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

Optimal. Leaf size=113 \[ \frac {7 \sqrt {5 x+3} (3 x+2)^3}{11 \sqrt {1-2 x}}+\frac {243}{220} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2+\frac {9 \sqrt {1-2 x} \sqrt {5 x+3} (11316 x+27269)}{7040}-\frac {184641 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{640 \sqrt {10}} \]

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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, 153, 147, 54, 216} \begin {gather*} \frac {7 \sqrt {5 x+3} (3 x+2)^3}{11 \sqrt {1-2 x}}+\frac {243}{220} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2+\frac {9 \sqrt {1-2 x} \sqrt {5 x+3} (11316 x+27269)}{7040}-\frac {184641 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{640 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(243*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/220 + (7*(2 + 3*x)^3*Sqrt[3 + 5*x])/(11*Sqrt[1 - 2*x]) + (9*Sqrt
[1 - 2*x]*Sqrt[3 + 5*x]*(27269 + 11316*x))/7040 - (184641*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(640*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 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 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}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx &=\frac {7 (2+3 x)^3 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}-\frac {1}{11} \int \frac {(2+3 x)^2 \left (222+\frac {729 x}{2}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {243}{220} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {7 (2+3 x)^3 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {1}{330} \int \frac {\left (-\frac {39033}{2}-\frac {127305 x}{4}\right ) (2+3 x)}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {243}{220} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {7 (2+3 x)^3 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} \sqrt {3+5 x} (27269+11316 x)}{7040}-\frac {184641 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1280}\\ &=\frac {243}{220} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {7 (2+3 x)^3 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} \sqrt {3+5 x} (27269+11316 x)}{7040}-\frac {184641 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{640 \sqrt {5}}\\ &=\frac {243}{220} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {7 (2+3 x)^3 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} \sqrt {3+5 x} (27269+11316 x)}{7040}-\frac {184641 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{640 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 83, normalized size = 0.73 \begin {gather*} \frac {-10 \sqrt {2 x-1} \sqrt {5 x+3} \left (19008 x^3+78408 x^2+196614 x-312365\right )-2031051 \sqrt {10} (2 x-1) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{70400 \sqrt {-(1-2 x)^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-10*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(-312365 + 196614*x + 78408*x^2 + 19008*x^3) - 2031051*Sqrt[10]*(-1 + 2*x)*A
rcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(70400*Sqrt[-(1 - 2*x)^2])

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IntegrateAlgebraic [A]  time = 0.18, size = 125, normalized size = 1.11 \begin {gather*} \frac {\sqrt {5 x+3} \left (\frac {50776531 (1-2 x)^3}{(5 x+3)^3}+\frac {54163920 (1-2 x)^2}{(5 x+3)^2}+\frac {17888916 (1-2 x)}{5 x+3}+1536640\right )}{7040 \sqrt {1-2 x} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^3}+\frac {184641 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{640 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[3 + 5*x]*(1536640 + (50776531*(1 - 2*x)^3)/(3 + 5*x)^3 + (54163920*(1 - 2*x)^2)/(3 + 5*x)^2 + (17888916*
(1 - 2*x))/(3 + 5*x)))/(7040*Sqrt[1 - 2*x]*(2 + (5*(1 - 2*x))/(3 + 5*x))^3) + (184641*ArcTan[(Sqrt[5/2]*Sqrt[1
- 2*x])/Sqrt[3 + 5*x]])/(640*Sqrt[10])

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fricas [A]  time = 1.64, size = 86, normalized size = 0.76 \begin {gather*} \frac {2031051 \, \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 (19008 \, x^{3} + 78408 \, x^{2} + 196614 \, x - 312365\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{140800 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/140800*(2031051*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*(19008*x^3 + 78408*x^2 + 196614*x - 312365)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2*x - 1)

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giac [A]  time = 1.09, size = 84, normalized size = 0.74 \begin {gather*} -\frac {184641}{6400} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (594 \, {\left (4 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} + 93 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 5179 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 50776531 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{4400000 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-184641/6400*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/4400000*(594*(4*(8*sqrt(5)*(5*x + 3) + 93*sqrt(5
))*(5*x + 3) + 5179*sqrt(5))*(5*x + 3) - 50776531*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)

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maple [A]  time = 0.02, size = 123, normalized size = 1.09 \begin {gather*} -\frac {\left (-380160 \sqrt {-10 x^{2}-x +3}\, x^{3}-1568160 \sqrt {-10 x^{2}-x +3}\, x^{2}+4062102 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-3932280 \sqrt {-10 x^{2}-x +3}\, x -2031051 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+6247300 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{140800 \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/140800*(-380160*(-10*x^2-x+3)^(1/2)*x^3+4062102*10^(1/2)*x*arcsin(20/11*x+1/11)-1568160*(-10*x^2-x+3)^(1/2)
*x^2-2031051*10^(1/2)*arcsin(20/11*x+1/11)-3932280*(-10*x^2-x+3)^(1/2)*x+6247300*(-10*x^2-x+3)^(1/2))*(5*x+3)^
(1/2)*(-2*x+1)^(1/2)/(2*x-1)/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.47, size = 82, normalized size = 0.73 \begin {gather*} \frac {27}{20} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {184641}{12800} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {999}{160} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {2187}{128} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {2401 \, \sqrt {-10 \, x^{2} - x + 3}}{88 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27/20*sqrt(-10*x^2 - x + 3)*x^2 - 184641/12800*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 999/160*sqrt(-10*x^2 -
x + 3)*x + 2187/128*sqrt(-10*x^2 - x + 3) - 2401/88*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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