Integrand size = 26, antiderivative size = 121 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{13/2}} \, dx=\frac {\left (b^2-4 a c\right )^3}{704 c^4 d (b d+2 c d x)^{11/2}}-\frac {3 \left (b^2-4 a c\right )^2}{448 c^4 d^3 (b d+2 c d x)^{7/2}}+\frac {b^2-4 a c}{64 c^4 d^5 (b d+2 c d x)^{3/2}}+\frac {\sqrt {b d+2 c d x}}{64 c^4 d^7} \] Output:
1/704*(-4*a*c+b^2)^3/c^4/d/(2*c*d*x+b*d)^(11/2)-3/448*(-4*a*c+b^2)^2/c^4/d ^3/(2*c*d*x+b*d)^(7/2)+1/64*(-4*a*c+b^2)/c^4/d^5/(2*c*d*x+b*d)^(3/2)+1/64* (2*c*d*x+b*d)^(1/2)/c^4/d^7
Time = 0.11 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{13/2}} \, dx=\frac {7 b^6-84 a b^4 c+336 a^2 b^2 c^2-448 a^3 c^3-33 b^4 (b+2 c x)^2+264 a b^2 c (b+2 c x)^2-528 a^2 c^2 (b+2 c x)^2+77 b^2 (b+2 c x)^4-308 a c (b+2 c x)^4+77 (b+2 c x)^6}{4928 c^4 d (d (b+2 c x))^{11/2}} \] Input:
Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(13/2),x]
Output:
(7*b^6 - 84*a*b^4*c + 336*a^2*b^2*c^2 - 448*a^3*c^3 - 33*b^4*(b + 2*c*x)^2 + 264*a*b^2*c*(b + 2*c*x)^2 - 528*a^2*c^2*(b + 2*c*x)^2 + 77*b^2*(b + 2*c *x)^4 - 308*a*c*(b + 2*c*x)^4 + 77*(b + 2*c*x)^6)/(4928*c^4*d*(d*(b + 2*c* x))^(11/2))
Time = 0.25 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1107, 2009}
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.
\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{13/2}} \, dx\) |
\(\Big \downarrow \) 1107 |
\(\displaystyle \int \left (\frac {3 \left (4 a c-b^2\right )}{64 c^3 d^4 (b d+2 c d x)^{5/2}}+\frac {3 \left (4 a c-b^2\right )^2}{64 c^3 d^2 (b d+2 c d x)^{9/2}}+\frac {\left (4 a c-b^2\right )^3}{64 c^3 (b d+2 c d x)^{13/2}}+\frac {1}{64 c^3 d^6 \sqrt {b d+2 c d x}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b^2-4 a c}{64 c^4 d^5 (b d+2 c d x)^{3/2}}-\frac {3 \left (b^2-4 a c\right )^2}{448 c^4 d^3 (b d+2 c d x)^{7/2}}+\frac {\left (b^2-4 a c\right )^3}{704 c^4 d (b d+2 c d x)^{11/2}}+\frac {\sqrt {b d+2 c d x}}{64 c^4 d^7}\) |
Input:
Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(13/2),x]
Output:
(b^2 - 4*a*c)^3/(704*c^4*d*(b*d + 2*c*d*x)^(11/2)) - (3*(b^2 - 4*a*c)^2)/( 448*c^4*d^3*(b*d + 2*c*d*x)^(7/2)) + (b^2 - 4*a*c)/(64*c^4*d^5*(b*d + 2*c* d*x)^(3/2)) + Sqrt[b*d + 2*c*d*x]/(64*c^4*d^7)
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; F reeQ[{a, b, c, d, e, m}, x] && EqQ[2*c*d - b*e, 0] && IGtQ[p, 0] && !(EqQ[ m, 3] && NeQ[p, 1])
Time = 0.94 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.07
method | result | size |
derivativedivides | \(\frac {\sqrt {2 c d x +b d}-\frac {d^{6} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}{11 \left (2 c d x +b d \right )^{\frac {11}{2}}}-\frac {3 d^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}{7 \left (2 c d x +b d \right )^{\frac {7}{2}}}-\frac {d^{2} \left (4 a c -b^{2}\right )}{\left (2 c d x +b d \right )^{\frac {3}{2}}}}{64 c^{4} d^{7}}\) | \(130\) |
