3.2615 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=108 \[ -\frac{4390 \sqrt{5 x+3}}{124509 \sqrt{1-2 x}}+\frac{3 \sqrt{5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)}-\frac{190 \sqrt{5 x+3}}{1617 (1-2 x)^{3/2}}-\frac{405 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{343 \sqrt{7}} \]

[Out]

(-190*Sqrt[3 + 5*x])/(1617*(1 - 2*x)^(3/2)) - (4390*Sqrt[3 + 5*x])/(124509*Sqrt[1 - 2*x]) + (3*Sqrt[3 + 5*x])/
(7*(1 - 2*x)^(3/2)*(2 + 3*x)) - (405*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(343*Sqrt[7])

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Rubi [A]  time = 0.0343168, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {103, 152, 12, 93, 204} \[ -\frac{4390 \sqrt{5 x+3}}{124509 \sqrt{1-2 x}}+\frac{3 \sqrt{5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)}-\frac{190 \sqrt{5 x+3}}{1617 (1-2 x)^{3/2}}-\frac{405 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{343 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-190*Sqrt[3 + 5*x])/(1617*(1 - 2*x)^(3/2)) - (4390*Sqrt[3 + 5*x])/(124509*Sqrt[1 - 2*x]) + (3*Sqrt[3 + 5*x])/
(7*(1 - 2*x)^(3/2)*(2 + 3*x)) - (405*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(343*Sqrt[7])

Rule 103

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[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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 152

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 12

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

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 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}{(1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x}} \, dx &=\frac{3 \sqrt{3+5 x}}{7 (1-2 x)^{3/2} (2+3 x)}+\frac{1}{7} \int \frac{-\frac{35}{2}-60 x}{(1-2 x)^{5/2} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{190 \sqrt{3+5 x}}{1617 (1-2 x)^{3/2}}+\frac{3 \sqrt{3+5 x}}{7 (1-2 x)^{3/2} (2+3 x)}-\frac{2 \int \frac{-\frac{655}{4}+1425 x}{(1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}} \, dx}{1617}\\ &=-\frac{190 \sqrt{3+5 x}}{1617 (1-2 x)^{3/2}}-\frac{4390 \sqrt{3+5 x}}{124509 \sqrt{1-2 x}}+\frac{3 \sqrt{3+5 x}}{7 (1-2 x)^{3/2} (2+3 x)}+\frac{4 \int \frac{147015}{8 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{124509}\\ &=-\frac{190 \sqrt{3+5 x}}{1617 (1-2 x)^{3/2}}-\frac{4390 \sqrt{3+5 x}}{124509 \sqrt{1-2 x}}+\frac{3 \sqrt{3+5 x}}{7 (1-2 x)^{3/2} (2+3 x)}+\frac{405}{686} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{190 \sqrt{3+5 x}}{1617 (1-2 x)^{3/2}}-\frac{4390 \sqrt{3+5 x}}{124509 \sqrt{1-2 x}}+\frac{3 \sqrt{3+5 x}}{7 (1-2 x)^{3/2} (2+3 x)}+\frac{405}{343} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{190 \sqrt{3+5 x}}{1617 (1-2 x)^{3/2}}-\frac{4390 \sqrt{3+5 x}}{124509 \sqrt{1-2 x}}+\frac{3 \sqrt{3+5 x}}{7 (1-2 x)^{3/2} (2+3 x)}-\frac{405 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{343 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0501094, size = 86, normalized size = 0.8 \[ -\frac{-7 \sqrt{5 x+3} \left (26340 x^2-39500 x+15321\right )-147015 \sqrt{7-14 x} \left (6 x^2+x-2\right ) \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{871563 (1-2 x)^{3/2} (3 x+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-7*Sqrt[3 + 5*x]*(15321 - 39500*x + 26340*x^2) - 147015*Sqrt[7 - 14*x]*(-2 + x + 6*x^2)*ArcTan[Sqrt[1 - 2*x]
/(Sqrt[7]*Sqrt[3 + 5*x])])/(871563*(1 - 2*x)^(3/2)*(2 + 3*x))

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Maple [B]  time = 0.016, size = 209, normalized size = 1.9 \begin{align*}{\frac{1}{ \left ( 3486252+5229378\,x \right ) \left ( 2\,x-1 \right ) ^{2}} \left ( 1764180\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-588060\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-735075\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+368760\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+294030\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -553000\,x\sqrt{-10\,{x}^{2}-x+3}+214494\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1743126*(1764180*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-588060*7^(1/2)*arctan(1/14*(
37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-735075*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+
368760*x^2*(-10*x^2-x+3)^(1/2)+294030*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-553000*x*(-10
*x^2-x+3)^(1/2)+214494*(-10*x^2-x+3)^(1/2))*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(2+3*x)/(2*x-1)^2/(-10*x^2-x+3)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{2}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.80646, size = 305, normalized size = 2.82 \begin{align*} -\frac{147015 \, \sqrt{7}{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, 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 ) - 14 \,{\left (26340 \, x^{2} - 39500 \, x + 15321\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{1743126 \,{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1743126*(147015*sqrt(7)*(12*x^3 - 4*x^2 - 5*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x
+ 1)/(10*x^2 + x - 3)) - 14*(26340*x^2 - 39500*x + 15321)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(12*x^3 - 4*x^2 - 5*x
+ 2)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Giac [B]  time = 2.4943, size = 313, normalized size = 2.9 \begin{align*} \frac{81}{9604} \, \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{594 \, \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 )}}{343 \,{\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 )}} - \frac{8 \,{\left (536 \, \sqrt{5}{\left (5 \, x + 3\right )} - 3333 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{3112725 \,{\left (2 \, x - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

81/9604*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)))) + 594/343*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) - 8/3112725*(536*sqrt(5)*(5*x
+ 3) - 3333*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2