Integrand size = 19, antiderivative size = 123 \[ \int \frac {\left (a+b x^2\right )^2}{(c+d x)^{5/2}} \, dx=-\frac {2 \left (b c^2+a d^2\right )^2}{3 d^5 (c+d x)^{3/2}}+\frac {8 b c \left (b c^2+a d^2\right )}{d^5 \sqrt {c+d x}}+\frac {4 b \left (3 b c^2+a d^2\right ) \sqrt {c+d x}}{d^5}-\frac {8 b^2 c (c+d x)^{3/2}}{3 d^5}+\frac {2 b^2 (c+d x)^{5/2}}{5 d^5} \] Output:
-2/3*(a*d^2+b*c^2)^2/d^5/(d*x+c)^(3/2)+8*b*c*(a*d^2+b*c^2)/d^5/(d*x+c)^(1/ 2)+4*b*(a*d^2+3*b*c^2)*(d*x+c)^(1/2)/d^5-8/3*b^2*c*(d*x+c)^(3/2)/d^5+2/5*b ^2*(d*x+c)^(5/2)/d^5
Time = 0.08 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b x^2\right )^2}{(c+d x)^{5/2}} \, dx=\frac {2 \left (-5 a^2 d^4+10 a b d^2 \left (8 c^2+12 c d x+3 d^2 x^2\right )+b^2 \left (128 c^4+192 c^3 d x+48 c^2 d^2 x^2-8 c d^3 x^3+3 d^4 x^4\right )\right )}{15 d^5 (c+d x)^{3/2}} \] Input:
Integrate[(a + b*x^2)^2/(c + d*x)^(5/2),x]
Output:
(2*(-5*a^2*d^4 + 10*a*b*d^2*(8*c^2 + 12*c*d*x + 3*d^2*x^2) + b^2*(128*c^4 + 192*c^3*d*x + 48*c^2*d^2*x^2 - 8*c*d^3*x^3 + 3*d^4*x^4)))/(15*d^5*(c + d *x)^(3/2))
Time = 0.46 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {476, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^2}{(c+d x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 476 |
\(\displaystyle \int \left (\frac {2 b \left (a d^2+3 b c^2\right )}{d^4 \sqrt {c+d x}}-\frac {4 b c \left (a d^2+b c^2\right )}{d^4 (c+d x)^{3/2}}+\frac {\left (a d^2+b c^2\right )^2}{d^4 (c+d x)^{5/2}}+\frac {b^2 (c+d x)^{3/2}}{d^4}-\frac {4 b^2 c \sqrt {c+d x}}{d^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4 b \sqrt {c+d x} \left (a d^2+3 b c^2\right )}{d^5}+\frac {8 b c \left (a d^2+b c^2\right )}{d^5 \sqrt {c+d x}}-\frac {2 \left (a d^2+b c^2\right )^2}{3 d^5 (c+d x)^{3/2}}+\frac {2 b^2 (c+d x)^{5/2}}{5 d^5}-\frac {8 b^2 c (c+d x)^{3/2}}{3 d^5}\) |
Input:
Int[(a + b*x^2)^2/(c + d*x)^(5/2),x]
Output:
(-2*(b*c^2 + a*d^2)^2)/(3*d^5*(c + d*x)^(3/2)) + (8*b*c*(b*c^2 + a*d^2))/( d^5*Sqrt[c + d*x]) + (4*b*(3*b*c^2 + a*d^2)*Sqrt[c + d*x])/d^5 - (8*b^2*c* (c + d*x)^(3/2))/(3*d^5) + (2*b^2*(c + d*x)^(5/2))/(5*d^5)
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.68
method | result | size |
risch | \(\frac {2 b \left (3 b \,x^{2} d^{2}-14 b c d x +30 a \,d^{2}+73 b \,c^{2}\right ) \sqrt {d x +c}}{15 d^{5}}-\frac {2 \left (-12 b c d x +a \,d^{2}-11 b \,c^{2}\right ) \left (a \,d^{2}+b \,c^{2}\right )}{3 d^{5} \left (d x +c \right )^{\frac {3}{2}}}\) | \(84\) |
