3.2412 \(\int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=209 \[ -\frac{3}{80} (3 x+2) (5 x+3)^{7/2} (1-2 x)^{7/2}-\frac{999 (5 x+3)^{7/2} (1-2 x)^{7/2}}{11200}-\frac{12041 (5 x+3)^{5/2} (1-2 x)^{7/2}}{38400}-\frac{132451 (5 x+3)^{3/2} (1-2 x)^{7/2}}{153600}-\frac{1456961 \sqrt{5 x+3} (1-2 x)^{7/2}}{819200}+\frac{16026571 \sqrt{5 x+3} (1-2 x)^{5/2}}{24576000}+\frac{176292281 \sqrt{5 x+3} (1-2 x)^{3/2}}{98304000}+\frac{1939215091 \sqrt{5 x+3} \sqrt{1-2 x}}{327680000}+\frac{21331366001 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{327680000 \sqrt{10}} \]

[Out]

(1939215091*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/327680000 + (176292281*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/98304000 + (160
26571*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/24576000 - (1456961*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/819200 - (132451*(1 -
2*x)^(7/2)*(3 + 5*x)^(3/2))/153600 - (12041*(1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/38400 - (999*(1 - 2*x)^(7/2)*(3 +
5*x)^(7/2))/11200 - (3*(1 - 2*x)^(7/2)*(2 + 3*x)*(3 + 5*x)^(7/2))/80 + (21331366001*ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]])/(327680000*Sqrt[10])

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Rubi [A]  time = 0.0755245, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ -\frac{3}{80} (3 x+2) (5 x+3)^{7/2} (1-2 x)^{7/2}-\frac{999 (5 x+3)^{7/2} (1-2 x)^{7/2}}{11200}-\frac{12041 (5 x+3)^{5/2} (1-2 x)^{7/2}}{38400}-\frac{132451 (5 x+3)^{3/2} (1-2 x)^{7/2}}{153600}-\frac{1456961 \sqrt{5 x+3} (1-2 x)^{7/2}}{819200}+\frac{16026571 \sqrt{5 x+3} (1-2 x)^{5/2}}{24576000}+\frac{176292281 \sqrt{5 x+3} (1-2 x)^{3/2}}{98304000}+\frac{1939215091 \sqrt{5 x+3} \sqrt{1-2 x}}{327680000}+\frac{21331366001 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{327680000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1939215091*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/327680000 + (176292281*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/98304000 + (160
26571*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/24576000 - (1456961*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/819200 - (132451*(1 -
2*x)^(7/2)*(3 + 5*x)^(3/2))/153600 - (12041*(1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/38400 - (999*(1 - 2*x)^(7/2)*(3 +
5*x)^(7/2))/11200 - (3*(1 - 2*x)^(7/2)*(2 + 3*x)*(3 + 5*x)^(7/2))/80 + (21331366001*ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]])/(327680000*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 (3+5 x)^{5/2} \, dx &=-\frac{3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}-\frac{1}{80} \int \left (-326-\frac{999 x}{2}\right ) (1-2 x)^{5/2} (3+5 x)^{5/2} \, dx\\ &=-\frac{999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac{3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac{12041 \int (1-2 x)^{5/2} (3+5 x)^{5/2} \, dx}{3200}\\ &=-\frac{12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac{999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac{3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac{132451 \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx}{15360}\\ &=-\frac{132451 (1-2 x)^{7/2} (3+5 x)^{3/2}}{153600}-\frac{12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac{999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac{3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac{1456961 \int (1-2 x)^{5/2} \sqrt{3+5 x} \, dx}{102400}\\ &=-\frac{1456961 (1-2 x)^{7/2} \sqrt{3+5 x}}{819200}-\frac{132451 (1-2 x)^{7/2} (3+5 x)^{3/2}}{153600}-\frac{12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac{999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac{3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac{16026571 \int \frac{(1-2 x)^{5/2}}{\sqrt{3+5 x}} \, dx}{1638400}\\ &=\frac{16026571 (1-2 x)^{5/2} \sqrt{3+5 x}}{24576000}-\frac{1456961 (1-2 x)^{7/2} \sqrt{3+5 x}}{819200}-\frac{132451 (1-2 x)^{7/2} (3+5 x)^{3/2}}{153600}-\frac{12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac{999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac{3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac{176292281 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{9830400}\\ &=\frac{176292281 (1-2 x)^{3/2} \sqrt{3+5 x}}{98304000}+\frac{16026571 (1-2 x)^{5/2} \sqrt{3+5 x}}{24576000}-\frac{1456961 (1-2 x)^{7/2} \sqrt{3+5 x}}{819200}-\frac{132451 (1-2 x)^{7/2} (3+5 x)^{3/2}}{153600}-\frac{12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac{999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac{3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac{1939215091 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{65536000}\\ &=\frac{1939215091 \sqrt{1-2 x} \sqrt{3+5 x}}{327680000}+\frac{176292281 (1-2 x)^{3/2} \sqrt{3+5 x}}{98304000}+\frac{16026571 (1-2 x)^{5/2} \sqrt{3+5 x}}{24576000}-\frac{1456961 (1-2 x)^{7/2} \sqrt{3+5 x}}{819200}-\frac{132451 (1-2 x)^{7/2} (3+5 x)^{3/2}}{153600}-\frac{12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac{999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac{3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac{21331366001 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{655360000}\\ &=\frac{1939215091 \sqrt{1-2 x} \sqrt{3+5 x}}{327680000}+\frac{176292281 (1-2 x)^{3/2} \sqrt{3+5 x}}{98304000}+\frac{16026571 (1-2 x)^{5/2} \sqrt{3+5 x}}{24576000}-\frac{1456961 (1-2 x)^{7/2} \sqrt{3+5 x}}{819200}-\frac{132451 (1-2 x)^{7/2} (3+5 x)^{3/2}}{153600}-\frac{12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac{999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac{3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac{21331366001 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{327680000 \sqrt{5}}\\ &=\frac{1939215091 \sqrt{1-2 x} \sqrt{3+5 x}}{327680000}+\frac{176292281 (1-2 x)^{3/2} \sqrt{3+5 x}}{98304000}+\frac{16026571 (1-2 x)^{5/2} \sqrt{3+5 x}}{24576000}-\frac{1456961 (1-2 x)^{7/2} \sqrt{3+5 x}}{819200}-\frac{132451 (1-2 x)^{7/2} (3+5 x)^{3/2}}{153600}-\frac{12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac{999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac{3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac{21331366001 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{327680000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0676276, size = 85, normalized size = 0.41 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (774144000000 x^7+1362124800000 x^6+97008640000 x^5-1013681408000 x^4-413675529600 x^3+252700365920 x^2+169330465940 x-22414998339\right )-447958686021 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{68812800000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-22414998339 + 169330465940*x + 252700365920*x^2 - 413675529600*x^3 - 1013681
408000*x^4 + 97008640000*x^5 + 1362124800000*x^6 + 774144000000*x^7) - 447958686021*Sqrt[10]*ArcSin[Sqrt[5/11]
*Sqrt[1 - 2*x]])/68812800000

