3.2562 \(\int \frac{(2+3 x)^4}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=113 \[ \frac{7 (3 x+2)^3}{11 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{37 \sqrt{1-2 x} (3 x+2)^2}{605 \sqrt{5 x+3}}+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} (72060 x+173063)}{96800}-\frac{35451 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{800 \sqrt{10}} \]

[Out]

(-37*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(605*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^3)/(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (3*Sq
rt[1 - 2*x]*Sqrt[3 + 5*x]*(173063 + 72060*x))/96800 - (35451*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(800*Sqrt[10])

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Rubi [A]  time = 0.0335403, antiderivative size = 113, 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 = {98, 150, 147, 54, 216} \[ \frac{7 (3 x+2)^3}{11 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{37 \sqrt{1-2 x} (3 x+2)^2}{605 \sqrt{5 x+3}}+\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} (72060 x+173063)}{96800}-\frac{35451 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{800 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^4/((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)),x]

[Out]

(-37*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(605*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^3)/(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (3*Sq
rt[1 - 2*x]*Sqrt[3 + 5*x]*(173063 + 72060*x))/96800 - (35451*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(800*Sqrt[10])

Rule 98

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

Rule 150

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

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)^4}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac{7 (2+3 x)^3}{11 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{1}{11} \int \frac{(2+3 x)^2 \left (152+\frac{519 x}{2}\right )}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac{37 \sqrt{1-2 x} (2+3 x)^2}{605 \sqrt{3+5 x}}+\frac{7 (2+3 x)^3}{11 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{2}{605} \int \frac{(2+3 x) \left (\frac{5487}{2}+\frac{18015 x}{4}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{37 \sqrt{1-2 x} (2+3 x)^2}{605 \sqrt{3+5 x}}+\frac{7 (2+3 x)^3}{11 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{3 \sqrt{1-2 x} \sqrt{3+5 x} (173063+72060 x)}{96800}-\frac{35451 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{1600}\\ &=-\frac{37 \sqrt{1-2 x} (2+3 x)^2}{605 \sqrt{3+5 x}}+\frac{7 (2+3 x)^3}{11 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{3 \sqrt{1-2 x} \sqrt{3+5 x} (173063+72060 x)}{96800}-\frac{35451 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{800 \sqrt{5}}\\ &=-\frac{37 \sqrt{1-2 x} (2+3 x)^2}{605 \sqrt{3+5 x}}+\frac{7 (2+3 x)^3}{11 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{3 \sqrt{1-2 x} \sqrt{3+5 x} (173063+72060 x)}{96800}-\frac{35451 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{800 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0668346, size = 78, normalized size = 0.69 \[ \frac{10 \left (-392040 x^3-1992870 x^2+2323271 x+2026687\right )+4289571 \sqrt{10-20 x} \sqrt{5 x+3} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{968000 \sqrt{1-2 x} \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^4/((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)),x]

[Out]

(10*(2026687 + 2323271*x - 1992870*x^2 - 392040*x^3) + 4289571*Sqrt[10 - 20*x]*Sqrt[3 + 5*x]*ArcSin[Sqrt[5/11]
*Sqrt[1 - 2*x]])/(968000*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])

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Maple [A]  time = 0.015, size = 137, normalized size = 1.2 \begin{align*} -{\frac{1}{3872000\,x-1936000}\sqrt{1-2\,x} \left ( 42895710\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-7840800\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+4289571\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-39857400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-12868713\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^4/(1-2*x)^(3/2)/(3+5*x)^(3/2),x)

[Out]

-1/1936000*(1-2*x)^(1/2)*(42895710*10^(1/2)*arcsin(20/11*x+1/11)*x^2-7840800*x^3*(-10*x^2-x+3)^(1/2)+4289571*1
0^(1/2)*arcsin(20/11*x+1/11)*x-39857400*x^2*(-10*x^2-x+3)^(1/2)-12868713*10^(1/2)*arcsin(20/11*x+1/11)+4646542
0*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))/(2*x-1)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 2.64587, size = 101, normalized size = 0.89 \begin{align*} -\frac{81 \, x^{3}}{20 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{1647 \, x^{2}}{80 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{35451}{16000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{2323271 \, x}{96800 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{2026687}{96800 \, \sqrt{-10 \, x^{2} - x + 3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^(3/2)/(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

-81/20*x^3/sqrt(-10*x^2 - x + 3) - 1647/80*x^2/sqrt(-10*x^2 - x + 3) + 35451/16000*sqrt(10)*arcsin(-20/11*x -
1/11) + 2323271/96800*x/sqrt(-10*x^2 - x + 3) + 2026687/96800/sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.83084, size = 305, normalized size = 2.7 \begin{align*} \frac{4289571 \, \sqrt{10}{\left (10 \, x^{2} + x - 3\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 (392040 \, x^{3} + 1992870 \, x^{2} - 2323271 \, x - 2026687\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{1936000 \,{\left (10 \, x^{2} + x - 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^(3/2)/(3+5*x)^(3/2),x, algorithm="fricas")

[Out]

1/1936000*(4289571*sqrt(10)*(10*x^2 + x - 3)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*
x^2 + x - 3)) + 20*(392040*x^3 + 1992870*x^2 - 2323271*x - 2026687)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(10*x^2 + x
- 3)

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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)**4/(1-2*x)**(3/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 2.46274, size = 177, normalized size = 1.57 \begin{align*} -\frac{35451}{8000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (6534 \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} + 197 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 21456431 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{12100000 \,{\left (2 \, x - 1\right )}} - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{151250 \, \sqrt{5 \, x + 3}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{75625 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^(3/2)/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

-35451/8000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/12100000*(6534*(12*sqrt(5)*(5*x + 3) + 197*sqrt(5
))*(5*x + 3) - 21456431*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) - 1/151250*sqrt(10)*(sqrt(2)*sqrt(-10
*x + 5) - sqrt(22))/sqrt(5*x + 3) + 2/75625*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))