Integrand size = 31, antiderivative size = 75 \[ \int \frac {a+b x^2}{x^4 \sqrt {-c+d x} \sqrt {c+d x}} \, dx=\frac {a \sqrt {-c+d x} \sqrt {c+d x}}{3 c^2 x^3}+\frac {\left (3 b c^2+2 a d^2\right ) \sqrt {-c+d x} \sqrt {c+d x}}{3 c^4 x} \] Output:
1/3*a*(d*x-c)^(1/2)*(d*x+c)^(1/2)/c^2/x^3+1/3*(2*a*d^2+3*b*c^2)*(d*x-c)^(1 /2)*(d*x+c)^(1/2)/c^4/x
Time = 0.12 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.72 \[ \int \frac {a+b x^2}{x^4 \sqrt {-c+d x} \sqrt {c+d x}} \, dx=\frac {\sqrt {-c+d x} \sqrt {c+d x} \left (3 b c^2 x^2+a \left (c^2+2 d^2 x^2\right )\right )}{3 c^4 x^3} \] Input:
Integrate[(a + b*x^2)/(x^4*Sqrt[-c + d*x]*Sqrt[c + d*x]),x]
Output:
(Sqrt[-c + d*x]*Sqrt[c + d*x]*(3*b*c^2*x^2 + a*(c^2 + 2*d^2*x^2)))/(3*c^4* x^3)
Time = 0.35 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {956, 106}
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.
\(\displaystyle \int \frac {a+b x^2}{x^4 \sqrt {d x-c} \sqrt {c+d x}} \, dx\) |
\(\Big \downarrow \) 956 |
\(\displaystyle \frac {1}{3} \left (\frac {2 a d^2}{c^2}+3 b\right ) \int \frac {1}{x^2 \sqrt {d x-c} \sqrt {c+d x}}dx+\frac {a \sqrt {d x-c} \sqrt {c+d x}}{3 c^2 x^3}\) |
\(\Big \downarrow \) 106 |
\(\displaystyle \frac {\sqrt {d x-c} \sqrt {c+d x} \left (\frac {2 a d^2}{c^2}+3 b\right )}{3 c^2 x}+\frac {a \sqrt {d x-c} \sqrt {c+d x}}{3 c^2 x^3}\) |
Input:
Int[(a + b*x^2)/(x^4*Sqrt[-c + d*x]*Sqrt[c + d*x]),x]
Output:
(a*Sqrt[-c + d*x]*Sqrt[c + d*x])/(3*c^2*x^3) + ((3*b + (2*a*d^2)/c^2)*Sqrt [-c + d*x]*Sqrt[c + d*x])/(3*c^2*x)
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] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0] && NeQ[m, -1]
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 = 0.13 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.65
method | result | size |
gosper | \(\frac {\sqrt {d x -c}\, \sqrt {d x +c}\, \left (2 a \,d^{2} x^{2}+3 b \,c^{2} x^{2}+a \,c^{2}\right )}{3 c^{4} x^{3}}\) | \(49\) |
default | \(\frac {\sqrt {d x -c}\, \sqrt {d x +c}\, \operatorname {csgn}\left (d \right )^{2} \left (2 a \,d^{2} x^{2}+3 b \,c^{2} x^{2}+a \,c^{2}\right )}{3 c^{4} x^{3}}\) | \(53\) |
risch | \(-\frac {\sqrt {d x +c}\, \left (-d x +c \right ) \left (2 a \,d^{2} x^{2}+3 b \,c^{2} x^{2}+a \,c^{2}\right )}{3 x^{3} c^{4} \sqrt {d x -c}}\) | \(55\) |
orering | \(-\frac {\sqrt {d x +c}\, \left (-d x +c \right ) \left (2 a \,d^{2} x^{2}+3 b \,c^{2} x^{2}+a \,c^{2}\right )}{3 x^{3} c^{4} \sqrt {d x -c}}\) | \(55\) |
Input:
int((b*x^2+a)/x^4/(d*x-c)^(1/2)/(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/3/c^4/x^3*(d*x-c)^(1/2)*(d*x+c)^(1/2)*(2*a*d^2*x^2+3*b*c^2*x^2+a*c^2)
Time = 0.09 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.89 \[ \int \frac {a+b x^2}{x^4 \sqrt {-c+d x} \sqrt {c+d x}} \, dx=\frac {{\left (3 \, b c^{2} d + 2 \, a d^{3}\right )} x^{3} + {\left (a c^{2} + {\left (3 \, b c^{2} + 2 \, a d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {d x - c}}{3 \, c^{4} x^{3}} \] Input:
integrate((b*x^2+a)/x^4/(d*x-c)^(1/2)/(d*x+c)^(1/2),x, algorithm="fricas")
Output:
1/3*((3*b*c^2*d + 2*a*d^3)*x^3 + (a*c^2 + (3*b*c^2 + 2*a*d^2)*x^2)*sqrt(d* x + c)*sqrt(d*x - c))/(c^4*x^3)
Result contains complex when optimal does not.
