3.2609 \(\int \frac {(2+3 x)^4}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx\)

Optimal. Leaf size=113 \[ \frac {7 \sqrt {5 x+3} (3 x+2)^3}{33 (1-2 x)^{3/2}}-\frac {1589 \sqrt {5 x+3} (3 x+2)^2}{726 \sqrt {1-2 x}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3} (2380020 x+5735477)}{193600}+\frac {392283 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1600 \sqrt {10}} \]

[Out]

392283/16000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+7/33*(2+3*x)^3*(3+5*x)^(1/2)/(1-2*x)^(3/2)-1589/726*
(2+3*x)^2*(3+5*x)^(1/2)/(1-2*x)^(1/2)-1/193600*(5735477+2380020*x)*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {98, 150, 147, 54, 216} \[ \frac {7 \sqrt {5 x+3} (3 x+2)^3}{33 (1-2 x)^{3/2}}-\frac {1589 \sqrt {5 x+3} (3 x+2)^2}{726 \sqrt {1-2 x}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3} (2380020 x+5735477)}{193600}+\frac {392283 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1600 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1589*(2 + 3*x)^2*Sqrt[3 + 5*x])/(726*Sqrt[1 - 2*x]) + (7*(2 + 3*x)^3*Sqrt[3 + 5*x])/(33*(1 - 2*x)^(3/2)) - (
Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(5735477 + 2380020*x))/193600 + (392283*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1600*Sq
rt[10])

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 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 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 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}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx &=\frac {7 (2+3 x)^3 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {(2+3 x)^2 \left (218+\frac {717 x}{2}\right )}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {1589 (2+3 x)^2 \sqrt {3+5 x}}{726 \sqrt {1-2 x}}+\frac {7 (2+3 x)^3 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {1}{363} \int \frac {\left (-\frac {36489}{2}-\frac {119001 x}{4}\right ) (2+3 x)}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {1589 (2+3 x)^2 \sqrt {3+5 x}}{726 \sqrt {1-2 x}}+\frac {7 (2+3 x)^3 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (5735477+2380020 x)}{193600}+\frac {392283 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{3200}\\ &=-\frac {1589 (2+3 x)^2 \sqrt {3+5 x}}{726 \sqrt {1-2 x}}+\frac {7 (2+3 x)^3 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (5735477+2380020 x)}{193600}+\frac {392283 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1600 \sqrt {5}}\\ &=-\frac {1589 (2+3 x)^2 \sqrt {3+5 x}}{726 \sqrt {1-2 x}}+\frac {7 (2+3 x)^3 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (5735477+2380020 x)}{193600}+\frac {392283 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1600 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 90, normalized size = 0.80 \[ \frac {10 \sqrt {2 x-1} \sqrt {5 x+3} \left (2352240 x^3+14544684 x^2-61036064 x+21305631\right )[phone] \sqrt {10} (1-2 x)^2 \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{5808000 \sqrt {1-2 x} (2 x-1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(21305631 - 61036064*x + 14544684*x^2 + 2352240*x^3) + 142398729*Sqrt[10]*(1
- 2*x)^2*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(5808000*Sqrt[1 - 2*x]*(-1 + 2*x)^(3/2))

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fricas [A]  time = 0.99, size = 96, normalized size = 0.85 \[ -\frac {142398729 \, \sqrt {10} {\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (2352240 \, x^{3} + 14544684 \, x^{2} - 61036064 \, x + 21305631\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{11616000 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/11616000*(142398729*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*(2352240*x^3 + 14544684*x^2 - 61036064*x + 21305631)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(4*
x^2 - 4*x + 1)

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giac [A]  time = 1.04, size = 84, normalized size = 0.74 \[ \frac {392283}{16000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (9801 \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} + 263 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 94936582 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 1566381795 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{72600000 \, {\left (2 \, x - 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

392283/16000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/72600000*(4*(9801*(12*sqrt(5)*(5*x + 3) + 263*sq
rt(5))*(5*x + 3) - 94936582*sqrt(5))*(5*x + 3) + 1566381795*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2

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maple [A]  time = 0.02, size = 137, normalized size = 1.21 \[ \frac {\left (-47044800 \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-290893680 \sqrt {-10 x^{2}-x +3}\, x^{2}-569594916 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-426112620 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{11616000 \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11616000*(569594916*10^(1/2)*x^2*arcsin(20/11*x+1/11)-47044800*(-10*x^2-x+3)^(1/2)*x^3-569594916*10^(1/2)*x*
arcsin(20/11*x+1/11)-290893680*(-10*x^2-x+3)^(1/2)*x^2+142398729*10^(1/2)*arcsin(20/11*x+1/11)[phone]*(-10
*x^2-x+3)^(1/2)*x-426112620*(-10*x^2-x+3)^(1/2))*(5*x+3)^(1/2)*(-2*x+1)^(1/2)/(2*x-1)^2/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.27, size = 91, normalized size = 0.81 \[ \frac {392283}{32000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {81}{80} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {11637}{1600} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {2401 \, \sqrt {-10 \, x^{2} - x + 3}}{264 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {55909 \, \sqrt {-10 \, x^{2} - x + 3}}{1452 \, {\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

392283/32000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 81/80*sqrt(-10*x^2 - x + 3)*x - 11637/1600*sqrt(-10*x^2
- x + 3) + 2401/264*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 55909/1452*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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