Integrand size = 26, antiderivative size = 39 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^6} \, dx=\frac {2 \left (a+b x+c x^2\right )^{5/2}}{5 \left (b^2-4 a c\right ) d^6 (b+2 c x)^5} \] Output:
2/5*(c*x^2+b*x+a)^(5/2)/(-4*a*c+b^2)/d^6/(2*c*x+b)^5
Time = 10.15 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^6} \, dx=\frac {2 (a+x (b+c x))^{5/2}}{5 \left (b^2-4 a c\right ) d^6 (b+2 c x)^5} \] Input:
Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^6,x]
Output:
(2*(a + x*(b + c*x))^(5/2))/(5*(b^2 - 4*a*c)*d^6*(b + 2*c*x)^5)
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.
\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^6} \, dx\) |
\(\Big \downarrow \) 1106 |
\(\displaystyle \frac {2 \left (a+b x+c x^2\right )^{5/2}}{5 d^6 \left (b^2-4 a c\right ) (b+2 c x)^5}\) |
Input:
Int[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^6,x]
Output:
(2*(a + b*x + c*x^2)^(5/2))/(5*(b^2 - 4*a*c)*d^6*(b + 2*c*x)^5)
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 = 1.46 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.97
method | result | size |
gosper | \(-\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5 \left (2 c x +b \right )^{5} d^{6} \left (4 a c -b^{2}\right )}\) | \(38\) |
orering | \(-\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} \left (2 c x +b \right )}{5 \left (4 a c -b^{2}\right ) \left (2 c d x +b d \right )^{6}}\) | \(44\) |
default | \(-\frac {\left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{80 d^{6} c^{5} \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{5}}\) | \(61\) |
trager | \(-\frac {2 \left (c^{2} x^{4}+2 b c \,x^{3}+2 a c \,x^{2}+b^{2} x^{2}+2 a b x +a^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{5 d^{6} \left (2 c x +b \right )^{5} \left (4 a c -b^{2}\right )}\) | \(75\) |
Input:
int((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^6,x,method=_RETURNVERBOSE)
Output:
-2/5*(c*x^2+b*x+a)^(5/2)/(2*c*x+b)^5/d^6/(4*a*c-b^2)
Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (35) = 70\).
Time = 1.08 (sec) , antiderivative size = 183, normalized size of antiderivative = 4.69 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^6} \, dx=\frac {2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt {c x^{2} + b x + a}}{5 \, {\left (32 \, {\left (b^{2} c^{5} - 4 \, a c^{6}\right )} d^{6} x^{5} + 80 \, {\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} d^{6} x^{4} + 80 \, {\left (b^{4} c^{3} - 4 \, a b^{2} c^{4}\right )} d^{6} x^{3} + 40 \, {\left (b^{5} c^{2} - 4 \, a b^{3} c^{3}\right )} d^{6} x^{2} + 10 \, {\left (b^{6} c - 4 \, a b^{4} c^{2}\right )} d^{6} x + {\left (b^{7} - 4 \, a b^{5} c\right )} d^{6}\right )}} \] Input:
integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^6,x, algorithm="fricas")
Output:
2/5*(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*sqrt(c*x^2 + b*x + a)/(32*(b^2*c^5 - 4*a*c^6)*d^6*x^5 + 80*(b^3*c^4 - 4*a*b*c^5)*d^6*x ^4 + 80*(b^4*c^3 - 4*a*b^2*c^4)*d^6*x^3 + 40*(b^5*c^2 - 4*a*b^3*c^3)*d^6*x ^2 + 10*(b^6*c - 4*a*b^4*c^2)*d^6*x + (b^7 - 4*a*b^5*c)*d^6)
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^6} \, dx=\frac {\int \frac {a \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac {b x \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac {c x^{2} \sqrt {a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx}{d^{6}} \] Input:
integrate((c*x**2+b*x+a)**(3/2)/(2*c*d*x+b*d)**6,x)
Output:
(Integral(a*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x + 60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c**4*x**4 + 192*b*c**5*x**5 + 64*c**6*x** 6), x) + Integral(b*x*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x + 60*b**4 *c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c**4*x**4 + 192*b*c**5*x**5 + 6 4*c**6*x**6), x) + Integral(c*x**2*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5* c*x + 60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c**4*x**4 + 192*b* c**5*x**5 + 64*c**6*x**6), x))/d**6
Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^6} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^6,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. 588 vs. \(2 (35) = 70\).
