3.2519 \(\int \frac {(2+3 x)^4 \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=139 \[ \frac {\sqrt {5 x+3} (3 x+2)^4}{\sqrt {1-2 x}}+\frac {27}{16} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^3+\frac {2203}{320} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2+\frac {\sqrt {1-2 x} \sqrt {5 x+3} (4618500 x+11129753)}{51200}-\frac {92108287 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{51200 \sqrt {10}} \]

[Out]

-92108287/512000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+(2+3*x)^4*(3+5*x)^(1/2)/(1-2*x)^(1/2)+2203/320*(
2+3*x)^2*(1-2*x)^(1/2)*(3+5*x)^(1/2)+27/16*(2+3*x)^3*(1-2*x)^(1/2)*(3+5*x)^(1/2)+1/51200*(11129753+4618500*x)*
(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ \frac {\sqrt {5 x+3} (3 x+2)^4}{\sqrt {1-2 x}}+\frac {27}{16} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^3+\frac {2203}{320} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2+\frac {\sqrt {1-2 x} \sqrt {5 x+3} (4618500 x+11129753)}{51200}-\frac {92108287 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{51200 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2203*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/320 + (27*Sqrt[1 - 2*x]*(2 + 3*x)^3*Sqrt[3 + 5*x])/16 + ((2 + 3
*x)^4*Sqrt[3 + 5*x])/Sqrt[1 - 2*x] + (Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(11129753 + 4618500*x))/51200 - (92108287*Ar
cSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(51200*Sqrt[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 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 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 \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx &=\frac {(2+3 x)^4 \sqrt {3+5 x}}{\sqrt {1-2 x}}-\int \frac {(2+3 x)^3 \left (41+\frac {135 x}{2}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {27}{16} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}+\frac {(2+3 x)^4 \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {1}{40} \int \frac {\left (-5035-\frac {33045 x}{4}\right ) (2+3 x)^2}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {2203}{320} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {27}{16} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}+\frac {(2+3 x)^4 \sqrt {3+5 x}}{\sqrt {1-2 x}}-\frac {\int \frac {(2+3 x) \left (\frac {1770165}{4}+\frac {5773125 x}{8}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1200}\\ &=\frac {2203}{320} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {27}{16} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}+\frac {(2+3 x)^4 \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} \sqrt {3+5 x} (11129753+4618500 x)}{51200}-\frac {92108287 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{102400}\\ &=\frac {2203}{320} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {27}{16} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}+\frac {(2+3 x)^4 \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} \sqrt {3+5 x} (11129753+4618500 x)}{51200}-\frac {92108287 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{51200 \sqrt {5}}\\ &=\frac {2203}{320} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {27}{16} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}+\frac {(2+3 x)^4 \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} \sqrt {3+5 x} (11129753+4618500 x)}{51200}-\frac {92108287 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{51200 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 88, normalized size = 0.63 \[ \frac {-10 \sqrt {2 x-1} \sqrt {5 x+3} \left (518400 x^4+2283840 x^3+5020200 x^2+9587886 x-14050073\right )-92108287 \sqrt {10} (2 x-1) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{512000 \sqrt {-(1-2 x)^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-10*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(-14050073 + 9587886*x + 5020200*x^2 + 2283840*x^3 + 518400*x^4) - 92108287*
Sqrt[10]*(-1 + 2*x)*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(512000*Sqrt[-(1 - 2*x)^2])

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fricas [A]  time = 1.18, size = 91, normalized size = 0.65 \[ \frac {92108287 \, \sqrt {10} {\left (2 \, x - 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (518400 \, x^{4} + 2283840 \, x^{3} + 5020200 \, x^{2} + 9587886 \, x - 14050073\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1024000 \, {\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1024000*(92108287*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*(518400*x^4 + 2283840*x^3 + 5020200*x^2 + 9587886*x - 14050073)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2*
x - 1)

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giac [A]  time = 0.99, size = 97, normalized size = 0.70 \[ -\frac {92108287}{512000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (6 \, {\left (12 \, {\left (8 \, {\left (36 \, \sqrt {5} {\left (5 \, x + 3\right )} + 361 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 28181 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 4651913 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 460541435 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{6400000 \, {\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-92108287/512000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/6400000*(6*(12*(8*(36*sqrt(5)*(5*x + 3) + 36
1*sqrt(5))*(5*x + 3) + 28181*sqrt(5))*(5*x + 3) + 4651913*sqrt(5))*(5*x + 3) - 460541435*sqrt(5))*sqrt(5*x + 3
)*sqrt(-10*x + 5)/(2*x - 1)

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maple [A]  time = 0.02, size = 140, normalized size = 1.01 \[ -\frac {\left (-10368000 \sqrt {-10 x^{2}-x +3}\, x^{4}-45676800 \sqrt {-10 x^{2}-x +3}\, x^{3}-100404000 \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-191757720 \sqrt {-10 x^{2}-x +3}\, x -92108287 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{1024000 \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1024000*(-10368000*(-10*x^2-x+3)^(1/2)*x^4-45676800*(-10*x^2-x+3)^(1/2)*x^3+184216574*10^(1/2)*x*arcsin(20/
11*x+1/11)-100404000*(-10*x^2-x+3)^(1/2)*x^2-92108287*10^(1/2)*arcsin(20/11*x+1/11)-191757720*(-10*x^2-x+3)^(1
/2)*x+281001460*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)*(5*x+3)^(1/2)/(2*x-1)/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.31, size = 94, normalized size = 0.68 \[ -\frac {81}{160} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {92108287}{1024000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {1557}{640} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {154953}{2560} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {6740553}{51200} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {2401 \, \sqrt {-10 \, x^{2} - x + 3}}{16 \, {\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81/160*(-10*x^2 - x + 3)^(3/2)*x - 92108287/1024000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 1557/640*(-10*x^
2 - x + 3)^(3/2) + 154953/2560*sqrt(-10*x^2 - x + 3)*x + 6740553/51200*sqrt(-10*x^2 - x + 3) - 2401/16*sqrt(-1
0*x^2 - x + 3)/(2*x - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (3 x + 2\right )^{4} \sqrt {5 x + 3}}{\left (1 - 2 x\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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