Time = 31.01 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.27 \[ \int \frac {a+b x^2}{x^4 \sqrt {-c+d x} \sqrt {c+d x}} \, dx=- \frac {a d^{3} {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {9}{4}, \frac {11}{4}, 1 & \frac {5}{2}, \frac {5}{2}, 3 \\2, \frac {9}{4}, \frac {5}{2}, \frac {11}{4}, 3 & 0 \end {matrix} \middle | {\frac {c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} c^{4}} - \frac {i a d^{3} {G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {3}{2}, \frac {7}{4}, 2, \frac {9}{4}, \frac {5}{2}, 1 & \\\frac {7}{4}, \frac {9}{4} & \frac {3}{2}, 2, 2, 0 \end {matrix} \middle | {\frac {c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} c^{4}} - \frac {b d {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {5}{4}, \frac {7}{4}, 1 & \frac {3}{2}, \frac {3}{2}, 2 \\1, \frac {5}{4}, \frac {3}{2}, \frac {7}{4}, 2 & 0 \end {matrix} \middle | {\frac {c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} c^{2}} - \frac {i b d {G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2}, 1 & \\\frac {3}{4}, \frac {5}{4} & \frac {1}{2}, 1, 1, 0 \end {matrix} \middle | {\frac {c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} c^{2}} \] Input:
integrate((b*x**2+a)/x**4/(d*x-c)**(1/2)/(d*x+c)**(1/2),x)
Output:
-a*d**3*meijerg(((9/4, 11/4, 1), (5/2, 5/2, 3)), ((2, 9/4, 5/2, 11/4, 3), (0,)), c**2/(d**2*x**2))/(4*pi**(3/2)*c**4) - I*a*d**3*meijerg(((3/2, 7/4, 2, 9/4, 5/2, 1), ()), ((7/4, 9/4), (3/2, 2, 2, 0)), c**2*exp_polar(2*I*pi )/(d**2*x**2))/(4*pi**(3/2)*c**4) - b*d*meijerg(((5/4, 7/4, 1), (3/2, 3/2, 2)), ((1, 5/4, 3/2, 7/4, 2), (0,)), c**2/(d**2*x**2))/(4*pi**(3/2)*c**2) - I*b*d*meijerg(((1/2, 3/4, 1, 5/4, 3/2, 1), ()), ((3/4, 5/4), (1/2, 1, 1, 0)), c**2*exp_polar(2*I*pi)/(d**2*x**2))/(4*pi**(3/2)*c**2)
Time = 0.12 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x^2}{x^4 \sqrt {-c+d x} \sqrt {c+d x}} \, dx=\frac {\sqrt {d^{2} x^{2} - c^{2}} b}{c^{2} x} + \frac {2 \, \sqrt {d^{2} x^{2} - c^{2}} a d^{2}}{3 \, c^{4} x} + \frac {\sqrt {d^{2} x^{2} - c^{2}} a}{3 \, c^{2} x^{3}} \] Input:
integrate((b*x^2+a)/x^4/(d*x-c)^(1/2)/(d*x+c)^(1/2),x, algorithm="maxima")