Time = 0.50 (sec) , antiderivative size = 588, normalized size of antiderivative = 15.08 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^6} \, dx=\frac {80 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{8} c^{4} + 320 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{7} b c^{\frac {7}{2}} + 560 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{6} b^{2} c^{3} + 560 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} b^{3} c^{\frac {5}{2}} + 360 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} b^{4} c^{2} - 80 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} a b^{2} c^{3} + 160 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} a^{2} c^{4} + 160 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b^{5} c^{\frac {3}{2}} - 160 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a b^{3} c^{\frac {5}{2}} + 320 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a^{2} b c^{\frac {7}{2}} + 50 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{6} c - 120 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a b^{4} c^{2} + 240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{2} b^{2} c^{3} + 10 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{7} \sqrt {c} - 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b^{5} c^{\frac {3}{2}} + 80 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} b^{3} c^{\frac {5}{2}} + b^{8} - 6 \, a b^{6} c + 16 \, a^{2} b^{4} c^{2} - 16 \, a^{3} b^{2} c^{3} + 16 \, a^{4} c^{4}}{80 \, {\left (2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b \sqrt {c} + b^{2} - 2 \, a c\right )}^{5} c^{\frac {5}{2}} d^{6}} \] Input:
integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^6,x, algorithm="giac")
Output:
1/80*(80*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*c^4 + 320*(sqrt(c)*x - sqrt (c*x^2 + b*x + a))^7*b*c^(7/2) + 560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6 *b^2*c^3 + 560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^3*c^(5/2) + 360*(sq rt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^4*c^2 - 80*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^2*c^3 + 160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*c^4 + 160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^5*c^(3/2) - 160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^3*c^(5/2) + 320*(sqrt(c)*x - sqrt(c*x^2 + b* x + a))^3*a^2*b*c^(7/2) + 50*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^6*c - 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^4*c^2 + 240*(sqrt(c)*x - sq rt(c*x^2 + b*x + a))^2*a^2*b^2*c^3 + 10*(sqrt(c)*x - sqrt(c*x^2 + b*x + a) )*b^7*sqrt(c) - 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^5*c^(3/2) + 80* (sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^3*c^(5/2) + b^8 - 6*a*b^6*c + 16 *a^2*b^4*c^2 - 16*a^3*b^2*c^3 + 16*a^4*c^4)/((2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*sqrt(c) + b^2 - 2* a*c)^5*c^(5/2)*d^6)
Time = 5.76 (sec) , antiderivative size = 1156, normalized size of antiderivative = 29.64 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^6} \, dx =\text {Too large to display} \] Input:
int((a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^6,x)
