3.2394 \(\int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=144 \[ -\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{6 (3 x+2)^2}+\frac {5 \sqrt {5 x+3} (1-2 x)^{3/2}}{4 (3 x+2)}+\frac {19}{18} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {118}{27} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {155}{108} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]

[Out]

118/135*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-155/108*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7
^(1/2)-1/6*(1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^2+5/4*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)+19/18*(1-2*x)^(1/2)*(
3+5*x)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {97, 149, 154, 157, 54, 216, 93, 204} \[ -\frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{6 (3 x+2)^2}+\frac {5 \sqrt {5 x+3} (1-2 x)^{3/2}}{4 (3 x+2)}+\frac {19}{18} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {118}{27} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {155}{108} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(19*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/18 - ((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(6*(2 + 3*x)^2) + (5*(1 - 2*x)^(3/2)*Sqr
t[3 + 5*x])/(4*(2 + 3*x)) + (118*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/27 - (155*Sqrt[7]*ArcTan[Sqrt[1 -
2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/108

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 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 149

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] && IntegerQ[m]

Rule 154

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] && 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} \sqrt {3+5 x}}{(2+3 x)^3} \, dx &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{6 (2+3 x)^2}+\frac {1}{6} \int \frac {\left (-\frac {25}{2}-30 x\right ) (1-2 x)^{3/2}}{(2+3 x)^2 \sqrt {3+5 x}} \, dx\\ &=-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{6 (2+3 x)^2}+\frac {5 (1-2 x)^{3/2} \sqrt {3+5 x}}{4 (2+3 x)}-\frac {1}{18} \int \frac {\left (-\frac {915}{4}-285 x\right ) \sqrt {1-2 x}}{(2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {19}{18} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{6 (2+3 x)^2}+\frac {5 (1-2 x)^{3/2} \sqrt {3+5 x}}{4 (2+3 x)}-\frac {1}{270} \int \frac {-\frac {14865}{4}-3540 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {19}{18} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{6 (2+3 x)^2}+\frac {5 (1-2 x)^{3/2} \sqrt {3+5 x}}{4 (2+3 x)}+\frac {118}{27} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx+\frac {1085}{216} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {19}{18} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{6 (2+3 x)^2}+\frac {5 (1-2 x)^{3/2} \sqrt {3+5 x}}{4 (2+3 x)}+\frac {1085}{108} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )+\frac {236 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{27 \sqrt {5}}\\ &=\frac {19}{18} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{6 (2+3 x)^2}+\frac {5 (1-2 x)^{3/2} \sqrt {3+5 x}}{4 (2+3 x)}+\frac {118}{27} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {155}{108} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 134, normalized size = 0.93 \[ \frac {15 \sqrt {-(1-2 x)^2} \sqrt {5 x+3} \left (48 x^2+435 x+236\right )-775 \sqrt {14 x-7} (3 x+2)^2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-472 \sqrt {10-20 x} (3 x+2)^2 \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{540 \sqrt {2 x-1} (3 x+2)^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(15*Sqrt[-(1 - 2*x)^2]*Sqrt[3 + 5*x]*(236 + 435*x + 48*x^2) - 472*Sqrt[10 - 20*x]*(2 + 3*x)^2*ArcSinh[Sqrt[5/1
1]*Sqrt[-1 + 2*x]] - 775*(2 + 3*x)^2*Sqrt[-7 + 14*x]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(540*Sqrt[
-1 + 2*x]*(2 + 3*x)^2)

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fricas [A]  time = 0.73, size = 147, normalized size = 1.02 \[ -\frac {472 \, \sqrt {5} \sqrt {2} {\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 775 \, \sqrt {7} {\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 30 \, {\left (48 \, x^{2} + 435 \, x + 236\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1080 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1080*(472*sqrt(5)*sqrt(2)*(9*x^2 + 12*x + 4)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x
+ 1)/(10*x^2 + x - 3)) + 775*sqrt(7)*(9*x^2 + 12*x + 4)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2
*x + 1)/(10*x^2 + x - 3)) - 30*(48*x^2 + 435*x + 236)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)

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giac [B]  time = 2.44, size = 338, normalized size = 2.35 \[ \frac {31}{432} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {59}{135} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {4}{135} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {77 \, \sqrt {10} {\left (17 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {13720 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {54880 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{54 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

31/432*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)))) + 59/135*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)))) + 4/135*sqrt(5)*sqrt
(5*x + 3)*sqrt(-10*x + 5) - 77/54*sqrt(10)*(17*((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)))^3 - 13720*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 54
880*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)^2

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maple [A]  time = 0.01, size = 208, normalized size = 1.44 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (4248 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+6975 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1440 \sqrt {-10 x^{2}-x +3}\, x^{2}+5664 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+9300 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+13050 \sqrt {-10 x^{2}-x +3}\, x +1888 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+3100 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+7080 \sqrt {-10 x^{2}-x +3}\right )}{1080 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1080*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(4248*10^(1/2)*x^2*arcsin(20/11*x+1/11)+6975*7^(1/2)*x^2*arctan(1/14*(37*x
+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+5664*10^(1/2)*x*arcsin(20/11*x+1/11)+9300*7^(1/2)*x*arctan(1/14*(37*x+20)*7^
(1/2)/(-10*x^2-x+3)^(1/2))+1440*(-10*x^2-x+3)^(1/2)*x^2+1888*10^(1/2)*arcsin(20/11*x+1/11)+3100*7^(1/2)*arctan
(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+13050*(-10*x^2-x+3)^(1/2)*x+7080*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+
3)^(1/2)/(3*x+2)^2

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maxima [A]  time = 1.35, size = 101, normalized size = 0.70 \[ \frac {59}{135} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {155}{216} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {13}{9} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {7 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{6 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {49 \, \sqrt {-10 \, x^{2} - x + 3}}{36 \, {\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

59/135*sqrt(10)*arcsin(20/11*x + 1/11) + 155/216*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 1
3/9*sqrt(-10*x^2 - x + 3) + 7/6*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 49/36*sqrt(-10*x^2 - x + 3)/(3*x
+ 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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