Integrand size = 31, antiderivative size = 119 \[ \int \frac {a+b x^2}{x^4 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {a}{3 c^2 x^3 \sqrt {-c+d x} \sqrt {c+d x}}+\frac {3 b c^2+4 a d^2}{3 c^4 x \sqrt {-c+d x} \sqrt {c+d x}}-\frac {2 d^2 \left (3 b c^2+4 a d^2\right ) x}{3 c^6 \sqrt {-c+d x} \sqrt {c+d x}} \]
1/3*a/c^2/x^3/(d*x-c)^(1/2)/(d*x+c)^(1/2)+1/3*(4*a*d^2+3*b*c^2)/c^4/x/(d*x -c)^(1/2)/(d*x+c)^(1/2)-2/3*d^2*(4*a*d^2+3*b*c^2)*x/c^6/(d*x-c)^(1/2)/(d*x +c)^(1/2)
Time = 0.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.65 \[ \int \frac {a+b x^2}{x^4 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {3 b c^2 x^2 \left (c^2-2 d^2 x^2\right )+a \left (c^4+4 c^2 d^2 x^2-8 d^4 x^4\right )}{3 c^6 x^3 \sqrt {-c+d x} \sqrt {c+d x}} \]
(3*b*c^2*x^2*(c^2 - 2*d^2*x^2) + a*(c^4 + 4*c^2*d^2*x^2 - 8*d^4*x^4))/(3*c ^6*x^3*Sqrt[-c + d*x]*Sqrt[c + d*x])
Time = 0.25 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.89, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {956, 114, 27, 41}
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.
\(\displaystyle \int \frac {a+b x^2}{x^4 (d x-c)^{3/2} (c+d x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 956 |
\(\displaystyle \frac {1}{3} \left (\frac {4 a d^2}{c^2}+3 b\right ) \int \frac {1}{x^2 (d x-c)^{3/2} (c+d x)^{3/2}}dx+\frac {a}{3 c^2 x^3 \sqrt {d x-c} \sqrt {c+d x}}\) |
\(\Big \downarrow \) 114 |
\(\displaystyle \frac {1}{3} \left (\frac {4 a d^2}{c^2}+3 b\right ) \left (\frac {\int \frac {2 d^2}{(d x-c)^{3/2} (c+d x)^{3/2}}dx}{c^2}+\frac {1}{c^2 x \sqrt {d x-c} \sqrt {c+d x}}\right )+\frac {a}{3 c^2 x^3 \sqrt {d x-c} \sqrt {c+d x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \left (\frac {4 a d^2}{c^2}+3 b\right ) \left (\frac {2 d^2 \int \frac {1}{(d x-c)^{3/2} (c+d x)^{3/2}}dx}{c^2}+\frac {1}{c^2 x \sqrt {d x-c} \sqrt {c+d x}}\right )+\frac {a}{3 c^2 x^3 \sqrt {d x-c} \sqrt {c+d x}}\) |
\(\Big \downarrow \) 41 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{c^2 x \sqrt {d x-c} \sqrt {c+d x}}-\frac {2 d^2 x}{c^4 \sqrt {d x-c} \sqrt {c+d x}}\right ) \left (\frac {4 a d^2}{c^2}+3 b\right )+\frac {a}{3 c^2 x^3 \sqrt {d x-c} \sqrt {c+d x}}\) |
a/(3*c^2*x^3*Sqrt[-c + d*x]*Sqrt[c + d*x]) + ((3*b + (4*a*d^2)/c^2)*(1/(c^ 2*x*Sqrt[-c + d*x]*Sqrt[c + d*x]) - (2*d^2*x)/(c^4*Sqrt[-c + d*x]*Sqrt[c + d*x])))/3
3.4.76.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> S imp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && Eq Q[b*c + a*d, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.) *(x_)^(non2_.))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^( m + 1)*(a1 + b1*x^(n/2))^(p + 1)*((a2 + b2*x^(n/2))^(p + 1)/(a1*a2*e*(m + 1 ))), x] + Simp[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(a1*a2*e^n*( m + 1)) Int[(e*x)^(m + n)*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, p}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || ( LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]
Time = 4.60 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.61
method | result | size |
gosper | \(\frac {-8 a \,d^{4} x^{4}-6 b \,c^{2} d^{2} x^{4}+4 a \,c^{2} d^{2} x^{2}+3 b \,c^{4} x^{2}+a \,c^{4}}{3 c^{6} x^{3} \sqrt {d x -c}\, \sqrt {d x +c}}\) | \(73\) |
default | \(-\frac {\sqrt {d x -c}\, \operatorname {csgn}\left (d \right )^{2} \left (-8 a \,d^{4} x^{4}-6 b \,c^{2} d^{2} x^{4}+4 a \,c^{2} d^{2} x^{2}+3 b \,c^{4} x^{2}+a \,c^{4}\right )}{3 c^{6} \left (-d x +c \right ) x^{3} \sqrt {d x +c}}\) | \(85\) |
