3.1153 \(\int (3-6 x)^{5/2} (2+4 x)^{5/2} \, dx\)

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

[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

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Rubi [A]  time = 0.0166831, antiderivative size = 100, normalized size of antiderivative = 1., 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} \[ 6 \sqrt{6} (1-2 x)^{5/2} x (2 x+1)^{5/2}+15 \sqrt{\frac{3}{2}} (1-2 x)^{3/2} x (2 x+1)^{3/2}+\frac{45}{2} \sqrt{\frac{3}{2}} \sqrt{1-2 x} x \sqrt{2 x+1}+\frac{45}{4} \sqrt{\frac{3}{2}} \sin ^{-1}(2 x) \]

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*}

Mathematica [A]  time = 0.0326457, size = 44, normalized size = 0.44 \[ \frac{3}{4} \sqrt{\frac{3}{2}} \left (2 x \sqrt{1-4 x^2} \left (128 x^4-104 x^2+33\right )+15 \sin ^{-1}(2 x)\right ) \]

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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Maple [A]  time = 0.006, size = 134, normalized size = 1.3 \begin{align*}{\frac{1}{24} \left ( 3-6\,x \right ) ^{{\frac{5}{2}}} \left ( 2+4\,x \right ) ^{{\frac{7}{2}}}}+{\frac{1}{8} \left ( 3-6\,x \right ) ^{{\frac{3}{2}}} \left ( 2+4\,x \right ) ^{{\frac{7}{2}}}}+{\frac{9}{32}\sqrt{3-6\,x} \left ( 2+4\,x \right ) ^{{\frac{7}{2}}}}-{\frac{3}{16} \left ( 2+4\,x \right ) ^{{\frac{5}{2}}}\sqrt{3-6\,x}}-{\frac{15}{16} \left ( 2+4\,x \right ) ^{{\frac{3}{2}}}\sqrt{3-6\,x}}-{\frac{45}{8}\sqrt{3-6\,x}\sqrt{2+4\,x}}+{\frac{45\,\arcsin \left ( 2\,x \right ) \sqrt{6}}{8}\sqrt{ \left ( 2+4\,x \right ) \left ( 3-6\,x \right ) }{\frac{1}{\sqrt{3-6\,x}}}{\frac{1}{\sqrt{2+4\,x}}}} \end{align*}

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 = 1.46279, size = 62, normalized size = 0.62 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.65272, size = 194, normalized size = 1.94 \begin{align*} \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{align*}

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

[Out]

Timed out

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Giac [A]  time = 1.10393, size = 174, normalized size = 1.74 \begin{align*} \frac{3}{8} \, \sqrt{3} \sqrt{2}{\left ({\left ({\left (2 \,{\left ({\left (8 \,{\left (2 \, x + 1\right )}{\left (x - 2\right )} + 39\right )}{\left (2 \, x + 1\right )} - 37\right )}{\left (2 \, x + 1\right )} + 31\right )}{\left (2 \, x + 1\right )} - 3\right )} \sqrt{2 \, x + 1} \sqrt{-2 \, x + 1} - 12 \,{\left ({\left (4 \,{\left (2 \, x + 1\right )}{\left (x - 1\right )} + 5\right )}{\left (2 \, x + 1\right )} - 1\right )} \sqrt{2 \, x + 1} \sqrt{-2 \, x + 1} + 48 \, \sqrt{2 \, x + 1} x \sqrt{-2 \, x + 1} + 30 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{2 \, x + 1}\right )\right )} \end{align*}

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]

3/8*sqrt(3)*sqrt(2)*(((2*((8*(2*x + 1)*(x - 2) + 39)*(2*x + 1) - 37)*(2*x + 1) + 31)*(2*x + 1) - 3)*sqrt(2*x +
1)*sqrt(-2*x + 1) - 12*((4*(2*x + 1)*(x - 1) + 5)*(2*x + 1) - 1)*sqrt(2*x + 1)*sqrt(-2*x + 1) + 48*sqrt(2*x +
1)*x*sqrt(-2*x + 1) + 30*arcsin(1/2*sqrt(2)*sqrt(2*x + 1)))