3.2532 \(\int \frac{(2+3 x) (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=94 \[ \frac{7 (5 x+3)^{5/2}}{11 \sqrt{1-2 x}}+\frac{173}{88} \sqrt{1-2 x} (5 x+3)^{3/2}+\frac{519}{32} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{5709 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{32 \sqrt{10}} \]

[Out]

(519*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/32 + (173*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/88 + (7*(3 + 5*x)^(5/2))/(11*Sqrt[1
- 2*x]) - (5709*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(32*Sqrt[10])

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Rubi [A]  time = 0.0217582, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {78, 50, 54, 216} \[ \frac{7 (5 x+3)^{5/2}}{11 \sqrt{1-2 x}}+\frac{173}{88} \sqrt{1-2 x} (5 x+3)^{3/2}+\frac{519}{32} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{5709 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{32 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(519*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/32 + (173*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/88 + (7*(3 + 5*x)^(5/2))/(11*Sqrt[1
- 2*x]) - (5709*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(32*Sqrt[10])

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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 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) (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx &=\frac{7 (3+5 x)^{5/2}}{11 \sqrt{1-2 x}}-\frac{173}{22} \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\\ &=\frac{173}{88} \sqrt{1-2 x} (3+5 x)^{3/2}+\frac{7 (3+5 x)^{5/2}}{11 \sqrt{1-2 x}}-\frac{519}{16} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=\frac{519}{32} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{173}{88} \sqrt{1-2 x} (3+5 x)^{3/2}+\frac{7 (3+5 x)^{5/2}}{11 \sqrt{1-2 x}}-\frac{5709}{64} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{519}{32} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{173}{88} \sqrt{1-2 x} (3+5 x)^{3/2}+\frac{7 (3+5 x)^{5/2}}{11 \sqrt{1-2 x}}-\frac{5709 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{32 \sqrt{5}}\\ &=\frac{519}{32} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{173}{88} \sqrt{1-2 x} (3+5 x)^{3/2}+\frac{7 (3+5 x)^{5/2}}{11 \sqrt{1-2 x}}-\frac{5709 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{32 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0299062, size = 64, normalized size = 0.68 \[ \frac{5709 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (120 x^2+490 x-891\right )}{320 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(-891 + 490*x + 120*x^2) + 5709*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(320*Sqrt
[1 - 2*x])

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Maple [A]  time = 0.01, size = 106, normalized size = 1.1 \begin{align*} -{\frac{1}{1280\,x-640} \left ( 11418\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-2400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-5709\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -9800\,x\sqrt{-10\,{x}^{2}-x+3}+17820\,\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)^(3/2)/(1-2*x)^(3/2),x)

[Out]

-1/640*(11418*10^(1/2)*arcsin(20/11*x+1/11)*x-2400*x^2*(-10*x^2-x+3)^(1/2)-5709*10^(1/2)*arcsin(20/11*x+1/11)-
9800*x*(-10*x^2-x+3)^(1/2)+17820*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2*x-1)/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 3.53646, size = 131, normalized size = 1.39 \begin{align*} -\frac{5709}{640} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{99}{32} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{7 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{4 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{8 \,{\left (2 \, x - 1\right )}} - \frac{231 \, \sqrt{-10 \, x^{2} - x + 3}}{8 \,{\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)^(3/2)/(1-2*x)^(3/2),x, algorithm="maxima")

[Out]

-5709/640*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 99/32*sqrt(-10*x^2 - x + 3) - 7/4*(-10*x^2 - x + 3)^(3/2)/(
4*x^2 - 4*x + 1) - 3/8*(-10*x^2 - x + 3)^(3/2)/(2*x - 1) - 231/8*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 1.78273, size = 243, normalized size = 2.59 \begin{align*} \frac{5709 \, \sqrt{10}{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \,{\left (120 \, x^{2} + 490 \, x - 891\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{640 \,{\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)^(3/2)/(1-2*x)^(3/2),x, algorithm="fricas")

[Out]

1/640*(5709*sqrt(10)*(2*x - 1)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
+ 20*(120*x^2 + 490*x - 891)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2*x - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19752, size = 96, normalized size = 1.02 \begin{align*} -\frac{5709}{320} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (2 \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} + 173 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 5709 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{800 \,{\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)^(3/2)/(1-2*x)^(3/2),x, algorithm="giac")

[Out]

-5709/320*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/800*(2*(12*sqrt(5)*(5*x + 3) + 173*sqrt(5))*(5*x +
3) - 5709*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)