default | \(\frac {\sqrt {2 c d x +b d}-\frac {d^{6} \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}{11 \left (2 c d x +b d \right )^{\frac {11}{2}}}-\frac {3 d^{4} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right )}{7 \left (2 c d x +b d \right )^{\frac {7}{2}}}-\frac {d^{2} \left (4 a c -b^{2}\right )}{\left (2 c d x +b d \right )^{\frac {3}{2}}}}{64 c^{4} d^{7}}\) | \(130\) |
gosper | \(-\frac {\left (2 c x +b \right ) \left (-77 x^{6} c^{6}-231 x^{5} b \,c^{5}+77 a \,c^{5} x^{4}-308 x^{4} b^{2} c^{4}+154 a b \,c^{4} x^{3}-231 b^{3} c^{3} x^{3}+33 a^{2} c^{4} x^{2}+99 a \,b^{2} c^{3} x^{2}-99 c^{2} x^{2} b^{4}+33 a^{2} b \,c^{3} x +22 x a \,b^{3} c^{2}-22 x c \,b^{5}+7 a^{3} c^{3}+3 a^{2} b^{2} c^{2}+2 a \,b^{4} c -2 b^{6}\right )}{77 c^{4} \left (2 c d x +b d \right )^{\frac {13}{2}}}\) | \(174\) |
orering | \(-\frac {\left (2 c x +b \right ) \left (-77 x^{6} c^{6}-231 x^{5} b \,c^{5}+77 a \,c^{5} x^{4}-308 x^{4} b^{2} c^{4}+154 a b \,c^{4} x^{3}-231 b^{3} c^{3} x^{3}+33 a^{2} c^{4} x^{2}+99 a \,b^{2} c^{3} x^{2}-99 c^{2} x^{2} b^{4}+33 a^{2} b \,c^{3} x +22 x a \,b^{3} c^{2}-22 x c \,b^{5}+7 a^{3} c^{3}+3 a^{2} b^{2} c^{2}+2 a \,b^{4} c -2 b^{6}\right )}{77 c^{4} \left (2 c d x +b d \right )^{\frac {13}{2}}}\) | \(174\) |
pseudoelliptic | \(\frac {77 x^{6} c^{6}+231 x^{5} b \,c^{5}-77 a \,c^{5} x^{4}+308 x^{4} b^{2} c^{4}-154 a b \,c^{4} x^{3}+231 b^{3} c^{3} x^{3}-33 a^{2} c^{4} x^{2}-99 a \,b^{2} c^{3} x^{2}+99 c^{2} x^{2} b^{4}-33 a^{2} b \,c^{3} x -22 x a \,b^{3} c^{2}+22 x c \,b^{5}-7 a^{3} c^{3}-3 a^{2} b^{2} c^{2}-2 a \,b^{4} c +2 b^{6}}{77 d^{6} \left (2 c x +b \right )^{5} \sqrt {d \left (2 c x +b \right )}\, c^{4}}\) | \(178\) |
trager | \(-\frac {\left (-77 x^{6} c^{6}-231 x^{5} b \,c^{5}+77 a \,c^{5} x^{4}-308 x^{4} b^{2} c^{4}+154 a b \,c^{4} x^{3}-231 b^{3} c^{3} x^{3}+33 a^{2} c^{4} x^{2}+99 a \,b^{2} c^{3} x^{2}-99 c^{2} x^{2} b^{4}+33 a^{2} b \,c^{3} x +22 x a \,b^{3} c^{2}-22 x c \,b^{5}+7 a^{3} c^{3}+3 a^{2} b^{2} c^{2}+2 a \,b^{4} c -2 b^{6}\right ) \sqrt {2 c d x +b d}}{77 d^{7} \left (2 c x +b \right )^{6} c^{4}}\) | \(179\) |
Input:
int((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^(13/2),x,method=_RETURNVERBOSE)
Output:
1/64/c^4/d^7*((2*c*d*x+b*d)^(1/2)-1/11*d^6*(64*a^3*c^3-48*a^2*b^2*c^2+12*a *b^4*c-b^6)/(2*c*d*x+b*d)^(11/2)-3/7*d^4*(16*a^2*c^2-8*a*b^2*c+b^4)/(2*c*d *x+b*d)^(7/2)-d^2*(4*a*c-b^2)/(2*c*d*x+b*d)^(3/2))
Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (105) = 210\).
