Integrand size = 13, antiderivative size = 145 \[ \int \frac {1}{\left (3 x-4 x^2\right )^{9/2}} \, dx=\frac {2}{21 (3-4 x)^{7/2} x^{7/2}}+\frac {4}{45 (3-4 x)^{5/2} x^{7/2}}+\frac {16}{135 (3-4 x)^{3/2} x^{7/2}}+\frac {32}{81 \sqrt {3-4 x} x^{7/2}}-\frac {256 \sqrt {3-4 x}}{1701 x^{7/2}}-\frac {2048 \sqrt {3-4 x}}{8505 x^{5/2}}-\frac {32768 \sqrt {3-4 x}}{76545 x^{3/2}}-\frac {262144 \sqrt {3-4 x}}{229635 \sqrt {x}} \] Output:
2/21/(3-4*x)^(7/2)/x^(7/2)+4/45/(3-4*x)^(5/2)/x^(7/2)+16/135/(3-4*x)^(3/2) /x^(7/2)+32/81/(3-4*x)^(1/2)/x^(7/2)-256/1701*(3-4*x)^(1/2)/x^(7/2)-2048/8 505*(3-4*x)^(1/2)/x^(5/2)-32768/76545*(3-4*x)^(1/2)/x^(3/2)-262144/229635* (3-4*x)^(1/2)/x^(1/2)
Time = 0.11 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.35 \[ \int \frac {1}{\left (3 x-4 x^2\right )^{9/2}} \, dx=-\frac {2 \left (10935+40824 x+217728 x^2+2903040 x^3-30965760 x^4+82575360 x^5-88080384 x^6+33554432 x^7\right )}{229635 (-x (-3+4 x))^{7/2}} \] Input:
Integrate[(3*x - 4*x^2)^(-9/2),x]
Output:
(-2*(10935 + 40824*x + 217728*x^2 + 2903040*x^3 - 30965760*x^4 + 82575360* x^5 - 88080384*x^6 + 33554432*x^7))/(229635*(-(x*(-3 + 4*x)))^(7/2))
Time = 0.35 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.68, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1089, 1089, 1089, 1088}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\left (3 x-4 x^2\right )^{9/2}} \, dx\) |
\(\Big \downarrow \) 1089 |
\(\displaystyle \frac {32}{21} \int \frac {1}{\left (3 x-4 x^2\right )^{7/2}}dx-\frac {2 (3-8 x)}{63 \left (3 x-4 x^2\right )^{7/2}}\) |
\(\Big \downarrow \) 1089 |
\(\displaystyle \frac {32}{21} \left (\frac {64}{45} \int \frac {1}{\left (3 x-4 x^2\right )^{5/2}}dx-\frac {2 (3-8 x)}{45 \left (3 x-4 x^2\right )^{5/2}}\right )-\frac {2 (3-8 x)}{63 \left (3 x-4 x^2\right )^{7/2}}\) |
\(\Big \downarrow \) 1089 |
\(\displaystyle \frac {32}{21} \left (\frac {64}{45} \left (\frac {32}{27} \int \frac {1}{\left (3 x-4 x^2\right )^{3/2}}dx-\frac {2 (3-8 x)}{27 \left (3 x-4 x^2\right )^{3/2}}\right )-\frac {2 (3-8 x)}{45 \left (3 x-4 x^2\right )^{5/2}}\right )-\frac {2 (3-8 x)}{63 \left (3 x-4 x^2\right )^{7/2}}\) |
\(\Big \downarrow \) 1088 |
\(\displaystyle \frac {32}{21} \left (\frac {64}{45} \left (-\frac {64 (3-8 x)}{243 \sqrt {3 x-4 x^2}}-\frac {2 (3-8 x)}{27 \left (3 x-4 x^2\right )^{3/2}}\right )-\frac {2 (3-8 x)}{45 \left (3 x-4 x^2\right )^{5/2}}\right )-\frac {2 (3-8 x)}{63 \left (3 x-4 x^2\right )^{7/2}}\) |
Input:
Int[(3*x - 4*x^2)^(-9/2),x]
Output:
(-2*(3 - 8*x))/(63*(3*x - 4*x^2)^(7/2)) + (32*((-2*(3 - 8*x))/(45*(3*x - 4 *x^2)^(5/2)) + (64*((-2*(3 - 8*x))/(27*(3*x - 4*x^2)^(3/2)) - (64*(3 - 8*x ))/(243*Sqrt[3*x - 4*x^2])))/45))/21
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre eQ[{a, b, c}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[3*p])
Time = 0.29 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.35
method | result | size |
