Integrand size = 20, antiderivative size = 119 \[ \int \frac {x^2 \left (a+b x^2\right )}{(c+d x)^{5/2}} \, dx=-\frac {2 c^2 \left (b c^2+a d^2\right )}{3 d^5 (c+d x)^{3/2}}+\frac {4 c \left (2 b c^2+a d^2\right )}{d^5 \sqrt {c+d x}}+\frac {2 \left (6 b c^2+a d^2\right ) \sqrt {c+d x}}{d^5}-\frac {8 b c (c+d x)^{3/2}}{3 d^5}+\frac {2 b (c+d x)^{5/2}}{5 d^5} \] Output:
-2/3*c^2*(a*d^2+b*c^2)/d^5/(d*x+c)^(3/2)+4*c*(a*d^2+2*b*c^2)/d^5/(d*x+c)^( 1/2)+2*(a*d^2+6*b*c^2)*(d*x+c)^(1/2)/d^5-8/3*b*c*(d*x+c)^(3/2)/d^5+2/5*b*( d*x+c)^(5/2)/d^5
Time = 0.06 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.71 \[ \int \frac {x^2 \left (a+b x^2\right )}{(c+d x)^{5/2}} \, dx=\frac {2 \left (5 a d^2 \left (8 c^2+12 c d x+3 d^2 x^2\right )+b \left (128 c^4+192 c^3 d x+48 c^2 d^2 x^2-8 c d^3 x^3+3 d^4 x^4\right )\right )}{15 d^5 (c+d x)^{3/2}} \] Input:
Integrate[(x^2*(a + b*x^2))/(c + d*x)^(5/2),x]
Output:
(2*(5*a*d^2*(8*c^2 + 12*c*d*x + 3*d^2*x^2) + b*(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 = 0.42 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {522, 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.
\(\displaystyle \int \frac {x^2 \left (a+b x^2\right )}{(c+d x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 522 |
\(\displaystyle \int \left (-\frac {2 \left (a c d^2+2 b c^3\right )}{d^4 (c+d x)^{3/2}}+\frac {a d^2+6 b c^2}{d^4 \sqrt {c+d x}}+\frac {a c^2 d^2+b c^4}{d^4 (c+d x)^{5/2}}+\frac {b (c+d x)^{3/2}}{d^4}-\frac {4 b c \sqrt {c+d x}}{d^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \sqrt {c+d x} \left (a d^2+6 b c^2\right )}{d^5}+\frac {4 c \left (a d^2+2 b c^2\right )}{d^5 \sqrt {c+d x}}-\frac {2 c^2 \left (a d^2+b c^2\right )}{3 d^5 (c+d x)^{3/2}}+\frac {2 b (c+d x)^{5/2}}{5 d^5}-\frac {8 b c (c+d x)^{3/2}}{3 d^5}\) |
Input:
Int[(x^2*(a + b*x^2))/(c + d*x)^(5/2),x]
Output:
(-2*c^2*(b*c^2 + a*d^2))/(3*d^5*(c + d*x)^(3/2)) + (4*c*(2*b*c^2 + a*d^2)) /(d^5*Sqrt[c + d*x]) + (2*(6*b*c^2 + a*d^2)*Sqrt[c + d*x])/d^5 - (8*b*c*(c + d*x)^(3/2))/(3*d^5) + (2*b*(c + d*x)^(5/2))/(5*d^5)
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.62
method | result | size |
pseudoelliptic | \(\frac {2 \left (\frac {b \,x^{2}}{5}+a \right ) x^{2} d^{4}+8 x \left (-\frac {2 b \,x^{2}}{15}+a \right ) c \,d^{3}+\frac {16 c^{2} \left (\frac {6 b \,x^{2}}{5}+a \right ) d^{2}}{3}+\frac {128 b \,c^{3} d x}{5}+\frac {256 b \,c^{4}}{15}}{\left (d x +c \right )^{\frac {3}{2}} d^{5}}\) | \(74\) |
risch | \(\frac {2 \left (3 b \,x^{2} d^{2}-14 b c d x +15 a \,d^{2}+73 b \,c^{2}\right ) \sqrt {d x +c}}{15 d^{5}}+\frac {2 c \left (6 a x \,d^{3}+12 b \,c^{2} d x +5 a \,d^{2} c +11 b \,c^{3}\right )}{3 d^{5} \left (d x +c \right )^{\frac {3}{2}}}\) | \(84\) |
gosper | \(\frac {\frac {2}{5} b \,x^{4} d^{4}-\frac {16}{15} c b \,d^{3} x^{3}+2 a \,d^{4} x^{2}+\frac {32}{5} b \,c^{2} d^{2} x^{2}+8 a c \,d^{3} x +\frac {128}{5} b \,c^{3} d x +\frac {16}{3} a \,c^{2} d^{2}+\frac {256}{15} b \,c^{4}}{\left (d x +c \right )^{\frac {3}{2}} d^{5}}\) | \(85\) |
