3.2585 \(\int \frac{\sqrt{3+5 x}}{(1-2 x)^{5/2} (2+3 x)} \, dx\)

Optimal. Leaf size=79 \[ \frac{4 (5 x+3)^{3/2}}{231 (1-2 x)^{3/2}}+\frac{6 \sqrt{5 x+3}}{49 \sqrt{1-2 x}}+\frac{6 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

[Out]

(6*Sqrt[3 + 5*x])/(49*Sqrt[1 - 2*x]) + (4*(3 + 5*x)^(3/2))/(231*(1 - 2*x)^(3/2)) + (6*ArcTan[Sqrt[1 - 2*x]/(Sq
rt[7]*Sqrt[3 + 5*x])])/(49*Sqrt[7])

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Rubi [A]  time = 0.0186877, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {96, 94, 93, 204} \[ \frac{4 (5 x+3)^{3/2}}{231 (1-2 x)^{3/2}}+\frac{6 \sqrt{5 x+3}}{49 \sqrt{1-2 x}}+\frac{6 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*Sqrt[3 + 5*x])/(49*Sqrt[1 - 2*x]) + (4*(3 + 5*x)^(3/2))/(231*(1 - 2*x)^(3/2)) + (6*ArcTan[Sqrt[1 - 2*x]/(Sq
rt[7]*Sqrt[3 + 5*x])])/(49*Sqrt[7])

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{\sqrt{3+5 x}}{(1-2 x)^{5/2} (2+3 x)} \, dx &=\frac{4 (3+5 x)^{3/2}}{231 (1-2 x)^{3/2}}+\frac{3}{7} \int \frac{\sqrt{3+5 x}}{(1-2 x)^{3/2} (2+3 x)} \, dx\\ &=\frac{6 \sqrt{3+5 x}}{49 \sqrt{1-2 x}}+\frac{4 (3+5 x)^{3/2}}{231 (1-2 x)^{3/2}}-\frac{3}{49} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=\frac{6 \sqrt{3+5 x}}{49 \sqrt{1-2 x}}+\frac{4 (3+5 x)^{3/2}}{231 (1-2 x)^{3/2}}-\frac{6}{49} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=\frac{6 \sqrt{3+5 x}}{49 \sqrt{1-2 x}}+\frac{4 (3+5 x)^{3/2}}{231 (1-2 x)^{3/2}}+\frac{6 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{49 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0378538, size = 71, normalized size = 0.9 \[ -\frac{2 \left (7 \sqrt{5 x+3} (128 x-141)+99 \sqrt{7-14 x} (2 x-1) \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )}{11319 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(7*Sqrt[3 + 5*x]*(-141 + 128*x) + 99*Sqrt[7 - 14*x]*(-1 + 2*x)*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x]
)]))/(11319*(1 - 2*x)^(3/2))

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Maple [B]  time = 0.014, size = 154, normalized size = 2. \begin{align*} -{\frac{1}{11319\, \left ( 2\,x-1 \right ) ^{2}} \left ( 396\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-396\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+99\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +1792\,x\sqrt{-10\,{x}^{2}-x+3}-1974\,\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)^(1/2)/(1-2*x)^(5/2)/(2+3*x),x)

[Out]

-1/11319*(396*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-396*7^(1/2)*arctan(1/14*(37*x+20)
*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+99*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+1792*x*(-10*x^2-
x+3)^(1/2)-1974*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2*x-1)^2/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 2.773, size = 117, normalized size = 1.48 \begin{align*} -\frac{3}{343} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{640 \, x}{1617 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{1}{1617 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{55 \, x}{21 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{11}{7 \,{\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)^(1/2)/(1-2*x)^(5/2)/(2+3*x),x, algorithm="maxima")

[Out]

-3/343*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 640/1617*x/sqrt(-10*x^2 - x + 3) - 1/1617/s
qrt(-10*x^2 - x + 3) + 55/21*x/(-10*x^2 - x + 3)^(3/2) + 11/7/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 1.52147, size = 250, normalized size = 3.16 \begin{align*} \frac{99 \, \sqrt{7}{\left (4 \, x^{2} - 4 \, x + 1\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 (128 \, x - 141\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{11319 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11319*(99*sqrt(7)*(4*x^2 - 4*x + 1)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x
- 3)) - 14*(128*x - 141)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)

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

[Out]

Timed out

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Giac [A]  time = 2.44863, size = 153, normalized size = 1.94 \begin{align*} -\frac{3}{3430} \, \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{2 \,{\left (128 \, \sqrt{5}{\left (5 \, x + 3\right )} - 1089 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{40425 \,{\left (2 \, x - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/3430*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)))) - 2/40425*(128*sqrt(5)*(5*x + 3) - 1089*sqrt(5))*sqrt(5
*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2