3.2561 \(\int \frac{(2+3 x)^5}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=142 \[ \frac{7 (3 x+2)^4}{11 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{37 \sqrt{1-2 x} (3 x+2)^3}{605 \sqrt{5 x+3}}+\frac{8463 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{12100}+\frac{21 \sqrt{1-2 x} \sqrt{5 x+3} (841380 x+2027201)}{1936000}-\frac{2911419 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{16000 \sqrt{10}} \]

[Out]

(-37*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(605*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^4)/(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (8463
*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/12100 + (21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2027201 + 841380*x))/193600
0 - (2911419*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(16000*Sqrt[10])

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Rubi [A]  time = 0.0426555, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {98, 150, 153, 147, 54, 216} \[ \frac{7 (3 x+2)^4}{11 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{37 \sqrt{1-2 x} (3 x+2)^3}{605 \sqrt{5 x+3}}+\frac{8463 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{12100}+\frac{21 \sqrt{1-2 x} \sqrt{5 x+3} (841380 x+2027201)}{1936000}-\frac{2911419 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{16000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-37*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(605*Sqrt[3 + 5*x]) + (7*(2 + 3*x)^4)/(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (8463
*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/12100 + (21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2027201 + 841380*x))/193600
0 - (2911419*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(16000*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 153

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

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)^5}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac{7 (2+3 x)^4}{11 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{1}{11} \int \frac{(2+3 x)^3 \left (215+\frac{729 x}{2}\right )}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac{37 \sqrt{1-2 x} (2+3 x)^3}{605 \sqrt{3+5 x}}+\frac{7 (2+3 x)^4}{11 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{2}{605} \int \frac{(2+3 x)^2 \left (3843+\frac{25389 x}{4}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{37 \sqrt{1-2 x} (2+3 x)^3}{605 \sqrt{3+5 x}}+\frac{7 (2+3 x)^4}{11 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{8463 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{12100}+\frac{\int \frac{\left (-\frac{1353933}{4}-\frac{4417245 x}{8}\right ) (2+3 x)}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{9075}\\ &=-\frac{37 \sqrt{1-2 x} (2+3 x)^3}{605 \sqrt{3+5 x}}+\frac{7 (2+3 x)^4}{11 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{8463 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{12100}+\frac{21 \sqrt{1-2 x} \sqrt{3+5 x} (2027201+841380 x)}{1936000}-\frac{2911419 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{32000}\\ &=-\frac{37 \sqrt{1-2 x} (2+3 x)^3}{605 \sqrt{3+5 x}}+\frac{7 (2+3 x)^4}{11 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{8463 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{12100}+\frac{21 \sqrt{1-2 x} \sqrt{3+5 x} (2027201+841380 x)}{1936000}-\frac{2911419 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{16000 \sqrt{5}}\\ &=-\frac{37 \sqrt{1-2 x} (2+3 x)^3}{605 \sqrt{3+5 x}}+\frac{7 (2+3 x)^4}{11 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{8463 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{12100}+\frac{21 \sqrt{1-2 x} \sqrt{3+5 x} (2027201+841380 x)}{1936000}-\frac{2911419 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{16000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0634608, size = 86, normalized size = 0.61 \[ \frac{352281699 \sqrt{10-20 x} (5 x+3) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (15681600 x^4+75663720 x^3+208989990 x^2-169670279 x-162727423\right )}{19360000 \sqrt{1-2 x} (5 x+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(-162727423 - 169670279*x + 208989990*x^2 + 75663720*x^3 + 15681600*x^4) + 352281699*Sqrt[1
0 - 20*x]*(3 + 5*x)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(19360000*Sqrt[1 - 2*x]*(3 + 5*x))

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Maple [A]  time = 0.014, size = 154, normalized size = 1.1 \begin{align*} -{\frac{1}{77440000\,x-38720000}\sqrt{1-2\,x} \left ( -313632000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-1513274400\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-4179799800\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-1056845097\,\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)^5/(1-2*x)^(3/2)/(3+5*x)^(3/2),x)

[Out]

-1/38720000*(1-2*x)^(1/2)*(-313632000*x^4*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x^2-151
3274400*x^3*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x-4179799800*x^2*(-10*x^2-x+3)^(1/2)-1
056845097*10^(1/2)*arcsin(20/11*x+1/11)[phone]*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 = 2.69552, size = 124, normalized size = 0.87 \begin{align*} -\frac{81 \, x^{4}}{10 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{15633 \, x^{3}}{400 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{172719 \, x^{2}}{1600 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{2911419}{320000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{169670279 \, x}{1936000 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{162727423}{1936000 \, \sqrt{-10 \, x^{2} - x + 3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81/10*x^4/sqrt(-10*x^2 - x + 3) - 15633/400*x^3/sqrt(-10*x^2 - x + 3) - 172719/1600*x^2/sqrt(-10*x^2 - x + 3)
+ 2911419/320000*sqrt(10)*arcsin(-20/11*x - 1/11) + 169670279/1936000*x/sqrt(-10*x^2 - x + 3) + 162727423/193
6000/sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.76574, size = 340, normalized size = 2.39 \begin{align*} \frac{352281699 \, \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 (15681600 \, x^{4} + 75663720 \, x^{3} + 208989990 \, x^{2} - 169670279 \, x - 162727423\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{38720000 \,{\left (10 \, x^{2} + x - 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/38720000*(352281699*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*(15681600*x^4 + 75663720*x^3 + 208989990*x^2 - 169670279*x - 162727423)*sqrt(5*x + 3)*sq
rt(-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)**5/(1-2*x)**(3/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 2.42519, size = 194, normalized size = 1.37 \begin{align*} -\frac{2911419}{160000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (6534 \,{\left (12 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + 97 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 16325 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 1761451247 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{242000000 \,{\left (2 \, x - 1\right )}} - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{756250 \, \sqrt{5 \, x + 3}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{378125 \,{\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)^5/(1-2*x)^(3/2)/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

-2911419/160000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/242000000*(6534*(12*(8*sqrt(5)*(5*x + 3) + 97
*sqrt(5))*(5*x + 3) + 16325*sqrt(5))*(5*x + 3) - 1761451247*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1) -
1/756250*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 2/378125*sqrt(10)*sqrt(5*x + 3)/(sqrt(
2)*sqrt(-10*x + 5) - sqrt(22))