risch | \(\frac {\sqrt {d x +c}\, \left (-d x +c \right ) \left (5 a \,d^{2} x^{2}+3 b \,c^{2} x^{2}+c^{2} a \right )}{3 c^{6} x^{3} \sqrt {d x -c}}-\frac {d^{2} \left (a \,d^{2}+b \,c^{2}\right ) x \sqrt {\left (d x -c \right ) \left (d x +c \right )}}{\sqrt {-\left (d x +c \right ) \left (-d x +c \right )}\, c^{6} \sqrt {d x -c}\, \sqrt {d x +c}}\) | \(122\) |
1/3/c^6/x^3/(d*x-c)^(1/2)/(d*x+c)^(1/2)*(-8*a*d^4*x^4-6*b*c^2*d^2*x^4+4*a* c^2*d^2*x^2+3*b*c^4*x^2+a*c^4)
Time = 0.27 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.11 \[ \int \frac {a+b x^2}{x^4 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=-\frac {2 \, {\left (3 \, b c^{2} d^{3} + 4 \, a d^{5}\right )} x^{5} - 2 \, {\left (3 \, b c^{4} d + 4 \, a c^{2} d^{3}\right )} x^{3} - {\left (a c^{4} - 2 \, {\left (3 \, b c^{2} d^{2} + 4 \, a d^{4}\right )} x^{4} + {\left (3 \, b c^{4} + 4 \, a c^{2} d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {d x - c}}{3 \, {\left (c^{6} d^{2} x^{5} - c^{8} x^{3}\right )}} \]
-1/3*(2*(3*b*c^2*d^3 + 4*a*d^5)*x^5 - 2*(3*b*c^4*d + 4*a*c^2*d^3)*x^3 - (a *c^4 - 2*(3*b*c^2*d^2 + 4*a*d^4)*x^4 + (3*b*c^4 + 4*a*c^2*d^2)*x^2)*sqrt(d *x + c)*sqrt(d*x - c))/(c^6*d^2*x^5 - c^8*x^3)
Timed out. \[ \int \frac {a+b x^2}{x^4 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.05 \[ \int \frac {a+b x^2}{x^4 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=-\frac {2 \, b d^{2} x}{\sqrt {d^{2} x^{2} - c^{2}} c^{4}} - \frac {8 \, a d^{4} x}{3 \, \sqrt {d^{2} x^{2} - c^{2}} c^{6}} + \frac {b}{\sqrt {d^{2} x^{2} - c^{2}} c^{2} x} + \frac {4 \, a d^{2}}{3 \, \sqrt {d^{2} x^{2} - c^{2}} c^{4} x} + \frac {a}{3 \, \sqrt {d^{2} x^{2} - c^{2}} c^{2} x^{3}} \]
-2*b*d^2*x/(sqrt(d^2*x^2 - c^2)*c^4) - 8/3*a*d^4*x/(sqrt(d^2*x^2 - c^2)*c^ 6) + b/(sqrt(d^2*x^2 - c^2)*c^2*x) + 4/3*a*d^2/(sqrt(d^2*x^2 - c^2)*c^4*x) + 1/3*a/(sqrt(d^2*x^2 - c^2)*c^2*x^3)
Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (101) = 202\).
Time = 0.40 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.03 \[ \int \frac {a+b x^2}{x^4 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=-\frac {{\left (b c^{2} d + a d^{3}\right )} \sqrt {d x + c}}{2 \, \sqrt {d x - c} c^{6}} - \frac {2 \, {\left (b c^{2} d + a d^{3}\right )}}{{\left ({\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2} + 2 \, c\right )} c^{5}} - \frac {8 \, {\left (3 \, b c^{2} d {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{8} + 3 \, a d^{3} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{8} + 24 \, b c^{4} d {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{4} + 48 \, a c^{2} d^{3} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{4} + 48 \, b c^{6} d + 80 \, a c^{4} d^{3}\right )}}{3 \, {\left ({\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{4} + 4 \, c^{2}\right )}^{3} c^{4}} \]
-1/2*(b*c^2*d + a*d^3)*sqrt(d*x + c)/(sqrt(d*x - c)*c^6) - 2*(b*c^2*d + a* d^3)/(((sqrt(d*x + c) - sqrt(d*x - c))^2 + 2*c)*c^5) - 8/3*(3*b*c^2*d*(sqr t(d*x + c) - sqrt(d*x - c))^8 + 3*a*d^3*(sqrt(d*x + c) - sqrt(d*x - c))^8 + 24*b*c^4*d*(sqrt(d*x + c) - sqrt(d*x - c))^4 + 48*a*c^2*d^3*(sqrt(d*x + c) - sqrt(d*x - c))^4 + 48*b*c^6*d + 80*a*c^4*d^3)/(((sqrt(d*x + c) - sqrt (d*x - c))^4 + 4*c^2)^3*c^4)
Time = 7.37 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.87 \[ \int \frac {a+b x^2}{x^4 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {\sqrt {d\,x-c}\,\left (\frac {a}{3\,c^2\,d}+\frac {x^2\,\left (3\,b\,c^4+4\,a\,c^2\,d^2\right )}{3\,c^6\,d}-\frac {x^4\,\left (6\,b\,c^2\,d^2+8\,a\,d^4\right )}{3\,c^6\,d}\right )}{x^4\,\sqrt {c+d\,x}-\frac {c\,x^3\,\sqrt {c+d\,x}}{d}} \]