3.2389 \(\int (1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=165 \[ -\frac{1}{20} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{193 (5 x+3)^{3/2} (1-2 x)^{7/2}}{2000}-\frac{7189 \sqrt{5 x+3} (1-2 x)^{7/2}}{32000}+\frac{79079 \sqrt{5 x+3} (1-2 x)^{5/2}}{960000}+\frac{869869 \sqrt{5 x+3} (1-2 x)^{3/2}}{3840000}+\frac{9568559 \sqrt{5 x+3} \sqrt{1-2 x}}{12800000}+\frac{105254149 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{12800000 \sqrt{10}} \]

[Out]

(9568559*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/12800000 + (869869*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/3840000 + (79079*(1 -
2*x)^(5/2)*Sqrt[3 + 5*x])/960000 - (7189*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/32000 - (193*(1 - 2*x)^(7/2)*(3 + 5*x)
^(3/2))/2000 - ((1 - 2*x)^(7/2)*(2 + 3*x)*(3 + 5*x)^(3/2))/20 + (105254149*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(
12800000*Sqrt[10])

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Rubi [A]  time = 0.0511839, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {90, 80, 50, 54, 216} \[ -\frac{1}{20} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{193 (5 x+3)^{3/2} (1-2 x)^{7/2}}{2000}-\frac{7189 \sqrt{5 x+3} (1-2 x)^{7/2}}{32000}+\frac{79079 \sqrt{5 x+3} (1-2 x)^{5/2}}{960000}+\frac{869869 \sqrt{5 x+3} (1-2 x)^{3/2}}{3840000}+\frac{9568559 \sqrt{5 x+3} \sqrt{1-2 x}}{12800000}+\frac{105254149 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{12800000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9568559*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/12800000 + (869869*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/3840000 + (79079*(1 -
2*x)^(5/2)*Sqrt[3 + 5*x])/960000 - (7189*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/32000 - (193*(1 - 2*x)^(7/2)*(3 + 5*x)
^(3/2))/2000 - ((1 - 2*x)^(7/2)*(2 + 3*x)*(3 + 5*x)^(3/2))/20 + (105254149*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(
12800000*Sqrt[10])

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
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 (1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x} \, dx &=-\frac{1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}-\frac{1}{60} \int \left (-186-\frac{579 x}{2}\right ) (1-2 x)^{5/2} \sqrt{3+5 x} \, dx\\ &=-\frac{193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac{1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac{7189 \int (1-2 x)^{5/2} \sqrt{3+5 x} \, dx}{4000}\\ &=-\frac{7189 (1-2 x)^{7/2} \sqrt{3+5 x}}{32000}-\frac{193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac{1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac{79079 \int \frac{(1-2 x)^{5/2}}{\sqrt{3+5 x}} \, dx}{64000}\\ &=\frac{79079 (1-2 x)^{5/2} \sqrt{3+5 x}}{960000}-\frac{7189 (1-2 x)^{7/2} \sqrt{3+5 x}}{32000}-\frac{193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac{1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac{869869 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{384000}\\ &=\frac{869869 (1-2 x)^{3/2} \sqrt{3+5 x}}{3840000}+\frac{79079 (1-2 x)^{5/2} \sqrt{3+5 x}}{960000}-\frac{7189 (1-2 x)^{7/2} \sqrt{3+5 x}}{32000}-\frac{193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac{1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac{9568559 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{2560000}\\ &=\frac{9568559 \sqrt{1-2 x} \sqrt{3+5 x}}{12800000}+\frac{869869 (1-2 x)^{3/2} \sqrt{3+5 x}}{3840000}+\frac{79079 (1-2 x)^{5/2} \sqrt{3+5 x}}{960000}-\frac{7189 (1-2 x)^{7/2} \sqrt{3+5 x}}{32000}-\frac{193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac{1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac{105254149 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{25600000}\\ &=\frac{9568559 \sqrt{1-2 x} \sqrt{3+5 x}}{12800000}+\frac{869869 (1-2 x)^{3/2} \sqrt{3+5 x}}{3840000}+\frac{79079 (1-2 x)^{5/2} \sqrt{3+5 x}}{960000}-\frac{7189 (1-2 x)^{7/2} \sqrt{3+5 x}}{32000}-\frac{193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac{1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac{105254149 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{12800000 \sqrt{5}}\\ &=\frac{9568559 \sqrt{1-2 x} \sqrt{3+5 x}}{12800000}+\frac{869869 (1-2 x)^{3/2} \sqrt{3+5 x}}{3840000}+\frac{79079 (1-2 x)^{5/2} \sqrt{3+5 x}}{960000}-\frac{7189 (1-2 x)^{7/2} \sqrt{3+5 x}}{32000}-\frac{193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac{1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac{105254149 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{12800000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0498672, size = 75, normalized size = 0.45 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (230400000 x^5+94464000 x^4-237187200 x^3-61262560 x^2+102523580 x+9303927\right )-315762447 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{384000000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(9303927 + 102523580*x - 61262560*x^2 - 237187200*x^3 + 94464000*x^4 + 2304000
00*x^5) - 315762447*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/384000000

