3.2388 \(\int (1-2 x)^{5/2} (2+3 x)^3 \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=172 \[ -\frac{3}{70} (3 x+2)^2 (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{3 (5 x+3)^{3/2} (26700 x+33857) (1-2 x)^{7/2}}{280000}-\frac{255169 \sqrt{5 x+3} (1-2 x)^{7/2}}{640000}+\frac{2806859 \sqrt{5 x+3} (1-2 x)^{5/2}}{19200000}+\frac{30875449 \sqrt{5 x+3} (1-2 x)^{3/2}}{76800000}+\frac{339629939 \sqrt{5 x+3} \sqrt{1-2 x}}{256000000}+\frac{3735929329 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{256000000 \sqrt{10}} \]

[Out]

(339629939*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/256000000 + (30875449*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/76800000 + (28068
59*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/19200000 - (255169*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/640000 - (3*(1 - 2*x)^(7/2
)*(2 + 3*x)^2*(3 + 5*x)^(3/2))/70 - (3*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2)*(33857 + 26700*x))/280000 + (3735929329
*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(256000000*Sqrt[10])

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Rubi [A]  time = 0.0524287, antiderivative size = 172, 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 = {100, 147, 50, 54, 216} \[ -\frac{3}{70} (3 x+2)^2 (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{3 (5 x+3)^{3/2} (26700 x+33857) (1-2 x)^{7/2}}{280000}-\frac{255169 \sqrt{5 x+3} (1-2 x)^{7/2}}{640000}+\frac{2806859 \sqrt{5 x+3} (1-2 x)^{5/2}}{19200000}+\frac{30875449 \sqrt{5 x+3} (1-2 x)^{3/2}}{76800000}+\frac{339629939 \sqrt{5 x+3} \sqrt{1-2 x}}{256000000}+\frac{3735929329 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{256000000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(339629939*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/256000000 + (30875449*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/76800000 + (28068
59*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/19200000 - (255169*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/640000 - (3*(1 - 2*x)^(7/2
)*(2 + 3*x)^2*(3 + 5*x)^(3/2))/70 - (3*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2)*(33857 + 26700*x))/280000 + (3735929329
*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(256000000*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)^{5/2} (2+3 x)^3 \sqrt{3+5 x} \, dx &=-\frac{3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{1}{70} \int \left (-253-\frac{801 x}{2}\right ) (1-2 x)^{5/2} (2+3 x) \sqrt{3+5 x} \, dx\\ &=-\frac{3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac{255169 \int (1-2 x)^{5/2} \sqrt{3+5 x} \, dx}{80000}\\ &=-\frac{255169 (1-2 x)^{7/2} \sqrt{3+5 x}}{640000}-\frac{3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac{2806859 \int \frac{(1-2 x)^{5/2}}{\sqrt{3+5 x}} \, dx}{1280000}\\ &=\frac{2806859 (1-2 x)^{5/2} \sqrt{3+5 x}}{19200000}-\frac{255169 (1-2 x)^{7/2} \sqrt{3+5 x}}{640000}-\frac{3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac{30875449 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{7680000}\\ &=\frac{30875449 (1-2 x)^{3/2} \sqrt{3+5 x}}{76800000}+\frac{2806859 (1-2 x)^{5/2} \sqrt{3+5 x}}{19200000}-\frac{255169 (1-2 x)^{7/2} \sqrt{3+5 x}}{640000}-\frac{3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac{339629939 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{51200000}\\ &=\frac{339629939 \sqrt{1-2 x} \sqrt{3+5 x}}{256000000}+\frac{30875449 (1-2 x)^{3/2} \sqrt{3+5 x}}{76800000}+\frac{2806859 (1-2 x)^{5/2} \sqrt{3+5 x}}{19200000}-\frac{255169 (1-2 x)^{7/2} \sqrt{3+5 x}}{640000}-\frac{3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac{3735929329 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{512000000}\\ &=\frac{339629939 \sqrt{1-2 x} \sqrt{3+5 x}}{256000000}+\frac{30875449 (1-2 x)^{3/2} \sqrt{3+5 x}}{76800000}+\frac{2806859 (1-2 x)^{5/2} \sqrt{3+5 x}}{19200000}-\frac{255169 (1-2 x)^{7/2} \sqrt{3+5 x}}{640000}-\frac{3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac{3735929329 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{256000000 \sqrt{5}}\\ &=\frac{339629939 \sqrt{1-2 x} \sqrt{3+5 x}}{256000000}+\frac{30875449 (1-2 x)^{3/2} \sqrt{3+5 x}}{76800000}+\frac{2806859 (1-2 x)^{5/2} \sqrt{3+5 x}}{19200000}-\frac{255169 (1-2 x)^{7/2} \sqrt{3+5 x}}{640000}-\frac{3}{70} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac{3 (1-2 x)^{7/2} (3+5 x)^{3/2} (33857+26700 x)}{280000}+\frac{3735929329 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{256000000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.179815, size = 89, normalized size = 0.52 \[ -\frac{10 \sqrt{5 x+3} \left (165888000000 x^7+111974400000 x^6-202397184000 x^5-117722284800 x^4+104908893760 x^3+49731440840 x^2-31177591322 x+679278531\right )[phone] \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{53760000000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(10*Sqrt[3 + 5*x]*(679278531 - 31177591322*x + 49731440840*x^2 + 104908893760*x^3 - 117722284800*x^4 - 202397
184000*x^5 + 111974400000*x^6 + 165888000000*x^7) + 78454515909*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x
]])/(53760000000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.01, size = 155, normalized size = 0.9 \begin{align*}{\frac{1}{107520000000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 1658880000000\,\sqrt{-10\,{x}^{2}-x+3}{x}^{6}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-1049379840000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-1701912768000\,{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}-13585570620\,\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)^3*(3+5*x)^(1/2),x)

