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

Optimal. Leaf size=165 \[ -\frac {1}{20} (3 x+2) (5 x+3)^{5/2} (1-2 x)^{5/2}-\frac {259 (5 x+3)^{5/2} (1-2 x)^{5/2}}{2000}-\frac {3101 (5 x+3)^{3/2} (1-2 x)^{5/2}}{6400}-\frac {34111 \sqrt {5 x+3} (1-2 x)^{5/2}}{25600}+\frac {375221 \sqrt {5 x+3} (1-2 x)^{3/2}}{512000}+\frac {12382293 \sqrt {5 x+3} \sqrt {1-2 x}}{5120000}+\frac {136205223 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5120000 \sqrt {10}} \]

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Rubi [A]  time = 0.05, antiderivative size = 165, normalized size of antiderivative = 1.00, 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 = {90, 80, 50, 54, 216} \begin {gather*} -\frac {1}{20} (3 x+2) (5 x+3)^{5/2} (1-2 x)^{5/2}-\frac {259 (5 x+3)^{5/2} (1-2 x)^{5/2}}{2000}-\frac {3101 (5 x+3)^{3/2} (1-2 x)^{5/2}}{6400}-\frac {34111 \sqrt {5 x+3} (1-2 x)^{5/2}}{25600}+\frac {375221 \sqrt {5 x+3} (1-2 x)^{3/2}}{512000}+\frac {12382293 \sqrt {5 x+3} \sqrt {1-2 x}}{5120000}+\frac {136205223 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5120000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(12382293*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/5120000 + (375221*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/512000 - (34111*(1 - 2
*x)^(5/2)*Sqrt[3 + 5*x])/25600 - (3101*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/6400 - (259*(1 - 2*x)^(5/2)*(3 + 5*x)^
(5/2))/2000 - ((1 - 2*x)^(5/2)*(2 + 3*x)*(3 + 5*x)^(5/2))/20 + (136205223*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5
120000*Sqrt[10])

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 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 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 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)^2 (3+5 x)^{3/2} \, dx &=-\frac {1}{20} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2}-\frac {1}{60} \int \left (-252-\frac {777 x}{2}\right ) (1-2 x)^{3/2} (3+5 x)^{3/2} \, dx\\ &=-\frac {259 (1-2 x)^{5/2} (3+5 x)^{5/2}}{2000}-\frac {1}{20} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2}+\frac {3101}{800} \int (1-2 x)^{3/2} (3+5 x)^{3/2} \, dx\\ &=-\frac {3101 (1-2 x)^{5/2} (3+5 x)^{3/2}}{6400}-\frac {259 (1-2 x)^{5/2} (3+5 x)^{5/2}}{2000}-\frac {1}{20} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2}+\frac {102333 \int (1-2 x)^{3/2} \sqrt {3+5 x} \, dx}{12800}\\ &=-\frac {34111 (1-2 x)^{5/2} \sqrt {3+5 x}}{25600}-\frac {3101 (1-2 x)^{5/2} (3+5 x)^{3/2}}{6400}-\frac {259 (1-2 x)^{5/2} (3+5 x)^{5/2}}{2000}-\frac {1}{20} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2}+\frac {375221 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{51200}\\ &=\frac {375221 (1-2 x)^{3/2} \sqrt {3+5 x}}{512000}-\frac {34111 (1-2 x)^{5/2} \sqrt {3+5 x}}{25600}-\frac {3101 (1-2 x)^{5/2} (3+5 x)^{3/2}}{6400}-\frac {259 (1-2 x)^{5/2} (3+5 x)^{5/2}}{2000}-\frac {1}{20} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2}+\frac {12382293 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{1024000}\\ &=\frac {12382293 \sqrt {1-2 x} \sqrt {3+5 x}}{5120000}+\frac {375221 (1-2 x)^{3/2} \sqrt {3+5 x}}{512000}-\frac {34111 (1-2 x)^{5/2} \sqrt {3+5 x}}{25600}-\frac {3101 (1-2 x)^{5/2} (3+5 x)^{3/2}}{6400}-\frac {259 (1-2 x)^{5/2} (3+5 x)^{5/2}}{2000}-\frac {1}{20} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2}+\frac {136205223 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{10240000}\\ &=\frac {12382293 \sqrt {1-2 x} \sqrt {3+5 x}}{5120000}+\frac {375221 (1-2 x)^{3/2} \sqrt {3+5 x}}{512000}-\frac {34111 (1-2 x)^{5/2} \sqrt {3+5 x}}{25600}-\frac {3101 (1-2 x)^{5/2} (3+5 x)^{3/2}}{6400}-\frac {259 (1-2 x)^{5/2} (3+5 x)^{5/2}}{2000}-\frac {1}{20} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2}+\frac {136205223 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{5120000 \sqrt {5}}\\ &=\frac {12382293 \sqrt {1-2 x} \sqrt {3+5 x}}{5120000}+\frac {375221 (1-2 x)^{3/2} \sqrt {3+5 x}}{512000}-\frac {34111 (1-2 x)^{5/2} \sqrt {3+5 x}}{25600}-\frac {3101 (1-2 x)^{5/2} (3+5 x)^{3/2}}{6400}-\frac {259 (1-2 x)^{5/2} (3+5 x)^{5/2}}{2000}-\frac {1}{20} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2}+\frac {136205223 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{5120000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 84, normalized size = 0.51 \begin {gather*} \frac {10 \sqrt {5 x+3} \left (153600000 x^6+188928000 x^5-77254400 x^4-160790720 x^3-8083480 x^2+54699134 x-8705457\right )[phone] \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{51200000 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[3 + 5*x]*(-8705457 + 54699134*x - 8083480*x^2 - 160790720*x^3 - 77254400*x^4 + 188928000*x^5 + 153600
000*x^6) + 136205223*Sqrt[-10 + 20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(51200000*Sqrt[1 - 2*x])