Output:
sqrt(d^2*x^2 - c^2)*b/(c^2*x) + 2/3*sqrt(d^2*x^2 - c^2)*a*d^2/(c^4*x) + 1/ 3*sqrt(d^2*x^2 - c^2)*a/(c^2*x^3)
Time = 0.14 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.68 \[ \int \frac {a+b x^2}{x^4 \sqrt {-c+d x} \sqrt {c+d x}} \, dx=\frac {8 \, {\left (3 \, b {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{8} + 24 \, b c^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{4} + 24 \, a d^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{4} + 48 \, b c^{4} + 32 \, a c^{2} d^{2}\right )} d}{3 \, {\left ({\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{4} + 4 \, c^{2}\right )}^{3}} \] Input:
integrate((b*x^2+a)/x^4/(d*x-c)^(1/2)/(d*x+c)^(1/2),x, algorithm="giac")
Output:
8/3*(3*b*(sqrt(d*x + c) - sqrt(d*x - c))^8 + 24*b*c^2*(sqrt(d*x + c) - sqr t(d*x - c))^4 + 24*a*d^2*(sqrt(d*x + c) - sqrt(d*x - c))^4 + 48*b*c^4 + 32 *a*c^2*d^2)*d/((sqrt(d*x + c) - sqrt(d*x - c))^4 + 4*c^2)^3
Time = 4.99 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.05 \[ \int \frac {a+b x^2}{x^4 \sqrt {-c+d x} \sqrt {c+d x}} \, dx=\frac {\sqrt {d\,x-c}\,\left (\frac {a}{3\,c}+\frac {x^2\,\left (3\,b\,c^3+2\,a\,c\,d^2\right )}{3\,c^4}+\frac {x^3\,\left (3\,b\,c^2\,d+2\,a\,d^3\right )}{3\,c^4}+\frac {a\,d\,x}{3\,c^2}\right )}{x^3\,\sqrt {c+d\,x}} \] Input:
int((a + b*x^2)/(x^4*(c + d*x)^(1/2)*(d*x - c)^(1/2)),x)
Output:
((d*x - c)^(1/2)*(a/(3*c) + (x^2*(3*b*c^3 + 2*a*c*d^2))/(3*c^4) + (x^3*(2* a*d^3 + 3*b*c^2*d))/(3*c^4) + (a*d*x)/(3*c^2)))/(x^3*(c + d*x)^(1/2))
Time = 0.22 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.24 \[ \int \frac {a+b x^2}{x^4 \sqrt {-c+d x} \sqrt {c+d x}} \, dx=\frac {\sqrt {d x +c}\, \sqrt {d x -c}\, a \,c^{2}+2 \sqrt {d x +c}\, \sqrt {d x -c}\, a \,d^{2} x^{2}+3 \sqrt {d x +c}\, \sqrt {d x -c}\, b \,c^{2} x^{2}-2 a \,d^{3} x^{3}-b \,c^{2} d \,x^{3}}{3 c^{4} x^{3}} \] Input:
int((b*x^2+a)/x^4/(d*x-c)^(1/2)/(d*x+c)^(1/2),x)
Output:
(sqrt(c + d*x)*sqrt( - c + d*x)*a*c**2 + 2*sqrt(c + d*x)*sqrt( - c + d*x)* a*d**2*x**2 + 3*sqrt(c + d*x)*sqrt( - c + d*x)*b*c**2*x**2 - 2*a*d**3*x**3 - b*c**2*d*x**3)/(3*c**4*x**3)