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

Optimal. Leaf size=142 \[ -\frac {128 \sqrt {1-2 x} (3 x+2)^3}{25 \sqrt {5 x+3}}-\frac {2 (1-2 x)^{3/2} (3 x+2)^3}{15 (5 x+3)^{3/2}}+\frac {378}{125} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2+\frac {21 \sqrt {1-2 x} \sqrt {5 x+3} (1140 x+853)}{10000}+\frac {13153 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{10000 \sqrt {10}} \]

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Rubi [A]  time = 0.04, antiderivative size = 142, normalized size of antiderivative = 1.00, 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 = {97, 150, 153, 147, 54, 216} \begin {gather*} -\frac {128 \sqrt {1-2 x} (3 x+2)^3}{25 \sqrt {5 x+3}}-\frac {2 (1-2 x)^{3/2} (3 x+2)^3}{15 (5 x+3)^{3/2}}+\frac {378}{125} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2+\frac {21 \sqrt {1-2 x} \sqrt {5 x+3} (1140 x+853)}{10000}+\frac {13153 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{10000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^3)/(15*(3 + 5*x)^(3/2)) - (128*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(25*Sqrt[3 + 5*x]) + (
378*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/125 + (21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(853 + 1140*x))/10000 + (13
153*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(10000*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 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 {(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^{5/2}} \, dx &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {(3-27 x) \sqrt {1-2 x} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac {128 \sqrt {1-2 x} (2+3 x)^3}{25 \sqrt {3+5 x}}+\frac {4}{75} \int \frac {\left (\frac {1029}{2}-1701 x\right ) (2+3 x)^2}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac {128 \sqrt {1-2 x} (2+3 x)^3}{25 \sqrt {3+5 x}}+\frac {378}{125} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {2 \int \frac {(2+3 x) \left (-1953+\frac {17955 x}{2}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1125}\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac {128 \sqrt {1-2 x} (2+3 x)^3}{25 \sqrt {3+5 x}}+\frac {378}{125} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} (853+1140 x)}{10000}+\frac {13153 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{20000}\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac {128 \sqrt {1-2 x} (2+3 x)^3}{25 \sqrt {3+5 x}}+\frac {378}{125} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} (853+1140 x)}{10000}+\frac {13153 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{10000 \sqrt {5}}\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac {128 \sqrt {1-2 x} (2+3 x)^3}{25 \sqrt {3+5 x}}+\frac {378}{125} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} (853+1140 x)}{10000}+\frac {13153 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{10000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 88, normalized size = 0.62 \begin {gather*} \frac {10 \left (216000 x^5+59400 x^4-320490 x^3-141425 x^2+67568 x+31171\right )+39459 \sqrt {20 x-10} (5 x+3)^{3/2} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{300000 \sqrt {1-2 x} (5 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*(31171 + 67568*x - 141425*x^2 - 320490*x^3 + 59400*x^4 + 216000*x^5) + 39459*(3 + 5*x)^(3/2)*Sqrt[-10 + 20
*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(300000*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))

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IntegrateAlgebraic [A]  time = 0.21, size = 160, normalized size = 1.13 \begin {gather*} \frac {-\frac {4000 (1-2 x)^{9/2}}{(5 x+3)^{9/2}}-\frac {237600 (1-2 x)^{7/2}}{(5 x+3)^{7/2}}+\frac {112245 (1-2 x)^{5/2}}{(5 x+3)^{5/2}}+\frac {1052240 (1-2 x)^{3/2}}{(5 x+3)^{3/2}}+\frac {157836 \sqrt {1-2 x}}{\sqrt {5 x+3}}}{30000 \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^3}-\frac {13153 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{10000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-4000*(1 - 2*x)^(9/2))/(3 + 5*x)^(9/2) - (237600*(1 - 2*x)^(7/2))/(3 + 5*x)^(7/2) + (112245*(1 - 2*x)^(5/2))
/(3 + 5*x)^(5/2) + (1052240*(1 - 2*x)^(3/2))/(3 + 5*x)^(3/2) + (157836*Sqrt[1 - 2*x])/Sqrt[3 + 5*x])/(30000*(2
+ (5*(1 - 2*x))/(3 + 5*x))^3) - (13153*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/(10000*Sqrt[10])

