3.2498 \(\int \frac{(2+3 x)^4}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=113 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^3}{55 \sqrt{5 x+3}}-\frac{21}{550} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2-\frac{21 \sqrt{1-2 x} \sqrt{5 x+3} (3660 x+8987)}{88000}+\frac{143283 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8000 \sqrt{10}} \]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(55*Sqrt[3 + 5*x]) - (21*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/550 - (21*Sqr
t[1 - 2*x]*Sqrt[3 + 5*x]*(8987 + 3660*x))/88000 + (143283*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(8000*Sqrt[10])

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Rubi [A]  time = 0.0322779, 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, 153, 147, 54, 216} \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^3}{55 \sqrt{5 x+3}}-\frac{21}{550} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2-\frac{21 \sqrt{1-2 x} \sqrt{5 x+3} (3660 x+8987)}{88000}+\frac{143283 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(55*Sqrt[3 + 5*x]) - (21*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/550 - (21*Sqr
t[1 - 2*x]*Sqrt[3 + 5*x]*(8987 + 3660*x))/88000 + (143283*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(8000*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 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} (3+5 x)^{3/2}} \, dx &=-\frac{2 \sqrt{1-2 x} (2+3 x)^3}{55 \sqrt{3+5 x}}-\frac{2}{55} \int \frac{\left (-42-\frac{63 x}{2}\right ) (2+3 x)^2}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^3}{55 \sqrt{3+5 x}}-\frac{21}{550} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}+\frac{1}{825} \int \frac{(2+3 x) \left (\frac{6111}{2}+\frac{19215 x}{4}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^3}{55 \sqrt{3+5 x}}-\frac{21}{550} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}-\frac{21 \sqrt{1-2 x} \sqrt{3+5 x} (8987+3660 x)}{88000}+\frac{143283 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{16000}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^3}{55 \sqrt{3+5 x}}-\frac{21}{550} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}-\frac{21 \sqrt{1-2 x} \sqrt{3+5 x} (8987+3660 x)}{88000}+\frac{143283 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{8000 \sqrt{5}}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^3}{55 \sqrt{3+5 x}}-\frac{21}{550} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}-\frac{21 \sqrt{1-2 x} \sqrt{3+5 x} (8987+3660 x)}{88000}+\frac{143283 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{8000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0405381, size = 69, normalized size = 0.61 \[ \frac{-10 \sqrt{1-2 x} \left (237600 x^3+849420 x^2+1477575 x+632101\right )-1576113 \sqrt{50 x+30} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{880000 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[1 - 2*x]*(632101 + 1477575*x + 849420*x^2 + 237600*x^3) - 1576113*Sqrt[30 + 50*x]*ArcSin[Sqrt[5/11]*
Sqrt[1 - 2*x]])/(880000*Sqrt[3 + 5*x])

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Maple [A]  time = 0.013, size = 116, normalized size = 1. \begin{align*}{\frac{1}{1760000} \left ( -4752000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+7880565\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-16988400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+4728339\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -29551500\,x\sqrt{-10\,{x}^{2}-x+3}-12642020\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\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/(3+5*x)^(3/2)/(1-2*x)^(1/2),x)

[Out]

1/1760000*(-4752000*x^3*(-10*x^2-x+3)^(1/2)+7880565*10^(1/2)*arcsin(20/11*x+1/11)*x-16988400*x^2*(-10*x^2-x+3)
^(1/2)+4728339*10^(1/2)*arcsin(20/11*x+1/11)-29551500*x*(-10*x^2-x+3)^(1/2)-12642020*(-10*x^2-x+3)^(1/2))*(1-2
*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 3.81278, size = 111, normalized size = 0.98 \begin{align*} -\frac{27}{50} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + \frac{143283}{160000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{3213}{2000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{95769}{40000} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{6875 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/50*sqrt(-10*x^2 - x + 3)*x^2 + 143283/160000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 3213/2000*sqrt(-10*x
^2 - x + 3)*x - 95769/40000*sqrt(-10*x^2 - x + 3) - 2/6875*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]  time = 1.79376, size = 285, normalized size = 2.52 \begin{align*} -\frac{1576113 \, \sqrt{10}{\left (5 \, 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 (237600 \, x^{3} + 849420 \, x^{2} + 1477575 \, x + 632101\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{1760000 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1760000*(1576113*sqrt(10)*(5*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 +
x - 3)) + 20*(237600*x^3 + 849420*x^2 + 1477575*x + 632101)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(5*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} \left (5 x + 3\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 2.02801, size = 167, normalized size = 1.48 \begin{align*} -\frac{27}{200000} \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + 71 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 2407 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{143283}{80000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{68750 \, \sqrt{5 \, x + 3}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{34375 \,{\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/(3+5*x)^(3/2)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

-27/200000*(4*(8*sqrt(5)*(5*x + 3) + 71*sqrt(5))*(5*x + 3) + 2407*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) + 143
283/80000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/68750*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))
/sqrt(5*x + 3) + 2/34375*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))