3.12.53 \(\int (3-6 x)^{5/2} (2+4 x)^{5/2} \, dx\) [1153]

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

[Out]

15/2*(1-2*x)^(3/2)*x*(1+2*x)^(3/2)*6^(1/2)+45/8*arcsin(2*x)*6^(1/2)+6*(1-2*x)^(5/2)*x*(1+2*x)^(5/2)*6^(1/2)+45
/4*x*6^(1/2)*(1-2*x)^(1/2)*(1+2*x)^(1/2)

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Rubi [A]
time = 0.01, 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, 222} \begin {gather*} \frac {45}{4} \sqrt {\frac {3}{2}} \text {ArcSin}(2 x)+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} \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 222

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 = 1.28, size = 193, normalized size = 1.93 \begin {gather*} -\frac {3 \sqrt {3-6 x} x \left (33-104 x^2+128 x^4\right ) \left (8119+45112 x+91052 x^2+80768 x^3+30160 x^4+3712 x^5+64 x^6-\sqrt {2+4 x} \left (5741+26158 x+41096 x^2+26224 x^3+6160 x^4+352 x^5\right )\right )}{2 \left (11482+52316 x+82192 x^2+52448 x^3+12320 x^4+704 x^5-\sqrt {2+4 x} \left (8119+28874 x+33304 x^2+14160 x^3+1840 x^4+32 x^5\right )\right )}+45 \sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {-\sqrt {2}+\sqrt {1+2 x}}{\sqrt {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 - 6*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 - Sqrt[2 + 4*x]*(5741 + 26158*x + 41096*x^2 + 26224*x^3 + 6160*x^4 + 352*x^5)))/(2*(11482 + 52316*x + 8
2192*x^2 + 52448*x^3 + 12320*x^4 + 704*x^5 - Sqrt[2 + 4*x]*(8119 + 28874*x + 33304*x^2 + 14160*x^3 + 1840*x^4
+ 32*x^5))) + 45*Sqrt[3/2]*ArcTan[(-Sqrt[2] + Sqrt[1 + 2*x])/Sqrt[1 - 2*x]]

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Maple [A]
time = 0.18, size = 134, normalized size = 1.34

method	result	size
risch	\(-\frac {3 x \left (128 x^{4}-104 x^{2}+33\right ) \left (2 x -1\right ) \left (1+2 x \right ) \sqrt {\left (2+4 x \right ) \left (3-6 x \right )}\, \sqrt {6}}{4 \sqrt {-\left (2 x -1\right ) \left (1+2 x \right )}\, \sqrt {3-6 x}\, \sqrt {2+4 x}}+\frac {45 \sqrt {\left (2+4 x \right ) \left (3-6 x \right )}\, \arcsin \left (2 x \right ) \sqrt {6}}{8 \sqrt {2+4 x}\, \sqrt {3-6 x}}\)	\(107\)
default	\(\frac {\left (3-6 x \right )^{\frac {5}{2}} \left (2+4 x \right )^{\frac {7}{2}}}{24}+\frac {\left (3-6 x \right )^{\frac {3}{2}} \left (2+4 x \right )^{\frac {7}{2}}}{8}+\frac {9 \sqrt {3-6 x}\, \left (2+4 x \right )^{\frac {7}{2}}}{32}-\frac {3 \left (2+4 x \right )^{\frac {5}{2}} \sqrt {3-6 x}}{16}-\frac {15 \left (2+4 x \right )^{\frac {3}{2}} \sqrt {3-6 x}}{16}-\frac {45 \sqrt {3-6 x}\, \sqrt {2+4 x}}{8}+\frac {45 \sqrt {\left (2+4 x \right ) \left (3-6 x \right )}\, \arcsin \left (2 x \right ) \sqrt {6}}{8 \sqrt {2+4 x}\, \sqrt {3-6 x}}\)	\(134\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)*arcsin(2*x)*6^(1/2)

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Maxima [A]
time = 0.51, 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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Fricas [A]
time = 1.00, 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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Sympy [F(-1)] Timed out
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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (70) = 140\).
time = 1.52, 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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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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