Integrand size = 26, antiderivative size = 39 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^8} \, dx=\frac {2 \left (a+b x+c x^2\right )^{7/2}}{7 \left (b^2-4 a c\right ) d^8 (b+2 c x)^7} \] Output:
2/7*(c*x^2+b*x+a)^(7/2)/(-4*a*c+b^2)/d^8/(2*c*x+b)^7
Time = 10.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^8} \, dx=\frac {2 (a+x (b+c x))^{7/2}}{7 \left (b^2-4 a c\right ) d^8 (b+2 c x)^7} \] Input:
Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^8,x]
Output:
(2*(a + x*(b + c*x))^(7/2))/(7*(b^2 - 4*a*c)*d^8*(b + 2*c*x)^7)
Time = 0.17 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {1106}
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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^8} \, dx\) |
| \(\Big \downarrow \) 1106 |
| \(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{7/2}}{7 d^8 \left (b^2-4 a c\right ) (b+2 c x)^7}\) |
Input:
Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^8,x]
Output:
(2*(a + b*x + c*x^2)^(7/2))/(7*(b^2 - 4*a*c)*d^8*(b + 2*c*x)^7)
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[2*c*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)* (b^2 - 4*a*c))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 0]
Time = 7.24 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.97
| method | result | size |
| gosper | \(-\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{7 \left (2 c x +b \right )^{7} d^{8} \left (4 a c -b^{2}\right )}\) | \(38\) |
| orering | \(-\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} \left (2 c x +b \right )}{7 \left (4 a c -b^{2}\right ) \left (2 c d x +b d \right )^{8}}\) | \(44\) |
| default | \(-\frac {\left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {7}{2}}}{448 d^{8} c^{7} \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{7}}\) | \(61\) |
| trager | \(-\frac {2 \left (c^{3} x^{6}+3 b \,c^{2} x^{5}+3 a \,c^{2} x^{4}+3 b^{2} c \,x^{4}+6 a b c \,x^{3}+b^{3} x^{3}+3 a^{2} c \,x^{2}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}\right ) \sqrt {c \,x^{2}+b x +a}}{7 d^{8} \left (2 c x +b \right )^{7} \left (4 a c -b^{2}\right )}\) | \(116\) |
Input:
int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^8,x,method=_RETURNVERBOSE)
Output:
-2/7*(c*x^2+b*x+a)^(7/2)/(2*c*x+b)^7/d^8/(4*a*c-b^2)
Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (35) = 70\).
Time = 5.37 (sec) , antiderivative size = 270, normalized size of antiderivative = 6.92 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^8} \, dx=\frac {2 \, {\left (c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, {\left (b^{2} c + a c^{2}\right )} x^{4} + 3 \, a^{2} b x + {\left (b^{3} + 6 \, a b c\right )} x^{3} + a^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} x^{2}\right )} \sqrt {c x^{2} + b x + a}}{7 \, {\left (128 \, {\left (b^{2} c^{7} - 4 \, a c^{8}\right )} d^{8} x^{7} + 448 \, {\left (b^{3} c^{6} - 4 \, a b c^{7}\right )} d^{8} x^{6} + 672 \, {\left (b^{4} c^{5} - 4 \, a b^{2} c^{6}\right )} d^{8} x^{5} + 560 \, {\left (b^{5} c^{4} - 4 \, a b^{3} c^{5}\right )} d^{8} x^{4} + 280 \, {\left (b^{6} c^{3} - 4 \, a b^{4} c^{4}\right )} d^{8} x^{3} + 84 \, {\left (b^{7} c^{2} - 4 \, a b^{5} c^{3}\right )} d^{8} x^{2} + 14 \, {\left (b^{8} c - 4 \, a b^{6} c^{2}\right )} d^{8} x + {\left (b^{9} - 4 \, a b^{7} c\right )} d^{8}\right )}} \] Input:
integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^8,x, algorithm="fricas")
Output:
