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

Optimal. Leaf size=115 \[ \frac {7 (1-2 x)^{3/2}}{3 (3 x+2) \sqrt {5 x+3}}-\frac {1111 \sqrt {1-2 x}}{15 \sqrt {5 x+3}}-\frac {8}{45} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {665}{9} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {98, 150, 157, 54, 216, 93, 204} \begin {gather*} \frac {7 (1-2 x)^{3/2}}{3 (3 x+2) \sqrt {5 x+3}}-\frac {1111 \sqrt {1-2 x}}{15 \sqrt {5 x+3}}-\frac {8}{45} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {665}{9} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1111*Sqrt[1 - 2*x])/(15*Sqrt[3 + 5*x]) + (7*(1 - 2*x)^(3/2))/(3*(2 + 3*x)*Sqrt[3 + 5*x]) - (8*Sqrt[2/5]*ArcS
in[Sqrt[2/11]*Sqrt[3 + 5*x]])/45 + (665*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/9

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 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

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 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 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

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)^{5/2}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx &=\frac {7 (1-2 x)^{3/2}}{3 (2+3 x) \sqrt {3+5 x}}+\frac {1}{3} \int \frac {\sqrt {1-2 x} \left (\frac {227}{2}+4 x\right )}{(2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac {1111 \sqrt {1-2 x}}{15 \sqrt {3+5 x}}+\frac {7 (1-2 x)^{3/2}}{3 (2+3 x) \sqrt {3+5 x}}+\frac {2}{15} \int \frac {-\frac {7769}{4}-4 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {1111 \sqrt {1-2 x}}{15 \sqrt {3+5 x}}+\frac {7 (1-2 x)^{3/2}}{3 (2+3 x) \sqrt {3+5 x}}-\frac {8}{45} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx-\frac {4655}{18} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {1111 \sqrt {1-2 x}}{15 \sqrt {3+5 x}}+\frac {7 (1-2 x)^{3/2}}{3 (2+3 x) \sqrt {3+5 x}}-\frac {4655}{9} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )-\frac {16 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{45 \sqrt {5}}\\ &=-\frac {1111 \sqrt {1-2 x}}{15 \sqrt {3+5 x}}+\frac {7 (1-2 x)^{3/2}}{3 (2+3 x) \sqrt {3+5 x}}-\frac {8}{45} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {665}{9} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\\ \end {align*}

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Mathematica [C]  time = 0.37, size = 196, normalized size = 1.70 \begin {gather*} \frac {-15450 \sqrt {22} \left (15 x^2+19 x+6\right ) \left (-(1-2 x)^2\right )^{3/2} \, _2F_1\left (\frac {3}{2},\frac {5}{2};\frac {7}{2};-\frac {5}{11} (2 x-1)\right )-1815 \sqrt {5 x+3} \left (3090 x^2+129817 x+82313\right ) \sqrt {-(1-2 x)^2}[phone] \sqrt {14 x-7} \left (15 x^2+19 x+6\right ) \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )+1267717 \sqrt {10-20 x} \left (15 x^2+19 x+6\right ) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{952875 \sqrt {2 x-1} (3 x+2) (5 x+3)} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-1815*Sqrt[-(1 - 2*x)^2]*Sqrt[3 + 5*x]*(82313 + 129817*x + 3090*x^2) + 1267717*Sqrt[10 - 20*x]*(6 + 19*x + 15
*x^2)*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]] + 70406875*Sqrt[-7 + 14*x]*(6 + 19*x + 15*x^2)*ArcTan[Sqrt[1 - 2*x]/(
Sqrt[7]*Sqrt[3 + 5*x])] - 15450*Sqrt[22]*(-(1 - 2*x)^2)^(3/2)*(6 + 19*x + 15*x^2)*Hypergeometric2F1[3/2, 5/2,
7/2, (-5*(-1 + 2*x))/11])/(952875*Sqrt[-1 + 2*x]*(2 + 3*x)*(3 + 5*x))

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IntegrateAlgebraic [A]  time = 0.18, size = 128, normalized size = 1.11 \begin {gather*} -\frac {11 \sqrt {1-2 x} \left (\frac {66 (1-2 x)}{5 x+3}+707\right )}{15 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )}+\frac {8}{45} \sqrt {\frac {2}{5}} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )+\frac {665}{9} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-11*Sqrt[1 - 2*x]*(707 + (66*(1 - 2*x))/(3 + 5*x)))/(15*Sqrt[3 + 5*x]*(7 + (1 - 2*x)/(3 + 5*x))) + (8*Sqrt[2/
5]*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/45 + (665*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5
*x])])/9

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fricas [A]  time = 1.35, size = 142, normalized size = 1.23 \begin {gather*} \frac {8 \, \sqrt {5} \sqrt {2} {\left (15 \, x^{2} + 19 \, x + 6\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 16625 \, \sqrt {7} {\left (15 \, x^{2} + 19 \, x + 6\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 30 \, {\left (3403 \, x + 2187\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{450 \, {\left (15 \, x^{2} + 19 \, x + 6\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/450*(8*sqrt(5)*sqrt(2)*(15*x^2 + 19*x + 6)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x +
1)/(10*x^2 + x - 3)) + 16625*sqrt(7)*(15*x^2 + 19*x + 6)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2
*x + 1)/(10*x^2 + x - 3)) - 30*(3403*x + 2187)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(15*x^2 + 19*x + 6)

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giac [B]  time = 2.10, size = 319, normalized size = 2.77 \begin {gather*} -\frac {133}{36} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {4}{225} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {121}{50} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {1078 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{3 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-133/36*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2
/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 4/225*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sq
rt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 121/50*sqrt(10)*((
sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) - 10
78/3*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) -
sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) -
sqrt(22)))^2 + 280)

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maple [B]  time = 0.02, size = 191, normalized size = 1.66 \begin {gather*} -\frac {\left (120 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+249375 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+152 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+315875 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+102090 \sqrt {-10 x^{2}-x +3}\, x +48 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+99750 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+65610 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{450 \left (3 x +2\right ) \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)^(5/2)/(3*x+2)^2/(5*x+3)^(3/2),x)

[Out]

-1/450*(120*10^(1/2)*x^2*arcsin(20/11*x+1/11)+249375*7^(1/2)*x^2*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(
1/2))+152*10^(1/2)*x*arcsin(20/11*x+1/11)+315875*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+
48*10^(1/2)*arcsin(20/11*x+1/11)+99750*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+102090*(-10*
x^2-x+3)^(1/2)*x+65610*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)/(3*x+2)/(-10*x^2-x+3)^(1/2)/(5*x+3)^(1/2)

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maxima [A]  time = 1.28, size = 103, normalized size = 0.90 \begin {gather*} -\frac {4}{225} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {665}{18} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {6806 \, x}{45 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {10699}{135 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {343}{27 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/225*sqrt(10)*arcsin(20/11*x + 1/11) - 665/18*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 68
06/45*x/sqrt(-10*x^2 - x + 3) - 10699/135/sqrt(-10*x^2 - x + 3) + 343/27/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-
10*x^2 - 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 )}^{5/2}}{{\left (3\,x+2\right )}^2\,{\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)^(5/2)/((3*x + 2)^2*(5*x + 3)^(3/2)),x)

[Out]

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

[Out]

Timed out

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