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fricas [A]  time = 1.35, size = 101, normalized size = 0.71 \begin {gather*} -\frac {39459 \, \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 (108000 \, x^{4} + 83700 \, x^{3} - 118395 \, x^{2} - 129910 \, x - 31171\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{600000 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/600000*(39459*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*(108000*x^4 + 83700*x^3 - 118395*x^2 - 129910*x - 31171)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(25
*x^2 + 30*x + 9)

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giac [A]  time = 1.52, size = 184, normalized size = 1.30 \begin {gather*} -\frac {9}{250000} \, {\left (4 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} - 65 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 265 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {1}{750000} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {2316 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} + \frac {13153}{100000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {579 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{46875 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/250000*(4*(8*sqrt(5)*(5*x + 3) - 65*sqrt(5))*(5*x + 3) - 265*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) - 1/750
000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 2316*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22
))/sqrt(5*x + 3)) + 13153/100000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/46875*sqrt(10)*(5*x + 3)^(3/
2)*(579*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3

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maple [A]  time = 0.01, size = 147, normalized size = 1.04 \begin {gather*} \frac {\left (-2160000 \sqrt {-10 x^{2}-x +3}\, x^{4}-1674000 \sqrt {-10 x^{2}-x +3}\, x^{3}+986475 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+2367900 \sqrt {-10 x^{2}-x +3}\, x^{2}+1183770 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+2598200 \sqrt {-10 x^{2}-x +3}\, x +355131 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+623420 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{600000 \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/600000*(-2160000*(-10*x^2-x+3)^(1/2)*x^4+986475*10^(1/2)*x^2*arcsin(20/11*x+1/11)-1674000*(-10*x^2-x+3)^(1/2
)*x^3+1183770*10^(1/2)*x*arcsin(20/11*x+1/11)+2367900*(-10*x^2-x+3)^(1/2)*x^2+355131*10^(1/2)*arcsin(20/11*x+1
/11)+2598200*(-10*x^2-x+3)^(1/2)*x+623420*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)/(-10*x^2-x+3)^(1/2)/(5*x+3)^(3/2
)

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maxima [C]  time = 1.25, size = 211, normalized size = 1.49 \begin {gather*} -\frac {35937}{1000000} i \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {23}{11}\right ) + \frac {7457}{250000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {9}{625} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {297}{2500} \, \sqrt {10 \, x^{2} + 23 \, x + \frac {51}{5}} x + \frac {6831}{50000} \, \sqrt {10 \, x^{2} + 23 \, x + \frac {51}{5}} + \frac {891}{12500} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{1875 \, {\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac {9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{625 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{1250 \, {\left (5 \, x + 3\right )}} - \frac {11 \, \sqrt {-10 \, x^{2} - x + 3}}{9375 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {877 \, \sqrt {-10 \, x^{2} - x + 3}}{9375 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-35937/1000000*I*sqrt(5)*sqrt(2)*arcsin(20/11*x + 23/11) + 7457/250000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11)
+ 9/625*(-10*x^2 - x + 3)^(3/2) + 297/2500*sqrt(10*x^2 + 23*x + 51/5)*x + 6831/50000*sqrt(10*x^2 + 23*x + 51/5
) + 891/12500*sqrt(-10*x^2 - x + 3) - 1/1875*(-10*x^2 - x + 3)^(3/2)/(125*x^3 + 225*x^2 + 135*x + 27) + 9/625*
(-10*x^2 - x + 3)^(3/2)/(25*x^2 + 30*x + 9) + 27/1250*(-10*x^2 - x + 3)^(3/2)/(5*x + 3) - 11/9375*sqrt(-10*x^2
- x + 3)/(25*x^2 + 30*x + 9) - 877/9375*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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