2/7*(c^3*x^6 + 3*b*c^2*x^5 + 3*(b^2*c + a*c^2)*x^4 + 3*a^2*b*x + (b^3 + 6* a*b*c)*x^3 + a^3 + 3*(a*b^2 + a^2*c)*x^2)*sqrt(c*x^2 + b*x + a)/(128*(b^2* c^7 - 4*a*c^8)*d^8*x^7 + 448*(b^3*c^6 - 4*a*b*c^7)*d^8*x^6 + 672*(b^4*c^5 - 4*a*b^2*c^6)*d^8*x^5 + 560*(b^5*c^4 - 4*a*b^3*c^5)*d^8*x^4 + 280*(b^6*c^ 3 - 4*a*b^4*c^4)*d^8*x^3 + 84*(b^7*c^2 - 4*a*b^5*c^3)*d^8*x^2 + 14*(b^8*c - 4*a*b^6*c^2)*d^8*x + (b^9 - 4*a*b^7*c)*d^8)
\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^8} \, dx=\frac {\int \frac {a^{2} \sqrt {a + b x + c x^{2}}}{b^{8} + 16 b^{7} c x + 112 b^{6} c^{2} x^{2} + 448 b^{5} c^{3} x^{3} + 1120 b^{4} c^{4} x^{4} + 1792 b^{3} c^{5} x^{5} + 1792 b^{2} c^{6} x^{6} + 1024 b c^{7} x^{7} + 256 c^{8} x^{8}}\, dx + \int \frac {b^{2} x^{2} \sqrt {a + b x + c x^{2}}}{b^{8} + 16 b^{7} c x + 112 b^{6} c^{2} x^{2} + 448 b^{5} c^{3} x^{3} + 1120 b^{4} c^{4} x^{4} + 1792 b^{3} c^{5} x^{5} + 1792 b^{2} c^{6} x^{6} + 1024 b c^{7} x^{7} + 256 c^{8} x^{8}}\, dx + \int \frac {c^{2} x^{4} \sqrt {a + b x + c x^{2}}}{b^{8} + 16 b^{7} c x + 112 b^{6} c^{2} x^{2} + 448 b^{5} c^{3} x^{3} + 1120 b^{4} c^{4} x^{4} + 1792 b^{3} c^{5} x^{5} + 1792 b^{2} c^{6} x^{6} + 1024 b c^{7} x^{7} + 256 c^{8} x^{8}}\, dx + \int \frac {2 a b x \sqrt {a + b x + c x^{2}}}{b^{8} + 16 b^{7} c x + 112 b^{6} c^{2} x^{2} + 448 b^{5} c^{3} x^{3} + 1120 b^{4} c^{4} x^{4} + 1792 b^{3} c^{5} x^{5} + 1792 b^{2} c^{6} x^{6} + 1024 b c^{7} x^{7} + 256 c^{8} x^{8}}\, dx + \int \frac {2 a c x^{2} \sqrt {a + b x + c x^{2}}}{b^{8} + 16 b^{7} c x + 112 b^{6} c^{2} x^{2} + 448 b^{5} c^{3} x^{3} + 1120 b^{4} c^{4} x^{4} + 1792 b^{3} c^{5} x^{5} + 1792 b^{2} c^{6} x^{6} + 1024 b c^{7} x^{7} + 256 c^{8} x^{8}}\, dx + \int \frac {2 b c x^{3} \sqrt {a + b x + c x^{2}}}{b^{8} + 16 b^{7} c x + 112 b^{6} c^{2} x^{2} + 448 b^{5} c^{3} x^{3} + 1120 b^{4} c^{4} x^{4} + 1792 b^{3} c^{5} x^{5} + 1792 b^{2} c^{6} x^{6} + 1024 b c^{7} x^{7} + 256 c^{8} x^{8}}\, dx}{d^{8}} \] Input:
integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**8,x)
Output:
(Integral(a**2*sqrt(a + b*x + c*x**2)/(b**8 + 16*b**7*c*x + 112*b**6*c**2* x**2 + 448*b**5*c**3*x**3 + 1120*b**4*c**4*x**4 + 1792*b**3*c**5*x**5 + 17 92*b**2*c**6*x**6 + 1024*b*c**7*x**7 + 256*c**8*x**8), x) + Integral(b**2* x**2*sqrt(a + b*x + c*x**2)/(b**8 + 16*b**7*c*x + 112*b**6*c**2*x**2 + 448 *b**5*c**3*x**3 + 1120*b**4*c**4*x**4 + 1792*b**3*c**5*x**5 + 1792*b**2*c* *6*x**6 + 1024*b*c**7*x**7 + 256*c**8*x**8), x) + Integral(c**2*x**4*sqrt( a + b*x + c*x**2)/(b**8 + 16*b**7*c*x + 112*b**6*c**2*x**2 + 448*b**5*c**3 *x**3 + 1120*b**4*c**4*x**4 + 1792*b**3*c**5*x**5 + 1792*b**2*c**6*x**6 + 1024*b*c**7*x**7 + 256*c**8*x**8), x) + Integral(2*a*b*x*sqrt(a + b*x + c* x**2)/(b**8 + 16*b**7*c*x + 112*b**6*c**2*x**2 + 448*b**5*c**3*x**3 + 1120 *b**4*c**4*x**4 + 1792*b**3*c**5*x**5 + 1792*b**2*c**6*x**6 + 1024*b*c**7* x**7 + 256*c**8*x**8), x) + Integral(2*a*c*x**2*sqrt(a + b*x + c*x**2)/(b* *8 + 16*b**7*c*x + 112*b**6*c**2*x**2 + 448*b**5*c**3*x**3 + 1120*b**4*c** 4*x**4 + 1792*b**3*c**5*x**5 + 1792*b**2*c**6*x**6 + 1024*b*c**7*x**7 + 25 6*c**8*x**8), x) + Integral(2*b*c*x**3*sqrt(a + b*x + c*x**2)/(b**8 + 16*b **7*c*x + 112*b**6*c**2*x**2 + 448*b**5*c**3*x**3 + 1120*b**4*c**4*x**4 + 1792*b**3*c**5*x**5 + 1792*b**2*c**6*x**6 + 1024*b*c**7*x**7 + 256*c**8*x* *8), x))/d**8
Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^8} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^8,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 1241 vs. \(2 (35) = 70\).
