3.2327 \(\int (1-2 x)^{3/2} (2+3 x)^3 \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=150 \[ -\frac{1}{20} (3 x+2)^2 (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{(5 x+3)^{3/2} (63120 x+88987) (1-2 x)^{5/2}}{160000}-\frac{339983 \sqrt{5 x+3} (1-2 x)^{5/2}}{384000}+\frac{3739813 \sqrt{5 x+3} (1-2 x)^{3/2}}{7680000}+\frac{41137943 \sqrt{5 x+3} \sqrt{1-2 x}}{25600000}+\frac{452517373 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{25600000 \sqrt{10}} \]

[Out]

(41137943*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/25600000 + (3739813*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/7680000 - (339983*(1
- 2*x)^(5/2)*Sqrt[3 + 5*x])/384000 - ((1 - 2*x)^(5/2)*(2 + 3*x)^2*(3 + 5*x)^(3/2))/20 - ((1 - 2*x)^(5/2)*(3 +
5*x)^(3/2)*(88987 + 63120*x))/160000 + (452517373*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(25600000*Sqrt[10])

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Rubi [A]  time = 0.0415341, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {100, 147, 50, 54, 216} \[ -\frac{1}{20} (3 x+2)^2 (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{(5 x+3)^{3/2} (63120 x+88987) (1-2 x)^{5/2}}{160000}-\frac{339983 \sqrt{5 x+3} (1-2 x)^{5/2}}{384000}+\frac{3739813 \sqrt{5 x+3} (1-2 x)^{3/2}}{7680000}+\frac{41137943 \sqrt{5 x+3} \sqrt{1-2 x}}{25600000}+\frac{452517373 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{25600000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(41137943*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/25600000 + (3739813*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/7680000 - (339983*(1
- 2*x)^(5/2)*Sqrt[3 + 5*x])/384000 - ((1 - 2*x)^(5/2)*(2 + 3*x)^2*(3 + 5*x)^(3/2))/20 - ((1 - 2*x)^(5/2)*(3 +
5*x)^(3/2)*(88987 + 63120*x))/160000 + (452517373*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(25600000*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 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 (1-2 x)^{3/2} (2+3 x)^3 \sqrt{3+5 x} \, dx &=-\frac{1}{20} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{1}{60} \int \left (-249-\frac{789 x}{2}\right ) (1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x} \, dx\\ &=-\frac{1}{20} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2} (88987+63120 x)}{160000}+\frac{339983 \int (1-2 x)^{3/2} \sqrt{3+5 x} \, dx}{64000}\\ &=-\frac{339983 (1-2 x)^{5/2} \sqrt{3+5 x}}{384000}-\frac{1}{20} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2} (88987+63120 x)}{160000}+\frac{3739813 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{768000}\\ &=\frac{3739813 (1-2 x)^{3/2} \sqrt{3+5 x}}{7680000}-\frac{339983 (1-2 x)^{5/2} \sqrt{3+5 x}}{384000}-\frac{1}{20} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2} (88987+63120 x)}{160000}+\frac{41137943 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{5120000}\\ &=\frac{41137943 \sqrt{1-2 x} \sqrt{3+5 x}}{25600000}+\frac{3739813 (1-2 x)^{3/2} \sqrt{3+5 x}}{7680000}-\frac{339983 (1-2 x)^{5/2} \sqrt{3+5 x}}{384000}-\frac{1}{20} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2} (88987+63120 x)}{160000}+\frac{452517373 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{51200000}\\ &=\frac{41137943 \sqrt{1-2 x} \sqrt{3+5 x}}{25600000}+\frac{3739813 (1-2 x)^{3/2} \sqrt{3+5 x}}{7680000}-\frac{339983 (1-2 x)^{5/2} \sqrt{3+5 x}}{384000}-\frac{1}{20} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2} (88987+63120 x)}{160000}+\frac{452517373 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{25600000 \sqrt{5}}\\ &=\frac{41137943 \sqrt{1-2 x} \sqrt{3+5 x}}{25600000}+\frac{3739813 (1-2 x)^{3/2} \sqrt{3+5 x}}{7680000}-\frac{339983 (1-2 x)^{5/2} \sqrt{3+5 x}}{384000}-\frac{1}{20} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2} (88987+63120 x)}{160000}+\frac{452517373 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{25600000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.164818, size = 84, normalized size = 0.56 \[ \frac{10 \sqrt{5 x+3} \left (1382400000 x^6+1810944000 x^5-634003200 x^4-1555668160 x^3-125580440 x^2+537385502 x-81405921\right )-1357552119 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{768000000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[3 + 5*x]*(-81405921 + 537385502*x - 125580440*x^2 - 1555668160*x^3 - 634003200*x^4 + 1810944000*x^5 +
1382400000*x^6) - 1357552119*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(768000000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.008, size = 138, normalized size = 0.9 \begin{align*}{\frac{1}{1536000000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -13824000000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-25021440000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-6170688000\,{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}-1628118420\,\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)^(3/2)*(2+3*x)^3*(3+5*x)^(1/2),x)

