3.2583 \(\int \frac{(2+3 x) \sqrt{3+5 x}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=74 \[ \frac{7 (5 x+3)^{3/2}}{33 (1-2 x)^{3/2}}-\frac{3 \sqrt{5 x+3}}{2 \sqrt{1-2 x}}+\frac{3}{2} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

[Out]

(-3*Sqrt[3 + 5*x])/(2*Sqrt[1 - 2*x]) + (7*(3 + 5*x)^(3/2))/(33*(1 - 2*x)^(3/2)) + (3*Sqrt[5/2]*ArcSin[Sqrt[2/1
1]*Sqrt[3 + 5*x]])/2

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Rubi [A]  time = 0.016146, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {78, 47, 54, 216} \[ \frac{7 (5 x+3)^{3/2}}{33 (1-2 x)^{3/2}}-\frac{3 \sqrt{5 x+3}}{2 \sqrt{1-2 x}}+\frac{3}{2} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Sqrt[3 + 5*x])/(2*Sqrt[1 - 2*x]) + (7*(3 + 5*x)^(3/2))/(33*(1 - 2*x)^(3/2)) + (3*Sqrt[5/2]*ArcSin[Sqrt[2/1
1]*Sqrt[3 + 5*x]])/2

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{(2+3 x) \sqrt{3+5 x}}{(1-2 x)^{5/2}} \, dx &=\frac{7 (3+5 x)^{3/2}}{33 (1-2 x)^{3/2}}-\frac{3}{2} \int \frac{\sqrt{3+5 x}}{(1-2 x)^{3/2}} \, dx\\ &=-\frac{3 \sqrt{3+5 x}}{2 \sqrt{1-2 x}}+\frac{7 (3+5 x)^{3/2}}{33 (1-2 x)^{3/2}}+\frac{15}{4} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{3 \sqrt{3+5 x}}{2 \sqrt{1-2 x}}+\frac{7 (3+5 x)^{3/2}}{33 (1-2 x)^{3/2}}+\frac{1}{2} \left (3 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )\\ &=-\frac{3 \sqrt{3+5 x}}{2 \sqrt{1-2 x}}+\frac{7 (3+5 x)^{3/2}}{33 (1-2 x)^{3/2}}+\frac{3}{2} \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )\\ \end{align*}

Mathematica [C]  time = 0.054108, size = 51, normalized size = 0.69 \[ \frac{363 \sqrt{22} \, _2F_1\left (-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};\frac{5}{11} (1-2 x)\right )+8 (5 x+3)^{3/2}}{660 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*(3 + 5*x)^(3/2) + 363*Sqrt[22]*Hypergeometric2F1[-3/2, -3/2, -1/2, (5*(1 - 2*x))/11])/(660*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.01, size = 103, normalized size = 1.4 \begin{align*}{\frac{1}{264\, \left ( 2\,x-1 \right ) ^{2}} \left ( 396\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-396\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+99\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +1072\,x\sqrt{-10\,{x}^{2}-x+3}-228\,\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((2+3*x)*(3+5*x)^(1/2)/(1-2*x)^(5/2),x)

[Out]

1/264*(396*10^(1/2)*arcsin(20/11*x+1/11)*x^2-396*10^(1/2)*arcsin(20/11*x+1/11)*x+99*10^(1/2)*arcsin(20/11*x+1/
11)+1072*x*(-10*x^2-x+3)^(1/2)-228*(-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 = 3.66334, size = 65, normalized size = 0.88 \begin{align*} \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{3 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{10 \, \sqrt{-10 \, x^{2} - x + 3}}{33 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 10/33*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 1.54992, size = 266, normalized size = 3.59 \begin{align*} -\frac{99 \, \sqrt{5} \sqrt{2}{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 4 \,{\left (268 \, x - 57\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{264 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/264*(99*sqrt(5)*sqrt(2)*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x +
1)/(10*x^2 + x - 3)) - 4*(268*x - 57)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (3 x + 2\right ) \sqrt{5 x + 3}}{\left (1 - 2 x\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 2.09578, size = 78, normalized size = 1.05 \begin{align*} \frac{3}{4} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (268 \, \sqrt{5}{\left (5 \, x + 3\right )} - 1089 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1650 \,{\left (2 \, x - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/1650*(268*sqrt(5)*(5*x + 3) - 1089*sqrt(5))*sqrt(5*x + 3)
*sqrt(-10*x + 5)/(2*x - 1)^2