3.2537 \(\int \frac{(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^4} \, dx\)

Optimal. Leaf size=151 \[ -\frac{415 \sqrt{1-2 x} \sqrt{5 x+3}}{8232 (3 x+2)}-\frac{145 \sqrt{1-2 x} \sqrt{5 x+3}}{588 (3 x+2)^2}-\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{3 (3 x+2)^3}+\frac{11 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^3}-\frac{2805 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

[Out]

(11*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^3) - (145*Sqrt
[1 - 2*x]*Sqrt[3 + 5*x])/(588*(2 + 3*x)^2) - (415*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(8232*(2 + 3*x)) - (2805*ArcTan
[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2744*Sqrt[7])

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Rubi [A]  time = 0.0500648, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {98, 151, 12, 93, 204} \[ -\frac{415 \sqrt{1-2 x} \sqrt{5 x+3}}{8232 (3 x+2)}-\frac{145 \sqrt{1-2 x} \sqrt{5 x+3}}{588 (3 x+2)^2}-\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{3 (3 x+2)^3}+\frac{11 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^3}-\frac{2805 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(11*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^3) - (145*Sqrt
[1 - 2*x]*Sqrt[3 + 5*x])/(588*(2 + 3*x)^2) - (415*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(8232*(2 + 3*x)) - (2805*ArcTan
[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2744*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 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)^{3/2} (2+3 x)^4} \, dx &=\frac{11 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^3}-\frac{1}{7} \int \frac{-239-\frac{815 x}{2}}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx\\ &=\frac{11 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^3}-\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{3 (2+3 x)^3}-\frac{1}{147} \int \frac{-\frac{2275}{2}-1960 x}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx\\ &=\frac{11 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^3}-\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{3 (2+3 x)^3}-\frac{145 \sqrt{1-2 x} \sqrt{3+5 x}}{588 (2+3 x)^2}-\frac{\int \frac{-\frac{12565}{4}-5075 x}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{2058}\\ &=\frac{11 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^3}-\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{3 (2+3 x)^3}-\frac{145 \sqrt{1-2 x} \sqrt{3+5 x}}{588 (2+3 x)^2}-\frac{415 \sqrt{1-2 x} \sqrt{3+5 x}}{8232 (2+3 x)}-\frac{\int -\frac{58905}{8 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{14406}\\ &=\frac{11 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^3}-\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{3 (2+3 x)^3}-\frac{145 \sqrt{1-2 x} \sqrt{3+5 x}}{588 (2+3 x)^2}-\frac{415 \sqrt{1-2 x} \sqrt{3+5 x}}{8232 (2+3 x)}+\frac{2805 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{5488}\\ &=\frac{11 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^3}-\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{3 (2+3 x)^3}-\frac{145 \sqrt{1-2 x} \sqrt{3+5 x}}{588 (2+3 x)^2}-\frac{415 \sqrt{1-2 x} \sqrt{3+5 x}}{8232 (2+3 x)}+\frac{2805 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{2744}\\ &=\frac{11 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^3}-\frac{2 \sqrt{1-2 x} \sqrt{3+5 x}}{3 (2+3 x)^3}-\frac{145 \sqrt{1-2 x} \sqrt{3+5 x}}{588 (2+3 x)^2}-\frac{415 \sqrt{1-2 x} \sqrt{3+5 x}}{8232 (2+3 x)}-\frac{2805 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{2744 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0552229, size = 90, normalized size = 0.6 \[ \frac{7 \sqrt{5 x+3} \left (2490 x^3+6135 x^2+3782 x+576\right )-2805 \sqrt{7-14 x} (3 x+2)^3 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{19208 \sqrt{1-2 x} (3 x+2)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*Sqrt[3 + 5*x]*(576 + 3782*x + 6135*x^2 + 2490*x^3) - 2805*Sqrt[7 - 14*x]*(2 + 3*x)^3*ArcTan[Sqrt[1 - 2*x]/(
Sqrt[7]*Sqrt[3 + 5*x])])/(19208*Sqrt[1 - 2*x]*(2 + 3*x)^3)

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Maple [B]  time = 0.014, size = 257, normalized size = 1.7 \begin{align*}{\frac{1}{38416\, \left ( 2+3\,x \right ) ^{3} \left ( 2\,x-1 \right ) } \left ( 151470\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+227205\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+50490\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-34860\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-56100\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-85890\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-22440\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -52948\,x\sqrt{-10\,{x}^{2}-x+3}-8064\,\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)^(3/2)/(2+3*x)^4,x)

[Out]

1/38416*(151470*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+227205*7^(1/2)*arctan(1/14*(37*
x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+50490*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-34
860*x^3*(-10*x^2-x+3)^(1/2)-56100*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-85890*x^2*(-10*
x^2-x+3)^(1/2)-22440*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-52948*x*(-10*x^2-x+3)^(1/2)-80
64*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3/(2*x-1)/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 3.64463, size = 285, normalized size = 1.89 \begin{align*} \frac{2805}{38416} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{2075 \, x}{12348 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{4415}{24696 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{1}{189 \,{\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{53}{756 \,{\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{275}{1176 \,{\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((3+5*x)^(3/2)/(1-2*x)^(3/2)/(2+3*x)^4,x, algorithm="maxima")

[Out]

2805/38416*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 2075/12348*x/sqrt(-10*x^2 - x + 3) + 44
15/24696/sqrt(-10*x^2 - x + 3) - 1/189/(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)) + 53/756/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)
*x + 4*sqrt(-10*x^2 - x + 3)) - 275/1176/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 1.8432, size = 339, normalized size = 2.25 \begin{align*} -\frac{2805 \, \sqrt{7}{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\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 (2490 \, x^{3} + 6135 \, x^{2} + 3782 \, x + 576\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{38416 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/38416*(2805*sqrt(7)*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqr
t(-2*x + 1)/(10*x^2 + x - 3)) + 14*(2490*x^3 + 6135*x^2 + 3782*x + 576)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(54*x^4
+ 81*x^3 + 18*x^2 - 20*x - 8)

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

[Out]

Exception raised: ValueError

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Giac [B]  time = 3.3482, size = 464, normalized size = 3.07 \begin{align*} \frac{561}{76832} \, \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{88 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{12005 \,{\left (2 \, x - 1\right )}} - \frac{11 \,{\left (1849 \, \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} + 1386560 \, \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} + 15601600 \, \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 )}}{9604 \,{\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((3+5*x)^(3/2)/(1-2*x)^(3/2)/(2+3*x)^4,x, algorithm="giac")

[Out]

561/76832*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)))) - 88/12005*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x
- 1) - 11/9604*(1849*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 + 1386560*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 + 15601600*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sq
rt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sq
rt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^3