3.2596 \(\int \frac{(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx\)

Optimal. Leaf size=144 \[ \frac{5 \sqrt{5 x+3}}{42 \sqrt{1-2 x}}-\frac{5 \sqrt{5 x+3}}{28 \sqrt{1-2 x} (3 x+2)}-\frac{3 \sqrt{5 x+3}}{14 \sqrt{1-2 x} (3 x+2)^2}+\frac{11 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}-\frac{5 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{28 \sqrt{7}} \]

[Out]

(5*Sqrt[3 + 5*x])/(42*Sqrt[1 - 2*x]) + (11*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^2) - (3*Sqrt[3 + 5*x])
/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (5*Sqrt[3 + 5*x])/(28*Sqrt[1 - 2*x]*(2 + 3*x)) - (5*ArcTan[Sqrt[1 - 2*x]/(Sq
rt[7]*Sqrt[3 + 5*x])])/(28*Sqrt[7])

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Rubi [A]  time = 0.0475858, antiderivative size = 144, 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} \[ \frac{5 \sqrt{5 x+3}}{42 \sqrt{1-2 x}}-\frac{5 \sqrt{5 x+3}}{28 \sqrt{1-2 x} (3 x+2)}-\frac{3 \sqrt{5 x+3}}{14 \sqrt{1-2 x} (3 x+2)^2}+\frac{11 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^2}-\frac{5 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{28 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*Sqrt[3 + 5*x])/(42*Sqrt[1 - 2*x]) + (11*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^2) - (3*Sqrt[3 + 5*x])
/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (5*Sqrt[3 + 5*x])/(28*Sqrt[1 - 2*x]*(2 + 3*x)) - (5*ArcTan[Sqrt[1 - 2*x]/(Sq
rt[7]*Sqrt[3 + 5*x])])/(28*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{(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^3} \, dx &=\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac{1}{21} \int \frac{-134-\frac{465 x}{2}}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt{3+5 x}} \, dx\\ &=\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac{3 \sqrt{3+5 x}}{14 \sqrt{1-2 x} (2+3 x)^2}-\frac{1}{294} \int \frac{-\frac{1435}{2}-1260 x}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}} \, dx\\ &=\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac{3 \sqrt{3+5 x}}{14 \sqrt{1-2 x} (2+3 x)^2}-\frac{5 \sqrt{3+5 x}}{28 \sqrt{1-2 x} (2+3 x)}-\frac{\int \frac{-\frac{11515}{4}-3675 x}{(1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}} \, dx}{2058}\\ &=\frac{5 \sqrt{3+5 x}}{42 \sqrt{1-2 x}}+\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac{3 \sqrt{3+5 x}}{14 \sqrt{1-2 x} (2+3 x)^2}-\frac{5 \sqrt{3+5 x}}{28 \sqrt{1-2 x} (2+3 x)}+\frac{\int \frac{56595}{8 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{79233}\\ &=\frac{5 \sqrt{3+5 x}}{42 \sqrt{1-2 x}}+\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac{3 \sqrt{3+5 x}}{14 \sqrt{1-2 x} (2+3 x)^2}-\frac{5 \sqrt{3+5 x}}{28 \sqrt{1-2 x} (2+3 x)}+\frac{5}{56} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=\frac{5 \sqrt{3+5 x}}{42 \sqrt{1-2 x}}+\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac{3 \sqrt{3+5 x}}{14 \sqrt{1-2 x} (2+3 x)^2}-\frac{5 \sqrt{3+5 x}}{28 \sqrt{1-2 x} (2+3 x)}+\frac{5}{28} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=\frac{5 \sqrt{3+5 x}}{42 \sqrt{1-2 x}}+\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac{3 \sqrt{3+5 x}}{14 \sqrt{1-2 x} (2+3 x)^2}-\frac{5 \sqrt{3+5 x}}{28 \sqrt{1-2 x} (2+3 x)}-\frac{5 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{28 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0574508, size = 95, normalized size = 0.66 \[ -\frac{7 \sqrt{5 x+3} \left (180 x^3+60 x^2-91 x-36\right )-15 \sqrt{7-14 x} (2 x-1) (3 x+2)^2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{588 (1-2 x)^{3/2} (3 x+2)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(7*Sqrt[3 + 5*x]*(-36 - 91*x + 60*x^2 + 180*x^3) - 15*Sqrt[7 - 14*x]*(-1 + 2*x)*(2 + 3*x)^2*ArcTan[Sqrt[1 - 2
*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(588*(1 - 2*x)^(3/2)*(2 + 3*x)^2)

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Maple [B]  time = 0.014, size = 257, normalized size = 1.8 \begin{align*}{\frac{1}{1176\, \left ( 2+3\,x \right ) ^{2} \left ( 2\,x-1 \right ) ^{2}} \left ( 540\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+180\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-345\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-2520\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-60\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-840\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+60\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +1274\,x\sqrt{-10\,{x}^{2}-x+3}+504\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1176*(540*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+180*7^(1/2)*arctan(1/14*(37*x+20)*7
^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-345*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-2520*x^3*(-
10*x^2-x+3)^(1/2)-60*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-840*x^2*(-10*x^2-x+3)^(1/2)+
60*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+1274*x*(-10*x^2-x+3)^(1/2)+504*(-10*x^2-x+3)^(1/
2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2/(2*x-1)^2/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.99966, size = 232, normalized size = 1.61 \begin{align*} \frac{5}{392} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{25 \, x}{42 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{5}{84 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{125 \, x}{126 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{1}{378 \,{\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{43}{756 \,{\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{205}{252 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/392*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 25/42*x/sqrt(-10*x^2 - x + 3) + 5/84/sqrt(-1
0*x^2 - x + 3) + 125/126*x/(-10*x^2 - x + 3)^(3/2) + 1/378/(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)) - 43/756/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) + 2
05/252/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 1.57379, size = 324, normalized size = 2.25 \begin{align*} -\frac{15 \, \sqrt{7}{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\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 (180 \, x^{3} + 60 \, x^{2} - 91 \, x - 36\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{1176 \,{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1176*(15*sqrt(7)*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2
*x + 1)/(10*x^2 + x - 3)) + 14*(180*x^3 + 60*x^2 - 91*x - 36)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(36*x^4 + 12*x^3 -
23*x^2 - 4*x + 4)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)**(3/2)/(1-2*x)**(5/2)/(2+3*x)**3,x)

[Out]

Exception raised: ValueError

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Giac [B]  time = 4.09849, size = 400, normalized size = 2.78 \begin{align*} \frac{1}{784} \, \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{8 \,{\left (157 \, \sqrt{5}{\left (5 \, x + 3\right )} - 1056 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{180075 \,{\left (2 \, x - 1\right )}^{2}} - \frac{33 \,{\left (83 \, \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} + 41720 \, \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 )}}{4802 \,{\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((3+5*x)^(3/2)/(1-2*x)^(5/2)/(2+3*x)^3,x, algorithm="giac")

[Out]

1/784*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)))) - 8/180075*(157*sqrt(5)*(5*x + 3) - 1056*sqrt(5))*sqrt(5*
x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 - 33/4802*(83*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 + 41720*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