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

Optimal. Leaf size=138 \[ -\frac {3}{50} (5 x+3)^{5/2} (1-2 x)^{5/2}-\frac {37}{160} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac {407}{640} \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {4477 \sqrt {5 x+3} (1-2 x)^{3/2}}{12800}+\frac {147741 \sqrt {5 x+3} \sqrt {1-2 x}}{128000}+\frac {1625151 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{128000 \sqrt {10}} \]

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Rubi [A]  time = 0.04, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {80, 50, 54, 216} \begin {gather*} -\frac {3}{50} (5 x+3)^{5/2} (1-2 x)^{5/2}-\frac {37}{160} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac {407}{640} \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {4477 \sqrt {5 x+3} (1-2 x)^{3/2}}{12800}+\frac {147741 \sqrt {5 x+3} \sqrt {1-2 x}}{128000}+\frac {1625151 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{128000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(147741*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/128000 + (4477*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/12800 - (407*(1 - 2*x)^(5/2
)*Sqrt[3 + 5*x])/640 - (37*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/160 - (3*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/50 + (16
25151*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(128000*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 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+5 x)^{3/2} \, dx &=-\frac {3}{50} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac {37}{20} \int (1-2 x)^{3/2} (3+5 x)^{3/2} \, dx\\ &=-\frac {37}{160} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {3}{50} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac {1221}{320} \int (1-2 x)^{3/2} \sqrt {3+5 x} \, dx\\ &=-\frac {407}{640} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {37}{160} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {3}{50} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac {4477 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{1280}\\ &=\frac {4477 (1-2 x)^{3/2} \sqrt {3+5 x}}{12800}-\frac {407}{640} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {37}{160} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {3}{50} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac {147741 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{25600}\\ &=\frac {147741 \sqrt {1-2 x} \sqrt {3+5 x}}{128000}+\frac {4477 (1-2 x)^{3/2} \sqrt {3+5 x}}{12800}-\frac {407}{640} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {37}{160} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {3}{50} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac {1625151 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{256000}\\ &=\frac {147741 \sqrt {1-2 x} \sqrt {3+5 x}}{128000}+\frac {4477 (1-2 x)^{3/2} \sqrt {3+5 x}}{12800}-\frac {407}{640} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {37}{160} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {3}{50} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac {1625151 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{128000 \sqrt {5}}\\ &=\frac {147741 \sqrt {1-2 x} \sqrt {3+5 x}}{128000}+\frac {4477 (1-2 x)^{3/2} \sqrt {3+5 x}}{12800}-\frac {407}{640} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {37}{160} (1-2 x)^{5/2} (3+5 x)^{3/2}-\frac {3}{50} (1-2 x)^{5/2} (3+5 x)^{5/2}+\frac {1625151 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{128000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 79, normalized size = 0.57 \begin {gather*} \frac {10 \sqrt {5 x+3} \left (1536000 x^5+723200 x^4-1474240 x^3-614360 x^2+582958 x-46809\right )+1625151 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{1280000 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[3 + 5*x]*(-46809 + 582958*x - 614360*x^2 - 1474240*x^3 + 723200*x^4 + 1536000*x^5) + 1625151*Sqrt[-10
+ 20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(1280000*Sqrt[1 - 2*x])

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IntegrateAlgebraic [A]  time = 0.23, size = 141, normalized size = 1.02 \begin {gather*} -\frac {14641 \sqrt {1-2 x} \left (\frac {69375 (1-2 x)^4}{(5 x+3)^4}+\frac {129500 (1-2 x)^3}{(5 x+3)^3}+\frac {84480 (1-2 x)^2}{(5 x+3)^2}-\frac {20720 (1-2 x)}{5 x+3}-1776\right )}{128000 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^5}-\frac {1625151 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{128000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-14641*Sqrt[1 - 2*x]*(-1776 + (69375*(1 - 2*x)^4)/(3 + 5*x)^4 + (129500*(1 - 2*x)^3)/(3 + 5*x)^3 + (84480*(1
- 2*x)^2)/(3 + 5*x)^2 - (20720*(1 - 2*x))/(3 + 5*x)))/(128000*Sqrt[3 + 5*x]*(2 + (5*(1 - 2*x))/(3 + 5*x))^5) -
(1625151*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/(128000*Sqrt[10])

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fricas [A]  time = 1.51, size = 77, normalized size = 0.56 \begin {gather*} -\frac {1}{128000} \, {\left (768000 \, x^{4} + 745600 \, x^{3} - 364320 \, x^{2} - 489340 \, x + 46809\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {1625151}{2560000} \, \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)*(3+5*x)^(3/2),x, algorithm="fricas")

[Out]

-1/128000*(768000*x^4 + 745600*x^3 - 364320*x^2 - 489340*x + 46809)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1625151/256
0000*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 = 1.38, size = 275, normalized size = 1.99 \begin {gather*} -\frac {1}{6400000} \, \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 {41}{1920000} \, \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}{60000} \, \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 {51}{2000} \, \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 {9}{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)*(3+5*x)^(3/2),x, algorithm="giac")

[Out]

-1/6400000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) - 506185)*(5*x + 3) + 4031895)*sqrt(
5*x + 3)*sqrt(-10*x + 5) + 10392195*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 41/1920000*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))) - 17/60000*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))) + 51/2000*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))) + 9/25*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 = 121, normalized size = 0.88 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-15360000 \sqrt {-10 x^{2}-x +3}\, x^{4}-14912000 \sqrt {-10 x^{2}-x +3}\, x^{3}+7286400 \sqrt {-10 x^{2}-x +3}\, x^{2}+9786800 \sqrt {-10 x^{2}-x +3}\, x +1625151 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-936180 \sqrt {-10 x^{2}-x +3}\right )}{2560000 \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)*(5*x+3)^(3/2),x)

[Out]

1/2560000*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(-15360000*(-10*x^2-x+3)^(1/2)*x^4-14912000*(-10*x^2-x+3)^(1/2)*x^3+728
6400*(-10*x^2-x+3)^(1/2)*x^2+1625151*10^(1/2)*arcsin(20/11*x+1/11)+9786800*(-10*x^2-x+3)^(1/2)*x-936180*(-10*x
^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.24, size = 84, normalized size = 0.61 \begin {gather*} -\frac {3}{50} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {37}{80} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {37}{1600} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {13431}{6400} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {1625151}{2560000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {13431}{128000} \, \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)*(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

-3/50*(-10*x^2 - x + 3)^(5/2) + 37/80*(-10*x^2 - x + 3)^(3/2)*x + 37/1600*(-10*x^2 - x + 3)^(3/2) + 13431/6400
*sqrt(-10*x^2 - x + 3)*x - 1625151/2560000*sqrt(10)*arcsin(-20/11*x - 1/11) + 13431/128000*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 )\,{\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)*(5*x + 3)^(3/2),x)

[Out]

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

[Out]

Timed out

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