3.2444 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=144 \[ \frac{(1-2 x)^{7/2}}{7 (3 x+2)^3 \sqrt{5 x+3}}+\frac{81 (1-2 x)^{5/2}}{28 (3 x+2)^2 \sqrt{5 x+3}}+\frac{4455 (1-2 x)^{3/2}}{56 (3 x+2) \sqrt{5 x+3}}-\frac{147015 \sqrt{1-2 x}}{56 \sqrt{5 x+3}}+\frac{147015 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{8 \sqrt{7}} \]

[Out]

(-147015*Sqrt[1 - 2*x])/(56*Sqrt[3 + 5*x]) + (1 - 2*x)^(7/2)/(7*(2 + 3*x)^3*Sqrt[3 + 5*x]) + (81*(1 - 2*x)^(5/
2))/(28*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (4455*(1 - 2*x)^(3/2))/(56*(2 + 3*x)*Sqrt[3 + 5*x]) + (147015*ArcTan[Sqrt
[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(8*Sqrt[7])

________________________________________________________________________________________

Rubi [A]  time = 0.0402417, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {96, 94, 93, 204} \[ \frac{(1-2 x)^{7/2}}{7 (3 x+2)^3 \sqrt{5 x+3}}+\frac{81 (1-2 x)^{5/2}}{28 (3 x+2)^2 \sqrt{5 x+3}}+\frac{4455 (1-2 x)^{3/2}}{56 (3 x+2) \sqrt{5 x+3}}-\frac{147015 \sqrt{1-2 x}}{56 \sqrt{5 x+3}}+\frac{147015 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{8 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(5/2)/((2 + 3*x)^4*(3 + 5*x)^(3/2)),x]

[Out]

(-147015*Sqrt[1 - 2*x])/(56*Sqrt[3 + 5*x]) + (1 - 2*x)^(7/2)/(7*(2 + 3*x)^3*Sqrt[3 + 5*x]) + (81*(1 - 2*x)^(5/
2))/(28*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (4455*(1 - 2*x)^(3/2))/(56*(2 + 3*x)*Sqrt[3 + 5*x]) + (147015*ArcTan[Sqrt
[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(8*Sqrt[7])

Rule 96

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[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 94

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[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

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)^{5/2}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx &=\frac{(1-2 x)^{7/2}}{7 (2+3 x)^3 \sqrt{3+5 x}}+\frac{81}{14} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^{3/2}} \, dx\\ &=\frac{(1-2 x)^{7/2}}{7 (2+3 x)^3 \sqrt{3+5 x}}+\frac{81 (1-2 x)^{5/2}}{28 (2+3 x)^2 \sqrt{3+5 x}}+\frac{4455}{56} \int \frac{(1-2 x)^{3/2}}{(2+3 x)^2 (3+5 x)^{3/2}} \, dx\\ &=\frac{(1-2 x)^{7/2}}{7 (2+3 x)^3 \sqrt{3+5 x}}+\frac{81 (1-2 x)^{5/2}}{28 (2+3 x)^2 \sqrt{3+5 x}}+\frac{4455 (1-2 x)^{3/2}}{56 (2+3 x) \sqrt{3+5 x}}+\frac{147015}{112} \int \frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac{147015 \sqrt{1-2 x}}{56 \sqrt{3+5 x}}+\frac{(1-2 x)^{7/2}}{7 (2+3 x)^3 \sqrt{3+5 x}}+\frac{81 (1-2 x)^{5/2}}{28 (2+3 x)^2 \sqrt{3+5 x}}+\frac{4455 (1-2 x)^{3/2}}{56 (2+3 x) \sqrt{3+5 x}}-\frac{147015}{16} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{147015 \sqrt{1-2 x}}{56 \sqrt{3+5 x}}+\frac{(1-2 x)^{7/2}}{7 (2+3 x)^3 \sqrt{3+5 x}}+\frac{81 (1-2 x)^{5/2}}{28 (2+3 x)^2 \sqrt{3+5 x}}+\frac{4455 (1-2 x)^{3/2}}{56 (2+3 x) \sqrt{3+5 x}}-\frac{147015}{8} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{147015 \sqrt{1-2 x}}{56 \sqrt{3+5 x}}+\frac{(1-2 x)^{7/2}}{7 (2+3 x)^3 \sqrt{3+5 x}}+\frac{81 (1-2 x)^{5/2}}{28 (2+3 x)^2 \sqrt{3+5 x}}+\frac{4455 (1-2 x)^{3/2}}{56 (2+3 x) \sqrt{3+5 x}}+\frac{147015 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{8 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0600482, size = 79, normalized size = 0.55 \[ \frac{147015 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{8 \sqrt{7}}-\frac{\sqrt{1-2 x} \left (578245 x^3+1143741 x^2+753654 x+165424\right )}{8 (3 x+2)^3 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^4*(3 + 5*x)^(3/2)),x]

[Out]

-(Sqrt[1 - 2*x]*(165424 + 753654*x + 1143741*x^2 + 578245*x^3))/(8*(2 + 3*x)^3*Sqrt[3 + 5*x]) + (147015*ArcTan
[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(8*Sqrt[7])

________________________________________________________________________________________

Maple [B]  time = 0.014, size = 250, normalized size = 1.7 \begin{align*} -{\frac{1}{112\, \left ( 2+3\,x \right ) ^{3}} \left ( 19847025\,\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}+8095430\,{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+16012374\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+3528360\,\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}+2315936\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(5/2)/(2+3*x)^4/(3+5*x)^(3/2),x)

[Out]

-1/112*(19847025*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+51602265*7^(1/2)*arctan(1/14*(
37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+50279130*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*
x^2+8095430*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+1601
2374*x^2*(-10*x^2-x+3)^(1/2)+3528360*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)+2315936*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(2+3*x)^3/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 3.182, size = 285, normalized size = 1.98 \begin{align*} -\frac{147015}{112} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{578245 \, x}{108 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{603743}{216 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{343}{81 \,{\left (27 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt{-10 \, x^{2} - x + 3} x + 8 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{10339}{324 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{87199}{216 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^4/(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

-147015/112*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 578245/108*x/sqrt(-10*x^2 - x + 3) - 6
03743/216/sqrt(-10*x^2 - x + 3) + 343/81/(27*sqrt(-10*x^2 - x + 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^2 + 36*sqr
t(-10*x^2 - x + 3)*x + 8*sqrt(-10*x^2 - x + 3)) + 10339/324/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x
+ 3)*x + 4*sqrt(-10*x^2 - x + 3)) + 87199/216/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

________________________________________________________________________________________

Fricas [A]  time = 1.81852, size = 365, normalized size = 2.53 \begin{align*} \frac{147015 \, \sqrt{7}{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\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 (578245 \, x^{3} + 1143741 \, x^{2} + 753654 \, x + 165424\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{112 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^4/(3+5*x)^(3/2),x, algorithm="fricas")

[Out]

1/112*(147015*sqrt(7)*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)
*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(578245*x^3 + 1143741*x^2 + 753654*x + 165424)*sqrt(5*x + 3)*sqrt(-2*x
+ 1))/(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)

________________________________________________________________________________________

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)**(5/2)/(2+3*x)**4/(3+5*x)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 2.36906, size = 509, normalized size = 3.53 \begin{align*} -\frac{29403}{224} \, \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{121}{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{121 \,{\left (993 \, \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 )}^{5} + 436800 \, \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} + 51352000 \, \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 )}}{4 \,{\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 )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^4/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

-29403/224*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)))) - 121/2*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))) - 121/4*(993*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)))^5 + 436800*sqrt(1
0)*((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 + 51352000*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)^3