3.2582 \(\int \frac{(2+3 x)^2 \sqrt{3+5 x}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=94 \[ -\frac{21 (5 x+3)^{3/2}}{11 \sqrt{1-2 x}}+\frac{49 (5 x+3)^{3/2}}{66 (1-2 x)^{3/2}}-\frac{519}{88} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{519 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8 \sqrt{10}} \]

[Out]

(-519*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/88 + (49*(3 + 5*x)^(3/2))/(66*(1 - 2*x)^(3/2)) - (21*(3 + 5*x)^(3/2))/(11*S
qrt[1 - 2*x]) + (519*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(8*Sqrt[10])

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Rubi [A]  time = 0.0267313, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {89, 78, 50, 54, 216} \[ -\frac{21 (5 x+3)^{3/2}}{11 \sqrt{1-2 x}}+\frac{49 (5 x+3)^{3/2}}{66 (1-2 x)^{3/2}}-\frac{519}{88} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{519 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-519*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/88 + (49*(3 + 5*x)^(3/2))/(66*(1 - 2*x)^(3/2)) - (21*(3 + 5*x)^(3/2))/(11*S
qrt[1 - 2*x]) + (519*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(8*Sqrt[10])

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

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)^2 \sqrt{3+5 x}}{(1-2 x)^{5/2}} \, dx &=\frac{49 (3+5 x)^{3/2}}{66 (1-2 x)^{3/2}}-\frac{1}{66} \int \frac{\sqrt{3+5 x} \left (\frac{1089}{2}+297 x\right )}{(1-2 x)^{3/2}} \, dx\\ &=\frac{49 (3+5 x)^{3/2}}{66 (1-2 x)^{3/2}}-\frac{21 (3+5 x)^{3/2}}{11 \sqrt{1-2 x}}+\frac{519}{44} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{519}{88} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{49 (3+5 x)^{3/2}}{66 (1-2 x)^{3/2}}-\frac{21 (3+5 x)^{3/2}}{11 \sqrt{1-2 x}}+\frac{519}{16} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{519}{88} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{49 (3+5 x)^{3/2}}{66 (1-2 x)^{3/2}}-\frac{21 (3+5 x)^{3/2}}{11 \sqrt{1-2 x}}+\frac{519 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{8 \sqrt{5}}\\ &=-\frac{519}{88} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{49 (3+5 x)^{3/2}}{66 (1-2 x)^{3/2}}-\frac{21 (3+5 x)^{3/2}}{11 \sqrt{1-2 x}}+\frac{519 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{8 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0543017, size = 69, normalized size = 0.73 \[ \frac{17127 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (1188 x^2-7712 x+2481\right )}{2640 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(2481 - 7712*x + 1188*x^2) + 17127*Sqrt[10 - 20*x]*(-1 + 2*x)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*
x]])/(2640*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.01, size = 120, normalized size = 1.3 \begin{align*}{\frac{1}{5280\, \left ( 2\,x-1 \right ) ^{2}} \left ( 68508\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-68508\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-23760\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+17127\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +154240\,x\sqrt{-10\,{x}^{2}-x+3}-49620\,\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)^2*(3+5*x)^(1/2)/(1-2*x)^(5/2),x)

[Out]

1/5280*(68508*10^(1/2)*arcsin(20/11*x+1/11)*x^2-68508*10^(1/2)*arcsin(20/11*x+1/11)*x-23760*x^2*(-10*x^2-x+3)^
(1/2)+17127*10^(1/2)*arcsin(20/11*x+1/11)+154240*x*(-10*x^2-x+3)^(1/2)-49620*(-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 [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((2+3*x)^2*(3+5*x)^(1/2)/(1-2*x)^(5/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 1.57679, size = 273, normalized size = 2.9 \begin{align*} -\frac{17127 \, \sqrt{10}{\left (4 \, x^{2} - 4 \, 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 (1188 \, x^{2} - 7712 \, x + 2481\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{5280 \,{\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)^2*(3+5*x)^(1/2)/(1-2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/5280*(17127*sqrt(10)*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2
+ x - 3)) + 20*(1188*x^2 - 7712*x + 2481)*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((2+3*x)**2*(3+5*x)**(1/2)/(1-2*x)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.7228, size = 96, normalized size = 1.02 \begin{align*} \frac{519}{80} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (297 \, \sqrt{5}{\left (5 \, x + 3\right )} - 11422 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 188397 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{33000 \,{\left (2 \, x - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

519/80*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/33000*(4*(297*sqrt(5)*(5*x + 3) - 11422*sqrt(5))*(5*x
+ 3) + 188397*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2