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Maple [A]  time = 0.009, size = 138, normalized size = 0.8 \begin{align*}{\frac{1}{768000000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 4608000000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-4743744000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-1225251200\,{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}[phone]\,\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((1-2*x)^(5/2)*(2+3*x)^2*(3+5*x)^(1/2),x)

[Out]

1/768000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(4608000000*x^5*(-10*x^2-x+3)^(1/2)[phone]*x^4*(-10*x^2-x+3)^(1/2)
-4743744000*x^3*(-10*x^2-x+3)^(1/2)-1225251200*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)[phone]*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 2.64744, size = 140, normalized size = 0.85 \begin{align*} -\frac{3}{5} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} - \frac{93}{500} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + \frac{18251}{40000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{27893}{480000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{869869}{640000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{105254149}{256000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{869869}{12800000} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/5*(-10*x^2 - x + 3)^(3/2)*x^3 - 93/500*(-10*x^2 - x + 3)^(3/2)*x^2 + 18251/40000*(-10*x^2 - x + 3)^(3/2)*x
+ 27893/480000*(-10*x^2 - x + 3)^(3/2) + 869869/640000*sqrt(-10*x^2 - x + 3)*x - 105254149/256000000*sqrt(10)*
arcsin(-20/11*x - 1/11) + 869869/12800000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.56109, size = 320, normalized size = 1.94 \begin{align*} \frac{1}{38400000} \,{\left (230400000 \, x^{5} + 94464000 \, x^{4} - 237187200 \, x^{3} - 61262560 \, x^{2} + 102523580 \, x + 9303927\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{105254149}{256000000} \, \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((1-2*x)^(5/2)*(2+3*x)^2*(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

1/38400000*(230400000*x^5 + 94464000*x^4 - 237187200*x^3 - 61262560*x^2 + 102523580*x + 9303927)*sqrt(5*x + 3)
*sqrt(-2*x + 1) - 105254149/256000000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(1
0*x^2 + x - 3))

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Sympy [A]  time = 143.037, size = 695, normalized size = 4.21 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

242*sqrt(5)*Piecewise((121*sqrt(2)*(-sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/121 + asin(sqrt(22)*sqrt
(5*x + 3)/11))/32, (x >= -3/5) & (x < 1/2)))/15625 + 1364*sqrt(5)*Piecewise((1331*sqrt(2)*(-sqrt(2)*(5 - 10*x)
**(3/2)*(5*x + 3)**(3/2)/3993 - sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/1936 + asin(sqrt(22)*sqrt(5*x
+ 3)/11)/16)/8, (x >= -3/5) & (x < 1/2)))/15625 + 1658*sqrt(5)*Piecewise((14641*sqrt(2)*(-sqrt(2)*(5 - 10*x)*
*(3/2)*(5*x + 3)**(3/2)/3993 - 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 + 5*asin(sqrt(22)*sqrt(5*x + 3)
/11)/128)/16, (x >= -3/5) & (x < 1/2)))/15625 - 744*sqrt(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)/3993 - sqrt(2)*sqrt(5 - 10*x)*(-20*x
- 1)*sqrt(5*x + 3)/7744 - 3*sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x + 3
)**2 - 5929)/3748096 + 7*asin(sqrt(22)*sqrt(5*x + 3)/11)/256)/32, (x >= -3/5) & (x < 1/2)))/15625 + 72*sqrt(5)
*Piecewise((1771561*sqrt(2)*(4*sqrt(2)*(5 - 10*x)**(5/2)*(5*x + 3)**(5/2)/805255 + sqrt(2)*(5 - 10*x)**(3/2)*(
-20*x - 1)**3*(5*x + 3)**(3/2)/85034928 - sqrt(2)*(5 - 10*x)**(3/2)*(5*x + 3)**(3/2)/3993 - sqrt(2)*sqrt(5 - 1
0*x)*(-20*x - 1)*sqrt(5*x + 3)/15488 - 13*sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)*(-12100*x - 128*(5*x + 3)**3 +
1056*(5*x + 3)**2 - 5929)/14992384 + 21*asin(sqrt(22)*sqrt(5*x + 3)/11)/1024)/64, (x >= -3/5) & (x < 1/2)))/15
625

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Giac [B]  time = 1.64639, size = 427, normalized size = 2.59 \begin{align*} \frac{3}{640000000} \, \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{1}{16000000} \, \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{23}{1920000} \, \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}{6000} \, \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}{100} \, \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((1-2*x)^(5/2)*(2+3*x)^2*(3+5*x)^(1/2),x, algorithm="giac")

[Out]

3/640000000*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))) + 1
/16000000*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))) - 23/1920000*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)*
sqrt(5*x + 3))) - 1/6000*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 363*sqrt(2)
*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1/100*sqrt(5)*(2*(20*x + 1)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2
)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))