\(\int \frac {\sqrt {-c+d x} \sqrt {c+d x} (a+b x^2)}{x^5} \, dx\) [6]

Optimal result

Integrand size = 31, antiderivative size = 118 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x^5} \, dx=-\frac {\left (4 b+\frac {a d^2}{c^2}\right ) \sqrt {-c+d x} \sqrt {c+d x}}{8 x^2}+\frac {a (-c+d x)^{3/2} (c+d x)^{3/2}}{4 c^2 x^4}+\frac {d^2 \left (4 b c^2+a d^2\right ) \arctan \left (\frac {\sqrt {-c+d x} \sqrt {c+d x}}{c}\right )}{8 c^3} \] Output:

-1/8*(4*b+a*d^2/c^2)*(d*x-c)^(1/2)*(d*x+c)^(1/2)/x^2+1/4*a*(d*x-c)^(3/2)*( 
d*x+c)^(3/2)/c^2/x^4+1/8*d^2*(a*d^2+4*b*c^2)*arctan((d*x-c)^(1/2)*(d*x+c)^ 
(1/2)/c)/c^3
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x^5} \, dx=\frac {c \sqrt {-c+d x} \sqrt {c+d x} \left (-2 a c^2-4 b c^2 x^2+a d^2 x^2\right )+2 d^2 \left (4 b c^2+a d^2\right ) x^4 \arctan \left (\frac {\sqrt {-c+d x}}{\sqrt {c+d x}}\right )}{8 c^3 x^4} \] Input:

Integrate[(Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2))/x^5,x]
 

Output:

(c*Sqrt[-c + d*x]*Sqrt[c + d*x]*(-2*a*c^2 - 4*b*c^2*x^2 + a*d^2*x^2) + 2*d 
^2*(4*b*c^2 + a*d^2)*x^4*ArcTan[Sqrt[-c + d*x]/Sqrt[c + d*x]])/(8*c^3*x^4)
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.20, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {956, 105, 105, 103, 218}

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.

\(\displaystyle \int \frac {\left (a+b x^2\right ) \sqrt {d x-c} \sqrt {c+d x}}{x^5} \, dx\)

\(\Big \downarrow \) 956

\(\displaystyle \frac {1}{4} \left (\frac {a d^2}{c^2}+4 b\right ) \int \frac {\sqrt {d x-c} \sqrt {c+d x}}{x^3}dx+\frac {a (d x-c)^{3/2} (c+d x)^{3/2}}{4 c^2 x^4}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {1}{4} \left (\frac {a d^2}{c^2}+4 b\right ) \left (\frac {1}{2} d \int \frac {\sqrt {c+d x}}{x^2 \sqrt {d x-c}}dx-\frac {\sqrt {d x-c} (c+d x)^{3/2}}{2 c x^2}\right )+\frac {a (d x-c)^{3/2} (c+d x)^{3/2}}{4 c^2 x^4}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {1}{4} \left (\frac {a d^2}{c^2}+4 b\right ) \left (\frac {1}{2} d \left (d \int \frac {1}{x \sqrt {d x-c} \sqrt {c+d x}}dx+\frac {\sqrt {d x-c} \sqrt {c+d x}}{c x}\right )-\frac {\sqrt {d x-c} (c+d x)^{3/2}}{2 c x^2}\right )+\frac {a (d x-c)^{3/2} (c+d x)^{3/2}}{4 c^2 x^4}\)

\(\Big \downarrow \) 103

\(\displaystyle \frac {1}{4} \left (\frac {a d^2}{c^2}+4 b\right ) \left (\frac {1}{2} d \left (d^2 \int \frac {1}{d c^2+d (d x-c) (c+d x)}d\left (\sqrt {d x-c} \sqrt {c+d x}\right )+\frac {\sqrt {d x-c} \sqrt {c+d x}}{c x}\right )-\frac {\sqrt {d x-c} (c+d x)^{3/2}}{2 c x^2}\right )+\frac {a (d x-c)^{3/2} (c+d x)^{3/2}}{4 c^2 x^4}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {1}{4} \left (\frac {a d^2}{c^2}+4 b\right ) \left (\frac {1}{2} d \left (\frac {d \arctan \left (\frac {\sqrt {d x-c} \sqrt {c+d x}}{c}\right )}{c}+\frac {\sqrt {d x-c} \sqrt {c+d x}}{c x}\right )-\frac {\sqrt {d x-c} (c+d x)^{3/2}}{2 c x^2}\right )+\frac {a (d x-c)^{3/2} (c+d x)^{3/2}}{4 c^2 x^4}\)

Input:

Int[(Sqrt[-c + d*x]*Sqrt[c + d*x]*(a + b*x^2))/x^5,x]
 

Output:

(a*(-c + d*x)^(3/2)*(c + d*x)^(3/2))/(4*c^2*x^4) + ((4*b + (a*d^2)/c^2)*(- 
1/2*(Sqrt[-c + d*x]*(c + d*x)^(3/2))/(c*x^2) + (d*((Sqrt[-c + d*x]*Sqrt[c 
+ d*x])/(c*x) + (d*ArcTan[(Sqrt[-c + d*x]*Sqrt[c + d*x])/c])/c))/2))/4
 

Defintions of rubi rules used

rule 103

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ 
))), x_] :> Simp[b*f   Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq 
rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d 
*e - f*(b*c + a*d), 0]
 

rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f)))   Int[(a 
+ b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, 
e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] 
 ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 956

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]
 

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.27

method	result	size
risch	\(\frac {\sqrt {d x +c}\, \left (-d x +c \right ) \left (-a \,d^{2} x^{2}+4 b \,c^{2} x^{2}+2 a \,c^{2}\right )}{8 x^{4} c^{2} \sqrt {d x -c}}-\frac {d^{2} \left (a \,d^{2}+4 b \,c^{2}\right ) \ln \left (\frac {-2 c^{2}+2 \sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}}{x}\right ) \sqrt {\left (d x -c \right ) \left (d x +c \right )}}{8 c^{2} \sqrt {-c^{2}}\, \sqrt {d x -c}\, \sqrt {d x +c}}\)	\(150\)
default	\(-\frac {\sqrt {d x -c}\, \sqrt {d x +c}\, \left (\ln \left (-\frac {2 \left (c^{2}-\sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\right )}{x}\right ) a \,d^{4} x^{4}+4 \ln \left (-\frac {2 \left (c^{2}-\sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\right )}{x}\right ) b \,c^{2} d^{2} x^{4}-\sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\, a \,d^{2} x^{2}+4 \sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\, b \,c^{2} x^{2}+2 \sqrt {-c^{2}}\, \sqrt {d^{2} x^{2}-c^{2}}\, a \,c^{2}\right )}{8 c^{2} \sqrt {d^{2} x^{2}-c^{2}}\, x^{4} \sqrt {-c^{2}}}\)	\(226\)

Input:

int((d*x-c)^(1/2)*(d*x+c)^(1/2)*(b*x^2+a)/x^5,x,method=_RETURNVERBOSE)
 

Output:

1/8*(d*x+c)^(1/2)*(-d*x+c)*(-a*d^2*x^2+4*b*c^2*x^2+2*a*c^2)/x^4/c^2/(d*x-c 
)^(1/2)-1/8*d^2*(a*d^2+4*b*c^2)/c^2/(-c^2)^(1/2)*ln((-2*c^2+2*(-c^2)^(1/2) 
*(d^2*x^2-c^2)^(1/2))/x)*((d*x-c)*(d*x+c))^(1/2)/(d*x-c)^(1/2)/(d*x+c)^(1/ 
2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x^5} \, dx=\frac {2 \, {\left (4 \, b c^{2} d^{2} + a d^{4}\right )} x^{4} \arctan \left (-\frac {d x - \sqrt {d x + c} \sqrt {d x - c}}{c}\right ) - {\left (2 \, a c^{3} + {\left (4 \, b c^{3} - a c d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {d x - c}}{8 \, c^{3} x^{4}} \] Input:

integrate((d*x-c)^(1/2)*(d*x+c)^(1/2)*(b*x^2+a)/x^5,x, algorithm="fricas")
 

Output:

1/8*(2*(4*b*c^2*d^2 + a*d^4)*x^4*arctan(-(d*x - sqrt(d*x + c)*sqrt(d*x - c 
))/c) - (2*a*c^3 + (4*b*c^3 - a*c*d^2)*x^2)*sqrt(d*x + c)*sqrt(d*x - c))/( 
c^3*x^4)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x^5} \, dx=\text {Timed out} \] Input:

integrate((d*x-c)**(1/2)*(d*x+c)**(1/2)*(b*x**2+a)/x**5,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.37 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x^5} \, dx=-\frac {b d^{2} \arcsin \left (\frac {c}{d {\left | x \right |}}\right )}{2 \, c} - \frac {a d^{4} \arcsin \left (\frac {c}{d {\left | x \right |}}\right )}{8 \, c^{3}} - \frac {\sqrt {d^{2} x^{2} - c^{2}} b d^{2}}{2 \, c^{2}} - \frac {\sqrt {d^{2} x^{2} - c^{2}} a d^{4}}{8 \, c^{4}} + \frac {{\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} b}{2 \, c^{2} x^{2}} + \frac {{\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} a d^{2}}{8 \, c^{4} x^{2}} + \frac {{\left (d^{2} x^{2} - c^{2}\right )}^{\frac {3}{2}} a}{4 \, c^{2} x^{4}} \] Input:

integrate((d*x-c)^(1/2)*(d*x+c)^(1/2)*(b*x^2+a)/x^5,x, algorithm="maxima")
 