Output:
(((b*((b*((b*((32*a*c^4)/(3*d^6*(4*a*c - b^2)^2*(32*a*c^3 - 8*b^2*c^2)) - (16*b^2*c^3)/(15*d^6*(4*a*c - b^2)^2*(32*a*c^3 - 8*b^2*c^2))))/(2*c) - (8* b*c^2*(6*a*c - b^2))/(3*d^6*(4*a*c - b^2)^2*(32*a*c^3 - 8*b^2*c^2))))/(2*c ) + (4*c*(36*a^2*c^2 - 4*b^4 + 12*a*b^2*c))/(15*d^6*(4*a*c - b^2)^2*(32*a* c^3 - 8*b^2*c^2))))/(2*c) - (8*a*b*c*(9*a*c - 2*b^2))/(15*d^6*(4*a*c - b^2 )^2*(32*a*c^3 - 8*b^2*c^2)))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^2 - (((b *((b*((2*c^2*(28*a*c - b^2))/(5*d^6*(4*a*c - b^2)*(48*a*c^3 - 12*b^2*c^2)) - (6*b^2*c^2)/(5*d^6*(4*a*c - b^2)*(48*a*c^3 - 12*b^2*c^2))))/(2*c) + (2* c*(5*b^3 - 28*a*b*c))/(5*d^6*(4*a*c - b^2)*(48*a*c^3 - 12*b^2*c^2))))/(2*c ) - (2*c*(5*a*b^2 - 24*a^2*c))/(5*d^6*(4*a*c - b^2)*(48*a*c^3 - 12*b^2*c^2 )))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x)^3 + (((8*a*c - b^2)/(40*c^2*d^6*( 4*a*c - b^2)^2) - b^2/(40*c^2*d^6*(4*a*c - b^2)^2))*(a + b*x + c*x^2)^(1/2 ))/(b + 2*c*x) - (((b*((b*((2*(10*a*c - b^2))/(15*d^6*(4*a*c - b^2)^3) - b ^2/(10*d^6*(4*a*c - b^2)^3)))/(2*c) + (4*b^3 - 20*a*b*c)/(15*c*d^6*(4*a*c - b^2)^3)))/(2*c) - (4*a*b^2 - 18*a^2*c)/(15*c*d^6*(4*a*c - b^2)^3))*(a + b*x + c*x^2)^(1/2))/(b + 2*c*x) + (((b*((b*((b*((2*c*(56*a*c^3 + 6*b^2*c^2 ))/(5*d^6*(4*a*c - b^2)*(64*a*c^3 - 16*b^2*c^2)) - (16*b^2*c^3)/(5*d^6*(4* a*c - b^2)*(64*a*c^3 - 16*b^2*c^2))))/(2*c) + (2*c*(11*b^3*c - 84*a*b*c^2) )/(5*d^6*(4*a*c - b^2)*(64*a*c^3 - 16*b^2*c^2))))/(2*c) + (2*c*(48*a^2*c^2 - 5*b^4 + 18*a*b^2*c))/(5*d^6*(4*a*c - b^2)*(64*a*c^3 - 16*b^2*c^2))))...
Time = 0.28 (sec) , antiderivative size = 310, normalized size of antiderivative = 7.95 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^6} \, dx=\frac {-32 \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{3}-64 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{3} x -64 \sqrt {c \,x^{2}+b x +a}\, a \,c^{4} x^{2}-32 \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{3} x^{2}-64 \sqrt {c \,x^{2}+b x +a}\, b \,c^{4} x^{3}-32 \sqrt {c \,x^{2}+b x +a}\, c^{5} x^{4}-\sqrt {c}\, b^{5}-10 \sqrt {c}\, b^{4} c x -40 \sqrt {c}\, b^{3} c^{2} x^{2}-80 \sqrt {c}\, b^{2} c^{3} x^{3}-80 \sqrt {c}\, b \,c^{4} x^{4}-32 \sqrt {c}\, c^{5} x^{5}}{80 c^{3} d^{6} \left (128 a \,c^{6} x^{5}-32 b^{2} c^{5} x^{5}+320 a b \,c^{5} x^{4}-80 b^{3} c^{4} x^{4}+320 a \,b^{2} c^{4} x^{3}-80 b^{4} c^{3} x^{3}+160 a \,b^{3} c^{3} x^{2}-40 b^{5} c^{2} x^{2}+40 a \,b^{4} c^{2} x -10 b^{6} c x +4 a \,b^{5} c -b^{7}\right )} \] Input:
int((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^6,x)
Output:
( - 32*sqrt(a + b*x + c*x**2)*a**2*c**3 - 64*sqrt(a + b*x + c*x**2)*a*b*c* *3*x - 64*sqrt(a + b*x + c*x**2)*a*c**4*x**2 - 32*sqrt(a + b*x + c*x**2)*b **2*c**3*x**2 - 64*sqrt(a + b*x + c*x**2)*b*c**4*x**3 - 32*sqrt(a + b*x + c*x**2)*c**5*x**4 - sqrt(c)*b**5 - 10*sqrt(c)*b**4*c*x - 40*sqrt(c)*b**3*c **2*x**2 - 80*sqrt(c)*b**2*c**3*x**3 - 80*sqrt(c)*b*c**4*x**4 - 32*sqrt(c) *c**5*x**5)/(80*c**3*d**6*(4*a*b**5*c + 40*a*b**4*c**2*x + 160*a*b**3*c**3 *x**2 + 320*a*b**2*c**4*x**3 + 320*a*b*c**5*x**4 + 128*a*c**6*x**5 - b**7 - 10*b**6*c*x - 40*b**5*c**2*x**2 - 80*b**4*c**3*x**3 - 80*b**3*c**4*x**4 - 32*b**2*c**5*x**5))