\(\int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\) [2527]

Optimal result

Integrand size = 26, antiderivative size = 151 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^3}-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{7 (2+3 x)^3}-\frac {5 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^2}+\frac {565 \sqrt {1-2 x} \sqrt {3+5 x}}{2744 (2+3 x)}-\frac {7435 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2744 \sqrt {7}} \]

[Out]

-7435/19208*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+2/7*(3+5*x)^(1/2)/(2+3*x)^3/(1-2*x)^(1/2)-
1/7*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3-5/196*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+565/2744*(1-2*x)^(1/2)*(
3+5*x)^(1/2)/(2+3*x)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {101, 156, 12, 95, 210} \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=-\frac {7435 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2744 \sqrt {7}}+\frac {565 \sqrt {1-2 x} \sqrt {5 x+3}}{2744 (3 x+2)}-\frac {5 \sqrt {1-2 x} \sqrt {5 x+3}}{196 (3 x+2)^2}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^3}+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3} \]

[In]

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

[Out]

(2*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)^3) - (5*Sqrt[1 -
2*x]*Sqrt[3 + 5*x])/(196*(2 + 3*x)^2) + (565*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2744*(2 + 3*x)) - (7435*ArcTan[Sqrt
[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2744*Sqrt[7])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 95

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 101

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[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 156

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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[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*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]

Rule 210

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^3}-\frac {2}{7} \int \frac {-\frac {53}{2}-45 x}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx \\ & = \frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^3}-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{7 (2+3 x)^3}-\frac {2}{147} \int \frac {-\frac {525}{4}-210 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx \\ & = \frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^3}-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{7 (2+3 x)^3}-\frac {5 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^2}-\frac {\int \frac {-\frac {5355}{8}-\frac {525 x}{2}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{1029} \\ & = \frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^3}-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{7 (2+3 x)^3}-\frac {5 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^2}+\frac {565 \sqrt {1-2 x} \sqrt {3+5 x}}{2744 (2+3 x)}-\frac {\int -\frac {156135}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{7203} \\ & = \frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^3}-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{7 (2+3 x)^3}-\frac {5 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^2}+\frac {565 \sqrt {1-2 x} \sqrt {3+5 x}}{2744 (2+3 x)}+\frac {7435 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{5488} \\ & = \frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^3}-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{7 (2+3 x)^3}-\frac {5 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^2}+\frac {565 \sqrt {1-2 x} \sqrt {3+5 x}}{2744 (2+3 x)}+\frac {7435 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{2744} \\ & = \frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^3}-\frac {\sqrt {1-2 x} \sqrt {3+5 x}}{7 (2+3 x)^3}-\frac {5 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^2}+\frac {565 \sqrt {1-2 x} \sqrt {3+5 x}}{2744 (2+3 x)}-\frac {7435 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2744 \sqrt {7}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\frac {7 \sqrt {3+5 x} \left (2512+3114 x-8055 x^2-10170 x^3\right )-7435 \sqrt {7-14 x} (2+3 x)^3 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{19208 \sqrt {1-2 x} (2+3 x)^3} \]

[In]

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

[Out]

(7*Sqrt[3 + 5*x]*(2512 + 3114*x - 8055*x^2 - 10170*x^3) - 7435*Sqrt[7 - 14*x]*(2 + 3*x)^3*ArcTan[Sqrt[1 - 2*x]
/(Sqrt[7]*Sqrt[3 + 5*x])])/(19208*Sqrt[1 - 2*x]*(2 + 3*x)^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(256\) vs. \(2(118)=236\).

Time = 1.20 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.70

method	result	size
default	\(\frac {\left (401490 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+602235 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+133830 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+142380 x^{3} \sqrt {-10 x^{2}-x +3}-148700 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +112770 x^{2} \sqrt {-10 x^{2}-x +3}-59480 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-43596 x \sqrt {-10 x^{2}-x +3}-35168 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{38416 \left (2+3 x \right )^{3} \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\)	\(257\)

[In]

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

[Out]

1/38416*(401490*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+602235*7^(1/2)*arctan(1/14*(37*
x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+133830*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+1
42380*x^3*(-10*x^2-x+3)^(1/2)-148700*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+112770*x^2*(
-10*x^2-x+3)^(1/2)-59480*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-43596*x*(-10*x^2-x+3)^(1/2
)-35168*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3/(-1+2*x)/(-10*x^2-x+3)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=-\frac {7435 \, \sqrt {7} {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (10170 \, x^{3} + 8055 \, x^{2} - 3114 \, x - 2512\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{38416 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \]

[In]

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

[Out]

-1/38416*(7435*sqrt(7)*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqr
t(-2*x + 1)/(10*x^2 + x - 3)) - 14*(10170*x^3 + 8055*x^2 - 3114*x - 2512)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(54*x^
4 + 81*x^3 + 18*x^2 - 20*x - 8)

Sympy [F]

\[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\int \frac {\sqrt {5 x + 3}}{\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{4}}\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.40 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\frac {7435}{38416} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {2825 \, x}{4116 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1145}{2744 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1}{63 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {23}{252 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {125}{1176 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]

[In]

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

[Out]

7435/38416*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 2825/4116*x/sqrt(-10*x^2 - x + 3) + 114
5/2744/sqrt(-10*x^2 - x + 3) + 1/63/(27*sqrt(-10*x^2 - x + 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^2 + 36*sqrt(-10
*x^2 - x + 3)*x + 8*sqrt(-10*x^2 - x + 3)) - 23/252/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x
+ 4*sqrt(-10*x^2 - x + 3)) - 125/1176/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 336 vs. \(2 (118) = 236\).

Time = 0.53 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.23 \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\frac {1487}{76832} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {16 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{12005 \, {\left (2 \, x - 1\right )}} - \frac {99 \, \sqrt {10} {\left (527 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 253120 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {36299200 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {145196800 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{9604 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]

[In]

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

[Out]

1487/76832*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)))) - 16/12005*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*
x - 1) - 99/9604*sqrt(10)*(527*((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)))^5 - 253120*((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 - 36299200*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 145196
800*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)^3

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx=\int \frac {\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^4} \,d x \]

[In]

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

[Out]

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