3.2586 \(\int \frac{\sqrt{3+5 x}}{(1-2 x)^{5/2} (2+3 x)^2} \, dx\)

Optimal. Leaf size=115 \[ \frac{850 \sqrt{5 x+3}}{11319 \sqrt{1-2 x}}-\frac{5 \sqrt{5 x+3}}{49 \sqrt{1-2 x} (3 x+2)}+\frac{2 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)}-\frac{75 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{343 \sqrt{7}} \]

[Out]

(850*Sqrt[3 + 5*x])/(11319*Sqrt[1 - 2*x]) + (2*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)) - (5*Sqrt[3 + 5*x
])/(49*Sqrt[1 - 2*x]*(2 + 3*x)) - (75*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(343*Sqrt[7])

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Rubi [A]  time = 0.0352297, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {99, 151, 152, 12, 93, 204} \[ \frac{850 \sqrt{5 x+3}}{11319 \sqrt{1-2 x}}-\frac{5 \sqrt{5 x+3}}{49 \sqrt{1-2 x} (3 x+2)}+\frac{2 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)}-\frac{75 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{343 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(850*Sqrt[3 + 5*x])/(11319*Sqrt[1 - 2*x]) + (2*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)) - (5*Sqrt[3 + 5*x
])/(49*Sqrt[1 - 2*x]*(2 + 3*x)) - (75*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(343*Sqrt[7])

Rule 99

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 151

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] && IntegerQ[m]

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{\sqrt{3+5 x}}{(1-2 x)^{5/2} (2+3 x)^2} \, dx &=\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)}-\frac{2}{21} \int \frac{-\frac{35}{2}-30 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)}-\frac{5 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)}-\frac{2}{147} \int \frac{-\frac{275}{4}-75 x}{(1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=\frac{850 \sqrt{3+5 x}}{11319 \sqrt{1-2 x}}+\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)}-\frac{5 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)}+\frac{4 \int \frac{2475}{8 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{11319}\\ &=\frac{850 \sqrt{3+5 x}}{11319 \sqrt{1-2 x}}+\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)}-\frac{5 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)}+\frac{75}{686} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=\frac{850 \sqrt{3+5 x}}{11319 \sqrt{1-2 x}}+\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)}-\frac{5 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)}+\frac{75}{343} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=\frac{850 \sqrt{3+5 x}}{11319 \sqrt{1-2 x}}+\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)}-\frac{5 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)}-\frac{75 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{343 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0466743, size = 86, normalized size = 0.75 \[ -\frac{7 \sqrt{5 x+3} \left (5100 x^2-1460 x-1623\right )-2475 \sqrt{7-14 x} \left (6 x^2+x-2\right ) \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{79233 (1-2 x)^{3/2} (3 x+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(7*Sqrt[3 + 5*x]*(-1623 - 1460*x + 5100*x^2) - 2475*Sqrt[7 - 14*x]*(-2 + x + 6*x^2)*ArcTan[Sqrt[1 - 2*x]/(Sqr
t[7]*Sqrt[3 + 5*x])])/(79233*(1 - 2*x)^(3/2)*(2 + 3*x))

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Maple [B]  time = 0.013, size = 209, normalized size = 1.8 \begin{align*}{\frac{1}{ \left ( 316932+475398\,x \right ) \left ( 2\,x-1 \right ) ^{2}} \left ( 29700\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-9900\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-12375\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-71400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+4950\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +20440\,x\sqrt{-10\,{x}^{2}-x+3}+22722\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

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)^2,x)

[Out]

1/158466*(29700*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-9900*7^(1/2)*arctan(1/14*(37*x+
20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-12375*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-71400*
x^2*(-10*x^2-x+3)^(1/2)+4950*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+20440*x*(-10*x^2-x+3)^
(1/2)+22722*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)/(2*x-1)^2/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 3.346, size = 163, normalized size = 1.42 \begin{align*} \frac{75}{4802} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{4250 \, x}{11319 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{625}{11319 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{100 \, x}{147 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{1}{63 \,{\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{215}{441 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}

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)^2,x, algorithm="maxima")

[Out]

75/4802*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 4250/11319*x/sqrt(-10*x^2 - x + 3) + 625/1
1319/sqrt(-10*x^2 - x + 3) + 100/147*x/(-10*x^2 - x + 3)^(3/2) - 1/63/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^
2 - x + 3)^(3/2)) + 215/441/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 1.52721, size = 297, normalized size = 2.58 \begin{align*} -\frac{2475 \, \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 (5100 \, x^{2} - 1460 \, x - 1623\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{158466 \,{\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((3+5*x)^(1/2)/(1-2*x)^(5/2)/(2+3*x)^2,x, algorithm="fricas")

[Out]

-1/158466*(2475*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*(5100*x^2 - 1460*x - 1623)*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((3+5*x)**(1/2)/(1-2*x)**(5/2)/(2+3*x)**2,x)

[Out]

Exception raised: ValueError

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Giac [B]  time = 2.86561, size = 313, normalized size = 2.72 \begin{align*} \frac{15}{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{198 \, \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 (163 \, \sqrt{5}{\left (5 \, x + 3\right )} - 1089 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{282975 \,{\left (2 \, x - 1\right )}^{2}} \end{align*}

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)^2,x, algorithm="giac")

[Out]

15/9604*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)))) - 198/343*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) - 8/282975*(163*sqrt(5)*(5*x +
3) - 1089*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2