3.2271 \(\int \frac{\sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=151 \[ \frac{625115 \sqrt{1-2 x} \sqrt{5 x+3}}{197568 (3 x+2)}+\frac{6005 \sqrt{1-2 x} \sqrt{5 x+3}}{14112 (3 x+2)^2}+\frac{37 \sqrt{1-2 x} \sqrt{5 x+3}}{504 (3 x+2)^3}-\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{12 (3 x+2)^4}-\frac{794365 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{21952 \sqrt{7}} \]

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12*(2 + 3*x)^4) + (37*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(504*(2 + 3*x)^3) + (6005*S
qrt[1 - 2*x]*Sqrt[3 + 5*x])/(14112*(2 + 3*x)^2) + (625115*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(197568*(2 + 3*x)) - (7
94365*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(21952*Sqrt[7])

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Rubi [A]  time = 0.0500091, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {97, 151, 12, 93, 204} \[ \frac{625115 \sqrt{1-2 x} \sqrt{5 x+3}}{197568 (3 x+2)}+\frac{6005 \sqrt{1-2 x} \sqrt{5 x+3}}{14112 (3 x+2)^2}+\frac{37 \sqrt{1-2 x} \sqrt{5 x+3}}{504 (3 x+2)^3}-\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{12 (3 x+2)^4}-\frac{794365 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{21952 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(12*(2 + 3*x)^4) + (37*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(504*(2 + 3*x)^3) + (6005*S
qrt[1 - 2*x]*Sqrt[3 + 5*x])/(14112*(2 + 3*x)^2) + (625115*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(197568*(2 + 3*x)) - (7
94365*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(21952*Sqrt[7])

Rule 97

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)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 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 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{1-2 x} \sqrt{3+5 x}}{(2+3 x)^5} \, dx &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{12 (2+3 x)^4}+\frac{1}{12} \int \frac{-\frac{1}{2}-10 x}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{12 (2+3 x)^4}+\frac{37 \sqrt{1-2 x} \sqrt{3+5 x}}{504 (2+3 x)^3}+\frac{1}{252} \int \frac{\frac{1015}{4}-370 x}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{12 (2+3 x)^4}+\frac{37 \sqrt{1-2 x} \sqrt{3+5 x}}{504 (2+3 x)^3}+\frac{6005 \sqrt{1-2 x} \sqrt{3+5 x}}{14112 (2+3 x)^2}+\frac{\int \frac{\frac{128305}{8}-\frac{30025 x}{2}}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{3528}\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{12 (2+3 x)^4}+\frac{37 \sqrt{1-2 x} \sqrt{3+5 x}}{504 (2+3 x)^3}+\frac{6005 \sqrt{1-2 x} \sqrt{3+5 x}}{14112 (2+3 x)^2}+\frac{625115 \sqrt{1-2 x} \sqrt{3+5 x}}{197568 (2+3 x)}+\frac{\int \frac{7149285}{16 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{24696}\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{12 (2+3 x)^4}+\frac{37 \sqrt{1-2 x} \sqrt{3+5 x}}{504 (2+3 x)^3}+\frac{6005 \sqrt{1-2 x} \sqrt{3+5 x}}{14112 (2+3 x)^2}+\frac{625115 \sqrt{1-2 x} \sqrt{3+5 x}}{197568 (2+3 x)}+\frac{794365 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{43904}\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{12 (2+3 x)^4}+\frac{37 \sqrt{1-2 x} \sqrt{3+5 x}}{504 (2+3 x)^3}+\frac{6005 \sqrt{1-2 x} \sqrt{3+5 x}}{14112 (2+3 x)^2}+\frac{625115 \sqrt{1-2 x} \sqrt{3+5 x}}{197568 (2+3 x)}+\frac{794365 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{21952}\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{12 (2+3 x)^4}+\frac{37 \sqrt{1-2 x} \sqrt{3+5 x}}{504 (2+3 x)^3}+\frac{6005 \sqrt{1-2 x} \sqrt{3+5 x}}{14112 (2+3 x)^2}+\frac{625115 \sqrt{1-2 x} \sqrt{3+5 x}}{197568 (2+3 x)}-\frac{794365 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{21952 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0552367, size = 79, normalized size = 0.52 \[ \frac{\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (1875345 x^3+3834760 x^2+2617388 x+594416\right )}{(3 x+2)^4}-794365 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{153664} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(594416 + 2617388*x + 3834760*x^2 + 1875345*x^3))/(2 + 3*x)^4 - 794365*Sqrt[7]
*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/153664

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Maple [B]  time = 0.011, size = 250, normalized size = 1.7 \begin{align*}{\frac{1}{307328\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 64343565\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+53686640\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}+8321824\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/307328*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(64343565*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+
171582840*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+171582840*7^(1/2)*arctan(1/14*(37*x+2
0)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+26254830*x^3*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^
(1/2)/(-10*x^2-x+3)^(1/2))*x+53686640*x^2*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(
-10*x^2-x+3)^(1/2))[phone]*x*(-10*x^2-x+3)^(1/2)+8321824*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^4

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Maxima [A]  time = 1.85055, size = 212, normalized size = 1.4 \begin{align*} \frac{794365}{307328} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{32825}{16464} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{28 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{185 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{392 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{19695 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{10976 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{242905 \, \sqrt{-10 \, x^{2} - x + 3}}{65856 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

794365/307328*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 32825/16464*sqrt(-10*x^2 - x + 3) +
3/28*(-10*x^2 - x + 3)^(3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 185/392*(-10*x^2 - x + 3)^(3/2)/(27*x^
3 + 54*x^2 + 36*x + 8) + 19695/10976*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 242905/65856*sqrt(-10*x^2 -
x + 3)/(3*x + 2)

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Fricas [A]  time = 1.82874, size = 367, normalized size = 2.43 \begin{align*} -\frac{794365 \, \sqrt{7}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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 (1875345 \, x^{3} + 3834760 \, x^{2} + 2617388 \, x + 594416\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{307328 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/307328*(794365*sqrt(7)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x +
3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(1875345*x^3 + 3834760*x^2 + 2617388*x + 594416)*sqrt(5*x + 3)*sqrt(-
2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 3.68028, size = 504, normalized size = 3.34 \begin{align*} \frac{121}{614656} \, \sqrt{5}{\left (1313 \, \sqrt{70} \sqrt{2}{\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{280 \, \sqrt{2}{\left (1313 \,{\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 )}^{7} - 1578920 \,{\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} - 374767680 \,{\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{28822976000 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} + \frac{115291904000 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\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 )}^{4}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

121/614656*sqrt(5)*(1313*sqrt(70)*sqrt(2)*(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)))) - 280*sqrt(2)*(1313*((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)))^7 - 1578920*((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 - 3747676
80*((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 - 28822976000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 115291904000*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)^4)