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IntegrateAlgebraic [A]  time = 0.28, size = 157, normalized size = 0.95 \begin {gather*} -\frac {14641 \sqrt {1-2 x} \left (\frac {29071875 (1-2 x)^5}{(5 x+3)^5}+\frac {65896250 (1-2 x)^4}{(5 x+3)^4}+\frac {60990200 (1-2 x)^3}{(5 x+3)^3}+\frac {25616080 (1-2 x)^2}{(5 x+3)^2}-\frac {4217360 (1-2 x)}{5 x+3}-297696\right )}{5120000 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^6}-\frac {136205223 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{5120000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-14641*Sqrt[1 - 2*x]*(-297696 + (29071875*(1 - 2*x)^5)/(3 + 5*x)^5 + (65896250*(1 - 2*x)^4)/(3 + 5*x)^4 + (60
990200*(1 - 2*x)^3)/(3 + 5*x)^3 + (25616080*(1 - 2*x)^2)/(3 + 5*x)^2 - (4217360*(1 - 2*x))/(3 + 5*x)))/(512000
0*Sqrt[3 + 5*x]*(2 + (5*(1 - 2*x))/(3 + 5*x))^6) - (136205223*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])
/(5120000*Sqrt[10])

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fricas [A]  time = 1.34, size = 82, normalized size = 0.50 \begin {gather*} -\frac {1}{5120000} \, {\left (76800000 \, x^{5} + 132864000 \, x^{4} + 27804800 \, x^{3} - 66492960 \, x^{2} - 37288220 \, x + 8705457\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {136205223}{102400000} \, \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(2+3*x)^2*(3+5*x)^(3/2),x, algorithm="fricas")

[Out]

-1/5120000*(76800000*x^5 + 132864000*x^4 + 27804800*x^3 - 66492960*x^2 - 37288220*x + 8705457)*sqrt(5*x + 3)*s
qrt(-2*x + 1) - 136205223/102400000*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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giac [B]  time = 1.26, size = 356, normalized size = 2.16 \begin {gather*} -\frac {3}{256000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x - 311\right )} {\left (5 \, x + 3\right )} + 46071\right )} {\left (5 \, x + 3\right )} - 775911\right )} {\left (5 \, x + 3\right )} + 15385695\right )} {\left (5 \, x + 3\right )} - 99422145\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 220189365 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {61}{64000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {1}{18750} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {17}{24000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {39}{500} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {18}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(2+3*x)^2*(3+5*x)^(3/2),x, algorithm="giac")

[Out]

-3/256000000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 311)*(5*x + 3) + 46071)*(5*x + 3) - 775911)*(5*x + 3) + 15385695
)*(5*x + 3) - 99422145)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 220189365*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))
- 61/64000000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) - 506185)*(5*x + 3) + 4031895)*s
qrt(5*x + 3)*sqrt(-10*x + 5) + 10392195*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 1/18750*sqrt(5)*(2*(4*(
8*(60*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 184305*sqrt(2)*arcsin(1/11
*sqrt(22)*sqrt(5*x + 3))) + 17/24000*sqrt(5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqrt(-10*x + 5)
+ 4785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 39/500*sqrt(5)*(2*(20*x - 23)*sqrt(5*x + 3)*sqrt(-10*x
+ 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 18/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(
5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))

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maple [A]  time = 0.01, size = 138, normalized size = 0.84 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-1536000000 \sqrt {-10 x^{2}-x +3}\, x^{5}-2657280000 \sqrt {-10 x^{2}-x +3}\, x^{4}-556096000 \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-174109140 \sqrt {-10 x^{2}-x +3}\right )}{102400000 \sqrt {-10 x^{2}-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x+1)^(3/2)*(3*x+2)^2*(5*x+3)^(3/2),x)

[Out]

1/102400000*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(-1536000000*(-10*x^2-x+3)^(1/2)*x^5-2657280000*(-10*x^2-x+3)^(1/2)*x
^4-556096000*(-10*x^2-x+3)^(1/2)*x^3+1329859200*(-10*x^2-x+3)^(1/2)*x^2+136205223*10^(1/2)*arcsin(20/11*x+1/11
)[phone]*(-10*x^2-x+3)^(1/2)*x-174109140*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.19, size = 99, normalized size = 0.60 \begin {gather*} -\frac {3}{20} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x - \frac {459}{2000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {3101}{3200} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {3101}{64000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {1125663}{256000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {136205223}{102400000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {1125663}{5120000} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(2+3*x)^2*(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

-3/20*(-10*x^2 - x + 3)^(5/2)*x - 459/2000*(-10*x^2 - x + 3)^(5/2) + 3101/3200*(-10*x^2 - x + 3)^(3/2)*x + 310
1/64000*(-10*x^2 - x + 3)^(3/2) + 1125663/256000*sqrt(-10*x^2 - x + 3)*x - 136205223/102400000*sqrt(10)*arcsin
(-20/11*x - 1/11) + 1125663/5120000*sqrt(-10*x^2 - x + 3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(3/2)*(3*x + 2)^2*(5*x + 3)^(3/2),x)

[Out]

int((1 - 2*x)^(3/2)*(3*x + 2)^2*(5*x + 3)^(3/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**(3/2)*(2+3*x)**2*(3+5*x)**(3/2),x)

[Out]

Timed out

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