pseudoelliptic | \(\frac {\frac {2 \left (3 d^{4} x^{4}-8 c \,d^{3} x^{3}+48 d^{2} c^{2} x^{2}+192 c^{3} d x +128 c^{4}\right ) b^{2}}{15}+\frac {32 d^{2} a \left (\frac {3}{8} d^{2} x^{2}+\frac {3}{2} c d x +c^{2}\right ) b}{3}-\frac {2 a^{2} d^{4}}{3}}{\left (d x +c \right )^{\frac {3}{2}} d^{5}}\) | \(91\) |
gosper | \(-\frac {2 \left (-3 b^{2} d^{4} x^{4}+8 b^{2} c \,d^{3} x^{3}-30 a b \,d^{4} x^{2}-48 d^{2} c^{2} x^{2} b^{2}-120 a b c \,d^{3} x -192 b^{2} c^{3} d x +5 a^{2} d^{4}-80 b \,c^{2} d^{2} a -128 b^{2} c^{4}\right )}{15 \left (d x +c \right )^{\frac {3}{2}} d^{5}}\) | \(106\) |
trager | \(-\frac {2 \left (-3 b^{2} d^{4} x^{4}+8 b^{2} c \,d^{3} x^{3}-30 a b \,d^{4} x^{2}-48 d^{2} c^{2} x^{2} b^{2}-120 a b c \,d^{3} x -192 b^{2} c^{3} d x +5 a^{2} d^{4}-80 b \,c^{2} d^{2} a -128 b^{2} c^{4}\right )}{15 \left (d x +c \right )^{\frac {3}{2}} d^{5}}\) | \(106\) |
orering | \(-\frac {2 \left (-3 b^{2} d^{4} x^{4}+8 b^{2} c \,d^{3} x^{3}-30 a b \,d^{4} x^{2}-48 d^{2} c^{2} x^{2} b^{2}-120 a b c \,d^{3} x -192 b^{2} c^{3} d x +5 a^{2} d^{4}-80 b \,c^{2} d^{2} a -128 b^{2} c^{4}\right )}{15 \left (d x +c \right )^{\frac {3}{2}} d^{5}}\) | \(106\) |
derivativedivides | \(\frac {\frac {2 b^{2} \left (d x +c \right )^{\frac {5}{2}}}{5}-\frac {8 b^{2} c \left (d x +c \right )^{\frac {3}{2}}}{3}+4 a b \,d^{2} \sqrt {d x +c}+12 b^{2} c^{2} \sqrt {d x +c}+\frac {8 b c \left (a \,d^{2}+b \,c^{2}\right )}{\sqrt {d x +c}}-\frac {2 \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}}{d^{5}}\) | \(117\) |
default | \(\frac {\frac {2 b^{2} \left (d x +c \right )^{\frac {5}{2}}}{5}-\frac {8 b^{2} c \left (d x +c \right )^{\frac {3}{2}}}{3}+4 a b \,d^{2} \sqrt {d x +c}+12 b^{2} c^{2} \sqrt {d x +c}+\frac {8 b c \left (a \,d^{2}+b \,c^{2}\right )}{\sqrt {d x +c}}-\frac {2 \left (a^{2} d^{4}+2 b \,c^{2} d^{2} a +b^{2} c^{4}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}}{d^{5}}\) | \(117\) |
Input:
int((b*x^2+a)^2/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/15*b*(3*b*d^2*x^2-14*b*c*d*x+30*a*d^2+73*b*c^2)/d^5*(d*x+c)^(1/2)-2/3*(- 12*b*c*d*x+a*d^2-11*b*c^2)*(a*d^2+b*c^2)/d^5/(d*x+c)^(3/2)
Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x^2\right )^2}{(c+d x)^{5/2}} \, dx=\frac {2 \, {\left (3 \, b^{2} d^{4} x^{4} - 8 \, b^{2} c d^{3} x^{3} + 128 \, b^{2} c^{4} + 80 \, a b c^{2} d^{2} - 5 \, a^{2} d^{4} + 6 \, {\left (8 \, b^{2} c^{2} d^{2} + 5 \, a b d^{4}\right )} x^{2} + 24 \, {\left (8 \, b^{2} c^{3} d + 5 \, a b c d^{3}\right )} x\right )} \sqrt {d x + c}}{15 \, {\left (d^{7} x^{2} + 2 \, c d^{6} x + c^{2} d^{5}\right )}} \] Input:
integrate((b*x^2+a)^2/(d*x+c)^(5/2),x, algorithm="fricas")
Output:
2/15*(3*b^2*d^4*x^4 - 8*b^2*c*d^3*x^3 + 128*b^2*c^4 + 80*a*b*c^2*d^2 - 5*a ^2*d^4 + 6*(8*b^2*c^2*d^2 + 5*a*b*d^4)*x^2 + 24*(8*b^2*c^3*d + 5*a*b*c*d^3 )*x)*sqrt(d*x + c)/(d^7*x^2 + 2*c*d^6*x + c^2*d^5)
Time = 1.70 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a+b x^2\right )^2}{(c+d x)^{5/2}} \, dx=\begin {cases} \frac {2 \left (- \frac {4 b^{2} c \left (c + d x\right )^{\frac {3}{2}}}{3 d^{4}} + \frac {b^{2} \left (c + d x\right )^{\frac {5}{2}}}{5 d^{4}} + \frac {4 b c \left (a d^{2} + b c^{2}\right )}{d^{4} \sqrt {c + d x}} + \frac {\sqrt {c + d x} \left (2 a b d^{2} + 6 b^{2} c^{2}\right )}{d^{4}} - \frac {\left (a d^{2} + b c^{2}\right )^{2}}{3 d^{4} \left (c + d x\right )^{\frac {3}{2}}}\right )}{d} & \text {for}\: d \neq 0 \\\frac {a^{2} x + \frac {2 a b x^{3}}{3} + \frac {b^{2} x^{5}}{5}}{c^{\frac {5}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((b*x**2+a)**2/(d*x+c)**(5/2),x)
Output:
Piecewise((2*(-4*b**2*c*(c + d*x)**(3/2)/(3*d**4) + b**2*(c + d*x)**(5/2)/ (5*d**4) + 4*b*c*(a*d**2 + b*c**2)/(d**4*sqrt(c + d*x)) + sqrt(c + d*x)*(2 *a*b*d**2 + 6*b**2*c**2)/d**4 - (a*d**2 + b*c**2)**2/(3*d**4*(c + d*x)**(3 /2)))/d, Ne(d, 0)), ((a**2*x + 2*a*b*x**3/3 + b**2*x**5/5)/c**(5/2), True) )
Time = 0.03 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^2\right )^2}{(c+d x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {3 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} - 20 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c + 30 \, {\left (3 \, b^{2} c^{2} + a b d^{2}\right )} \sqrt {d x + c}}{d^{4}} - \frac {5 \, {\left (b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4} - 12 \, {\left (b^{2} c^{3} + a b c d^{2}\right )} {\left (d x + c\right )}\right )}}{{\left (d x + c\right )}^{\frac {3}{2}} d^{4}}\right )}}{15 \, d} \] Input:
integrate((b*x^2+a)^2/(d*x+c)^(5/2),x, algorithm="maxima")
Output:
2/15*((3*(d*x + c)^(5/2)*b^2 - 20*(d*x + c)^(3/2)*b^2*c + 30*(3*b^2*c^2 + a*b*d^2)*sqrt(d*x + c))/d^4 - 5*(b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4 - 12*(b ^2*c^3 + a*b*c*d^2)*(d*x + c))/((d*x + c)^(3/2)*d^4))/d
