3.2345 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=151 \[ \frac{11 \sqrt{1-2 x} (5 x+3)^{5/2}}{8 (3 x+2)^3}+\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{4 (3 x+2)^4}-\frac{121 \sqrt{1-2 x} (5 x+3)^{3/2}}{224 (3 x+2)^2}-\frac{3993 \sqrt{1-2 x} \sqrt{5 x+3}}{3136 (3 x+2)}-\frac{43923 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3136 \sqrt{7}} \]

[Out]

(-3993*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3136*(2 + 3*x)) - (121*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(224*(2 + 3*x)^2) +
((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(4*(2 + 3*x)^4) + (11*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(8*(2 + 3*x)^3) - (439
23*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(3136*Sqrt[7])

________________________________________________________________________________________

Rubi [A]  time = 0.0417811, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {94, 93, 204} \[ \frac{11 \sqrt{1-2 x} (5 x+3)^{5/2}}{8 (3 x+2)^3}+\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{4 (3 x+2)^4}-\frac{121 \sqrt{1-2 x} (5 x+3)^{3/2}}{224 (3 x+2)^2}-\frac{3993 \sqrt{1-2 x} \sqrt{5 x+3}}{3136 (3 x+2)}-\frac{43923 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3136 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3993*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3136*(2 + 3*x)) - (121*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(224*(2 + 3*x)^2) +
((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(4*(2 + 3*x)^4) + (11*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(8*(2 + 3*x)^3) - (439
23*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(3136*Sqrt[7])

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)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx &=\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{4 (2+3 x)^4}+\frac{33}{8} \int \frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx\\ &=\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{4 (2+3 x)^4}+\frac{11 \sqrt{1-2 x} (3+5 x)^{5/2}}{8 (2+3 x)^3}+\frac{121}{16} \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=-\frac{121 \sqrt{1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}+\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{4 (2+3 x)^4}+\frac{11 \sqrt{1-2 x} (3+5 x)^{5/2}}{8 (2+3 x)^3}+\frac{3993}{448} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=-\frac{3993 \sqrt{1-2 x} \sqrt{3+5 x}}{3136 (2+3 x)}-\frac{121 \sqrt{1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}+\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{4 (2+3 x)^4}+\frac{11 \sqrt{1-2 x} (3+5 x)^{5/2}}{8 (2+3 x)^3}+\frac{43923 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{6272}\\ &=-\frac{3993 \sqrt{1-2 x} \sqrt{3+5 x}}{3136 (2+3 x)}-\frac{121 \sqrt{1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}+\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{4 (2+3 x)^4}+\frac{11 \sqrt{1-2 x} (3+5 x)^{5/2}}{8 (2+3 x)^3}+\frac{43923 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{3136}\\ &=-\frac{3993 \sqrt{1-2 x} \sqrt{3+5 x}}{3136 (2+3 x)}-\frac{121 \sqrt{1-2 x} (3+5 x)^{3/2}}{224 (2+3 x)^2}+\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{4 (2+3 x)^4}+\frac{11 \sqrt{1-2 x} (3+5 x)^{5/2}}{8 (2+3 x)^3}-\frac{43923 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{3136 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0569035, size = 79, normalized size = 0.52 \[ \frac{\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (100159 x^3+213240 x^2+145940 x+32400\right )}{(3 x+2)^4}-43923 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{21952} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(32400 + 145940*x + 213240*x^2 + 100159*x^3))/(2 + 3*x)^4 - 43923*Sqrt[7]*ArcT
an[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/21952

________________________________________________________________________________________

Maple [B]  time = 0.01, size = 250, normalized size = 1.7 \begin{align*}{\frac{1}{43904\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 3557763\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+9487368\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+9487368\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+1402226\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+4216608\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+2985360\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+702768\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +2043160\,x\sqrt{-10\,{x}^{2}-x+3}+453600\,\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)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^5,x)

[Out]

1/43904*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(3557763*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+94
87368*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+9487368*7^(1/2)*arctan(1/14*(37*x+20)*7^(
1/2)/(-10*x^2-x+3)^(1/2))*x^2+1402226*x^3*(-10*x^2-x+3)^(1/2)+4216608*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-
10*x^2-x+3)^(1/2))*x+2985360*x^2*(-10*x^2-x+3)^(1/2)+702768*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3
)^(1/2))+2043160*x*(-10*x^2-x+3)^(1/2)+453600*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^4

________________________________________________________________________________________

Maxima [A]  time = 2.12719, size = 251, normalized size = 1.66 \begin{align*} \frac{8245}{16464} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{28 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{111 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{392 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{4947 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{10976 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{67155}{10976} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{43923}{43904} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{59169}{21952} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{19573 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{65856 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8245/16464*(-10*x^2 - x + 3)^(3/2) + 3/28*(-10*x^2 - x + 3)^(5/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 1
11/392*(-10*x^2 - x + 3)^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 4947/10976*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x
+ 4) + 67155/10976*sqrt(-10*x^2 - x + 3)*x + 43923/43904*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x
+ 2)) - 59169/21952*sqrt(-10*x^2 - x + 3) + 19573/65856*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

________________________________________________________________________________________

Fricas [A]  time = 1.58721, size = 359, normalized size = 2.38 \begin{align*} -\frac{43923 \, \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 (100159 \, x^{3} + 213240 \, x^{2} + 145940 \, x + 32400\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{43904 \,{\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)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^5,x, algorithm="fricas")

[Out]

-1/43904*(43923*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*(100159*x^3 + 213240*x^2 + 145940*x + 32400)*sqrt(5*x + 3)*sqrt(-2*x +
1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 3.5637, size = 512, normalized size = 3.39 \begin{align*} \frac{43923}{439040} \, \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{14641 \,{\left (3 \, \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 )}^{7} + 3080 \, \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} - 862400 \, \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} - 65856000 \, \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 )}}{1568 \,{\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

43923/439040*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(2
2))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 14641/1568*(3*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)))^7 + 3080*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 - 862400*
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 - 65856000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*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)*sq
rt(-10*x + 5) - sqrt(22)))^2 + 280)^4