3.2332 \(\int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=115 \[ -\frac {\sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)}-\frac {4}{9} \sqrt {5 x+3} \sqrt {1-2 x}-\frac {107}{27} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {41}{27} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]

[Out]

-107/135*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-41/27*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^
(1/2)-1/3*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)-4/9*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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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 = {97, 154, 157, 54, 216, 93, 204} \[ -\frac {\sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)}-\frac {4}{9} \sqrt {5 x+3} \sqrt {1-2 x}-\frac {107}{27} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {41}{27} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/9 - ((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(3*(2 + 3*x)) - (107*Sqrt[2/5]*ArcSin[Sqr
t[2/11]*Sqrt[3 + 5*x]])/27 - (41*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/27

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 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)^{3/2} \sqrt {3+5 x}}{(2+3 x)^2} \, dx &=-\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{3 (2+3 x)}+\frac {1}{3} \int \frac {\left (-\frac {13}{2}-20 x\right ) \sqrt {1-2 x}}{(2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {4}{9} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{3 (2+3 x)}+\frac {1}{45} \int \frac {-\frac {235}{2}-535 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {4}{9} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{3 (2+3 x)}-\frac {107}{27} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx+\frac {287}{54} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {4}{9} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{3 (2+3 x)}+\frac {287}{27} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )-\frac {214 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{27 \sqrt {5}}\\ &=-\frac {4}{9} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{3 (2+3 x)}-\frac {107}{27} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {41}{27} \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 = 0.19, size = 128, normalized size = 1.11 \[ \frac {107 \sqrt {10-20 x} (3 x+2) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-5 \left (3 \sqrt {-(1-2 x)^2} \sqrt {5 x+3} (6 x+11)+41 (3 x+2) \sqrt {14 x-7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )}{135 \sqrt {2 x-1} (3 x+2)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(107*Sqrt[10 - 20*x]*(2 + 3*x)*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]] - 5*(3*Sqrt[-(1 - 2*x)^2]*Sqrt[3 + 5*x]*(11
+ 6*x) + 41*(2 + 3*x)*Sqrt[-7 + 14*x]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))/(135*Sqrt[-1 + 2*x]*(2 +
3*x))

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fricas [A]  time = 0.91, size = 127, normalized size = 1.10 \[ \frac {107 \, \sqrt {5} \sqrt {2} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 205 \, \sqrt {7} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 30 \, {\left (6 \, x + 11\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{270 \, {\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/270*(107*sqrt(5)*sqrt(2)*(3*x + 2)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x
^2 + x - 3)) - 205*sqrt(7)*(3*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x
- 3)) - 30*(6*x + 11)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(3*x + 2)

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giac [B]  time = 1.70, size = 279, normalized size = 2.43 \[ \frac {41}{540} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {107}{270} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {2}{45} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {154 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{9 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

41/540*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)))) - 107/270*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((s
qrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 2/45*sqrt(5)*sqrt
(5*x + 3)*sqrt(-10*x + 5) - 154/9*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 [A]  time = 0.01, size = 146, normalized size = 1.27 \[ -\frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (321 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-615 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+180 \sqrt {-10 x^{2}-x +3}\, x +214 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-410 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+330 \sqrt {-10 x^{2}-x +3}\right )}{270 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/270*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(321*10^(1/2)*x*arcsin(20/11*x+1/11)-615*7^(1/2)*x*arctan(1/14*(37*x+20)*7
^(1/2)/(-10*x^2-x+3)^(1/2))+214*10^(1/2)*arcsin(20/11*x+1/11)-410*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x
^2-x+3)^(1/2))+180*(-10*x^2-x+3)^(1/2)*x+330*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(3*x+2)

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maxima [A]  time = 1.25, size = 75, normalized size = 0.65 \[ -\frac {107}{270} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {41}{54} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {2}{9} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {7 \, \sqrt {-10 \, x^{2} - x + 3}}{9 \, {\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-107/270*sqrt(10)*arcsin(20/11*x + 1/11) + 41/54*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 2
/9*sqrt(-10*x^2 - x + 3) - 7/9*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \sqrt {5 x + 3}}{\left (3 x + 2\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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