3.2531 \(\int \frac{(2+3 x)^2 (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=116 \[ \frac{3}{20} \sqrt{1-2 x} (5 x+3)^{5/2}+\frac{49 (5 x+3)^{5/2}}{22 \sqrt{1-2 x}}+\frac{14057 \sqrt{1-2 x} (5 x+3)^{3/2}}{1760}+\frac{42171}{640} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{463881 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{640 \sqrt{10}} \]

[Out]

(42171*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/640 + (14057*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/1760 + (49*(3 + 5*x)^(5/2))/(2
2*Sqrt[1 - 2*x]) + (3*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/20 - (463881*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(640*Sqrt[
10])

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Rubi [A]  time = 0.0298161, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {89, 80, 50, 54, 216} \[ \frac{3}{20} \sqrt{1-2 x} (5 x+3)^{5/2}+\frac{49 (5 x+3)^{5/2}}{22 \sqrt{1-2 x}}+\frac{14057 \sqrt{1-2 x} (5 x+3)^{3/2}}{1760}+\frac{42171}{640} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{463881 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{640 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(42171*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/640 + (14057*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/1760 + (49*(3 + 5*x)^(5/2))/(2
2*Sqrt[1 - 2*x]) + (3*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/20 - (463881*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(640*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 80

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

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 (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx &=\frac{49 (3+5 x)^{5/2}}{22 \sqrt{1-2 x}}-\frac{1}{22} \int \frac{(3+5 x)^{3/2} \left (\frac{1343}{2}+99 x\right )}{\sqrt{1-2 x}} \, dx\\ &=\frac{49 (3+5 x)^{5/2}}{22 \sqrt{1-2 x}}+\frac{3}{20} \sqrt{1-2 x} (3+5 x)^{5/2}-\frac{14057}{440} \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\\ &=\frac{14057 \sqrt{1-2 x} (3+5 x)^{3/2}}{1760}+\frac{49 (3+5 x)^{5/2}}{22 \sqrt{1-2 x}}+\frac{3}{20} \sqrt{1-2 x} (3+5 x)^{5/2}-\frac{42171}{320} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=\frac{42171}{640} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{14057 \sqrt{1-2 x} (3+5 x)^{3/2}}{1760}+\frac{49 (3+5 x)^{5/2}}{22 \sqrt{1-2 x}}+\frac{3}{20} \sqrt{1-2 x} (3+5 x)^{5/2}-\frac{463881 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{1280}\\ &=\frac{42171}{640} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{14057 \sqrt{1-2 x} (3+5 x)^{3/2}}{1760}+\frac{49 (3+5 x)^{5/2}}{22 \sqrt{1-2 x}}+\frac{3}{20} \sqrt{1-2 x} (3+5 x)^{5/2}-\frac{463881 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{640 \sqrt{5}}\\ &=\frac{42171}{640} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{14057 \sqrt{1-2 x} (3+5 x)^{3/2}}{1760}+\frac{49 (3+5 x)^{5/2}}{22 \sqrt{1-2 x}}+\frac{3}{20} \sqrt{1-2 x} (3+5 x)^{5/2}-\frac{463881 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{640 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0355326, size = 69, normalized size = 0.59 \[ \frac{463881 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (4800 x^3+18840 x^2+45538 x-71199\right )}{6400 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(-71199 + 45538*x + 18840*x^2 + 4800*x^3) + 463881*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1
- 2*x]])/(6400*Sqrt[1 - 2*x])

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Maple [A]  time = 0.012, size = 123, normalized size = 1.1 \begin{align*} -{\frac{1}{25600\,x-12800} \left ( -96000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+927762\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-376800\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-463881\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -910760\,x\sqrt{-10\,{x}^{2}-x+3}+1423980\,\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)^(3/2)/(1-2*x)^(3/2),x)

[Out]

-1/12800*(-96000*x^3*(-10*x^2-x+3)^(1/2)+927762*10^(1/2)*arcsin(20/11*x+1/11)*x-376800*x^2*(-10*x^2-x+3)^(1/2)
-463881*10^(1/2)*arcsin(20/11*x+1/11)-910760*x*(-10*x^2-x+3)^(1/2)+1423980*(-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 [C]  time = 2.43785, size = 208, normalized size = 1.79 \begin{align*} -\frac{23793}{640} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{11979}{12800} i \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x - \frac{21}{11}\right ) - \frac{3}{8} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{99}{32} \, \sqrt{10 \, x^{2} - 21 \, x + 8} x - \frac{2079}{640} \, \sqrt{10 \, x^{2} - 21 \, x + 8} + \frac{693}{32} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{49 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{8 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac{21 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{8 \,{\left (2 \, x - 1\right )}} - \frac{1617 \, \sqrt{-10 \, x^{2} - x + 3}}{16 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-23793/640*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 11979/12800*I*sqrt(5)*sqrt(2)*arcsin(20/11*x - 21/11) - 3/
8*(-10*x^2 - x + 3)^(3/2) + 99/32*sqrt(10*x^2 - 21*x + 8)*x - 2079/640*sqrt(10*x^2 - 21*x + 8) + 693/32*sqrt(-
10*x^2 - x + 3) - 49/8*(-10*x^2 - x + 3)^(3/2)/(4*x^2 - 4*x + 1) - 21/8*(-10*x^2 - x + 3)^(3/2)/(2*x - 1) - 16
17/16*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 1.84515, size = 271, normalized size = 2.34 \begin{align*} \frac{463881 \, \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 (4800 \, x^{3} + 18840 \, x^{2} + 45538 \, x - 71199\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{12800 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12800*(463881*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*(4800*x^3 + 18840*x^2 + 45538*x - 71199)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2*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)**(3/2)/(1-2*x)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.23109, size = 113, normalized size = 0.97 \begin{align*} -\frac{463881}{6400} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (2 \,{\left (12 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + 85 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 14057 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 463881 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{16000 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-463881/6400*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/16000*(2*(12*(8*sqrt(5)*(5*x + 3) + 85*sqrt(5))*
(5*x + 3) + 14057*sqrt(5))*(5*x + 3) - 463881*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)