3.2399 \(\int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=194 \[ -\frac{3}{80} (3 x+2)^2 (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac{735439 (5 x+3)^{3/2} (1-2 x)^{7/2}}{1280000}-\frac{9 (5 x+3)^{5/2} (13480 x+18399) (1-2 x)^{7/2}}{448000}-\frac{24269487 \sqrt{5 x+3} (1-2 x)^{7/2}}{20480000}+\frac{88988119 \sqrt{5 x+3} (1-2 x)^{5/2}}{204800000}+\frac{978869309 \sqrt{5 x+3} (1-2 x)^{3/2}}{819200000}+\frac{32302687197 \sqrt{5 x+3} \sqrt{1-2 x}}{8192000000}+\frac{355329559167 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8192000000 \sqrt{10}} \]

[Out]

(32302687197*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/8192000000 + (978869309*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/819200000 + (
88988119*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/204800000 - (24269487*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/20480000 - (73543
9*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/1280000 - (3*(1 - 2*x)^(7/2)*(2 + 3*x)^2*(3 + 5*x)^(5/2))/80 - (9*(1 - 2*x)
^(7/2)*(3 + 5*x)^(5/2)*(18399 + 13480*x))/448000 + (355329559167*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(8192000000
*Sqrt[10])

________________________________________________________________________________________

Rubi [A]  time = 0.064418, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{3}{80} (3 x+2)^2 (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac{735439 (5 x+3)^{3/2} (1-2 x)^{7/2}}{1280000}-\frac{9 (5 x+3)^{5/2} (13480 x+18399) (1-2 x)^{7/2}}{448000}-\frac{24269487 \sqrt{5 x+3} (1-2 x)^{7/2}}{20480000}+\frac{88988119 \sqrt{5 x+3} (1-2 x)^{5/2}}{204800000}+\frac{978869309 \sqrt{5 x+3} (1-2 x)^{3/2}}{819200000}+\frac{32302687197 \sqrt{5 x+3} \sqrt{1-2 x}}{8192000000}+\frac{355329559167 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{8192000000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(32302687197*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/8192000000 + (978869309*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/819200000 + (
88988119*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/204800000 - (24269487*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/20480000 - (73543
9*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/1280000 - (3*(1 - 2*x)^(7/2)*(2 + 3*x)^2*(3 + 5*x)^(5/2))/80 - (9*(1 - 2*x)
^(7/2)*(3 + 5*x)^(5/2)*(18399 + 13480*x))/448000 + (355329559167*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(8192000000
*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 (3+5 x)^{3/2} \, dx &=-\frac{3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac{1}{80} \int \left (-323-\frac{1011 x}{2}\right ) (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2} \, dx\\ &=-\frac{3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac{9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac{735439 \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx}{128000}\\ &=-\frac{735439 (1-2 x)^{7/2} (3+5 x)^{3/2}}{1280000}-\frac{3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac{9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac{24269487 \int (1-2 x)^{5/2} \sqrt{3+5 x} \, dx}{2560000}\\ &=-\frac{24269487 (1-2 x)^{7/2} \sqrt{3+5 x}}{20480000}-\frac{735439 (1-2 x)^{7/2} (3+5 x)^{3/2}}{1280000}-\frac{3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac{9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac{266964357 \int \frac{(1-2 x)^{5/2}}{\sqrt{3+5 x}} \, dx}{40960000}\\ &=\frac{88988119 (1-2 x)^{5/2} \sqrt{3+5 x}}{204800000}-\frac{24269487 (1-2 x)^{7/2} \sqrt{3+5 x}}{20480000}-\frac{735439 (1-2 x)^{7/2} (3+5 x)^{3/2}}{1280000}-\frac{3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac{9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac{978869309 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{81920000}\\ &=\frac{978869309 (1-2 x)^{3/2} \sqrt{3+5 x}}{819200000}+\frac{88988119 (1-2 x)^{5/2} \sqrt{3+5 x}}{204800000}-\frac{24269487 (1-2 x)^{7/2} \sqrt{3+5 x}}{20480000}-\frac{735439 (1-2 x)^{7/2} (3+5 x)^{3/2}}{1280000}-\frac{3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac{9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac{32302687197 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{1638400000}\\ &=\frac{32302687197 \sqrt{1-2 x} \sqrt{3+5 x}}{8192000000}+\frac{978869309 (1-2 x)^{3/2} \sqrt{3+5 x}}{819200000}+\frac{88988119 (1-2 x)^{5/2} \sqrt{3+5 x}}{204800000}-\frac{24269487 (1-2 x)^{7/2} \sqrt{3+5 x}}{20480000}-\frac{735439 (1-2 x)^{7/2} (3+5 x)^{3/2}}{1280000}-\frac{3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac{9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac{355329559167 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{16384000000}\\ &=\frac{32302687197 \sqrt{1-2 x} \sqrt{3+5 x}}{8192000000}+\frac{978869309 (1-2 x)^{3/2} \sqrt{3+5 x}}{819200000}+\frac{88988119 (1-2 x)^{5/2} \sqrt{3+5 x}}{204800000}-\frac{24269487 (1-2 x)^{7/2} \sqrt{3+5 x}}{20480000}-\frac{735439 (1-2 x)^{7/2} (3+5 x)^{3/2}}{1280000}-\frac{3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac{9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac{355329559167 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{8192000000 \sqrt{5}}\\ &=\frac{32302687197 \sqrt{1-2 x} \sqrt{3+5 x}}{8192000000}+\frac{978869309 (1-2 x)^{3/2} \sqrt{3+5 x}}{819200000}+\frac{88988119 (1-2 x)^{5/2} \sqrt{3+5 x}}{204800000}-\frac{24269487 (1-2 x)^{7/2} \sqrt{3+5 x}}{20480000}-\frac{735439 (1-2 x)^{7/2} (3+5 x)^{3/2}}{1280000}-\frac{3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac{9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac{355329559167 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{8192000000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.193644, size = 94, normalized size = 0.48 \[ -\frac{10 \sqrt{5 x+3} \left (7741440000000 x^8+10340352000000 x^7-5488281600000 x^6-11337362944000 x^5+569714643200 x^4+4956975460160 x^3+580113118440 x^2-1173301694402 x+115416461871\right )[phone] \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{573440000000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(10*Sqrt[3 + 5*x]*(115416461871 - 1173301694402*x + 580113118440*x^2 + 4956975460160*x^3 + 569714643200*x^4 -
11337362944000*x^5 - 5488281600000*x^6 + 10340352000000*x^7 + 7741440000000*x^8) + 2487306914169*Sqrt[10 - 20
*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(573440000000*Sqrt[1 - 2*x])