trager | \(\frac {\frac {2}{5} b \,x^{4} d^{4}-\frac {16}{15} c b \,d^{3} x^{3}+2 a \,d^{4} x^{2}+\frac {32}{5} b \,c^{2} d^{2} x^{2}+8 a c \,d^{3} x +\frac {128}{5} b \,c^{3} d x +\frac {16}{3} a \,c^{2} d^{2}+\frac {256}{15} b \,c^{4}}{\left (d x +c \right )^{\frac {3}{2}} d^{5}}\) | \(85\) |
orering | \(\frac {\frac {2}{5} b \,x^{4} d^{4}-\frac {16}{15} c b \,d^{3} x^{3}+2 a \,d^{4} x^{2}+\frac {32}{5} b \,c^{2} d^{2} x^{2}+8 a c \,d^{3} x +\frac {128}{5} b \,c^{3} d x +\frac {16}{3} a \,c^{2} d^{2}+\frac {256}{15} b \,c^{4}}{\left (d x +c \right )^{\frac {3}{2}} d^{5}}\) | \(85\) |
derivativedivides | \(\frac {\frac {2 b \left (d x +c \right )^{\frac {5}{2}}}{5}-\frac {8 b c \left (d x +c \right )^{\frac {3}{2}}}{3}+2 a \,d^{2} \sqrt {d x +c}+12 b \,c^{2} \sqrt {d x +c}+\frac {4 c \left (a \,d^{2}+2 b \,c^{2}\right )}{\sqrt {d x +c}}-\frac {2 c^{2} \left (a \,d^{2}+b \,c^{2}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}}{d^{5}}\) | \(98\) |
default | \(\frac {\frac {2 b \left (d x +c \right )^{\frac {5}{2}}}{5}-\frac {8 b c \left (d x +c \right )^{\frac {3}{2}}}{3}+2 a \,d^{2} \sqrt {d x +c}+12 b \,c^{2} \sqrt {d x +c}+\frac {4 c \left (a \,d^{2}+2 b \,c^{2}\right )}{\sqrt {d x +c}}-\frac {2 c^{2} \left (a \,d^{2}+b \,c^{2}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}}{d^{5}}\) | \(98\) |
Input:
int(x^2*(b*x^2+a)/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
Output:
16/3/(d*x+c)^(3/2)*(3/8*(1/5*b*x^2+a)*x^2*d^4+3/2*x*(-2/15*b*x^2+a)*c*d^3+ c^2*(6/5*b*x^2+a)*d^2+24/5*b*c^3*d*x+16/5*b*c^4)/d^5
Time = 0.10 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.90 \[ \int \frac {x^2 \left (a+b x^2\right )}{(c+d x)^{5/2}} \, dx=\frac {2 \, {\left (3 \, b d^{4} x^{4} - 8 \, b c d^{3} x^{3} + 128 \, b c^{4} + 40 \, a c^{2} d^{2} + 3 \, {\left (16 \, b c^{2} d^{2} + 5 \, a d^{4}\right )} x^{2} + 12 \, {\left (16 \, b c^{3} d + 5 \, a c d^{3}\right )} x\right )} \sqrt {d x + c}}{15 \, {\left (d^{7} x^{2} + 2 \, c d^{6} x + c^{2} d^{5}\right )}} \] Input:
integrate(x^2*(b*x^2+a)/(d*x+c)^(5/2),x, algorithm="fricas")
Output:
2/15*(3*b*d^4*x^4 - 8*b*c*d^3*x^3 + 128*b*c^4 + 40*a*c^2*d^2 + 3*(16*b*c^2 *d^2 + 5*a*d^4)*x^2 + 12*(16*b*c^3*d + 5*a*c*d^3)*x)*sqrt(d*x + c)/(d^7*x^ 2 + 2*c*d^6*x + c^2*d^5)
Time = 1.40 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.16 \[ \int \frac {x^2 \left (a+b x^2\right )}{(c+d x)^{5/2}} \, dx=\begin {cases} \frac {2 \left (- \frac {4 b c \left (c + d x\right )^{\frac {3}{2}}}{3 d^{2}} + \frac {b \left (c + d x\right )^{\frac {5}{2}}}{5 d^{2}} - \frac {c^{2} \left (a d^{2} + b c^{2}\right )}{3 d^{2} \left (c + d x\right )^{\frac {3}{2}}} + \frac {2 c \left (a d^{2} + 2 b c^{2}\right )}{d^{2} \sqrt {c + d x}} + \frac {\sqrt {c + d x} \left (a d^{2} + 6 b c^{2}\right )}{d^{2}}\right )}{d^{3}} & \text {for}\: d \neq 0 \\\frac {\frac {a x^{3}}{3} + \frac {b x^{5}}{5}}{c^{\frac {5}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate(x**2*(b*x**2+a)/(d*x+c)**(5/2),x)
Output:
Piecewise((2*(-4*b*c*(c + d*x)**(3/2)/(3*d**2) + b*(c + d*x)**(5/2)/(5*d** 2) - c**2*(a*d**2 + b*c**2)/(3*d**2*(c + d*x)**(3/2)) + 2*c*(a*d**2 + 2*b* c**2)/(d**2*sqrt(c + d*x)) + sqrt(c + d*x)*(a*d**2 + 6*b*c**2)/d**2)/d**3, Ne(d, 0)), ((a*x**3/3 + b*x**5/5)/c**(5/2), True))