Output:

-1/2*b*d^2*arcsin(c/(d*abs(x)))/c - 1/8*a*d^4*arcsin(c/(d*abs(x)))/c^3 - 1 
/2*sqrt(d^2*x^2 - c^2)*b*d^2/c^2 - 1/8*sqrt(d^2*x^2 - c^2)*a*d^4/c^4 + 1/2 
*(d^2*x^2 - c^2)^(3/2)*b/(c^2*x^2) + 1/8*(d^2*x^2 - c^2)^(3/2)*a*d^2/(c^4* 
x^2) + 1/4*(d^2*x^2 - c^2)^(3/2)*a/(c^2*x^4)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (100) = 200\).

Time = 0.20 (sec) , antiderivative size = 324, normalized size of antiderivative = 2.75 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x^5} \, dx=-\frac {\frac {{\left (4 \, b c^{2} d^{3} + a d^{5}\right )} \arctan \left (\frac {{\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2}}{2 \, c}\right )}{c^{3}} - \frac {2 \, {\left (4 \, b c^{2} d^{3} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{14} - a d^{5} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{14} + 16 \, b c^{4} d^{3} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{10} + 28 \, a c^{2} d^{5} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{10} - 64 \, b c^{6} d^{3} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{6} - 112 \, a c^{4} d^{5} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{6} - 256 \, b c^{8} d^{3} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2} + 64 \, a c^{6} d^{5} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2}\right )}}{{\left ({\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{4} + 4 \, c^{2}\right )}^{4} c^{2}}}{4 \, d} \] Input:

integrate((d*x-c)^(1/2)*(d*x+c)^(1/2)*(b*x^2+a)/x^5,x, algorithm="giac")
 

Output:

-1/4*((4*b*c^2*d^3 + a*d^5)*arctan(1/2*(sqrt(d*x + c) - sqrt(d*x - c))^2/c 
)/c^3 - 2*(4*b*c^2*d^3*(sqrt(d*x + c) - sqrt(d*x - c))^14 - a*d^5*(sqrt(d* 
x + c) - sqrt(d*x - c))^14 + 16*b*c^4*d^3*(sqrt(d*x + c) - sqrt(d*x - c))^ 
10 + 28*a*c^2*d^5*(sqrt(d*x + c) - sqrt(d*x - c))^10 - 64*b*c^6*d^3*(sqrt( 
d*x + c) - sqrt(d*x - c))^6 - 112*a*c^4*d^5*(sqrt(d*x + c) - sqrt(d*x - c) 
)^6 - 256*b*c^8*d^3*(sqrt(d*x + c) - sqrt(d*x - c))^2 + 64*a*c^6*d^5*(sqrt 
(d*x + c) - sqrt(d*x - c))^2)/(((sqrt(d*x + c) - sqrt(d*x - c))^4 + 4*c^2) 
^4*c^2))/d
 

Mupad [B] (verification not implemented)

Time = 18.88 (sec) , antiderivative size = 1004, normalized size of antiderivative = 8.51 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x^5} \, dx =\text {Too large to display} \] Input:

int(((a + b*x^2)*(c + d*x)^(1/2)*(d*x - c)^(1/2))/x^5,x)
 

Output:

