3.2453 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=101 \[ \frac{(1-2 x)^{5/2}}{(3 x+2) (5 x+3)^{3/2}}-\frac{55 (1-2 x)^{3/2}}{3 (5 x+3)^{3/2}}+\frac{385 \sqrt{1-2 x}}{\sqrt{5 x+3}}-385 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(-55*(1 - 2*x)^(3/2))/(3*(3 + 5*x)^(3/2)) + (1 - 2*x)^(5/2)/((2 + 3*x)*(3 + 5*x)^(3/2)) + (385*Sqrt[1 - 2*x])/
Sqrt[3 + 5*x] - 385*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]

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Rubi [A]  time = 0.0304279, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {94, 93, 204} \[ \frac{(1-2 x)^{5/2}}{(3 x+2) (5 x+3)^{3/2}}-\frac{55 (1-2 x)^{3/2}}{3 (5 x+3)^{3/2}}+\frac{385 \sqrt{1-2 x}}{\sqrt{5 x+3}}-385 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-55*(1 - 2*x)^(3/2))/(3*(3 + 5*x)^(3/2)) + (1 - 2*x)^(5/2)/((2 + 3*x)*(3 + 5*x)^(3/2)) + (385*Sqrt[1 - 2*x])/
Sqrt[3 + 5*x] - 385*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]

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)^2 (3+5 x)^{5/2}} \, dx &=\frac{(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^{3/2}}+\frac{55}{2} \int \frac{(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac{55 (1-2 x)^{3/2}}{3 (3+5 x)^{3/2}}+\frac{(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^{3/2}}-\frac{385}{2} \int \frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac{55 (1-2 x)^{3/2}}{3 (3+5 x)^{3/2}}+\frac{(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^{3/2}}+\frac{385 \sqrt{1-2 x}}{\sqrt{3+5 x}}+\frac{2695}{2} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{55 (1-2 x)^{3/2}}{3 (3+5 x)^{3/2}}+\frac{(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^{3/2}}+\frac{385 \sqrt{1-2 x}}{\sqrt{3+5 x}}+2695 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{55 (1-2 x)^{3/2}}{3 (3+5 x)^{3/2}}+\frac{(1-2 x)^{5/2}}{(2+3 x) (3+5 x)^{3/2}}+\frac{385 \sqrt{1-2 x}}{\sqrt{3+5 x}}-385 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0383784, size = 92, normalized size = 0.91 \[ \frac{\sqrt{1-2 x} \left (17667 x^2+21988 x+6823\right )-1155 \sqrt{7} \sqrt{5 x+3} \left (15 x^2+19 x+6\right ) \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3 (3 x+2) (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(6823 + 21988*x + 17667*x^2) - 1155*Sqrt[7]*Sqrt[3 + 5*x]*(6 + 19*x + 15*x^2)*ArcTan[Sqrt[1 - 2
*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(3*(2 + 3*x)*(3 + 5*x)^(3/2))

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Maple [B]  time = 0.012, size = 202, normalized size = 2. \begin{align*}{\frac{1}{12+18\,x} \left ( 86625\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+161700\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+100485\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+35334\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+20790\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +43976\,x\sqrt{-10\,{x}^{2}-x+3}+13646\,\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)^(5/2)/(2+3*x)^2/(3+5*x)^(5/2),x)

[Out]

1/6*(86625*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+161700*7^(1/2)*arctan(1/14*(37*x+20)
*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+100485*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+35334*x^
2*(-10*x^2-x+3)^(1/2)+20790*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+43976*x*(-10*x^2-x+3)^(
1/2)+13646*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(2+3*x)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [A]  time = 2.02714, size = 186, normalized size = 1.84 \begin{align*} \frac{385}{2} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{3926 \, x}{5 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{16 \, x^{2}}{45 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{30743}{75 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{133642 \, x}{675 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{2401}{81 \,{\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{217433}{2025 \,{\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)^(5/2)/(2+3*x)^2/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

385/2*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 3926/5*x/sqrt(-10*x^2 - x + 3) - 16/45*x^2/(
-10*x^2 - x + 3)^(3/2) + 30743/75/sqrt(-10*x^2 - x + 3) + 133642/675*x/(-10*x^2 - x + 3)^(3/2) + 2401/81/(3*(-
10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) - 217433/2025/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 1.9082, size = 302, normalized size = 2.99 \begin{align*} -\frac{1155 \, \sqrt{7}{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\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 ) - 2 \,{\left (17667 \, x^{2} + 21988 \, x + 6823\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{6 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(1155*sqrt(7)*(75*x^3 + 140*x^2 + 87*x + 18)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)
/(10*x^2 + x - 3)) - 2*(17667*x^2 + 21988*x + 6823)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(75*x^3 + 140*x^2 + 87*x + 1
8)

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

[Out]

Timed out

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Giac [B]  time = 2.20965, size = 423, normalized size = 4.19 \begin{align*} -\frac{11}{1200} \, \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{77}{4} \, \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{77}{5} \, \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{1078 \, \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 )}}{{\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-11/1200*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 + 77/4*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)))) + 77/5*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))) + 1078*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)