3.2325 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=137 \[ \frac{608185 \sqrt{1-2 x}}{924 \sqrt{5 x+3}}-\frac{6095 \sqrt{1-2 x}}{84 (5 x+3)^{3/2}}+\frac{243 \sqrt{1-2 x}}{28 (3 x+2) (5 x+3)^{3/2}}+\frac{\sqrt{1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{126513 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{28 \sqrt{7}} \]

[Out]

(-6095*Sqrt[1 - 2*x])/(84*(3 + 5*x)^(3/2)) + Sqrt[1 - 2*x]/(2*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (243*Sqrt[1 - 2*x
])/(28*(2 + 3*x)*(3 + 5*x)^(3/2)) + (608185*Sqrt[1 - 2*x])/(924*Sqrt[3 + 5*x]) - (126513*ArcTan[Sqrt[1 - 2*x]/
(Sqrt[7]*Sqrt[3 + 5*x])])/(28*Sqrt[7])

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Rubi [A]  time = 0.0469776, antiderivative size = 137, normalized size of antiderivative = 1., 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 = {99, 151, 152, 12, 93, 204} \[ \frac{608185 \sqrt{1-2 x}}{924 \sqrt{5 x+3}}-\frac{6095 \sqrt{1-2 x}}{84 (5 x+3)^{3/2}}+\frac{243 \sqrt{1-2 x}}{28 (3 x+2) (5 x+3)^{3/2}}+\frac{\sqrt{1-2 x}}{2 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{126513 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{28 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6095*Sqrt[1 - 2*x])/(84*(3 + 5*x)^(3/2)) + Sqrt[1 - 2*x]/(2*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (243*Sqrt[1 - 2*x
])/(28*(2 + 3*x)*(3 + 5*x)^(3/2)) + (608185*Sqrt[1 - 2*x])/(924*Sqrt[3 + 5*x]) - (126513*ArcTan[Sqrt[1 - 2*x]/
(Sqrt[7]*Sqrt[3 + 5*x])])/(28*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{1-2 x}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx &=\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)^{3/2}}-\frac{1}{2} \int \frac{-\frac{41}{2}+30 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx\\ &=\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{243 \sqrt{1-2 x}}{28 (2+3 x) (3+5 x)^{3/2}}-\frac{1}{14} \int \frac{-\frac{7577}{4}+2430 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac{6095 \sqrt{1-2 x}}{84 (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{243 \sqrt{1-2 x}}{28 (2+3 x) (3+5 x)^{3/2}}+\frac{1}{231} \int \frac{-\frac{855283}{8}+\frac{201135 x}{2}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac{6095 \sqrt{1-2 x}}{84 (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{243 \sqrt{1-2 x}}{28 (2+3 x) (3+5 x)^{3/2}}+\frac{608185 \sqrt{1-2 x}}{924 \sqrt{3+5 x}}-\frac{2 \int -\frac{45924219}{16 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{2541}\\ &=-\frac{6095 \sqrt{1-2 x}}{84 (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{243 \sqrt{1-2 x}}{28 (2+3 x) (3+5 x)^{3/2}}+\frac{608185 \sqrt{1-2 x}}{924 \sqrt{3+5 x}}+\frac{126513}{56} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{6095 \sqrt{1-2 x}}{84 (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{243 \sqrt{1-2 x}}{28 (2+3 x) (3+5 x)^{3/2}}+\frac{608185 \sqrt{1-2 x}}{924 \sqrt{3+5 x}}+\frac{126513}{28} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{6095 \sqrt{1-2 x}}{84 (3+5 x)^{3/2}}+\frac{\sqrt{1-2 x}}{2 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{243 \sqrt{1-2 x}}{28 (2+3 x) (3+5 x)^{3/2}}+\frac{608185 \sqrt{1-2 x}}{924 \sqrt{3+5 x}}-\frac{126513 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{28 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0643506, size = 79, normalized size = 0.58 \[ \frac{\sqrt{1-2 x} \left (27368325 x^3+52308690 x^2+33277877 x+7046540\right )}{924 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{126513 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{28 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(7046540 + 33277877*x + 52308690*x^2 + 27368325*x^3))/(924*(2 + 3*x)^2*(3 + 5*x)^(3/2)) - (1265
13*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(28*Sqrt[7])

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Maple [B]  time = 0.013, size = 250, normalized size = 1.8 \begin{align*}{\frac{1}{12936\, \left ( 2+3\,x \right ) ^{2}} \left ( 939359025\,\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+732321660\,{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}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12936*(939359025*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+2379709530*7^(1/2)*arctan(1/
14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+2258636589*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(
1/2))*x^2+383156550*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+732321660*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))+46
5890278*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(2+3*x)^2/(-10*x^2-x+3)^(1/2)/(3+5*x
)^(3/2)

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Maxima [A]  time = 1.95239, size = 232, normalized size = 1.69 \begin{align*} \frac{126513}{392} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{608185 \, x}{462 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{635003}{924 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1985 \, x}{6 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{49}{18 \,{\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{1645}{36 \,{\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{6433}{36 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

126513/392*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 608185/462*x/sqrt(-10*x^2 - x + 3) + 63
5003/924/sqrt(-10*x^2 - x + 3) + 1985/6*x/(-10*x^2 - x + 3)^(3/2) + 49/18/(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)) + 1645/36/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x
+ 3)^(3/2)) - 6433/36/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 1.59723, size = 378, normalized size = 2.76 \begin{align*} -\frac{4174929 \, \sqrt{7}{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\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 (27368325 \, x^{3} + 52308690 \, x^{2} + 33277877 \, x + 7046540\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{12936 \,{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12936*(4174929*sqrt(7)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x
+ 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(27368325*x^3 + 52308690*x^2 + 33277877*x + 7046540)*sqrt(5*x + 3)*
sqrt(-2*x + 1))/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 2.64839, size = 509, normalized size = 3.72 \begin{align*} -\frac{1}{129360} \, \sqrt{5}{\left (1225 \, \sqrt{2}{\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} - 4174929 \, \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 )} - 2910600 \, \sqrt{2}{\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 )} - \frac{2744280 \, \sqrt{2}{\left (151 \,{\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{36120 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} - \frac{144480 \, \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 )}^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/129360*sqrt(5)*(1225*sqrt(2)*((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 - 4174929*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)))) - 2910600*sqrt(2)*((sq
rt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) - 2744
280*sqrt(2)*(151*((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 + 36120*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 144480*sqrt(5*x + 3)/(sqrt(2)*sq
rt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqr
t(-10*x + 5) - sqrt(22)))^2 + 280)^2)