3.2458 \(\int \frac{(2+3 x)^4 \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\)
Optimal. Leaf size=135 \[ -\frac{3}{50} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^3-\frac{987 \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^2}{4000}-\frac{21 \sqrt{1-2 x} (5 x+3)^{3/2} (92040 x+194923)}{640000}-\frac{97032047 \sqrt{1-2 x} \sqrt{5 x+3}}{2560000}+\frac{1067352517 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2560000 \sqrt{10}} \]
[Out]
(-97032047*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/2560000 - (987*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(3/2))/4000 - (3*Sq
rt[1 - 2*x]*(2 + 3*x)^3*(3 + 5*x)^(3/2))/50 - (21*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)*(194923 + 92040*x))/640000 + (
1067352517*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2560000*Sqrt[10])
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Rubi [A] time = 0.0394758, antiderivative size = 135, 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 =
{100, 153, 147, 50, 54, 216} \[ -\frac{3}{50} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^3-\frac{987 \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^2}{4000}-\frac{21 \sqrt{1-2 x} (5 x+3)^{3/2} (92040 x+194923)}{640000}-\frac{97032047 \sqrt{1-2 x} \sqrt{5 x+3}}{2560000}+\frac{1067352517 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2560000 \sqrt{10}} \]
Antiderivative was successfully verified.
[In]
Int[((2 + 3*x)^4*Sqrt[3 + 5*x])/Sqrt[1 - 2*x],x]
[Out]
(-97032047*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/2560000 - (987*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(3/2))/4000 - (3*Sq
rt[1 - 2*x]*(2 + 3*x)^3*(3 + 5*x)^(3/2))/50 - (21*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)*(194923 + 92040*x))/640000 + (
1067352517*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2560000*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 \frac{(2+3 x)^4 \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx &=-\frac{3}{50} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{3/2}-\frac{1}{50} \int \frac{\left (-308-\frac{987 x}{2}\right ) (2+3 x)^2 \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{987 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}{4000}-\frac{3}{50} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{3/2}+\frac{\int \frac{(2+3 x) \sqrt{3+5 x} \left (\frac{75929}{2}+\frac{241605 x}{4}\right )}{\sqrt{1-2 x}} \, dx}{2000}\\ &=-\frac{987 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}{4000}-\frac{3}{50} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{3/2}-\frac{21 \sqrt{1-2 x} (3+5 x)^{3/2} (194923+92040 x)}{640000}+\frac{97032047 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx}{1280000}\\ &=-\frac{97032047 \sqrt{1-2 x} \sqrt{3+5 x}}{2560000}-\frac{987 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}{4000}-\frac{3}{50} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{3/2}-\frac{21 \sqrt{1-2 x} (3+5 x)^{3/2} (194923+92040 x)}{640000}+\frac{1067352517 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{5120000}\\ &=-\frac{97032047 \sqrt{1-2 x} \sqrt{3+5 x}}{2560000}-\frac{987 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}{4000}-\frac{3}{50} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{3/2}-\frac{21 \sqrt{1-2 x} (3+5 x)^{3/2} (194923+92040 x)}{640000}+\frac{1067352517 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{2560000 \sqrt{5}}\\ &=-\frac{97032047 \sqrt{1-2 x} \sqrt{3+5 x}}{2560000}-\frac{987 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}{4000}-\frac{3}{50} \sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{3/2}-\frac{21 \sqrt{1-2 x} (3+5 x)^{3/2} (194923+92040 x)}{640000}+\frac{1067352517 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{2560000 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.049347, size = 79, normalized size = 0.59 \[ \frac{10 \sqrt{5 x+3} \left (41472000 x^5+143942400 x^4+209949120 x^3+180193080 x^2+151669786 x-157419203\right )-1067352517 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25600000 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
[In]
Integrate[((2 + 3*x)^4*Sqrt[3 + 5*x])/Sqrt[1 - 2*x],x]
[Out]
(10*Sqrt[3 + 5*x]*(-157419203 + 151669786*x + 180193080*x^2 + 209949120*x^3 + 143942400*x^4 + 41472000*x^5) -
1067352517*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(25600000*Sqrt[1 - 2*x])
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Maple [A] time = 0.011, size = 121, normalized size = 0.9 \begin{align*}{\frac{1}{51200000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -414720000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-1646784000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-2922883200\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+1067352517\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -3263372400\,x\sqrt{-10\,{x}^{2}-x+3}-3148384060\,\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*(3+5*x)^(1/2)/(1-2*x)^(1/2),x)
