3.2368 \(\int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 \sqrt {3+5 x}} \, dx\)

Optimal. Leaf size=122 \[ \frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{7 (3 x+2)^3}+\frac {59 \sqrt {5 x+3} (1-2 x)^{3/2}}{28 (3 x+2)^2}+\frac {1947 \sqrt {5 x+3} \sqrt {1-2 x}}{56 (3 x+2)}-\frac {21417 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{56 \sqrt {7}} \]

[Out]

-21417/392*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+1/7*(1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^3+5
9/28*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^2+1947/56*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {96, 94, 93, 204} \[ \frac {\sqrt {5 x+3} (1-2 x)^{5/2}}{7 (3 x+2)^3}+\frac {59 \sqrt {5 x+3} (1-2 x)^{3/2}}{28 (3 x+2)^2}+\frac {1947 \sqrt {5 x+3} \sqrt {1-2 x}}{56 (3 x+2)}-\frac {21417 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{56 \sqrt {7}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(3/2)/((2 + 3*x)^4*Sqrt[3 + 5*x]),x]

[Out]

((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(7*(2 + 3*x)^3) + (59*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(28*(2 + 3*x)^2) + (1947*
Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(56*(2 + 3*x)) - (21417*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(56*Sqrt[7
])

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 94

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 + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

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])

Rubi steps

\begin {align*} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 \sqrt {3+5 x}} \, dx &=\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{7 (2+3 x)^3}+\frac {59}{14} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^3 \sqrt {3+5 x}} \, dx\\ &=\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{7 (2+3 x)^3}+\frac {59 (1-2 x)^{3/2} \sqrt {3+5 x}}{28 (2+3 x)^2}+\frac {1947}{56} \int \frac {\sqrt {1-2 x}}{(2+3 x)^2 \sqrt {3+5 x}} \, dx\\ &=\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{7 (2+3 x)^3}+\frac {59 (1-2 x)^{3/2} \sqrt {3+5 x}}{28 (2+3 x)^2}+\frac {1947 \sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)}+\frac {21417}{112} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{7 (2+3 x)^3}+\frac {59 (1-2 x)^{3/2} \sqrt {3+5 x}}{28 (2+3 x)^2}+\frac {1947 \sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)}+\frac {21417}{56} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=\frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{7 (2+3 x)^3}+\frac {59 (1-2 x)^{3/2} \sqrt {3+5 x}}{28 (2+3 x)^2}+\frac {1947 \sqrt {1-2 x} \sqrt {3+5 x}}{56 (2+3 x)}-\frac {21417 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{56 \sqrt {7}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 74, normalized size = 0.61 \[ \frac {1}{392} \left (\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (16847 x^2+23214 x+8032\right )}{(3 x+2)^3}-21417 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^(3/2)/((2 + 3*x)^4*Sqrt[3 + 5*x]),x]

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(8032 + 23214*x + 16847*x^2))/(2 + 3*x)^3 - 21417*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]
/(Sqrt[7]*Sqrt[3 + 5*x])])/392

________________________________________________________________________________________

fricas [A]  time = 0.91, size = 101, normalized size = 0.83 \[ -\frac {21417 \, \sqrt {7} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (16847 \, x^{2} + 23214 \, x + 8032\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{784 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

-1/784*(21417*sqrt(7)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1
)/(10*x^2 + x - 3)) - 14*(16847*x^2 + 23214*x + 8032)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x +
8)

________________________________________________________________________________________

giac [B]  time = 1.92, size = 310, normalized size = 2.54 \[ \frac {21417}{7840} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {121 \, \sqrt {10} {\left (383 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 132160 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {13876800 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {55507200 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{28 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^(1/2),x, algorithm="giac")

[Out]

21417/7840*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)))) + 121/28*sqrt(10)*(383*((sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 + 132160*((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 + 13876800*(sqrt(2)*
sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 55507200*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sq
rt(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 + 28
0)^3

________________________________________________________________________________________

maple [B]  time = 0.01, size = 202, normalized size = 1.66 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (578259 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1156518 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+235858 \sqrt {-10 x^{2}-x +3}\, x^{2}+771012 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+324996 \sqrt {-10 x^{2}-x +3}\, x +171336 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+112448 \sqrt {-10 x^{2}-x +3}\right )}{784 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x+1)^(3/2)/(3*x+2)^4/(5*x+3)^(1/2),x)

[Out]

1/784*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(578259*7^(1/2)*x^3*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+1156
518*7^(1/2)*x^2*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+771012*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1
/2)/(-10*x^2-x+3)^(1/2))+235858*(-10*x^2-x+3)^(1/2)*x^2+171336*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-
x+3)^(1/2))+324996*(-10*x^2-x+3)^(1/2)*x+112448*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(3*x+2)^3

________________________________________________________________________________________

maxima [A]  time = 1.40, size = 107, normalized size = 0.88 \[ \frac {21417}{784} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {7 \, \sqrt {-10 \, x^{2} - x + 3}}{9 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {161 \, \sqrt {-10 \, x^{2} - x + 3}}{36 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {16847 \, \sqrt {-10 \, x^{2} - x + 3}}{504 \, {\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

21417/784*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 7/9*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x
^2 + 36*x + 8) + 161/36*sqrt(-10*x^2 - x + 3)/(9*x^2 + 12*x + 4) + 16847/504*sqrt(-10*x^2 - x + 3)/(3*x + 2)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^4\,\sqrt {5\,x+3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(3/2)/((3*x + 2)^4*(5*x + 3)^(1/2)),x)

[Out]

int((1 - 2*x)^(3/2)/((3*x + 2)^4*(5*x + 3)^(1/2)), x)

________________________________________________________________________________________

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)**4/(3+5*x)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________