3.26.87 \(\int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx\) [2587]

Optimal. Leaf size=144 \[ \frac {415 \sqrt {3+5 x}}{22638 \sqrt {1-2 x}}+\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {\sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {5 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)}-\frac {765 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1372 \sqrt {7}} \]

[Out]

-765/9604*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+2/21*(3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^2+4
15/22638*(3+5*x)^(1/2)/(1-2*x)^(1/2)-1/14*(3+5*x)^(1/2)/(2+3*x)^2/(1-2*x)^(1/2)+5/196*(3+5*x)^(1/2)/(2+3*x)/(1
-2*x)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {101, 156, 157, 12, 95, 210} \begin {gather*} -\frac {765 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}}+\frac {415 \sqrt {5 x+3}}{22638 \sqrt {1-2 x}}+\frac {5 \sqrt {5 x+3}}{196 \sqrt {1-2 x} (3 x+2)}-\frac {\sqrt {5 x+3}}{14 \sqrt {1-2 x} (3 x+2)^2}+\frac {2 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(415*Sqrt[3 + 5*x])/(22638*Sqrt[1 - 2*x]) + (2*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^2) - Sqrt[3 + 5*x]
/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (5*Sqrt[3 + 5*x])/(196*Sqrt[1 - 2*x]*(2 + 3*x)) - (765*ArcTan[Sqrt[1 - 2*x]/
(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*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 157

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] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

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*} \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {2}{21} \int \frac {-\frac {53}{2}-45 x}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {\sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}-\frac {1}{147} \int \frac {-\frac {595}{4}-210 x}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {\sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {5 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)}-\frac {\int \frac {-\frac {3955}{8}+\frac {525 x}{2}}{(1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}} \, dx}{1029}\\ &=\frac {415 \sqrt {3+5 x}}{22638 \sqrt {1-2 x}}+\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {\sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {5 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)}+\frac {2 \int \frac {176715}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{79233}\\ &=\frac {415 \sqrt {3+5 x}}{22638 \sqrt {1-2 x}}+\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {\sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {5 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)}+\frac {765 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2744}\\ &=\frac {415 \sqrt {3+5 x}}{22638 \sqrt {1-2 x}}+\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {\sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {5 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)}+\frac {765 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{1372}\\ &=\frac {415 \sqrt {3+5 x}}{22638 \sqrt {1-2 x}}+\frac {2 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {\sqrt {3+5 x}}{14 \sqrt {1-2 x} (2+3 x)^2}+\frac {5 \sqrt {3+5 x}}{196 \sqrt {1-2 x} (2+3 x)}-\frac {765 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1372 \sqrt {7}}\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 95, normalized size = 0.66 \begin {gather*} -\frac {7 \sqrt {3+5 x} \left (-6708-8633 x+19380 x^2+14940 x^3\right )-25245 \sqrt {7-14 x} (-1+2 x) (2+3 x)^2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{316932 (1-2 x)^{3/2} (2+3 x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/316932*(7*Sqrt[3 + 5*x]*(-6708 - 8633*x + 19380*x^2 + 14940*x^3) - 25245*Sqrt[7 - 14*x]*(-1 + 2*x)*(2 + 3*x
)^2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/((1 - 2*x)^(3/2)*(2 + 3*x)^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(256\) vs. \(2(111)=222\).
time = 0.09, size = 257, normalized size = 1.78

method	result	size
default	\(\frac {\left (908820 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+302940 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-580635 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-209160 x^{3} \sqrt {-10 x^{2}-x +3}-100980 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x -271320 x^{2} \sqrt {-10 x^{2}-x +3}+100980 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+120862 x \sqrt {-10 x^{2}-x +3}+93912 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{633864 \left (2+3 x \right )^{2} \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\)	\(257\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/633864*(908820*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+302940*7^(1/2)*arctan(1/14*(37
*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-580635*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-
209160*x^3*(-10*x^2-x+3)^(1/2)-100980*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-271320*x^2*
(-10*x^2-x+3)^(1/2)+100980*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+120862*x*(-10*x^2-x+3)^(
1/2)+93912*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2/(-1+2*x)^2/(-10*x^2-x+3)^(1/2)

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Maxima [A]
time = 0.53, size = 172, normalized size = 1.19 \begin {gather*} \frac {765}{19208} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {2075 \, x}{22638 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {4415}{45276 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {125 \, x}{294 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {1}{126 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {23}{252 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {5}{1764 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

765/19208*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 2075/22638*x/sqrt(-10*x^2 - x + 3) + 441
5/45276/sqrt(-10*x^2 - x + 3) + 125/294*x/(-10*x^2 - x + 3)^(3/2) - 1/126/(9*(-10*x^2 - x + 3)^(3/2)*x^2 + 12*
(-10*x^2 - x + 3)^(3/2)*x + 4*(-10*x^2 - x + 3)^(3/2)) + 23/252/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x
+ 3)^(3/2)) - 5/1764/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]
time = 0.46, size = 116, normalized size = 0.81 \begin {gather*} -\frac {25245 \, \sqrt {7} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, 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 ) + 14 \, {\left (14940 \, x^{3} + 19380 \, x^{2} - 8633 \, x - 6708\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{633864 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/633864*(25245*sqrt(7)*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sq
rt(-2*x + 1)/(10*x^2 + x - 3)) + 14*(14940*x^3 + 19380*x^2 - 8633*x - 6708)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(36*
x^4 + 12*x^3 - 23*x^2 - 4*x + 4)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (111) = 222\).
time = 1.12, size = 291, normalized size = 2.02 \begin {gather*} \frac {153}{38416} \, \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 {8 \, {\left (524 \, \sqrt {5} {\left (5 \, x + 3\right )} - 3267 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1980825 \, {\left (2 \, x - 1\right )}^{2}} - \frac {297 \, \sqrt {10} {\left (19 \, {\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 {840 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {3360 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{4802 \, {\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

153/38416*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)))) - 8/1980825*(524*sqrt(5)*(5*x + 3) - 3267*sqrt(5))*sq
rt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 - 297/4802*sqrt(10)*(19*((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 - 840*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqr
t(5*x + 3) + 3360*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/s
qrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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