3.2386 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=137 \[ \frac{17825 \sqrt{1-2 x}}{12 \sqrt{5 x+3}}-\frac{655 \sqrt{1-2 x}}{4 (5 x+3)^{3/2}}+\frac{235 \sqrt{1-2 x}}{12 (3 x+2) (5 x+3)^{3/2}}+\frac{7 \sqrt{1-2 x}}{6 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{40787 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{4 \sqrt{7}} \]

[Out]

(-655*Sqrt[1 - 2*x])/(4*(3 + 5*x)^(3/2)) + (7*Sqrt[1 - 2*x])/(6*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (235*Sqrt[1 - 2
*x])/(12*(2 + 3*x)*(3 + 5*x)^(3/2)) + (17825*Sqrt[1 - 2*x])/(12*Sqrt[3 + 5*x]) - (40787*ArcTan[Sqrt[1 - 2*x]/(
Sqrt[7]*Sqrt[3 + 5*x])])/(4*Sqrt[7])

________________________________________________________________________________________

Rubi [A]  time = 0.0474379, 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 = {98, 151, 152, 12, 93, 204} \[ \frac{17825 \sqrt{1-2 x}}{12 \sqrt{5 x+3}}-\frac{655 \sqrt{1-2 x}}{4 (5 x+3)^{3/2}}+\frac{235 \sqrt{1-2 x}}{12 (3 x+2) (5 x+3)^{3/2}}+\frac{7 \sqrt{1-2 x}}{6 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{40787 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{4 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-655*Sqrt[1 - 2*x])/(4*(3 + 5*x)^(3/2)) + (7*Sqrt[1 - 2*x])/(6*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (235*Sqrt[1 - 2
*x])/(12*(2 + 3*x)*(3 + 5*x)^(3/2)) + (17825*Sqrt[1 - 2*x])/(12*Sqrt[3 + 5*x]) - (40787*ArcTan[Sqrt[1 - 2*x]/(
Sqrt[7]*Sqrt[3 + 5*x])])/(4*Sqrt[7])

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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{(1-2 x)^{3/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx &=\frac{7 \sqrt{1-2 x}}{6 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{1}{6} \int \frac{\frac{279}{2}-202 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{6 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{235 \sqrt{1-2 x}}{12 (2+3 x) (3+5 x)^{3/2}}+\frac{1}{42} \int \frac{\frac{51303}{4}-16450 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac{655 \sqrt{1-2 x}}{4 (3+5 x)^{3/2}}+\frac{7 \sqrt{1-2 x}}{6 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{235 \sqrt{1-2 x}}{12 (2+3 x) (3+5 x)^{3/2}}-\frac{1}{693} \int \frac{\frac{5790477}{8}-\frac{1361745 x}{2}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac{655 \sqrt{1-2 x}}{4 (3+5 x)^{3/2}}+\frac{7 \sqrt{1-2 x}}{6 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{235 \sqrt{1-2 x}}{12 (2+3 x) (3+5 x)^{3/2}}+\frac{17825 \sqrt{1-2 x}}{12 \sqrt{3+5 x}}+\frac{2 \int \frac{310919301}{16 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{7623}\\ &=-\frac{655 \sqrt{1-2 x}}{4 (3+5 x)^{3/2}}+\frac{7 \sqrt{1-2 x}}{6 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{235 \sqrt{1-2 x}}{12 (2+3 x) (3+5 x)^{3/2}}+\frac{17825 \sqrt{1-2 x}}{12 \sqrt{3+5 x}}+\frac{40787}{8} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{655 \sqrt{1-2 x}}{4 (3+5 x)^{3/2}}+\frac{7 \sqrt{1-2 x}}{6 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{235 \sqrt{1-2 x}}{12 (2+3 x) (3+5 x)^{3/2}}+\frac{17825 \sqrt{1-2 x}}{12 \sqrt{3+5 x}}+\frac{40787}{4} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{655 \sqrt{1-2 x}}{4 (3+5 x)^{3/2}}+\frac{7 \sqrt{1-2 x}}{6 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{235 \sqrt{1-2 x}}{12 (2+3 x) (3+5 x)^{3/2}}+\frac{17825 \sqrt{1-2 x}}{12 \sqrt{3+5 x}}-\frac{40787 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{4 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0643185, size = 79, normalized size = 0.58 \[ \frac{\sqrt{1-2 x} \left (802125 x^3+1533090 x^2+975325 x+206524\right )}{12 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{40787 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{4 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(206524 + 975325*x + 1533090*x^2 + 802125*x^3))/(12*(2 + 3*x)^2*(3 + 5*x)^(3/2)) - (40787*ArcTa
n[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(4*Sqrt[7])

________________________________________________________________________________________

Maple [B]  time = 0.012, size = 250, normalized size = 1.8 \begin{align*}{\frac{1}{168\, \left ( 2+3\,x \right ) ^{2}} \left ( 27531225\,\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+21463260\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+4404996\,\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}+2891336\,\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)^(3/2)/(2+3*x)^3/(3+5*x)^(5/2),x)

[Out]

1/168*(27531225*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+69745770*7^(1/2)*arctan(1/14*(3
7*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+66197301*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x
^2+11229750*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+2146
3260*x^2*(-10*x^2-x+3)^(1/2)+4404996*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*x*(-1
0*x^2-x+3)^(1/2)+2891336*(-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)

________________________________________________________________________________________

Maxima [A]  time = 1.60757, size = 232, normalized size = 1.69 \begin{align*} \frac{40787}{56} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{17825 \, x}{6 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{18611}{12 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{13439 \, x}{18 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{343}{54 \,{\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{11123}{108 \,{\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{1613}{4 \,{\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)^(3/2)/(2+3*x)^3/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

40787/56*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 17825/6*x/sqrt(-10*x^2 - x + 3) + 18611/1
2/sqrt(-10*x^2 - x + 3) + 13439/18*x/(-10*x^2 - x + 3)^(3/2) + 343/54/(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)) + 11123/108/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x +
3)^(3/2)) - 1613/4/(-10*x^2 - x + 3)^(3/2)

________________________________________________________________________________________

Fricas [A]  time = 1.54918, size = 366, normalized size = 2.67 \begin{align*} -\frac{122361 \, \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 (802125 \, x^{3} + 1533090 \, x^{2} + 975325 \, x + 206524\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{168 \,{\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)^(3/2)/(2+3*x)^3/(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

-1/168*(122361*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*(802125*x^3 + 1533090*x^2 + 975325*x + 206524)*sqrt(5*x + 3)*sqrt(-2*x
+ 1))/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)

________________________________________________________________________________________

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)**(3/2)/(2+3*x)**3/(3+5*x)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 3.01484, size = 509, normalized size = 3.72 \begin{align*} -\frac{1}{48} \, \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 )}^{3} + \frac{40787}{560} \, \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{101}{2} \, \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 )} + \frac{165 \,{\left (89 \, \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 )}^{3} + 21224 \, \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 )}\right )}}{2 \,{\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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*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)))^3 + 40787/560*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)))) + 101/2*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))) + 165/2*(89*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)))^3
+ 21224*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)^2