3.2382 \(\int \frac {(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=96 \[ -\frac {2 (1-2 x)^{5/2}}{165 (5 x+3)^{3/2}}-\frac {38 (1-2 x)^{3/2}}{165 \sqrt {5 x+3}}-\frac {38}{275} \sqrt {5 x+3} \sqrt {1-2 x}-\frac {19}{25} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ) \]

[Out]

-2/165*(1-2*x)^(5/2)/(3+5*x)^(3/2)-19/125*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-38/165*(1-2*x)^(3/2)/(3
+5*x)^(1/2)-38/275*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {78, 47, 50, 54, 216} \[ -\frac {2 (1-2 x)^{5/2}}{165 (5 x+3)^{3/2}}-\frac {38 (1-2 x)^{3/2}}{165 \sqrt {5 x+3}}-\frac {38}{275} \sqrt {5 x+3} \sqrt {1-2 x}-\frac {19}{25} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(5/2))/(165*(3 + 5*x)^(3/2)) - (38*(1 - 2*x)^(3/2))/(165*Sqrt[3 + 5*x]) - (38*Sqrt[1 - 2*x]*Sqrt
[3 + 5*x])/275 - (19*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/25

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

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 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

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+5 x)^{5/2}} \, dx &=-\frac {2 (1-2 x)^{5/2}}{165 (3+5 x)^{3/2}}+\frac {19}{33} \int \frac {(1-2 x)^{3/2}}{(3+5 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2}}{165 (3+5 x)^{3/2}}-\frac {38 (1-2 x)^{3/2}}{165 \sqrt {3+5 x}}-\frac {38}{55} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2}}{165 (3+5 x)^{3/2}}-\frac {38 (1-2 x)^{3/2}}{165 \sqrt {3+5 x}}-\frac {38}{275} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {19}{25} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2}}{165 (3+5 x)^{3/2}}-\frac {38 (1-2 x)^{3/2}}{165 \sqrt {3+5 x}}-\frac {38}{275} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {38 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{25 \sqrt {5}}\\ &=-\frac {2 (1-2 x)^{5/2}}{165 (3+5 x)^{3/2}}-\frac {38 (1-2 x)^{3/2}}{165 \sqrt {3+5 x}}-\frac {38}{275} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {19}{25} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 59, normalized size = 0.61 \[ -\frac {2 (1-2 x)^{5/2} \left (19 \sqrt {22} (5 x+3)^{3/2} \, _2F_1\left (\frac {3}{2},\frac {5}{2};\frac {7}{2};-\frac {5}{11} (2 x-1)\right )+121\right )}{19965 (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(5/2)*(121 + 19*Sqrt[22]*(3 + 5*x)^(3/2)*Hypergeometric2F1[3/2, 5/2, 7/2, (-5*(-1 + 2*x))/11]))/
(19965*(3 + 5*x)^(3/2))

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fricas [A]  time = 1.32, size = 97, normalized size = 1.01 \[ \frac {57 \, \sqrt {5} \sqrt {2} {\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 20 \, {\left (45 \, x^{2} + 145 \, x + 73\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/750*(57*sqrt(5)*sqrt(2)*(25*x^2 + 30*x + 9)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x +
1)/(10*x^2 + x - 3)) - 20*(45*x^2 + 145*x + 73)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)

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giac [B]  time = 1.59, size = 158, normalized size = 1.65 \[ -\frac {1}{30000} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {732 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} - \frac {6}{625} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {19}{125} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {183 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{1875 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 732*(sqrt(2)*sqrt(-10*x + 5) - sqr
t(22))/sqrt(5*x + 3)) - 6/625*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 19/125*sqrt(10)*arcsin(1/11*sqrt(22)*sqr
t(5*x + 3)) + 1/1875*sqrt(10)*(5*x + 3)^(3/2)*(183*(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 = 113, normalized size = 1.18 \[ -\frac {\left (1425 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+900 \sqrt {-10 x^{2}-x +3}\, x^{2}+1710 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+2900 \sqrt {-10 x^{2}-x +3}\, x +513 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1460 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{750 \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/750*(1425*10^(1/2)*x^2*arcsin(20/11*x+1/11)+1710*10^(1/2)*x*arcsin(20/11*x+1/11)+900*(-10*x^2-x+3)^(1/2)*x^
2+513*10^(1/2)*arcsin(20/11*x+1/11)+2900*(-10*x^2-x+3)^(1/2)*x+1460*(-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 [A]  time = 1.37, size = 119, normalized size = 1.24 \[ -\frac {19}{250} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{75 \, {\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{25 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {11 \, \sqrt {-10 \, x^{2} - x + 3}}{375 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {283 \, \sqrt {-10 \, x^{2} - x + 3}}{375 \, {\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-19/250*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 1/75*(-10*x^2 - x + 3)^(3/2)/(125*x^3 + 225*x^2 + 135*x + 27)
+ 3/25*(-10*x^2 - x + 3)^(3/2)/(25*x^2 + 30*x + 9) - 11/375*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) - 283/3
75*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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