((a*(-c)^(1/2)*d^4)/(1024*c^(7/2)) + (a*(-c)^(1/2)*d^4*((c + d*x)^(1/2) - 
c^(1/2))^2)/(128*c^(7/2)*((-c)^(1/2) - (d*x - c)^(1/2))^2) + (11*a*(-c)^(1 
/2)*d^4*((c + d*x)^(1/2) - c^(1/2))^4)/(512*c^(7/2)*((-c)^(1/2) - (d*x - c 
)^(1/2))^4) + (7*a*(-c)^(1/2)*d^4*((c + d*x)^(1/2) - c^(1/2))^6)/(256*c^(7 
/2)*((-c)^(1/2) - (d*x - c)^(1/2))^6) - (239*a*(-c)^(1/2)*d^4*((c + d*x)^( 
1/2) - c^(1/2))^8)/(1024*c^(7/2)*((-c)^(1/2) - (d*x - c)^(1/2))^8) + (a*(- 
c)^(1/2)*d^4*((c + d*x)^(1/2) - c^(1/2))^10)/(256*c^(7/2)*((-c)^(1/2) - (d 
*x - c)^(1/2))^10))/(((c + d*x)^(1/2) - c^(1/2))^4/((-c)^(1/2) - (d*x - c) 
^(1/2))^4 + (4*((c + d*x)^(1/2) - c^(1/2))^6)/((-c)^(1/2) - (d*x - c)^(1/2 
))^6 + (6*((c + d*x)^(1/2) - c^(1/2))^8)/((-c)^(1/2) - (d*x - c)^(1/2))^8 
+ (4*((c + d*x)^(1/2) - c^(1/2))^10)/((-c)^(1/2) - (d*x - c)^(1/2))^10 + ( 
(c + d*x)^(1/2) - c^(1/2))^12/((-c)^(1/2) - (d*x - c)^(1/2))^12) - ((b*(-c 
)^(1/2)*d^2)/(32*c^(3/2)) + (b*(-c)^(1/2)*d^2*((c + d*x)^(1/2) - c^(1/2))^ 
2)/(16*c^(3/2)*((-c)^(1/2) - (d*x - c)^(1/2))^2) - (15*b*(-c)^(1/2)*d^2*(( 
c + d*x)^(1/2) - c^(1/2))^4)/(32*c^(3/2)*((-c)^(1/2) - (d*x - c)^(1/2))^4) 
)/(((c + d*x)^(1/2) - c^(1/2))^2/((-c)^(1/2) - (d*x - c)^(1/2))^2 + (2*((c 
 + d*x)^(1/2) - c^(1/2))^4)/((-c)^(1/2) - (d*x - c)^(1/2))^4 + ((c + d*x)^ 
(1/2) - c^(1/2))^6/((-c)^(1/2) - (d*x - c)^(1/2))^6) + (a*(-c)^(1/2)*d^4*l 
og(((c + d*x)^(1/2) - c^(1/2))/((-c)^(1/2) - (d*x - c)^(1/2))))/(8*c^(7/2) 
) + (b*(-c)^(1/2)*d^2*log(((c + d*x)^(1/2) - c^(1/2))/((-c)^(1/2) - (d*...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.81 \[ \int \frac {\sqrt {-c+d x} \sqrt {c+d x} \left (a+b x^2\right )}{x^5} \, dx=\frac {2 \mathit {atan} \left (\frac {\sqrt {d x -c}+\sqrt {d x +c}-\sqrt {c}}{\sqrt {c}}\right ) a \,d^{4} x^{4}+8 \mathit {atan} \left (\frac {\sqrt {d x -c}+\sqrt {d x +c}-\sqrt {c}}{\sqrt {c}}\right ) b \,c^{2} d^{2} x^{4}-2 \mathit {atan} \left (\frac {\sqrt {d x -c}+\sqrt {d x +c}+\sqrt {c}}{\sqrt {c}}\right ) a \,d^{4} x^{4}-8 \mathit {atan} \left (\frac {\sqrt {d x -c}+\sqrt {d x +c}+\sqrt {c}}{\sqrt {c}}\right ) b \,c^{2} d^{2} x^{4}-2 \sqrt {d x +c}\, \sqrt {d x -c}\, a \,c^{3}+\sqrt {d x +c}\, \sqrt {d x -c}\, a c \,d^{2} x^{2}-4 \sqrt {d x +c}\, \sqrt {d x -c}\, b \,c^{3} x^{2}}{8 c^{3} x^{4}} \] Input:

int((d*x-c)^(1/2)*(d*x+c)^(1/2)*(b*x^2+a)/x^5,x)
 

Output:

(2*atan((sqrt( - c + d*x) + sqrt(c + d*x) - sqrt(c))/sqrt(c))*a*d**4*x**4 
+ 8*atan((sqrt( - c + d*x) + sqrt(c + d*x) - sqrt(c))/sqrt(c))*b*c**2*d**2 
*x**4 - 2*atan((sqrt( - c + d*x) + sqrt(c + d*x) + sqrt(c))/sqrt(c))*a*d** 
4*x**4 - 8*atan((sqrt( - c + d*x) + sqrt(c + d*x) + sqrt(c))/sqrt(c))*b*c* 
*2*d**2*x**4 - 2*sqrt(c + d*x)*sqrt( - c + d*x)*a*c**3 + sqrt(c + d*x)*sqr 
t( - c + d*x)*a*c*d**2*x**2 - 4*sqrt(c + d*x)*sqrt( - c + d*x)*b*c**3*x**2 
)/(8*c**3*x**4)