meijerg | \(-\frac {2 \sqrt {3}\, \left (\frac {33554432}{2187} x^{7}-\frac {29360128}{729} x^{6}+\frac {9175040}{243} x^{5}-\frac {1146880}{81} x^{4}+\frac {35840}{27} x^{3}+\frac {896}{9} x^{2}+\frac {56}{3} x +5\right )}{8505 x^{\frac {7}{2}} \left (-\frac {4 x}{3}+1\right )^{\frac {7}{2}}}\) | \(51\) |
gosper | \(\frac {2 x \left (4 x -3\right ) \left (33554432 x^{7}-88080384 x^{6}+82575360 x^{5}-30965760 x^{4}+2903040 x^{3}+217728 x^{2}+40824 x +10935\right )}{229635 \left (-4 x^{2}+3 x \right )^{\frac {9}{2}}}\) | \(55\) |
orering | \(\frac {2 x \left (4 x -3\right ) \left (33554432 x^{7}-88080384 x^{6}+82575360 x^{5}-30965760 x^{4}+2903040 x^{3}+217728 x^{2}+40824 x +10935\right )}{229635 \left (-4 x^{2}+3 x \right )^{\frac {9}{2}}}\) | \(55\) |
trager | \(-\frac {2 \left (33554432 x^{7}-88080384 x^{6}+82575360 x^{5}-30965760 x^{4}+2903040 x^{3}+217728 x^{2}+40824 x +10935\right ) \sqrt {-4 x^{2}+3 x}}{229635 x^{4} \left (4 x -3\right )^{4}}\) | \(59\) |
default | \(-\frac {2 \left (-8 x +3\right )}{63 \left (-4 x^{2}+3 x \right )^{\frac {7}{2}}}-\frac {64 \left (-8 x +3\right )}{945 \left (-4 x^{2}+3 x \right )^{\frac {5}{2}}}-\frac {4096 \left (-8 x +3\right )}{25515 \left (-4 x^{2}+3 x \right )^{\frac {3}{2}}}-\frac {131072 \left (-8 x +3\right )}{229635 \sqrt {-4 x^{2}+3 x}}\) | \(74\) |
Input:
int(1/(-4*x^2+3*x)^(9/2),x,method=_RETURNVERBOSE)
Output:
-2/8505/x^(7/2)*3^(1/2)*(33554432/2187*x^7-29360128/729*x^6+9175040/243*x^ 5-1146880/81*x^4+35840/27*x^3+896/9*x^2+56/3*x+5)/(-4/3*x+1)^(7/2)
Time = 0.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.52 \[ \int \frac {1}{\left (3 x-4 x^2\right )^{9/2}} \, dx=-\frac {2 \, {\left (33554432 \, x^{7} - 88080384 \, x^{6} + 82575360 \, x^{5} - 30965760 \, x^{4} + 2903040 \, x^{3} + 217728 \, x^{2} + 40824 \, x + 10935\right )} \sqrt {-4 \, x^{2} + 3 \, x}}{229635 \, {\left (256 \, x^{8} - 768 \, x^{7} + 864 \, x^{6} - 432 \, x^{5} + 81 \, x^{4}\right )}} \] Input:
integrate(1/(-4*x^2+3*x)^(9/2),x, algorithm="fricas")
Output:
-2/229635*(33554432*x^7 - 88080384*x^6 + 82575360*x^5 - 30965760*x^4 + 290 3040*x^3 + 217728*x^2 + 40824*x + 10935)*sqrt(-4*x^2 + 3*x)/(256*x^8 - 768 *x^7 + 864*x^6 - 432*x^5 + 81*x^4)
\[ \int \frac {1}{\left (3 x-4 x^2\right )^{9/2}} \, dx=\int \frac {1}{\left (- 4 x^{2} + 3 x\right )^{\frac {9}{2}}}\, dx \] Input:
integrate(1/(-4*x**2+3*x)**(9/2),x)
Output:
Integral((-4*x**2 + 3*x)**(-9/2), x)
Time = 0.03 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (3 x-4 x^2\right )^{9/2}} \, dx=\frac {1048576 \, x}{229635 \, \sqrt {-4 \, x^{2} + 3 \, x}} - \frac {131072}{76545 \, \sqrt {-4 \, x^{2} + 3 \, x}} + \frac {32768 \, x}{25515 \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {3}{2}}} - \frac {4096}{8505 \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {3}{2}}} + \frac {512 \, x}{945 \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {5}{2}}} - \frac {64}{315 \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {5}{2}}} + \frac {16 \, x}{63 \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {7}{2}}} - \frac {2}{21 \, {\left (-4 \, x^{2} + 3 \, x\right )}^{\frac {7}{2}}} \] Input:
