3.2349 \(\int (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=194 \[ -\frac{3}{80} (1-2 x)^{5/2} (3 x+2)^2 (5 x+3)^{7/2}-\frac{9 (1-2 x)^{5/2} (16120 x+25043) (5 x+3)^{7/2}}{448000}-\frac{306029 (1-2 x)^{5/2} (5 x+3)^{5/2}}{256000}-\frac{3366319 (1-2 x)^{5/2} (5 x+3)^{3/2}}{819200}-\frac{37029509 (1-2 x)^{5/2} \sqrt{5 x+3}}{3276800}+\frac{407324599 (1-2 x)^{3/2} \sqrt{5 x+3}}{65536000}+\frac{13441711767 \sqrt{1-2 x} \sqrt{5 x+3}}{655360000}+\frac{147858829437 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{655360000 \sqrt{10}} \]

[Out]

(13441711767*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/655360000 + (407324599*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/65536000 - (37
029509*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/3276800 - (3366319*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/819200 - (306029*(1
- 2*x)^(5/2)*(3 + 5*x)^(5/2))/256000 - (3*(1 - 2*x)^(5/2)*(2 + 3*x)^2*(3 + 5*x)^(7/2))/80 - (9*(1 - 2*x)^(5/2)
*(3 + 5*x)^(7/2)*(25043 + 16120*x))/448000 + (147858829437*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(655360000*Sqrt[1
0])

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Rubi [A]  time = 0.0653889, 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} \[ -\frac{3}{80} (1-2 x)^{5/2} (3 x+2)^2 (5 x+3)^{7/2}-\frac{9 (1-2 x)^{5/2} (16120 x+25043) (5 x+3)^{7/2}}{448000}-\frac{306029 (1-2 x)^{5/2} (5 x+3)^{5/2}}{256000}-\frac{3366319 (1-2 x)^{5/2} (5 x+3)^{3/2}}{819200}-\frac{37029509 (1-2 x)^{5/2} \sqrt{5 x+3}}{3276800}+\frac{407324599 (1-2 x)^{3/2} \sqrt{5 x+3}}{65536000}+\frac{13441711767 \sqrt{1-2 x} \sqrt{5 x+3}}{655360000}+\frac{147858829437 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{655360000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(13441711767*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/655360000 + (407324599*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/65536000 - (37
029509*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/3276800 - (3366319*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/819200 - (306029*(1
- 2*x)^(5/2)*(3 + 5*x)^(5/2))/256000 - (3*(1 - 2*x)^(5/2)*(2 + 3*x)^2*(3 + 5*x)^(7/2))/80 - (9*(1 - 2*x)^(5/2)
*(3 + 5*x)^(7/2)*(25043 + 16120*x))/448000 + (147858829437*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(655360000*Sqrt[1
0])

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 (3+5 x)^{5/2} \, dx &=-\frac{3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{1}{80} \int \left (-389-\frac{1209 x}{2}\right ) (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2} \, dx\\ &=-\frac{3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{9 (1-2 x)^{5/2} (3+5 x)^{7/2} (25043+16120 x)}{448000}+\frac{306029 \int (1-2 x)^{3/2} (3+5 x)^{5/2} \, dx}{25600}\\ &=-\frac{306029 (1-2 x)^{5/2} (3+5 x)^{5/2}}{256000}-\frac{3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{9 (1-2 x)^{5/2} (3+5 x)^{7/2} (25043+16120 x)}{448000}+\frac{3366319 \int (1-2 x)^{3/2} (3+5 x)^{3/2} \, dx}{102400}\\ &=-\frac{3366319 (1-2 x)^{5/2} (3+5 x)^{3/2}}{819200}-\frac{306029 (1-2 x)^{5/2} (3+5 x)^{5/2}}{256000}-\frac{3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{9 (1-2 x)^{5/2} (3+5 x)^{7/2} (25043+16120 x)}{448000}+\frac{111088527 \int (1-2 x)^{3/2} \sqrt{3+5 x} \, dx}{1638400}\\ &=-\frac{37029509 (1-2 x)^{5/2} \sqrt{3+5 x}}{3276800}-\frac{3366319 (1-2 x)^{5/2} (3+5 x)^{3/2}}{819200}-\frac{306029 (1-2 x)^{5/2} (3+5 x)^{5/2}}{256000}-\frac{3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{9 (1-2 x)^{5/2} (3+5 x)^{7/2} (25043+16120 x)}{448000}+\frac{407324599 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{6553600}\\ &=\frac{407324599 (1-2 x)^{3/2} \sqrt{3+5 x}}{65536000}-\frac{37029509 (1-2 x)^{5/2} \sqrt{3+5 x}}{3276800}-\frac{3366319 (1-2 x)^{5/2} (3+5 x)^{3/2}}{819200}-\frac{306029 (1-2 x)^{5/2} (3+5 x)^{5/2}}{256000}-\frac{3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{9 (1-2 x)^{5/2} (3+5 x)^{7/2} (25043+16120 x)}{448000}+\frac{13441711767 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{131072000}\\ &=\frac{13441711767 \sqrt{1-2 x} \sqrt{3+5 x}}{655360000}+\frac{407324599 (1-2 x)^{3/2} \sqrt{3+5 x}}{65536000}-\frac{37029509 (1-2 x)^{5/2} \sqrt{3+5 x}}{3276800}-\frac{3366319 (1-2 x)^{5/2} (3+5 x)^{3/2}}{819200}-\frac{306029 (1-2 x)^{5/2} (3+5 x)^{5/2}}{256000}-\frac{3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{9 (1-2 x)^{5/2} (3+5 x)^{7/2} (25043+16120 x)}{448000}+\frac{147858829437 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{1310720000}\\ &=\frac{13441711767 \sqrt{1-2 x} \sqrt{3+5 x}}{655360000}+\frac{407324599 (1-2 x)^{3/2} \sqrt{3+5 x}}{65536000}-\frac{37029509 (1-2 x)^{5/2} \sqrt{3+5 x}}{3276800}-\frac{3366319 (1-2 x)^{5/2} (3+5 x)^{3/2}}{819200}-\frac{306029 (1-2 x)^{5/2} (3+5 x)^{5/2}}{256000}-\frac{3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{9 (1-2 x)^{5/2} (3+5 x)^{7/2} (25043+16120 x)}{448000}+\frac{147858829437 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{655360000 \sqrt{5}}\\ &=\frac{13441711767 \sqrt{1-2 x} \sqrt{3+5 x}}{655360000}+\frac{407324599 (1-2 x)^{3/2} \sqrt{3+5 x}}{65536000}-\frac{37029509 (1-2 x)^{5/2} \sqrt{3+5 x}}{3276800}-\frac{3366319 (1-2 x)^{5/2} (3+5 x)^{3/2}}{819200}-\frac{306029 (1-2 x)^{5/2} (3+5 x)^{5/2}}{256000}-\frac{3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{9 (1-2 x)^{5/2} (3+5 x)^{7/2} (25043+16120 x)}{448000}+\frac{147858829437 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{655360000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.207039, size = 94, normalized size = 0.48 \[ \frac{10 \sqrt{5 x+3} \left (1548288000000 x^8+4014489600000 x^7+2714081280000 x^6-1370011136000 x^5-2412933395200 x^4-588662541760 x^3+472622713160 x^2+370542366022 x-116041578381\right )-1035011806059 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{45875200000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[3 + 5*x]*(-116041578381 + 370542366022*x + 472622713160*x^2 - 588662541760*x^3 - 2412933395200*x^4 -
1370011136000*x^5 + 2714081280000*x^6 + 4014489600000*x^7 + 1548288000000*x^8) - 1035011806059*Sqrt[10 - 20*x]
*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(45875200000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.008, size = 172, normalized size = 0.9 \begin{align*}{\frac{1}{91750400000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -15482880000000\,\sqrt{-10\,{x}^{2}-x+3}{x}^{7}-47886336000000\,\sqrt{-10\,{x}^{2}-x+3}{x}^{6}-51083980800000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-11841879040000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{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}-2320831567620\,\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)^(5/2),x)