Time = 0.12 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^2\right )^2}{(c+d x)^{5/2}} \, dx=\frac {2 \, {\left (12 \, {\left (d x + c\right )} b^{2} c^{3} - b^{2} c^{4} + 12 \, {\left (d x + c\right )} a b c d^{2} - 2 \, a b c^{2} d^{2} - a^{2} d^{4}\right )}}{3 \, {\left (d x + c\right )}^{\frac {3}{2}} d^{5}} + \frac {2 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{2} d^{20} - 20 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c d^{20} + 90 \, \sqrt {d x + c} b^{2} c^{2} d^{20} + 30 \, \sqrt {d x + c} a b d^{22}\right )}}{15 \, d^{25}} \] Input:
integrate((b*x^2+a)^2/(d*x+c)^(5/2),x, algorithm="giac")
Output:
2/3*(12*(d*x + c)*b^2*c^3 - b^2*c^4 + 12*(d*x + c)*a*b*c*d^2 - 2*a*b*c^2*d ^2 - a^2*d^4)/((d*x + c)^(3/2)*d^5) + 2/15*(3*(d*x + c)^(5/2)*b^2*d^20 - 2 0*(d*x + c)^(3/2)*b^2*c*d^20 + 90*sqrt(d*x + c)*b^2*c^2*d^20 + 30*sqrt(d*x + c)*a*b*d^22)/d^25
Time = 8.54 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^2\right )^2}{(c+d x)^{5/2}} \, dx=\frac {2\,b^2\,{\left (c+d\,x\right )}^{5/2}}{5\,d^5}-\frac {\frac {2\,a^2\,d^4}{3}+\frac {2\,b^2\,c^4}{3}-\left (8\,b^2\,c^3+8\,a\,b\,c\,d^2\right )\,\left (c+d\,x\right )+\frac {4\,a\,b\,c^2\,d^2}{3}}{d^5\,{\left (c+d\,x\right )}^{3/2}}+\frac {\left (12\,b^2\,c^2+4\,a\,b\,d^2\right )\,\sqrt {c+d\,x}}{d^5}-\frac {8\,b^2\,c\,{\left (c+d\,x\right )}^{3/2}}{3\,d^5} \] Input:
int((a + b*x^2)^2/(c + d*x)^(5/2),x)
Output:
(2*b^2*(c + d*x)^(5/2))/(5*d^5) - ((2*a^2*d^4)/3 + (2*b^2*c^4)/3 - (8*b^2* c^3 + 8*a*b*c*d^2)*(c + d*x) + (4*a*b*c^2*d^2)/3)/(d^5*(c + d*x)^(3/2)) + ((12*b^2*c^2 + 4*a*b*d^2)*(c + d*x)^(1/2))/d^5 - (8*b^2*c*(c + d*x)^(3/2)) /(3*d^5)
Time = 0.22 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^2\right )^2}{(c+d x)^{5/2}} \, dx=\frac {\frac {2}{5} b^{2} d^{4} x^{4}-\frac {16}{15} b^{2} c \,d^{3} x^{3}+4 a b \,d^{4} x^{2}+\frac {32}{5} b^{2} c^{2} d^{2} x^{2}+16 a b c \,d^{3} x +\frac {128}{5} b^{2} c^{3} d x -\frac {2}{3} a^{2} d^{4}+\frac {32}{3} a b \,c^{2} d^{2}+\frac {256}{15} b^{2} c^{4}}{\sqrt {d x +c}\, d^{5} \left (d x +c \right )} \] Input:
int((b*x^2+a)^2/(d*x+c)^(5/2),x)
Output:
(2*( - 5*a**2*d**4 + 80*a*b*c**2*d**2 + 120*a*b*c*d**3*x + 30*a*b*d**4*x** 2 + 128*b**2*c**4 + 192*b**2*c**3*d*x + 48*b**2*c**2*d**2*x**2 - 8*b**2*c* d**3*x**3 + 3*b**2*d**4*x**4))/(15*sqrt(c + d*x)*d**5*(c + d*x))