integrate(1/(-4*x^2+3*x)^(9/2),x, algorithm="maxima")
Output:
1048576/229635*x/sqrt(-4*x^2 + 3*x) - 131072/76545/sqrt(-4*x^2 + 3*x) + 32 768/25515*x/(-4*x^2 + 3*x)^(3/2) - 4096/8505/(-4*x^2 + 3*x)^(3/2) + 512/94 5*x/(-4*x^2 + 3*x)^(5/2) - 64/315/(-4*x^2 + 3*x)^(5/2) + 16/63*x/(-4*x^2 + 3*x)^(7/2) - 2/21/(-4*x^2 + 3*x)^(7/2)
Time = 0.13 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.41 \[ \int \frac {1}{\left (3 x-4 x^2\right )^{9/2}} \, dx=-\frac {2 \, {\left (8 \, {\left (16 \, {\left (8 \, {\left (32 \, {\left (8 \, {\left (16 \, {\left (8 \, x - 21\right )} x + 315\right )} x - 945\right )} x + 2835\right )} x + 1701\right )} x + 5103\right )} x + 10935\right )} \sqrt {-4 \, x^{2} + 3 \, x}}{229635 \, {\left (4 \, x^{2} - 3 \, x\right )}^{4}} \] Input:
integrate(1/(-4*x^2+3*x)^(9/2),x, algorithm="giac")
Output:
-2/229635*(8*(16*(8*(32*(8*(16*(8*x - 21)*x + 315)*x - 945)*x + 2835)*x + 1701)*x + 5103)*x + 10935)*sqrt(-4*x^2 + 3*x)/(4*x^2 - 3*x)^4
Time = 8.93 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.48 \[ \int \frac {1}{\left (3 x-4 x^2\right )^{9/2}} \, dx=\frac {\frac {16\,x}{63}-\frac {2}{21}}{{\left (3\,x-4\,x^2\right )}^{7/2}}+\frac {\frac {512\,x}{945}-\frac {64}{315}}{{\left (3\,x-4\,x^2\right )}^{5/2}}+\frac {\frac {32768\,x}{25515}-\frac {4096}{8505}}{{\left (3\,x-4\,x^2\right )}^{3/2}}+\frac {\frac {1048576\,x}{229635}-\frac {131072}{76545}}{\sqrt {3\,x-4\,x^2}} \] Input:
int(1/(3*x - 4*x^2)^(9/2),x)
Output:
((16*x)/63 - 2/21)/(3*x - 4*x^2)^(7/2) + ((512*x)/945 - 64/315)/(3*x - 4*x ^2)^(5/2) + ((32768*x)/25515 - 4096/8505)/(3*x - 4*x^2)^(3/2) + ((1048576* x)/229635 - 131072/76545)/(3*x - 4*x^2)^(1/2)
Time = 0.21 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (3 x-4 x^2\right )^{9/2}} \, dx=\frac {\frac {33554432 \sqrt {-4 x +3}\, i \,x^{7}}{229635}-\frac {8388608 \sqrt {-4 x +3}\, i \,x^{6}}{25515}+\frac {2097152 \sqrt {-4 x +3}\, i \,x^{5}}{8505}-\frac {524288 \sqrt {-4 x +3}\, i \,x^{4}}{8505}+\frac {67108864 \sqrt {x}\, x^{7}}{229635}-\frac {8388608 \sqrt {x}\, x^{6}}{10935}+\frac {524288 \sqrt {x}\, x^{5}}{729}-\frac {65536 \sqrt {x}\, x^{4}}{243}+\frac {2048 \sqrt {x}\, x^{3}}{81}+\frac {256 \sqrt {x}\, x^{2}}{135}+\frac {16 \sqrt {x}\, x}{45}+\frac {2 \sqrt {x}}{21}}{\sqrt {-4 x +3}\, x^{4} \left (64 x^{3}-144 x^{2}+108 x -27\right )} \] Input:
int(1/(-4*x^2+3*x)^(9/2),x)
Output:
(2*(16777216*sqrt( - 4*x + 3)*i*x**7 - 37748736*sqrt( - 4*x + 3)*i*x**6 + 28311552*sqrt( - 4*x + 3)*i*x**5 - 7077888*sqrt( - 4*x + 3)*i*x**4 + 33554 432*sqrt(x)*x**7 - 88080384*sqrt(x)*x**6 + 82575360*sqrt(x)*x**5 - 3096576 0*sqrt(x)*x**4 + 2903040*sqrt(x)*x**3 + 217728*sqrt(x)*x**2 + 40824*sqrt(x )*x + 10935*sqrt(x)))/(229635*sqrt( - 4*x + 3)*x**4*(64*x**3 - 144*x**2 + 108*x - 27))