[Out]

1/91750400000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-15482880000000*(-10*x^2-x+3)^(1/2)*x^7-47886336000000*(-10*x^2-x+3
)^(1/2)*x^6-51083980800000*x^5*(-10*x^2-x+3)^(1/2)-11841879040000*x^4*(-10*x^2-x+3)^(1/2)[phone]*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]
85200*x*(-10*x^2-x+3)^(1/2)-2320831567620*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 2.6188, size = 180, normalized size = 0.93 \begin{align*} -\frac{27}{16} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{3} - \frac{2187}{448} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{2} - \frac{100119}{17920} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x - \frac{5653247}{1792000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{3366319}{409600} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{3366319}{8192000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{1221973797}{32768000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{147858829437}{13107200000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{1221973797}{655360000} \, \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)^(5/2),x, algorithm="maxima")

[Out]

-27/16*(-10*x^2 - x + 3)^(5/2)*x^3 - 2187/448*(-10*x^2 - x + 3)^(5/2)*x^2 - 100119/17920*(-10*x^2 - x + 3)^(5/
2)*x - 5653247/1792000*(-10*x^2 - x + 3)^(5/2) + 3366319/409600*(-10*x^2 - x + 3)^(3/2)*x + 3366319/8192000*(-
10*x^2 - x + 3)^(3/2) + 1221973797/32768000*sqrt(-10*x^2 - x + 3)*x - 147858829437/13107200000*sqrt(10)*arcsin
(-20/11*x - 1/11) + 1221973797/655360000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.51755, size = 414, normalized size = 2.13 \begin{align*} -\frac{1}{4587520000} \,{\left (774144000000 \, x^{7} + 2394316800000 \, x^{6} + 2554199040000 \, x^{5} + 592093952000 \, x^{4} - 910419721600 \, x^{3} - 749541131680 \, x^{2} - 138459209260 \, x + 116041578381\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{147858829437}{13107200000} \, \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)^(5/2),x, algorithm="fricas")

[Out]

-1/4587520000*(774144000000*x^7 + 2394316800000*x^6 + 2554199040000*x^5 + 592093952000*x^4 - 910419721600*x^3
- 749541131680*x^2 - 138459209260*x + 116041578381)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 147858829437/13107200000*sq
rt(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)**(3/2)*(2+3*x)**3*(3+5*x)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 2.2316, 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)^(3/2)*(2+3*x)^3*(3+5*x)^(5/2),x, algorithm="giac")

[Out]

-9/1146880000000*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))) - 243/71680000000*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))) - 56
1/1280000000*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))) -
769/192000000*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))) + 319/960000*sqrt(5)*(2*(4*(8*(
60*x - 71)*(5*x + 3) + 2179)*(5*x + 3) - 4125)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 45375*sqrt(2)*arcsin(1/11*sqrt(
22)*sqrt(5*x + 3))) + 7/400*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))) + 9/50*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)))