3.11.84 \(\int (3-6 x)^{5/2} (2+4 x)^{5/2} \, dx\)

Optimal. Leaf size=100 \[ 6 \sqrt {6} (1-2 x)^{5/2} x (2 x+1)^{5/2}+15 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (2 x+1)^{3/2}+\frac {45}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {2 x+1}+\frac {45}{4} \sqrt {\frac {3}{2}} \sin ^{-1}(2 x) \]

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Rubi [A]  time = 0.02, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {38, 41, 216} \begin {gather*} 6 \sqrt {6} (1-2 x)^{5/2} x (2 x+1)^{5/2}+15 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (2 x+1)^{3/2}+\frac {45}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {2 x+1}+\frac {45}{4} \sqrt {\frac {3}{2}} \sin ^{-1}(2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 - 6*x)^(5/2)*(2 + 4*x)^(5/2),x]

[Out]

(45*Sqrt[3/2]*Sqrt[1 - 2*x]*x*Sqrt[1 + 2*x])/2 + 15*Sqrt[3/2]*(1 - 2*x)^(3/2)*x*(1 + 2*x)^(3/2) + 6*Sqrt[6]*(1
- 2*x)^(5/2)*x*(1 + 2*x)^(5/2) + (45*Sqrt[3/2]*ArcSin[2*x])/4

Rule 38

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(x*(a + b*x)^m*(c + d*x)^m)/(2*m + 1)
, x] + Dist[(2*a*c*m)/(2*m + 1), Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0]

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 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 (3-6 x)^{5/2} (2+4 x)^{5/2} \, dx &=6 \sqrt {6} (1-2 x)^{5/2} x (1+2 x)^{5/2}+5 \int (3-6 x)^{3/2} (2+4 x)^{3/2} \, dx\\ &=15 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+6 \sqrt {6} (1-2 x)^{5/2} x (1+2 x)^{5/2}+\frac {45}{2} \int \sqrt {3-6 x} \sqrt {2+4 x} \, dx\\ &=\frac {45}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {1+2 x}+15 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+6 \sqrt {6} (1-2 x)^{5/2} x (1+2 x)^{5/2}+\frac {135}{2} \int \frac {1}{\sqrt {3-6 x} \sqrt {2+4 x}} \, dx\\ &=\frac {45}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {1+2 x}+15 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+6 \sqrt {6} (1-2 x)^{5/2} x (1+2 x)^{5/2}+\frac {135}{2} \int \frac {1}{\sqrt {6-24 x^2}} \, dx\\ &=\frac {45}{2} \sqrt {\frac {3}{2}} \sqrt {1-2 x} x \sqrt {1+2 x}+15 \sqrt {\frac {3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+6 \sqrt {6} (1-2 x)^{5/2} x (1+2 x)^{5/2}+\frac {45}{4} \sqrt {\frac {3}{2}} \sin ^{-1}(2 x)\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 44, normalized size = 0.44 \begin {gather*} \frac {3}{4} \sqrt {\frac {3}{2}} \left (2 x \sqrt {1-4 x^2} \left (128 x^4-104 x^2+33\right )+15 \sin ^{-1}(2 x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 - 6*x)^(5/2)*(2 + 4*x)^(5/2),x]

[Out]

(3*Sqrt[3/2]*(2*x*Sqrt[1 - 4*x^2]*(33 - 104*x^2 + 128*x^4) + 15*ArcSin[2*x]))/4

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IntegrateAlgebraic [B]  time = 0.97, size = 229, normalized size = 2.29 \begin {gather*} \frac {48 \sqrt {6} \sqrt {1-2 x} x \sqrt {2 x+1} \left (128 x^4-104 x^2+33\right ) \left (-352 x^5-6160 x^4-26224 x^3-41096 x^2-26158 x-5741\right )+48 \sqrt {3} \sqrt {1-2 x} x \left (128 x^4-104 x^2+33\right ) \left (64 x^6+3712 x^5+30160 x^4+80768 x^3+91052 x^2+45112 x+8119\right )}{-22528 x^5-394240 x^4-1678336 x^3-2630144 x^2+\sqrt {2} \sqrt {2 x+1} \left (1024 x^5+58880 x^4+453120 x^3+1065728 x^2+923968 x+259808\right )-1674112 x-367424}+45 \sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {\sqrt {2 x+1}-\sqrt {2}}{\sqrt {1-2 x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(3 - 6*x)^(5/2)*(2 + 4*x)^(5/2),x]