Time = 0.03 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.83 \[ \int \frac {x^2 \left (a+b x^2\right )}{(c+d x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {3 \, {\left (d x + c\right )}^{\frac {5}{2}} b - 20 \, {\left (d x + c\right )}^{\frac {3}{2}} b c + 15 \, {\left (6 \, b c^{2} + a d^{2}\right )} \sqrt {d x + c}}{d^{2}} - \frac {5 \, {\left (b c^{4} + a c^{2} d^{2} - 6 \, {\left (2 \, b c^{3} + a c d^{2}\right )} {\left (d x + c\right )}\right )}}{{\left (d x + c\right )}^{\frac {3}{2}} d^{2}}\right )}}{15 \, d^{3}} \] Input:
integrate(x^2*(b*x^2+a)/(d*x+c)^(5/2),x, algorithm="maxima")
Output:
2/15*((3*(d*x + c)^(5/2)*b - 20*(d*x + c)^(3/2)*b*c + 15*(6*b*c^2 + a*d^2) *sqrt(d*x + c))/d^2 - 5*(b*c^4 + a*c^2*d^2 - 6*(2*b*c^3 + a*c*d^2)*(d*x + c))/((d*x + c)^(3/2)*d^2))/d^3
Time = 0.13 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.96 \[ \int \frac {x^2 \left (a+b x^2\right )}{(c+d x)^{5/2}} \, dx=\frac {2 \, {\left (12 \, {\left (d x + c\right )} b c^{3} - b c^{4} + 6 \, {\left (d x + c\right )} a c d^{2} - a c^{2} d^{2}\right )}}{3 \, {\left (d x + c\right )}^{\frac {3}{2}} d^{5}} + \frac {2 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} b d^{20} - 20 \, {\left (d x + c\right )}^{\frac {3}{2}} b c d^{20} + 90 \, \sqrt {d x + c} b c^{2} d^{20} + 15 \, \sqrt {d x + c} a d^{22}\right )}}{15 \, d^{25}} \] Input:
integrate(x^2*(b*x^2+a)/(d*x+c)^(5/2),x, algorithm="giac")
Output:
2/3*(12*(d*x + c)*b*c^3 - b*c^4 + 6*(d*x + c)*a*c*d^2 - a*c^2*d^2)/((d*x + c)^(3/2)*d^5) + 2/15*(3*(d*x + c)^(5/2)*b*d^20 - 20*(d*x + c)^(3/2)*b*c*d ^20 + 90*sqrt(d*x + c)*b*c^2*d^20 + 15*sqrt(d*x + c)*a*d^22)/d^25
Time = 8.71 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.85 \[ \int \frac {x^2 \left (a+b x^2\right )}{(c+d x)^{5/2}} \, dx=\frac {2\,b\,{\left (c+d\,x\right )}^{5/2}}{5\,d^5}-\frac {\frac {2\,b\,c^4}{3}-\left (8\,b\,c^3+4\,a\,c\,d^2\right )\,\left (c+d\,x\right )+\frac {2\,a\,c^2\,d^2}{3}}{d^5\,{\left (c+d\,x\right )}^{3/2}}+\frac {\left (12\,b\,c^2+2\,a\,d^2\right )\,\sqrt {c+d\,x}}{d^5}-\frac {8\,b\,c\,{\left (c+d\,x\right )}^{3/2}}{3\,d^5} \] Input:
int((x^2*(a + b*x^2))/(c + d*x)^(5/2),x)
Output:
(2*b*(c + d*x)^(5/2))/(5*d^5) - ((2*b*c^4)/3 - (8*b*c^3 + 4*a*c*d^2)*(c + d*x) + (2*a*c^2*d^2)/3)/(d^5*(c + d*x)^(3/2)) + ((2*a*d^2 + 12*b*c^2)*(c + d*x)^(1/2))/d^5 - (8*b*c*(c + d*x)^(3/2))/(3*d^5)
Time = 0.18 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.77 \[ \int \frac {x^2 \left (a+b x^2\right )}{(c+d x)^{5/2}} \, dx=\frac {\frac {2}{5} b \,d^{4} x^{4}-\frac {16}{15} b c \,d^{3} x^{3}+2 a \,d^{4} x^{2}+\frac {32}{5} b \,c^{2} d^{2} x^{2}+8 a c \,d^{3} x +\frac {128}{5} b \,c^{3} d x +\frac {16}{3} a \,c^{2} d^{2}+\frac {256}{15} b \,c^{4}}{\sqrt {d x +c}\, d^{5} \left (d x +c \right )} \] Input:
int(x^2*(b*x^2+a)/(d*x+c)^(5/2),x)
Output:
(2*(40*a*c**2*d**2 + 60*a*c*d**3*x + 15*a*d**4*x**2 + 128*b*c**4 + 192*b*c **3*d*x + 48*b*c**2*d**2*x**2 - 8*b*c*d**3*x**3 + 3*b*d**4*x**4))/(15*sqrt (c + d*x)*d**5*(c + d*x))