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

Optimal. Leaf size=135 \[ -\frac {2 (1-2 x)^{3/2} (3 x+2)^3}{5 \sqrt {5 x+3}}+\frac {27}{100} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^2-\frac {63 (35-8 x) (1-2 x)^{3/2} \sqrt {5 x+3}}{16000}+\frac {35511 \sqrt {1-2 x} \sqrt {5 x+3}}{160000}+\frac {390621 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{160000 \sqrt {10}} \]

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Rubi [A]  time = 0.04, antiderivative size = 135, 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, 153, 147, 50, 54, 216} \begin {gather*} -\frac {2 (1-2 x)^{3/2} (3 x+2)^3}{5 \sqrt {5 x+3}}+\frac {27}{100} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^2-\frac {63 (35-8 x) (1-2 x)^{3/2} \sqrt {5 x+3}}{16000}+\frac {35511 \sqrt {1-2 x} \sqrt {5 x+3}}{160000}+\frac {390621 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{160000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^3)/(5*Sqrt[3 + 5*x]) + (35511*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/160000 - (63*(35 - 8*
x)*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/16000 + (27*(1 - 2*x)^(3/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/100 + (390621*ArcSin[
Sqrt[2/11]*Sqrt[3 + 5*x]])/(160000*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 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)^{3/2}} \, dx &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^3}{5 \sqrt {3+5 x}}+\frac {2}{5} \int \frac {(3-27 x) \sqrt {1-2 x} (2+3 x)^2}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^3}{5 \sqrt {3+5 x}}+\frac {27}{100} (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {1}{100} \int \frac {\sqrt {1-2 x} (2+3 x) \left (-105+\frac {63 x}{2}\right )}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^3}{5 \sqrt {3+5 x}}-\frac {63 (35-8 x) (1-2 x)^{3/2} \sqrt {3+5 x}}{16000}+\frac {27}{100} (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}+\frac {35511 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{32000}\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^3}{5 \sqrt {3+5 x}}+\frac {35511 \sqrt {1-2 x} \sqrt {3+5 x}}{160000}-\frac {63 (35-8 x) (1-2 x)^{3/2} \sqrt {3+5 x}}{16000}+\frac {27}{100} (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}+\frac {390621 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{320000}\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^3}{5 \sqrt {3+5 x}}+\frac {35511 \sqrt {1-2 x} \sqrt {3+5 x}}{160000}-\frac {63 (35-8 x) (1-2 x)^{3/2} \sqrt {3+5 x}}{16000}+\frac {27}{100} (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}+\frac {390621 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{160000 \sqrt {5}}\\ &=-\frac {2 (1-2 x)^{3/2} (2+3 x)^3}{5 \sqrt {3+5 x}}+\frac {35511 \sqrt {1-2 x} \sqrt {3+5 x}}{160000}-\frac {63 (35-8 x) (1-2 x)^{3/2} \sqrt {3+5 x}}{16000}+\frac {27}{100} (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}+\frac {390621 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{160000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 88, normalized size = 0.65 \begin {gather*} \frac {10 \left (864000 x^5+446400 x^4-1014120 x^3-346790 x^2+223559 x+46783\right )+390621 \sqrt {5 x+3} \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{1600000 \sqrt {1-2 x} \sqrt {5 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*(46783 + 223559*x - 346790*x^2 - 1014120*x^3 + 446400*x^4 + 864000*x^5) + 390621*Sqrt[3 + 5*x]*Sqrt[-10 +
20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(1600000*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])

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IntegrateAlgebraic [A]  time = 0.67, size = 137, normalized size = 1.01 \begin {gather*} \frac {\sqrt {11-2 (5 x+3)} \left (-3456 \sqrt {5} (5 x+3)^4+23904 \sqrt {5} (5 x+3)^3+28980 \sqrt {5} (5 x+3)^2-128915 \sqrt {5} (5 x+3)-5632 \sqrt {5}\right )}{4000000 \sqrt {5 x+3}}-\frac {390621 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{80000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[11 - 2*(3 + 5*x)]*(-5632*Sqrt[5] - 128915*Sqrt[5]*(3 + 5*x) + 28980*Sqrt[5]*(3 + 5*x)^2 + 23904*Sqrt[5]*
(3 + 5*x)^3 - 3456*Sqrt[5]*(3 + 5*x)^4))/(4000000*Sqrt[3 + 5*x]) - (390621*ArcTan[(Sqrt[2]*Sqrt[3 + 5*x])/(Sqr
t[11] - Sqrt[11 - 2*(3 + 5*x)])])/(80000*Sqrt[10])