[Out]

1/107520000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(1658880000000*(-10*x^2-x+3)^(1/2)*x^6+1949184000000*x^5*(-10*x^2-x
+3)^(1/2)-1049379840000*x^4*(-10*x^2-x+3)^(1/2)-1701912768000*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)-13585570620*(-10*x^2
-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.93945, size = 163, normalized size = 0.95 \begin{align*} -\frac{54}{35} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{4} - \frac{1161}{700} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + \frac{47529}{70000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + \frac{5697497}{5600000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{5531929}{67200000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{30875449}{12800000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{3735929329}{5120000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{30875449}{256000000} \, \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)^3*(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

-54/35*(-10*x^2 - x + 3)^(3/2)*x^4 - 1161/700*(-10*x^2 - x + 3)^(3/2)*x^3 + 47529/70000*(-10*x^2 - x + 3)^(3/2
)*x^2 + 5697497/5600000*(-10*x^2 - x + 3)^(3/2)*x - 5531929/67200000*(-10*x^2 - x + 3)^(3/2) + 30875449/128000
00*sqrt(-10*x^2 - x + 3)*x - 3735929329/5120000000*sqrt(10)*arcsin(-20/11*x - 1/11) + 30875449/256000000*sqrt(
-10*x^2 - x + 3)

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Fricas [A]  time = 1.46185, size = 367, normalized size = 2.13 \begin{align*} \frac{1}{5376000000} \,{\left (82944000000 \, x^{6} + 97459200000 \, x^{5} - 52468992000 \, x^{4} - 85095638400 \, x^{3} + 9906627680 \, x^{2} + 29819034260 \, x - 679278531\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{3735929329}{5120000000} \, \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)^3*(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

1/5376000000*(82944000000*x^6 + 97459200000*x^5 - 52468992000*x^4 - 85095638400*x^3 + 9906627680*x^2 + 2981903
4260*x - 679278531)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 3735929329/5120000000*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((1-2*x)**(5/2)*(2+3*x)**3*(3+5*x)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 2.04691, size = 548, normalized size = 3.19 \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)^3*(3+5*x)^(1/2),x, algorithm="giac")

[Out]

9/89600000000*sqrt(5)*(2*(4*(8*(4*(16*(20*(120*x - 359)*(5*x + 3) + 63769)*(5*x + 3) - 3968469)*(5*x + 3) + 33
617829)*(5*x + 3) - 276044685)*(5*x + 3) + 87356115)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 960917265*sqrt(2)*arcsin(
1/11*sqrt(22)*sqrt(5*x + 3))) + 9/640000000*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))) - 3/12800000*sqrt(5)*(2*(4*(8*(12*(80*x - 143)*(5*x + 3) + 9773)*(5*x + 3) - 1364
05)*(5*x + 3) + 60555)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 666105*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 2
9/960000*sqrt(5)*(2*(4*(8*(60*x - 71)*(5*x + 3) + 2179)*(5*x + 3) - 4125)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 4537
5*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))) + 1/50*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)))