[Out]

(48*Sqrt[6]*Sqrt[1 - 2*x]*x*Sqrt[1 + 2*x]*(33 - 104*x^2 + 128*x^4)*(-5741 - 26158*x - 41096*x^2 - 26224*x^3 -
6160*x^4 - 352*x^5) + 48*Sqrt[3]*Sqrt[1 - 2*x]*x*(33 - 104*x^2 + 128*x^4)*(8119 + 45112*x + 91052*x^2 + 80768*
x^3 + 30160*x^4 + 3712*x^5 + 64*x^6))/(-367424 - 1674112*x - 2630144*x^2 - 1678336*x^3 - 394240*x^4 - 22528*x^
5 + Sqrt[2]*Sqrt[1 + 2*x]*(259808 + 923968*x + 1065728*x^2 + 453120*x^3 + 58880*x^4 + 1024*x^5)) + 45*Sqrt[3/2
]*ArcTan[(-Sqrt[2] + Sqrt[1 + 2*x])/Sqrt[1 - 2*x]]

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fricas [A]  time = 1.21, size = 65, normalized size = 0.65 \begin {gather*} \frac {3}{4} \, {\left (128 \, x^{5} - 104 \, x^{3} + 33 \, x\right )} \sqrt {4 \, x + 2} \sqrt {-6 \, x + 3} - \frac {45}{8} \, \sqrt {3} \sqrt {2} \arctan \left (\frac {\sqrt {3} \sqrt {2} \sqrt {4 \, x + 2} \sqrt {-6 \, x + 3}}{12 \, x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-6*x)^(5/2)*(4*x+2)^(5/2),x, algorithm="fricas")

[Out]

3/4*(128*x^5 - 104*x^3 + 33*x)*sqrt(4*x + 2)*sqrt(-6*x + 3) - 45/8*sqrt(3)*sqrt(2)*arctan(1/12*sqrt(3)*sqrt(2)
*sqrt(4*x + 2)*sqrt(-6*x + 3)/x)

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giac [B]  time = 1.24, size = 227, normalized size = 2.27 \begin {gather*} \frac {3}{40} \, \sqrt {3} \sqrt {2} {\left ({\left ({\left (2 \, {\left ({\left (8 \, {\left (5 \, x - 13\right )} {\left (2 \, x + 1\right )} + 321\right )} {\left (2 \, x + 1\right )} - 451\right )} {\left (2 \, x + 1\right )} + 745\right )} {\left (2 \, x + 1\right )} - 405\right )} \sqrt {2 \, x + 1} \sqrt {-2 \, x + 1} + 2 \, {\left ({\left (2 \, {\left (3 \, {\left (8 \, x - 17\right )} {\left (2 \, x + 1\right )} + 133\right )} {\left (2 \, x + 1\right )} - 295\right )} {\left (2 \, x + 1\right )} + 195\right )} \sqrt {2 \, x + 1} \sqrt {-2 \, x + 1} - 20 \, {\left ({\left (4 \, {\left (3 \, x - 5\right )} {\left (2 \, x + 1\right )} + 43\right )} {\left (2 \, x + 1\right )} - 39\right )} \sqrt {2 \, x + 1} \sqrt {-2 \, x + 1} - 80 \, {\left ({\left (4 \, x - 5\right )} {\left (2 \, x + 1\right )} + 9\right )} \sqrt {2 \, x + 1} \sqrt {-2 \, x + 1} + 240 \, \sqrt {2 \, x + 1} {\left (x - 1\right )} \sqrt {-2 \, x + 1} + 240 \, \sqrt {2 \, x + 1} \sqrt {-2 \, x + 1} + 150 \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {2 \, x + 1}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-6*x)^(5/2)*(4*x+2)^(5/2),x, algorithm="giac")