Time = 0.09 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.08 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{13/2}} \, dx=\frac {{\left (77 \, c^{6} x^{6} + 231 \, b c^{5} x^{5} + 2 \, b^{6} - 2 \, a b^{4} c - 3 \, a^{2} b^{2} c^{2} - 7 \, a^{3} c^{3} + 77 \, {\left (4 \, b^{2} c^{4} - a c^{5}\right )} x^{4} + 77 \, {\left (3 \, b^{3} c^{3} - 2 \, a b c^{4}\right )} x^{3} + 33 \, {\left (3 \, b^{4} c^{2} - 3 \, a b^{2} c^{3} - a^{2} c^{4}\right )} x^{2} + 11 \, {\left (2 \, b^{5} c - 2 \, a b^{3} c^{2} - 3 \, a^{2} b c^{3}\right )} x\right )} \sqrt {2 \, c d x + b d}}{77 \, {\left (64 \, c^{10} d^{7} x^{6} + 192 \, b c^{9} d^{7} x^{5} + 240 \, b^{2} c^{8} d^{7} x^{4} + 160 \, b^{3} c^{7} d^{7} x^{3} + 60 \, b^{4} c^{6} d^{7} x^{2} + 12 \, b^{5} c^{5} d^{7} x + b^{6} c^{4} d^{7}\right )}} \] Input:
integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^(13/2),x, algorithm="fricas")
Output:
1/77*(77*c^6*x^6 + 231*b*c^5*x^5 + 2*b^6 - 2*a*b^4*c - 3*a^2*b^2*c^2 - 7*a ^3*c^3 + 77*(4*b^2*c^4 - a*c^5)*x^4 + 77*(3*b^3*c^3 - 2*a*b*c^4)*x^3 + 33* (3*b^4*c^2 - 3*a*b^2*c^3 - a^2*c^4)*x^2 + 11*(2*b^5*c - 2*a*b^3*c^2 - 3*a^ 2*b*c^3)*x)*sqrt(2*c*d*x + b*d)/(64*c^10*d^7*x^6 + 192*b*c^9*d^7*x^5 + 240 *b^2*c^8*d^7*x^4 + 160*b^3*c^7*d^7*x^3 + 60*b^4*c^6*d^7*x^2 + 12*b^5*c^5*d ^7*x + b^6*c^4*d^7)
Leaf count of result is larger than twice the leaf count of optimal. 1975 vs. \(2 (116) = 232\).
Time = 1.49 (sec) , antiderivative size = 1975, normalized size of antiderivative = 16.32 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{13/2}} \, dx=\text {Too large to display} \] Input:
integrate((c*x**2+b*x+a)**3/(2*c*d*x+b*d)**(13/2),x)
Output:
Piecewise((-7*a**3*c**3*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5* c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480 *b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) - 3* a**2*b**2*c**2*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7 *x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c** 8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) - 33*a**2*b*c **3*x*sqrt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620 *b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x* *4 + 14784*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) - 33*a**2*c**4*x**2*sq rt(b*d + 2*c*d*x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c* *6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 147 84*b*c**9*d**7*x**5 + 4928*c**10*d**7*x**6) - 2*a*b**4*c*sqrt(b*d + 2*c*d* x)/(77*b**6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7* x**5 + 4928*c**10*d**7*x**6) - 22*a*b**3*c**2*x*sqrt(b*d + 2*c*d*x)/(77*b* *6*c**4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b** 3*c**7*d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 49 28*c**10*d**7*x**6) - 99*a*b**2*c**3*x**2*sqrt(b*d + 2*c*d*x)/(77*b**6*c** 4*d**7 + 924*b**5*c**5*d**7*x + 4620*b**4*c**6*d**7*x**2 + 12320*b**3*c**7 *d**7*x**3 + 18480*b**2*c**8*d**7*x**4 + 14784*b*c**9*d**7*x**5 + 4928*...
Time = 0.03 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{13/2}} \, dx=\frac {\frac {77 \, \sqrt {2 \, c d x + b d}}{c^{3} d^{6}} + \frac {77 \, {\left (2 \, c d x + b d\right )}^{4} {\left (b^{2} - 4 \, a c\right )} - 33 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} {\left (2 \, c d x + b d\right )}^{2} d^{2} + 7 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{4}}{{\left (2 \, c d x + b d\right )}^{\frac {11}{2}} c^{3} d^{4}}}{4928 \, c d} \] Input:
integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^(13/2),x, algorithm="maxima")
Output:
1/4928*(77*sqrt(2*c*d*x + b*d)/(c^3*d^6) + (77*(2*c*d*x + b*d)^4*(b^2 - 4* a*c) - 33*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*(2*c*d*x + b*d)^2*d^2 + 7*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*d^4)/((2*c*d*x + b*d)^(11/2)*c^3 *d^4))/(c*d)
Time = 0.15 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{13/2}} \, dx=\frac {\sqrt {2 \, c d x + b d}}{64 \, c^{4} d^{7}} + \frac {7 \, b^{6} d^{4} - 84 \, a b^{4} c d^{4} + 336 \, a^{2} b^{2} c^{2} d^{4} - 448 \, a^{3} c^{3} d^{4} - 33 \, {\left (2 \, c d x + b d\right )}^{2} b^{4} d^{2} + 264 \, {\left (2 \, c d x + b d\right )}^{2} a b^{2} c d^{2} - 528 \, {\left (2 \, c d x + b d\right )}^{2} a^{2} c^{2} d^{2} + 77 \, {\left (2 \, c d x + b d\right )}^{4} b^{2} - 308 \, {\left (2 \, c d x + b d\right )}^{4} a c}{4928 \, {\left (2 \, c d x + b d\right )}^{\frac {11}{2}} c^{4} d^{5}} \] Input:
integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^(13/2),x, algorithm="giac")
Output:
1/64*sqrt(2*c*d*x + b*d)/(c^4*d^7) + 1/4928*(7*b^6*d^4 - 84*a*b^4*c*d^4 + 336*a^2*b^2*c^2*d^4 - 448*a^3*c^3*d^4 - 33*(2*c*d*x + b*d)^2*b^4*d^2 + 264 *(2*c*d*x + b*d)^2*a*b^2*c*d^2 - 528*(2*c*d*x + b*d)^2*a^2*c^2*d^2 + 77*(2 *c*d*x + b*d)^4*b^2 - 308*(2*c*d*x + b*d)^4*a*c)/((2*c*d*x + b*d)^(11/2)*c ^4*d^5)
Time = 5.32 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.21 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{13/2}} \, dx=\frac {\sqrt {b\,d+2\,c\,d\,x}}{64\,c^4\,d^7}-\frac {{\left (b\,d+2\,c\,d\,x\right )}^2\,\left (\frac {48\,a^2\,c^2\,d^2}{7}-\frac {24\,a\,b^2\,c\,d^2}{7}+\frac {3\,b^4\,d^2}{7}\right )+{\left (b\,d+2\,c\,d\,x\right )}^4\,\left (4\,a\,c-b^2\right )-\frac {b^6\,d^4}{11}+\frac {64\,a^3\,c^3\,d^4}{11}-\frac {48\,a^2\,b^2\,c^2\,d^4}{11}+\frac {12\,a\,b^4\,c\,d^4}{11}}{64\,c^4\,d^5\,{\left (b\,d+2\,c\,d\,x\right )}^{11/2}} \] Input:
int((a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^(13/2),x)
Output:
(b*d + 2*c*d*x)^(1/2)/(64*c^4*d^7) - ((b*d + 2*c*d*x)^2*((3*b^4*d^2)/7 + ( 48*a^2*c^2*d^2)/7 - (24*a*b^2*c*d^2)/7) + (b*d + 2*c*d*x)^4*(4*a*c - b^2) - (b^6*d^4)/11 + (64*a^3*c^3*d^4)/11 - (48*a^2*b^2*c^2*d^4)/11 + (12*a*b^4 *c*d^4)/11)/(64*c^4*d^5*(b*d + 2*c*d*x)^(11/2))
Time = 0.20 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.83 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{13/2}} \, dx=\frac {\sqrt {d}\, \left (77 c^{6} x^{6}+231 b \,c^{5} x^{5}-77 a \,c^{5} x^{4}+308 b^{2} c^{4} x^{4}-154 a b \,c^{4} x^{3}+231 b^{3} c^{3} x^{3}-33 a^{2} c^{4} x^{2}-99 a \,b^{2} c^{3} x^{2}+99 b^{4} c^{2} x^{2}-33 a^{2} b \,c^{3} x -22 a \,b^{3} c^{2} x +22 b^{5} c x -7 a^{3} c^{3}-3 a^{2} b^{2} c^{2}-2 a \,b^{4} c +2 b^{6}\right )}{77 \sqrt {2 c x +b}\, c^{4} d^{7} \left (32 c^{5} x^{5}+80 b \,c^{4} x^{4}+80 b^{2} c^{3} x^{3}+40 b^{3} c^{2} x^{2}+10 b^{4} c x +b^{5}\right )} \] Input:
int((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^(13/2),x)
Output:
(sqrt(d)*( - 7*a**3*c**3 - 3*a**2*b**2*c**2 - 33*a**2*b*c**3*x - 33*a**2*c **4*x**2 - 2*a*b**4*c - 22*a*b**3*c**2*x - 99*a*b**2*c**3*x**2 - 154*a*b*c **4*x**3 - 77*a*c**5*x**4 + 2*b**6 + 22*b**5*c*x + 99*b**4*c**2*x**2 + 231 *b**3*c**3*x**3 + 308*b**2*c**4*x**4 + 231*b*c**5*x**5 + 77*c**6*x**6))/(7 7*sqrt(b + 2*c*x)*c**4*d**7*(b**5 + 10*b**4*c*x + 40*b**3*c**2*x**2 + 80*b **2*c**3*x**3 + 80*b*c**4*x**4 + 32*c**5*x**5))