3.2489 \(\int \frac{(2+3 x)^4}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=113 \[ -\frac{3}{40} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^3-\frac{259}{800} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2-\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (77820 x+187559)}{128000}+\frac{10866247 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{128000 \sqrt{10}} \]

[Out]

(-259*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/800 - (3*Sqrt[1 - 2*x]*(2 + 3*x)^3*Sqrt[3 + 5*x])/40 - (7*Sqrt[
1 - 2*x]*Sqrt[3 + 5*x]*(187559 + 77820*x))/128000 + (10866247*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(128000*Sqrt[1
0])

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Rubi [A]  time = 0.0312711, 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 = {100, 153, 147, 54, 216} \[ -\frac{3}{40} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^3-\frac{259}{800} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2-\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (77820 x+187559)}{128000}+\frac{10866247 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{128000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-259*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/800 - (3*Sqrt[1 - 2*x]*(2 + 3*x)^3*Sqrt[3 + 5*x])/40 - (7*Sqrt[
1 - 2*x]*Sqrt[3 + 5*x]*(187559 + 77820*x))/128000 + (10866247*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(128000*Sqrt[1
0])

Rule 100

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

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)^4}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx &=-\frac{3}{40} \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}-\frac{1}{40} \int \frac{\left (-238-\frac{777 x}{2}\right ) (2+3 x)^2}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{259}{800} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}-\frac{3}{40} \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}+\frac{\int \frac{(2+3 x) \left (\frac{41769}{2}+\frac{136185 x}{4}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{1200}\\ &=-\frac{259}{800} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}-\frac{3}{40} \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}-\frac{7 \sqrt{1-2 x} \sqrt{3+5 x} (187559+77820 x)}{128000}+\frac{10866247 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{256000}\\ &=-\frac{259}{800} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}-\frac{3}{40} \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}-\frac{7 \sqrt{1-2 x} \sqrt{3+5 x} (187559+77820 x)}{128000}+\frac{10866247 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{128000 \sqrt{5}}\\ &=-\frac{259}{800} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}-\frac{3}{40} \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}-\frac{7 \sqrt{1-2 x} \sqrt{3+5 x} (187559+77820 x)}{128000}+\frac{10866247 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{128000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0442879, size = 74, normalized size = 0.65 \[ \frac{30 \sqrt{5 x+3} \left (172800 x^4+507840 x^3+627960 x^2+574442 x-518491\right )-10866247 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1280000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(30*Sqrt[3 + 5*x]*(-518491 + 574442*x + 627960*x^2 + 507840*x^3 + 172800*x^4) - 10866247*Sqrt[10 - 20*x]*ArcSi
n[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1280000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.012, size = 104, normalized size = 0.9 \begin{align*}{\frac{1}{2560000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -5184000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-17827200\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -27752400\,x\sqrt{-10\,{x}^{2}-x+3}-31109460\,\sqrt{-10\,{x}^{2}-x+3} \right ){\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)^4/(1-2*x)^(1/2)/(3+5*x)^(1/2),x)

[Out]

1/2560000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-5184000*x^3*(-10*x^2-x+3)^(1/2)-17827200*x^2*(-10*x^2-x+3)^(1/2)+10866
247*10^(1/2)*arcsin(20/11*x+1/11)-27752400*x*(-10*x^2-x+3)^(1/2)-31109460*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(
1/2)

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Maxima [A]  time = 2.17275, size = 101, normalized size = 0.89 \begin{align*} -\frac{81}{40} \, \sqrt{-10 \, x^{2} - x + 3} x^{3} - \frac{5571}{800} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} - \frac{69381}{6400} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{10866247}{2560000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{1555473}{128000} \, \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)^(1/2)/(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

-81/40*sqrt(-10*x^2 - x + 3)*x^3 - 5571/800*sqrt(-10*x^2 - x + 3)*x^2 - 69381/6400*sqrt(-10*x^2 - x + 3)*x - 1
0866247/2560000*sqrt(10)*arcsin(-20/11*x - 1/11) - 1555473/128000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.82351, size = 259, normalized size = 2.29 \begin{align*} -\frac{3}{128000} \,{\left (86400 \, x^{3} + 297120 \, x^{2} + 462540 \, x + 518491\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{10866247}{2560000} \, \sqrt{10} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/128000*(86400*x^3 + 297120*x^2 + 462540*x + 518491)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 10866247/2560000*sqrt(10
)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (3 x + 2\right )^{4}}{\sqrt{1 - 2 x} \sqrt{5 x + 3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 2.52467, size = 85, normalized size = 0.75 \begin{align*} -\frac{1}{6400000} \, \sqrt{5}{\left (6 \,{\left (12 \,{\left (8 \,{\left (180 \, x + 403\right )}{\left (5 \, x + 3\right )} + 16609\right )}{\left (5 \, x + 3\right )} + 1646339\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 54331235 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6400000*sqrt(5)*(6*(12*(8*(180*x + 403)*(5*x + 3) + 16609)*(5*x + 3) + 1646339)*sqrt(5*x + 3)*sqrt(-10*x +
5) - 54331235*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))