[Out]

3/40*sqrt(3)*sqrt(2)*(((2*((8*(5*x - 13)*(2*x + 1) + 321)*(2*x + 1) - 451)*(2*x + 1) + 745)*(2*x + 1) - 405)*s
qrt(2*x + 1)*sqrt(-2*x + 1) + 2*((2*(3*(8*x - 17)*(2*x + 1) + 133)*(2*x + 1) - 295)*(2*x + 1) + 195)*sqrt(2*x
+ 1)*sqrt(-2*x + 1) - 20*((4*(3*x - 5)*(2*x + 1) + 43)*(2*x + 1) - 39)*sqrt(2*x + 1)*sqrt(-2*x + 1) - 80*((4*x
- 5)*(2*x + 1) + 9)*sqrt(2*x + 1)*sqrt(-2*x + 1) + 240*sqrt(2*x + 1)*(x - 1)*sqrt(-2*x + 1) + 240*sqrt(2*x +
1)*sqrt(-2*x + 1) + 150*arcsin(1/2*sqrt(2)*sqrt(2*x + 1)))

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maple [A]  time = 0.01, size = 134, normalized size = 1.34 \begin {gather*} \frac {45 \sqrt {\left (4 x +2\right ) \left (-6 x +3\right )}\, \sqrt {6}\, \arcsin \left (2 x \right )}{8 \sqrt {4 x +2}\, \sqrt {-6 x +3}}+\frac {\left (-6 x +3\right )^{\frac {5}{2}} \left (4 x +2\right )^{\frac {7}{2}}}{24}+\frac {\left (-6 x +3\right )^{\frac {3}{2}} \left (4 x +2\right )^{\frac {7}{2}}}{8}+\frac {9 \sqrt {-6 x +3}\, \left (4 x +2\right )^{\frac {7}{2}}}{32}-\frac {3 \left (4 x +2\right )^{\frac {5}{2}} \sqrt {-6 x +3}}{16}-\frac {15 \left (4 x +2\right )^{\frac {3}{2}} \sqrt {-6 x +3}}{16}-\frac {45 \sqrt {-6 x +3}\, \sqrt {4 x +2}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3-6*x)^(5/2)*(2+4*x)^(5/2),x)

[Out]

1/24*(3-6*x)^(5/2)*(2+4*x)^(7/2)+1/8*(3-6*x)^(3/2)*(2+4*x)^(7/2)+9/32*(3-6*x)^(1/2)*(2+4*x)^(7/2)-3/16*(2+4*x)
^(5/2)*(3-6*x)^(1/2)-15/16*(2+4*x)^(3/2)*(3-6*x)^(1/2)-45/8*(3-6*x)^(1/2)*(2+4*x)^(1/2)+45/8*((2+4*x)*(3-6*x))
^(1/2)/(2+4*x)^(1/2)/(3-6*x)^(1/2)*6^(1/2)*arcsin(2*x)

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maxima [A]  time = 2.86, size = 46, normalized size = 0.46 \begin {gather*} \frac {1}{6} \, {\left (-24 \, x^{2} + 6\right )}^{\frac {5}{2}} x + \frac {5}{4} \, {\left (-24 \, x^{2} + 6\right )}^{\frac {3}{2}} x + \frac {45}{4} \, \sqrt {-24 \, x^{2} + 6} x + \frac {45}{8} \, \sqrt {6} \arcsin \left (2 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-6*x)^(5/2)*(4*x+2)^(5/2),x, algorithm="maxima")

[Out]

1/6*(-24*x^2 + 6)^(5/2)*x + 5/4*(-24*x^2 + 6)^(3/2)*x + 45/4*sqrt(-24*x^2 + 6)*x + 45/8*sqrt(6)*arcsin(2*x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (4\,x+2\right )}^{5/2}\,{\left (3-6\,x\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x + 2)^(5/2)*(3 - 6*x)^(5/2),x)

[Out]

int((4*x + 2)^(5/2)*(3 - 6*x)^(5/2), 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((3-6*x)**(5/2)*(4*x+2)**(5/2),x)

[Out]

Timed out

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