[Out]

1/1536000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-13824000000*x^5*(-10*x^2-x+3)^(1/2)-25021440000*x^4*(-10*x^2-x+3)^(
1/2)-6170688000*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)-1628118420*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 3.84764, size = 140, normalized size = 0.93 \begin{align*} \frac{9}{10} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + \frac{1539}{1000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + \frac{41427}{80000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{385939}{960000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{3739813}{1280000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{452517373}{512000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{3739813}{25600000} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/10*(-10*x^2 - x + 3)^(3/2)*x^3 + 1539/1000*(-10*x^2 - x + 3)^(3/2)*x^2 + 41427/80000*(-10*x^2 - x + 3)^(3/2)
*x - 385939/960000*(-10*x^2 - x + 3)^(3/2) + 3739813/1280000*sqrt(-10*x^2 - x + 3)*x - 452517373/512000000*sqr
t(10)*arcsin(-20/11*x - 1/11) + 3739813/25600000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.57091, size = 327, normalized size = 2.18 \begin{align*} -\frac{1}{76800000} \,{\left (691200000 \, x^{5} + 1251072000 \, x^{4} + 308534400 \, x^{3} - 623566880 \, x^{2} - 374573660 \, x + 81405921\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{452517373}{512000000} \, \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)^(3/2)*(2+3*x)^3*(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

-1/76800000*(691200000*x^5 + 1251072000*x^4 + 308534400*x^3 - 623566880*x^2 - 374573660*x + 81405921)*sqrt(5*x
+ 3)*sqrt(-2*x + 1) - 452517373/512000000*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 [A]  time = 90.3526, size = 695, normalized size = 4.63 \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)**(3/2)*(2+3*x)**3*(3+5*x)**(1/2),x)

[Out]

22*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 + 194*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 + 558*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)*sqr
t(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 + 486*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 - 108*sqrt(5)*P
iecewise((1771561*sqrt(2)*(4*sqrt(2)*(5 - 10*x)**(5/2)*(5*x + 3)**(5/2)/805255 + sqrt(2)*(5 - 10*x)**(3/2)*(-2
0*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 - 10*
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 + 10
56*(5*x + 3)**2 - 5929)/14992384 + 21*asin(sqrt(22)*sqrt(5*x + 3)/11)/1024)/64, (x >= -3/5) & (x < 1/2)))/1562
5

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Giac [B]  time = 2.34564, size = 427, normalized size = 2.85 \begin{align*} -\frac{9}{1280000000} \, \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{27}{64000000} \, \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{3}{320000} \, \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}{1200} \, \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}{50} \, \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)^(3/2)*(2+3*x)^3*(3+5*x)^(1/2),x, algorithm="giac")

[Out]

-9/1280000000*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))) -
27/64000000*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))) - 3/320000*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))) + 1/1200*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))) + 1/50*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)))