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fricas [A]  time = 1.53, size = 91, normalized size = 0.67 \begin {gather*} -\frac {390621 \, \sqrt {10} {\left (5 \, x + 3\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 (432000 \, x^{4} + 439200 \, x^{3} - 287460 \, x^{2} - 317125 \, x - 46783\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3200000 \, {\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)^(3/2),x, algorithm="fricas")

[Out]

-1/3200000*(390621*sqrt(10)*(5*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x
- 3)) + 20*(432000*x^4 + 439200*x^3 - 287460*x^2 - 317125*x - 46783)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(5*x + 3)

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giac [A]  time = 1.51, size = 137, normalized size = 1.01 \begin {gather*} -\frac {1}{4000000} \, {\left (36 \, {\left (8 \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} - 83 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 805 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 128915 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {390621}{1600000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {11 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{31250 \, \sqrt {5 \, x + 3}} + \frac {22 \, \sqrt {10} \sqrt {5 \, x + 3}}{15625 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\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)^(3/2),x, algorithm="giac")

[Out]

-1/4000000*(36*(8*(12*sqrt(5)*(5*x + 3) - 83*sqrt(5))*(5*x + 3) - 805*sqrt(5))*(5*x + 3) + 128915*sqrt(5))*sqr
t(5*x + 3)*sqrt(-10*x + 5) + 390621/1600000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 11/31250*sqrt(10)*(
sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 22/15625*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) -
sqrt(22))

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maple [A]  time = 0.01, size = 133, normalized size = 0.99 \begin {gather*} \frac {\left (-8640000 \sqrt {-10 x^{2}-x +3}\, x^{4}-8784000 \sqrt {-10 x^{2}-x +3}\, x^{3}+5749200 \sqrt {-10 x^{2}-x +3}\, x^{2}+1953105 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+6342500 \sqrt {-10 x^{2}-x +3}\, x +1171863 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+935660 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{3200000 \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \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)^(3/2),x)

[Out]

1/3200000*(-8640000*(-10*x^2-x+3)^(1/2)*x^4-8784000*(-10*x^2-x+3)^(1/2)*x^3+1953105*10^(1/2)*x*arcsin(20/11*x+
1/11)+5749200*(-10*x^2-x+3)^(1/2)*x^2+1171863*10^(1/2)*arcsin(20/11*x+1/11)+6342500*(-10*x^2-x+3)^(1/2)*x+9356
60*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)/(-10*x^2-x+3)^(1/2)/(5*x+3)^(1/2)

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maxima [C]  time = 1.53, size = 184, normalized size = 1.36 \begin {gather*} \frac {27}{500} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {35937}{1000000} i \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {23}{11}\right ) + \frac {1378113}{16000000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {171}{10000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {297}{2500} \, \sqrt {10 \, x^{2} + 23 \, x + \frac {51}{5}} x + \frac {9801}{40000} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {6831}{50000} \, \sqrt {10 \, x^{2} + 23 \, x + \frac {51}{5}} + \frac {28809}{800000} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{625 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{1250 \, {\left (5 \, x + 3\right )}} - \frac {33 \, \sqrt {-10 \, x^{2} - x + 3}}{3125 \, {\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)^(3/2),x, algorithm="maxima")

[Out]

27/500*(-10*x^2 - x + 3)^(3/2)*x - 35937/1000000*I*sqrt(5)*sqrt(2)*arcsin(20/11*x + 23/11) + 1378113/16000000*
sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 171/10000*(-10*x^2 - x + 3)^(3/2) + 297/2500*sqrt(10*x^2 + 23*x + 51/
5)*x + 9801/40000*sqrt(-10*x^2 - x + 3)*x + 6831/50000*sqrt(10*x^2 + 23*x + 51/5) + 28809/800000*sqrt(-10*x^2
- x + 3) + 1/625*(-10*x^2 - x + 3)^(3/2)/(25*x^2 + 30*x + 9) + 9/1250*(-10*x^2 - x + 3)^(3/2)/(5*x + 3) - 33/3
125*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 )}^{3/2}} \,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)^(3/2),x)

[Out]

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

[Out]

Timed out

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