Time = 0.70 (sec) , antiderivative size = 1241, normalized size of antiderivative = 31.82 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^8} \, dx=\text {Too large to display} \] Input:
integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^8,x, algorithm="giac")
Output:
1/448*(448*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*c^6 + 2688*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*b*c^(11/2) + 7392*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*b^2*c^5 + 12320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^3*c^(9/2) + 14000*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^4*c^4 - 1120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a*b^2*c^5 + 2240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^2*c^6 + 11648*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^5*c^(7/2) - 4480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^3*c^(9/2) + 8960*(sqrt(c)* x - sqrt(c*x^2 + b*x + a))^7*a^2*b*c^(11/2) + 7448*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^6*c^3 - 7840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^4* c^4 + 15680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*b^2*c^5 + 3752*(sqrt (c)*x - sqrt(c*x^2 + b*x + a))^5*b^7*c^(5/2) - 7840*(sqrt(c)*x - sqrt(c*x^ 2 + b*x + a))^5*a*b^5*c^(7/2) + 15680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^ 5*a^2*b^3*c^(9/2) + 1484*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^8*c^2 - 4 984*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^6*c^3 + 10304*(sqrt(c)*x - s qrt(c*x^2 + b*x + a))^4*a^2*b^4*c^4 - 1344*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*b^2*c^5 + 1344*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^4*c^6 + 448*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^9*c^(3/2) - 2128*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^7*c^(5/2) + 4928*(sqrt(c)*x - sqrt(c*x^2 + b* x + a))^3*a^2*b^5*c^(7/2) - 2688*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^3 *b^3*c^(9/2) + 2688*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^4*b*c^(11/2...
Time = 8.00 (sec) , antiderivative size = 3088, normalized size of antiderivative = 79.18 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^8} \, dx=\text {Too large to display} \] Input:
int((a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^8,x)
Output:
((a/(56*c^2*d^8*(4*a*c - b^2)^2) - b^2/(224*c^3*d^8*(4*a*c - b^2)^2))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) + (((b*((b*((b*((b*((b*((4*c^4*(40*a*c + 11*b^2))/(7*d^8*(4*a*c - b^2)*(96*a*c^3 - 24*b^2*c^2)) - (24*b^2*c^4)/(7*d ^8*(4*a*c - b^2)*(96*a*c^3 - 24*b^2*c^2))))/(2*c) - (10*b*c^3*(40*a*c - 3* b^2))/(7*d^8*(4*a*c - b^2)*(96*a*c^3 - 24*b^2*c^2))))/(2*c) + (768*a^2*c^4 - 82*b^4*c^2 + 416*a*b^2*c^3)/(14*d^8*(4*a*c - b^2)*(96*a*c^3 - 24*b^2*c^ 2))))/(2*c) + (b*c*(7*b^4 - 1152*a^2*c^2 + 176*a*b^2*c))/(14*d^8*(4*a*c - b^2)*(96*a*c^3 - 24*b^2*c^2))))/(2*c) + (7*b^6 + 480*a^3*c^3 + 216*a^2*b^2 *c^2 - 98*a*b^4*c)/(14*d^8*(4*a*c - b^2)*(96*a*c^3 - 24*b^2*c^2))))/(2*c) - (7*a*b^5 - 84*a^2*b^3*c + 240*a^3*b*c^2)/(14*d^8*(4*a*c - b^2)*(96*a*c^3 - 24*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^6 + (((b*((b*((10*a* c - b^2)/(70*c*d^8*(4*a*c - b^2)^3) - (3*b^2)/(280*c*d^8*(4*a*c - b^2)^3)) )/(2*c) - (b*(5*a*c - b^2))/(35*c^2*d^8*(4*a*c - b^2)^3)))/(2*c) + (15*b^4 + 384*a^2*c^2 - 152*a*b^2*c)/(1120*c^3*d^8*(4*a*c - b^2)^3))*(a + b*x + c *x^2)^(1/2))/(b + 2*c*x) - (((b*((b*((b*((b*((10*c^3*(8*a*c + b^2))/(7*d^8 *(4*a*c - b^2)*(80*a*c^3 - 20*b^2*c^2)) - (10*b^2*c^3)/(7*d^8*(4*a*c - b^2 )*(80*a*c^3 - 20*b^2*c^2))))/(2*c) - (20*b*c^2*(8*a*c - b^2))/(7*d^8*(4*a* c - b^2)*(80*a*c^3 - 20*b^2*c^2))))/(2*c) + (384*a^2*c^3 - 21*b^4*c + 48*a *b^2*c^2)/(14*d^8*(4*a*c - b^2)*(80*a*c^3 - 20*b^2*c^2))))/(2*c) - (7*b^5 + 384*a^2*b*c^2 - 112*a*b^3*c)/(14*d^8*(4*a*c - b^2)*(80*a*c^3 - 20*b^2...
