3.2401 \(\int (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=160 \[ -\frac{1}{20} (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac{63}{400} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{2079 \sqrt{5 x+3} (1-2 x)^{7/2}}{6400}+\frac{7623 \sqrt{5 x+3} (1-2 x)^{5/2}}{64000}+\frac{83853 \sqrt{5 x+3} (1-2 x)^{3/2}}{256000}+\frac{2767149 \sqrt{5 x+3} \sqrt{1-2 x}}{2560000}+\frac{30438639 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2560000 \sqrt{10}} \]

[Out]

(2767149*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/2560000 + (83853*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/256000 + (7623*(1 - 2*x)
^(5/2)*Sqrt[3 + 5*x])/64000 - (2079*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/6400 - (63*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))
/400 - ((1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/20 + (30438639*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2560000*Sqrt[10])

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Rubi [A]  time = 0.0481522, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {80, 50, 54, 216} \[ -\frac{1}{20} (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac{63}{400} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac{2079 \sqrt{5 x+3} (1-2 x)^{7/2}}{6400}+\frac{7623 \sqrt{5 x+3} (1-2 x)^{5/2}}{64000}+\frac{83853 \sqrt{5 x+3} (1-2 x)^{3/2}}{256000}+\frac{2767149 \sqrt{5 x+3} \sqrt{1-2 x}}{2560000}+\frac{30438639 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2560000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2767149*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/2560000 + (83853*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/256000 + (7623*(1 - 2*x)
^(5/2)*Sqrt[3 + 5*x])/64000 - (2079*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/6400 - (63*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))
/400 - ((1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/20 + (30438639*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2560000*Sqrt[10])

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) (3+5 x)^{3/2} \, dx &=-\frac{1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac{63}{40} \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx\\ &=-\frac{63}{400} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac{1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac{2079}{800} \int (1-2 x)^{5/2} \sqrt{3+5 x} \, dx\\ &=-\frac{2079 (1-2 x)^{7/2} \sqrt{3+5 x}}{6400}-\frac{63}{400} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac{1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac{22869 \int \frac{(1-2 x)^{5/2}}{\sqrt{3+5 x}} \, dx}{12800}\\ &=\frac{7623 (1-2 x)^{5/2} \sqrt{3+5 x}}{64000}-\frac{2079 (1-2 x)^{7/2} \sqrt{3+5 x}}{6400}-\frac{63}{400} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac{1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac{83853 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{25600}\\ &=\frac{83853 (1-2 x)^{3/2} \sqrt{3+5 x}}{256000}+\frac{7623 (1-2 x)^{5/2} \sqrt{3+5 x}}{64000}-\frac{2079 (1-2 x)^{7/2} \sqrt{3+5 x}}{6400}-\frac{63}{400} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac{1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac{2767149 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{512000}\\ &=\frac{2767149 \sqrt{1-2 x} \sqrt{3+5 x}}{2560000}+\frac{83853 (1-2 x)^{3/2} \sqrt{3+5 x}}{256000}+\frac{7623 (1-2 x)^{5/2} \sqrt{3+5 x}}{64000}-\frac{2079 (1-2 x)^{7/2} \sqrt{3+5 x}}{6400}-\frac{63}{400} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac{1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac{30438639 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{5120000}\\ &=\frac{2767149 \sqrt{1-2 x} \sqrt{3+5 x}}{2560000}+\frac{83853 (1-2 x)^{3/2} \sqrt{3+5 x}}{256000}+\frac{7623 (1-2 x)^{5/2} \sqrt{3+5 x}}{64000}-\frac{2079 (1-2 x)^{7/2} \sqrt{3+5 x}}{6400}-\frac{63}{400} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac{1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac{30438639 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{2560000 \sqrt{5}}\\ &=\frac{2767149 \sqrt{1-2 x} \sqrt{3+5 x}}{2560000}+\frac{83853 (1-2 x)^{3/2} \sqrt{3+5 x}}{256000}+\frac{7623 (1-2 x)^{5/2} \sqrt{3+5 x}}{64000}-\frac{2079 (1-2 x)^{7/2} \sqrt{3+5 x}}{6400}-\frac{63}{400} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac{1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac{30438639 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{2560000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0473972, size = 75, normalized size = 0.47 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (25600000 x^5+8448000 x^4-25526400 x^3-5162720 x^2+10406460 x+717399\right )-30438639 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{25600000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(717399 + 10406460*x - 5162720*x^2 - 25526400*x^3 + 8448000*x^4 + 25600000*x^5
) - 30438639*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/25600000

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Maple [A]  time = 0.008, size = 138, normalized size = 0.9 \begin{align*}{\frac{1}{51200000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 512000000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-510528000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-103254400\,{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}[phone]\,\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+5*x)^(3/2),x)

[Out]

1/51200000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(512000000*x^5*(-10*x^2-x+3)^(1/2)[phone]*x^4*(-10*x^2-x+3)^(1/2)-51
0528000*x^3*(-10*x^2-x+3)^(1/2)-103254400*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)+20812
9200*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 4.95365, size = 134, normalized size = 0.84 \begin{align*} \frac{1}{10} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x + \frac{13}{1000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{693}{1600} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{693}{32000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{251559}{128000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{30438639}{51200000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{251559}{2560000} \, \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+5*x)^(3/2),x, algorithm="maxima")

[Out]

1/10*(-10*x^2 - x + 3)^(5/2)*x + 13/1000*(-10*x^2 - x + 3)^(5/2) + 693/1600*(-10*x^2 - x + 3)^(3/2)*x + 693/32
000*(-10*x^2 - x + 3)^(3/2) + 251559/128000*sqrt(-10*x^2 - x + 3)*x - 30438639/51200000*sqrt(10)*arcsin(-20/11
*x - 1/11) + 251559/2560000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.48396, size = 308, normalized size = 1.92 \begin{align*} \frac{1}{2560000} \,{\left (25600000 \, x^{5} + 8448000 \, x^{4} - 25526400 \, x^{3} - 5162720 \, x^{2} + 10406460 \, x + 717399\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{30438639}{51200000} \, \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+5*x)^(3/2),x, algorithm="fricas")

[Out]

1/2560000*(25600000*x^5 + 8448000*x^4 - 25526400*x^3 - 5162720*x^2 + 10406460*x + 717399)*sqrt(5*x + 3)*sqrt(-
2*x + 1) - 30438639/51200000*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+5*x)**(3/2),x)

[Out]

Timed out

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Giac [B]  time = 1.90109, size = 427, normalized size = 2.67 \begin{align*} \frac{1}{128000000} \, \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{1}{12000000} \, \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{37}{1920000} \, \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}{4800} \, \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{3}{200} \, \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)^(5/2)*(2+3*x)*(3+5*x)^(3/2),x, algorithm="giac")

[Out]

1/128000000*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))) + 1
/12000000*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))) - 37/1920000*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))) - 1/4800*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))) + 3/200*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)))