \sqrt {1-2 x} \sqrt {3+5 x}}{1126400}-\frac {3315}{352} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac {123 (2+3 x)^3 (3+5 x)^{3/2}}{22 \sqrt {1-2 x}}+\frac {(2+3 x)^4 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2} (22868329+10798680 x)}{281600}+\frac {1626211523 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{102400 \sqrt {5}}\\ &=-\frac {1626211523 \sqrt {1-2 x} \sqrt {3+5 x}}{1126400}-\frac {3315}{352} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac {123 (2+3 x)^3 (3+5 x)^{3/2}}{22 \sqrt {1-2 x}}+\frac {(2+3 x)^4 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2} (22868329+10798680 x)}{281600}+\frac {1626211523 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{102400 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 100, normalized size = 0.61 \[ \frac {10 \sqrt {2 x-1} \sqrt {5 x+3} \left (15552000 x^5+83548800 x^4+236669040 x^3+633940524 x^2-2034703904 x+739060191\right )[phone] \sqrt {10} (1-2 x)^2 \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{3072000 \sqrt {1-2 x} (2 x-1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(739060191 - 2034703904*x + 633940524*x^2 + 236669040*x^3 + 83548800*x^4 + 15
552000*x^5) + 4878634569*Sqrt[10]*(1 - 2*x)^2*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(3072000*Sqrt[1 - 2*x]*(-1 +
2*x)^(3/2))

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fricas [A]  time = 1.26, size = 106, normalized size = 0.65 \[ -\frac {4878634569 \, \sqrt {10} {\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (15552000 \, x^{5} + 83548800 \, x^{4} + 236669040 \, x^{3} + 633940524 \, x^{2} - 2034703904 \, x + 739060191\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{6144000 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6144000*(4878634569*sqrt(10)*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)
/(10*x^2 + x - 3)) + 20*(15552000*x^5 + 83548800*x^4 + 236669040*x^3 + 633940524*x^2 - 2034703904*x + 73906019
1)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)

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giac [A]  time = 1.27, size = 110, normalized size = 0.67 \[ \frac {1626211523}{1024000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (9 \, {\left (12 \, {\left (8 \, {\left (36 \, \sqrt {5} {\left (5 \, x + 3\right )} + 427 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 42657 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 9855815 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 3252423046 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 53664980259 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{38400000 \, {\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1626211523/1024000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/38400000*(4*(9*(12*(8*(36*sqrt(5)*(5*x + 3
) + 427*sqrt(5))*(5*x + 3) + 42657*sqrt(5))*(5*x + 3) + 9855815*sqrt(5))*(5*x + 3) - 3252423046*sqrt(5))*(5*x
+ 3) + 53664980259*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2

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maple [A]  time = 0.01, size = 171, normalized size = 1.04 \[ \frac {\left (-311040000 \sqrt {-10 x^{2}-x +3}\, x^{5}-1670976000 \sqrt {-10 x^{2}-x +3}\, x^{4}-4733380800 \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-12678810480 \sqrt {-10 x^{2}-x +3}\, x^{2}-19514538276 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-14781203820 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{6144000 \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6144000*(-311040000*(-10*x^2-x+3)^(1/2)*x^5-1670976000*(-10*x^2-x+3)^(1/2)*x^4+19514538276*10^(1/2)*x^2*arcs
in(20/11*x+1/11)-4733380800*(-10*x^2-x+3)^(1/2)*x^3-19514538276*10^(1/2)*x*arcsin(20/11*x+1/11)-12678810480*(-
10*x^2-x+3)^(1/2)*x^2+4878634569*10^(1/2)*arcsin(20/11*x+1/11)[phone]*(-10*x^2-x+3)^(1/2)*x-14781203820*(
-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)*(5*x+3)^(1/2)/(2*x-1)^2/(-10*x^2-x+3)^(1/2)

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maxima [C]  time = 1.48, size = 241, normalized size = 1.47 \[ \frac {81}{64} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {1666460963}{2048000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {251559}{12800} i \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x - \frac {21}{11}\right ) + \frac {10161}{1280} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {2079}{32} \, \sqrt {10 \, x^{2} - 21 \, x + 8} x + \frac {29403}{5120} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {43659}{640} \, \sqrt {10 \, x^{2} - 21 \, x + 8} - \frac {34897797}{102400} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {2401 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{96 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac {1029 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{8 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {1323 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{32 \, {\left (2 \, x - 1\right )}} + \frac {26411 \, \sqrt {-10 \, x^{2} - x + 3}}{192 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {491519 \, \sqrt {-10 \, x^{2} - x + 3}}{192 \, {\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

81/64*(-10*x^2 - x + 3)^(3/2)*x + 1666460963/2048000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 251559/12800*I*s
qrt(5)*sqrt(2)*arcsin(20/11*x - 21/11) + 10161/1280*(-10*x^2 - x + 3)^(3/2) - 2079/32*sqrt(10*x^2 - 21*x + 8)*
x + 29403/5120*sqrt(-10*x^2 - x + 3)*x + 43659/640*sqrt(10*x^2 - 21*x + 8) - 34897797/102400*sqrt(-10*x^2 - x
+ 3) - 2401/96*(-10*x^2 - x + 3)^(3/2)/(8*x^3 - 12*x^2 + 6*x - 1) + 1029/8*(-10*x^2 - x + 3)^(3/2)/(4*x^2 - 4*
x + 1) + 1323/32*(-10*x^2 - x + 3)^(3/2)/(2*x - 1) + 26411/192*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 49151
9/192*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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