Time = 0.35 (sec) , antiderivative size = 472, normalized size of antiderivative = 12.10 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^8} \, dx=\frac {-128 \sqrt {c \,x^{2}+b x +a}\, a^{3} c^{4}-384 \sqrt {c \,x^{2}+b x +a}\, a^{2} b \,c^{4} x -384 \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{5} x^{2}-384 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c^{4} x^{2}-768 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{5} x^{3}-384 \sqrt {c \,x^{2}+b x +a}\, a \,c^{6} x^{4}-128 \sqrt {c \,x^{2}+b x +a}\, b^{3} c^{4} x^{3}-384 \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{5} x^{4}-384 \sqrt {c \,x^{2}+b x +a}\, b \,c^{6} x^{5}-128 \sqrt {c \,x^{2}+b x +a}\, c^{7} x^{6}-\sqrt {c}\, b^{7}-14 \sqrt {c}\, b^{6} c x -84 \sqrt {c}\, b^{5} c^{2} x^{2}-280 \sqrt {c}\, b^{4} c^{3} x^{3}-560 \sqrt {c}\, b^{3} c^{4} x^{4}-672 \sqrt {c}\, b^{2} c^{5} x^{5}-448 \sqrt {c}\, b \,c^{6} x^{6}-128 \sqrt {c}\, c^{7} x^{7}}{448 c^{4} d^{8} \left (512 a \,c^{8} x^{7}-128 b^{2} c^{7} x^{7}+1792 a b \,c^{7} x^{6}-448 b^{3} c^{6} x^{6}+2688 a \,b^{2} c^{6} x^{5}-672 b^{4} c^{5} x^{5}+2240 a \,b^{3} c^{5} x^{4}-560 b^{5} c^{4} x^{4}+1120 a \,b^{4} c^{4} x^{3}-280 b^{6} c^{3} x^{3}+336 a \,b^{5} c^{3} x^{2}-84 b^{7} c^{2} x^{2}+56 a \,b^{6} c^{2} x -14 b^{8} c x +4 a \,b^{7} c -b^{9}\right )} \] Input:
int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^8,x)
Output:
( - 128*sqrt(a + b*x + c*x**2)*a**3*c**4 - 384*sqrt(a + b*x + c*x**2)*a**2 *b*c**4*x - 384*sqrt(a + b*x + c*x**2)*a**2*c**5*x**2 - 384*sqrt(a + b*x + c*x**2)*a*b**2*c**4*x**2 - 768*sqrt(a + b*x + c*x**2)*a*b*c**5*x**3 - 384 *sqrt(a + b*x + c*x**2)*a*c**6*x**4 - 128*sqrt(a + b*x + c*x**2)*b**3*c**4 *x**3 - 384*sqrt(a + b*x + c*x**2)*b**2*c**5*x**4 - 384*sqrt(a + b*x + c*x **2)*b*c**6*x**5 - 128*sqrt(a + b*x + c*x**2)*c**7*x**6 - sqrt(c)*b**7 - 1 4*sqrt(c)*b**6*c*x - 84*sqrt(c)*b**5*c**2*x**2 - 280*sqrt(c)*b**4*c**3*x** 3 - 560*sqrt(c)*b**3*c**4*x**4 - 672*sqrt(c)*b**2*c**5*x**5 - 448*sqrt(c)* b*c**6*x**6 - 128*sqrt(c)*c**7*x**7)/(448*c**4*d**8*(4*a*b**7*c + 56*a*b** 6*c**2*x + 336*a*b**5*c**3*x**2 + 1120*a*b**4*c**4*x**3 + 2240*a*b**3*c**5 *x**4 + 2688*a*b**2*c**6*x**5 + 1792*a*b*c**7*x**6 + 512*a*c**8*x**7 - b** 9 - 14*b**8*c*x - 84*b**7*c**2*x**2 - 280*b**6*c**3*x**3 - 560*b**5*c**4*x **4 - 672*b**4*c**5*x**5 - 448*b**3*c**6*x**6 - 128*b**2*c**7*x**7))