3.2262 \(\int \sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=157 \[ -\frac{1}{20} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^3-\frac{333 (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^2}{2000}-\frac{7 (1-2 x)^{3/2} (5 x+3)^{3/2} (140652 x+231223)}{640000}-\frac{34069301 (1-2 x)^{3/2} \sqrt{5 x+3}}{5120000}+\frac{374762311 \sqrt{1-2 x} \sqrt{5 x+3}}{51200000}+\frac{4122385421 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{51200000 \sqrt{10}} \]

[Out]

(374762311*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/51200000 - (34069301*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/5120000 - (333*(1
- 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^(3/2))/2000 - ((1 - 2*x)^(3/2)*(2 + 3*x)^3*(3 + 5*x)^(3/2))/20 - (7*(1 - 2*
x)^(3/2)*(3 + 5*x)^(3/2)*(231223 + 140652*x))/640000 + (4122385421*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(51200000
*Sqrt[10])

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Rubi [A]  time = 0.0561765, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {100, 153, 147, 50, 54, 216} \[ -\frac{1}{20} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^3-\frac{333 (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^2}{2000}-\frac{7 (1-2 x)^{3/2} (5 x+3)^{3/2} (140652 x+231223)}{640000}-\frac{34069301 (1-2 x)^{3/2} \sqrt{5 x+3}}{5120000}+\frac{374762311 \sqrt{1-2 x} \sqrt{5 x+3}}{51200000}+\frac{4122385421 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{51200000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(374762311*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/51200000 - (34069301*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/5120000 - (333*(1
- 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^(3/2))/2000 - ((1 - 2*x)^(3/2)*(2 + 3*x)^3*(3 + 5*x)^(3/2))/20 - (7*(1 - 2*
x)^(3/2)*(3 + 5*x)^(3/2)*(231223 + 140652*x))/640000 + (4122385421*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(51200000
*Sqrt[10])

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 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 \sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x} \, dx &=-\frac{1}{20} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}-\frac{1}{60} \int \left (-312-\frac{999 x}{2}\right ) \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x} \, dx\\ &=-\frac{333 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}{2000}-\frac{1}{20} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}+\frac{\int \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x} \left (\frac{77385}{2}+\frac{246141 x}{4}\right ) \, dx}{3000}\\ &=-\frac{333 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}{2000}-\frac{1}{20} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}-\frac{7 (1-2 x)^{3/2} (3+5 x)^{3/2} (231223+140652 x)}{640000}+\frac{34069301 \int \sqrt{1-2 x} \sqrt{3+5 x} \, dx}{1280000}\\ &=-\frac{34069301 (1-2 x)^{3/2} \sqrt{3+5 x}}{5120000}-\frac{333 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}{2000}-\frac{1}{20} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}-\frac{7 (1-2 x)^{3/2} (3+5 x)^{3/2} (231223+140652 x)}{640000}+\frac{374762311 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{10240000}\\ &=\frac{374762311 \sqrt{1-2 x} \sqrt{3+5 x}}{51200000}-\frac{34069301 (1-2 x)^{3/2} \sqrt{3+5 x}}{5120000}-\frac{333 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}{2000}-\frac{1}{20} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}-\frac{7 (1-2 x)^{3/2} (3+5 x)^{3/2} (231223+140652 x)}{640000}+\frac{4122385421 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{102400000}\\ &=\frac{374762311 \sqrt{1-2 x} \sqrt{3+5 x}}{51200000}-\frac{34069301 (1-2 x)^{3/2} \sqrt{3+5 x}}{5120000}-\frac{333 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}{2000}-\frac{1}{20} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}-\frac{7 (1-2 x)^{3/2} (3+5 x)^{3/2} (231223+140652 x)}{640000}+\frac{4122385421 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{51200000 \sqrt{5}}\\ &=\frac{374762311 \sqrt{1-2 x} \sqrt{3+5 x}}{51200000}-\frac{34069301 (1-2 x)^{3/2} \sqrt{3+5 x}}{5120000}-\frac{333 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}{2000}-\frac{1}{20} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}-\frac{7 (1-2 x)^{3/2} (3+5 x)^{3/2} (231223+140652 x)}{640000}+\frac{4122385421 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{51200000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0590225, size = 84, normalized size = 0.54 \[ \frac{-10 \sqrt{5 x+3} \left (1382400000 x^6+3746304000 x^5+3260908800 x^4+198117440 x^3-1377410040 x^2-1082027818 x+518122939\right )-4122385421 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{512000000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(518122939 - 1082027818*x - 1377410040*x^2 + 198117440*x^3 + 3260908800*x^4 + 3746304000*x^
5 + 1382400000*x^6) - 4122385421*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(512000000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.016, size = 138, normalized size = 0.9 \begin{align*}{\frac{1}{1024000000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 13824000000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-10362458780\,\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]