________________________________________________________________________________________

Maple [A]  time = 0.009, size = 172, normalized size = 0.9 \begin{align*}{\frac{1}{1146880000000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 77414400000000\,\sqrt{-10\,{x}^{2}-x+3}{x}^{7}[phone]\,\sqrt{-10\,{x}^{2}-x+3}{x}^{6}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-105287357440000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-46946532288000\,{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}-2308329237420\,\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)^(3/2),x)

[Out]

1/1146880000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(77414400000000*(-10*x^2-x+3)^(1/2)*x^7+142110720000000*(-10*x^2-x
+3)^(1/2)*x^6+16172544000000*x^5*(-10*x^2-x+3)^(1/2)-105287357440000*x^4*(-10*x^2-x+3)^(1/2)-46946532288000*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)+18849
375413200*x*(-10*x^2-x+3)^(1/2)-2308329237420*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 3.06342, size = 180, normalized size = 0.93 \begin{align*} \frac{27}{40} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{3} + \frac{6183}{5600} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{2} + \frac{71331}{224000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x - \frac{6491477}{22400000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{8089829}{5120000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{8089829}{102400000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{2936607927}{409600000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{355329559167}{163840000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{2936607927}{8192000000} \, \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)^(3/2),x, algorithm="maxima")

[Out]

27/40*(-10*x^2 - x + 3)^(5/2)*x^3 + 6183/5600*(-10*x^2 - x + 3)^(5/2)*x^2 + 71331/224000*(-10*x^2 - x + 3)^(5/
2)*x - 6491477/22400000*(-10*x^2 - x + 3)^(5/2) + 8089829/5120000*(-10*x^2 - x + 3)^(3/2)*x + 8089829/10240000
0*(-10*x^2 - x + 3)^(3/2) + 2936607927/409600000*sqrt(-10*x^2 - x + 3)*x - 355329559167/163840000000*sqrt(10)*
arcsin(-20/11*x - 1/11) + 2936607927/8192000000*sqrt(-10*x^2 - x + 3)

________________________________________________________________________________________

Fricas [A]  time = 1.50377, size = 420, normalized size = 2.16 \begin{align*} \frac{1}{57344000000} \,{\left (3870720000000 \, x^{7} + 7105536000000 \, x^{6} + 808627200000 \, x^{5} - 5264367872000 \, x^{4} - 2347326614400 \, x^{3} + 1304824422880 \, x^{2} + 942468770660 \, x - 115416461871\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{355329559167}{163840000000} \, \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)^(3/2),x, algorithm="fricas")

[Out]

1/57344000000*(3870720000000*x^7 + 7105536000000*x^6 + 808627200000*x^5 - 5264367872000*x^4 - 2347326614400*x^
3 + 1304824422880*x^2 + 942468770660*x - 115416461871)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 355329559167/16384000000
0*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

________________________________________________________________________________________

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)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.95286, size = 682, normalized size = 3.52 \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)^(3/2),x, algorithm="giac")

[Out]

9/2867200000000*sqrt(5)*(2*(4*(8*(4*(16*(20*(24*(140*x - 503)*(5*x + 3) + 125723)*(5*x + 3) - 12366397)*(5*x +
3) + 575611497)*(5*x + 3) - 3898324857)*(5*x + 3) + 26381882625)*(5*x + 3) - 12293622495)*sqrt(5*x + 3)*sqrt(
-10*x + 5) + 135229847445*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 9/11200000000*sqrt(5)*(2*(4*(8*(4*(16
*(20*(120*x - 359)*(5*x + 3) + 63769)*(5*x + 3) - 3968469)*(5*x + 3) + 33617829)*(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))) + 33/25
60000000*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))) - 17/7
680000*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))) - 77/960000*sqrt(5)*(2*(4*(8*(60*x - 7
1)*(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))) + 13/6000*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))) + 3/50*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)))