3.2454 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=137 \[ \frac{3 (1-2 x)^{7/2}}{14 (3 x+2)^2 (5 x+3)^{3/2}}+\frac{239 (1-2 x)^{5/2}}{28 (3 x+2) (5 x+3)^{3/2}}-\frac{13145 (1-2 x)^{3/2}}{84 (5 x+3)^{3/2}}+\frac{13145 \sqrt{1-2 x}}{4 \sqrt{5 x+3}}-\frac{13145}{4} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(-13145*(1 - 2*x)^(3/2))/(84*(3 + 5*x)^(3/2)) + (3*(1 - 2*x)^(7/2))/(14*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (239*(1
- 2*x)^(5/2))/(28*(2 + 3*x)*(3 + 5*x)^(3/2)) + (13145*Sqrt[1 - 2*x])/(4*Sqrt[3 + 5*x]) - (13145*Sqrt[7]*ArcTa
n[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/4

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Rubi [A]  time = 0.038895, antiderivative size = 137, 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} \[ \frac{3 (1-2 x)^{7/2}}{14 (3 x+2)^2 (5 x+3)^{3/2}}+\frac{239 (1-2 x)^{5/2}}{28 (3 x+2) (5 x+3)^{3/2}}-\frac{13145 (1-2 x)^{3/2}}{84 (5 x+3)^{3/2}}+\frac{13145 \sqrt{1-2 x}}{4 \sqrt{5 x+3}}-\frac{13145}{4} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-13145*(1 - 2*x)^(3/2))/(84*(3 + 5*x)^(3/2)) + (3*(1 - 2*x)^(7/2))/(14*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (239*(1
- 2*x)^(5/2))/(28*(2 + 3*x)*(3 + 5*x)^(3/2)) + (13145*Sqrt[1 - 2*x])/(4*Sqrt[3 + 5*x]) - (13145*Sqrt[7]*ArcTa
n[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/4

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)^3 (3+5 x)^{5/2}} \, dx &=\frac{3 (1-2 x)^{7/2}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{239}{28} \int \frac{(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^{5/2}} \, dx\\ &=\frac{3 (1-2 x)^{7/2}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{239 (1-2 x)^{5/2}}{28 (2+3 x) (3+5 x)^{3/2}}+\frac{13145}{56} \int \frac{(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac{13145 (1-2 x)^{3/2}}{84 (3+5 x)^{3/2}}+\frac{3 (1-2 x)^{7/2}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{239 (1-2 x)^{5/2}}{28 (2+3 x) (3+5 x)^{3/2}}-\frac{13145}{8} \int \frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac{13145 (1-2 x)^{3/2}}{84 (3+5 x)^{3/2}}+\frac{3 (1-2 x)^{7/2}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{239 (1-2 x)^{5/2}}{28 (2+3 x) (3+5 x)^{3/2}}+\frac{13145 \sqrt{1-2 x}}{4 \sqrt{3+5 x}}+\frac{92015}{8} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{13145 (1-2 x)^{3/2}}{84 (3+5 x)^{3/2}}+\frac{3 (1-2 x)^{7/2}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{239 (1-2 x)^{5/2}}{28 (2+3 x) (3+5 x)^{3/2}}+\frac{13145 \sqrt{1-2 x}}{4 \sqrt{3+5 x}}+\frac{92015}{4} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{13145 (1-2 x)^{3/2}}{84 (3+5 x)^{3/2}}+\frac{3 (1-2 x)^{7/2}}{14 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{239 (1-2 x)^{5/2}}{28 (2+3 x) (3+5 x)^{3/2}}+\frac{13145 \sqrt{1-2 x}}{4 \sqrt{3+5 x}}-\frac{13145}{4} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0637707, size = 78, normalized size = 0.57 \[ \frac{1}{12} \left (\frac{\sqrt{1-2 x} \left (1809585 x^3+3458634 x^2+2200321 x+465916\right )}{(3 x+2)^2 (5 x+3)^{3/2}}-39435 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[1 - 2*x]*(465916 + 2200321*x + 3458634*x^2 + 1809585*x^3))/((2 + 3*x)^2*(3 + 5*x)^(3/2)) - 39435*Sqrt[7
]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/12

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Maple [B]  time = 0.015, size = 250, normalized size = 1.8 \begin{align*}{\frac{1}{24\, \left ( 2+3\,x \right ) ^{2}} \left ( 8872875\,\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}+3619170\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+8991180\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+6917268\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+1419660\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +4400642\,x\sqrt{-10\,{x}^{2}-x+3}+931832\,\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)^3/(3+5*x)^(5/2),x)

[Out]

1/24*(8872875*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+22477950*7^(1/2)*arctan(1/14*(37*
x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+21334335*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2
+3619170*x^3*(-10*x^2-x+3)^(1/2)+8991180*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+6917268*
x^2*(-10*x^2-x+3)^(1/2)+1419660*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+4400642*x*(-10*x^2-
x+3)^(1/2)+931832*(-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 = 4.07088, size = 232, normalized size = 1.69 \begin{align*} \frac{13145}{8} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{40213 \, x}{6 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{69977}{20 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{454757 \, x}{270 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{2401}{162 \,{\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{25039}{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{1473541}{1620 \,{\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)^3/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

13145/8*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 40213/6*x/sqrt(-10*x^2 - x + 3) + 69977/20
/sqrt(-10*x^2 - x + 3) + 454757/270*x/(-10*x^2 - x + 3)^(3/2) + 2401/162/(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)) + 25039/108/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 -
x + 3)^(3/2)) - 1473541/1620/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 1.79313, size = 365, normalized size = 2.66 \begin{align*} -\frac{39435 \, \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 ) - 2 \,{\left (1809585 \, x^{3} + 3458634 \, x^{2} + 2200321 \, x + 465916\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{24 \,{\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)^(5/2)/(2+3*x)^3/(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

-1/24*(39435*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)) - 2*(1809585*x^3 + 3458634*x^2 + 2200321*x + 465916)*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)**(5/2)/(2+3*x)**3/(3+5*x)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 2.15648, size = 509, normalized size = 3.72 \begin{align*} -\frac{11}{240} \, \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{2629}{16} \, \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{1133}{10} \, \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{77 \,{\left (437 \, \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} + 103880 \, \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)^(5/2)/(2+3*x)^3/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

-11/240*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 + 2629/16*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)))) + 1133/10*sqrt(10)*((sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) + 77/2*(437*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 + 103880*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