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Maple [A]  time = 0.008, size = 172, normalized size = 0.8 \begin{align*}{\frac{1}{137625600000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 15482880000000\,\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}-20273628160000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-8273510592000\,{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}-448299966780\,\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)^(5/2),x)

[Out]

1/137625600000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(15482880000000*(-10*x^2-x+3)^(1/2)*x^7+27242496000000*(-10*x^2-x+3
)^(1/2)*x^6+1940172800000*x^5*(-10*x^2-x+3)^(1/2)-20273628160000*x^4*(-10*x^2-x+3)^(1/2)-8273510592000*x^3*(-1
0*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]
0*x*(-10*x^2-x+3)^(1/2)-448299966780*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 4.95692, size = 173, normalized size = 0.83 \begin{align*} -\frac{9}{80} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x - \frac{1839}{11200} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}} + \frac{12041}{19200} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x + \frac{12041}{384000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{1456961}{614400} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{1456961}{12288000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{176292281}{16384000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{21331366001}{6553600000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{176292281}{327680000} \, \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)^(5/2),x, algorithm="maxima")

[Out]

-9/80*(-10*x^2 - x + 3)^(7/2)*x - 1839/11200*(-10*x^2 - x + 3)^(7/2) + 12041/19200*(-10*x^2 - x + 3)^(5/2)*x +
12041/384000*(-10*x^2 - x + 3)^(5/2) + 1456961/614400*(-10*x^2 - x + 3)^(3/2)*x + 1456961/12288000*(-10*x^2 -
x + 3)^(3/2) + 176292281/16384000*sqrt(-10*x^2 - x + 3)*x - 21331366001/6553600000*sqrt(10)*arcsin(-20/11*x -
1/11) + 176292281/327680000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.74394, size = 408, normalized size = 1.95 \begin{align*} \frac{1}{6881280000} \,{\left (774144000000 \, x^{7} + 1362124800000 \, x^{6} + 97008640000 \, x^{5} - 1013681408000 \, x^{4} - 413675529600 \, x^{3} + 252700365920 \, x^{2} + 169330465940 \, x - 22414998339\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{21331366001}{6553600000} \, \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)^(5/2),x, algorithm="fricas")

[Out]

1/6881280000*(774144000000*x^7 + 1362124800000*x^6 + 97008640000*x^5 - 1013681408000*x^4 - 413675529600*x^3 +
252700365920*x^2 + 169330465940*x - 22414998339)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 21331366001/6553600000*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)**2*(3+5*x)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 2.5947, size = 682, normalized size = 3.26 \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)^(5/2),x, algorithm="giac")

[Out]

3/573440000000*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))) + 23/17920000000*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))) + 109/7
680000000*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))) - 341
/96000000*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))) - 227/1920000*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))) + 7/2000*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))) + 9/100*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)))