[Out]
1/51200000*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(-414720000*x^4*(-10*x^2-x+3)^(1/2)-1646784000*x^3*(-10*x^2-x+3)^(1/2)-
2922883200*x^2*(-10*x^2-x+3)^(1/2)+1067352517*10^(1/2)*arcsin(20/11*x+1/11)-3263372400*x*(-10*x^2-x+3)^(1/2)-3
148384060*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)
________________________________________________________________________________________
Maxima [A] time = 3.22366, size = 122, normalized size = 0.9 \begin{align*} \frac{81}{100} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + \frac{25083}{8000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{1067352517}{51200000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{180423}{32000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{8640723}{128000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{200720723}{2560000} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2+3*x)^4*(3+5*x)^(1/2)/(1-2*x)^(1/2),x, algorithm="maxima")
[Out]
81/100*(-10*x^2 - x + 3)^(3/2)*x^2 + 25083/8000*(-10*x^2 - x + 3)^(3/2)*x + 1067352517/51200000*sqrt(5)*sqrt(2
)*arcsin(20/11*x + 1/11) + 180423/32000*(-10*x^2 - x + 3)^(3/2) - 8640723/128000*sqrt(-10*x^2 - x + 3)*x - 200
720723/2560000*sqrt(-10*x^2 - x + 3)
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Fricas [A] time = 1.88017, size = 301, normalized size = 2.23 \begin{align*} -\frac{1}{2560000} \,{\left (20736000 \, x^{4} + 82339200 \, x^{3} + 146144160 \, x^{2} + 163168620 \, x + 157419203\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{1067352517}{51200000} \, \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*(3+5*x)^(1/2)/(1-2*x)^(1/2),x, algorithm="fricas")
[Out]
-1/2560000*(20736000*x^4 + 82339200*x^3 + 146144160*x^2 + 163168620*x + 157419203)*sqrt(5*x + 3)*sqrt(-2*x + 1
) - 1067352517/51200000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)
)
________________________________________________________________________________________
Sympy [A] time = 47.548, size = 665, normalized size = 4.93 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2+3*x)**4*(3+5*x)**(1/2)/(1-2*x)**(1/2),x)
[Out]
2*sqrt(5)*Piecewise((11*sqrt(2)*(-sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)/22 + asin(sqrt(22)*sqrt(5*x + 3)/11)/2)
/4, (x >= -3/5) & (x < 1/2)))/3125 + 24*sqrt(5)*Piecewise((121*sqrt(2)*(sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqr
t(5*x + 3)/968 - sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)/22 + 3*asin(sqrt(22)*sqrt(5*x + 3)/11)/8)/8, (x >= -3/5)
& (x < 1/2)))/3125 + 108*sqrt(5)*Piecewise((1331*sqrt(2)*(sqrt(2)*(5 - 10*x)**(3/2)*(5*x + 3)**(3/2)/3993 + 3
*sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/1936 - sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)/22 + 5*asin(sqrt
(22)*sqrt(5*x + 3)/11)/16)/16, (x >= -3/5) & (x < 1/2)))/3125 + 216*sqrt(5)*Piecewise((14641*sqrt(2)*(2*sqrt(2
)*(5 - 10*x)**(3/2)*(5*x + 3)**(3/2)/3993 + 7*sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/3872 + sqrt(2)*
sqrt(5 - 10*x)*sqrt(5*x + 3)*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x + 3)**2 - 5929)/1874048 - sqrt(2)*sqrt(5
- 10*x)*sqrt(5*x + 3)/22 + 35*asin(sqrt(22)*sqrt(5*x + 3)/11)/128)/32, (x >= -3/5) & (x < 1/2)))/3125 + 162*s
qrt(5)*Piecewise((161051*sqrt(2)*(-2*sqrt(2)*(5 - 10*x)**(5/2)*(5*x + 3)**(5/2)/805255 + sqrt(2)*(5 - 10*x)**(
3/2)*(5*x + 3)**(3/2)/1331 + 15*sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/7744 + 5*sqrt(2)*sqrt(5 - 10*
x)*sqrt(5*x + 3)*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x + 3)**2 - 5929)/3748096 - sqrt(2)*sqrt(5 - 10*x)*sqr
t(5*x + 3)/22 + 63*asin(sqrt(22)*sqrt(5*x + 3)/11)/256)/64, (x >= -3/5) & (x < 1/2)))/3125
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Giac [A] time = 2.46964, size = 97, normalized size = 0.72 \begin{align*} -\frac{1}{128000000} \, \sqrt{5}{\left (2 \,{\left (12 \,{\left (24 \,{\left (12 \,{\left (240 \, x + 521\right )}{\left (5 \, x + 3\right )} + 29669\right )}{\left (5 \, x + 3\right )} + 4900505\right )}{\left (5 \, x + 3\right )} + 485160235\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 5336762585 \, \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*(3+5*x)^(1/2)/(1-2*x)^(1/2),x, algorithm="giac")
[Out]
-1/128000000*sqrt(5)*(2*(12*(24*(12*(240*x + 521)*(5*x + 3) + 29669)*(5*x + 3) + 4900505)*(5*x + 3) + 48516023
5)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 5336762585*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))