1/1024000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(13824000000*x^5*(-10*x^2-x+3)^(1/2)[phone]*x^4*(-10*x^2-x+3)^(1
/2)[phone]*x^3*(-10*x^2-x+3)^(1/2)[phone]*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x
+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)-10362458780*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.89181, size = 140, normalized size = 0.89 \begin{align*} -\frac{27}{20} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} - \frac{8397}{2000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} - \frac{853821}{160000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{2300801}{640000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{34069301}{2560000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{4122385421}{1024000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{34069301}{51200000} \, \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]

-27/20*(-10*x^2 - x + 3)^(3/2)*x^3 - 8397/2000*(-10*x^2 - x + 3)^(3/2)*x^2 - 853821/160000*(-10*x^2 - x + 3)^(
3/2)*x - 2300801/640000*(-10*x^2 - x + 3)^(3/2) + 34069301/2560000*sqrt(-10*x^2 - x + 3)*x - 4122385421/102400
0000*sqrt(10)*arcsin(-20/11*x - 1/11) + 34069301/51200000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 2.09385, size = 331, normalized size = 2.11 \begin{align*} \frac{1}{51200000} \,{\left (691200000 \, x^{5} + 2218752000 \, x^{4} + 2739830400 \, x^{3} + 1468973920 \, x^{2} + 45781940 \, x - 518122939\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{4122385421}{1024000000} \, \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]

1/51200000*(691200000*x^5 + 2218752000*x^4 + 2739830400*x^3 + 1468973920*x^2 + 45781940*x - 518122939)*sqrt(5*
x + 3)*sqrt(-2*x + 1) - 4122385421/1024000000*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(-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)**(1/2)*(3+5*x)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 1.91315, size = 427, normalized size = 2.72 \begin{align*} \frac{27}{2560000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (100 \, x - 239\right )}{\left (5 \, x + 3\right )} + 27999\right )}{\left (5 \, x + 3\right )} - 318159\right )}{\left (5 \, x + 3\right )} + 3237255\right )}{\left (5 \, x + 3\right )} - 2656665\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 29223315 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{9}{8000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 143\right )}{\left (5 \, x + 3\right )} + 9773\right )}{\left (5 \, x + 3\right )} - 136405\right )}{\left (5 \, x + 3\right )} + 60555\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 666105 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{9}{80000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{250} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{25} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \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]

27/2560000000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 239)*(5*x + 3) + 27999)*(5*x + 3) - 318159)*(5*x + 3) + 3237255
)*(5*x + 3) - 2656665)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 29223315*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) +
9/8000000*sqrt(5)*(2*(4*(8*(12*(80*x - 143)*(5*x + 3) + 9773)*(5*x + 3) - 136405)*(5*x + 3) + 60555)*sqrt(5*x
+ 3)*sqrt(-10*x + 5) - 666105*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 9/80000*sqrt(5)*(2*(4*(8*(60*x -
71)*(5*x + 3) + 2179)*(5*x + 3) - 4125)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 45375*sqrt(2)*arcsin(1/11*sqrt(22)*sq
rt(5*x + 3))) + 1/250*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 363*sqrt(2)*ar
csin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1/25*sqrt(5)*(2*(20*x + 1)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*ar
csin(1/11*sqrt(22)*sqrt(5*x + 3)))