\(\int \frac {a+b x^2}{x^2 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx\) [374]

Optimal result

Integrand size = 31, antiderivative size = 67 \[ \int \frac {a+b x^2}{x^2 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {a}{c^2 x \sqrt {-c+d x} \sqrt {c+d x}}-\frac {\left (b c^2+2 a d^2\right ) x}{c^4 \sqrt {-c+d x} \sqrt {c+d x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 67, 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 = {465, 39} \[ \int \frac {a+b x^2}{x^2 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {a}{c^2 x \sqrt {d x-c} \sqrt {c+d x}}-\frac {x \left (2 a d^2+b c^2\right )}{c^4 \sqrt {d x-c} \sqrt {c+d x}} \]

[In]

Int[(a + b*x^2)/(x^2*(-c + d*x)^(3/2)*(c + d*x)^(3/2)),x]

[Out]

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

Rule 39

Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> Simp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d
*x]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0]

Rule 465

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] + Dist[(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] && Eq
Q[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]

Rubi steps \begin{align*} \text {integral}& = \frac {a}{c^2 x \sqrt {-c+d x} \sqrt {c+d x}}+\left (b+\frac {2 a d^2}{c^2}\right ) \int \frac {1}{(-c+d x)^{3/2} (c+d x)^{3/2}} \, dx \\ & = \frac {a}{c^2 x \sqrt {-c+d x} \sqrt {c+d x}}-\frac {\left (b c^2+2 a d^2\right ) x}{c^4 \sqrt {-c+d x} \sqrt {c+d x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x^2}{x^2 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {-b c^2 x^2+a \left (c^2-2 d^2 x^2\right )}{c^4 x \sqrt {-c+d x} \sqrt {c+d x}} \]

[In]

Integrate[(a + b*x^2)/(x^2*(-c + d*x)^(3/2)*(c + d*x)^(3/2)),x]

[Out]

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

Maple [A] (verified)

Time = 4.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.72

method	result	size
gosper	\(\frac {-2 a \,d^{2} x^{2}-b \,c^{2} x^{2}+c^{2} a}{c^{4} x \sqrt {d x -c}\, \sqrt {d x +c}}\)	\(48\)
default	\(\frac {\left (2 a \,d^{2} x^{2}+b \,c^{2} x^{2}-c^{2} a \right ) \sqrt {d x -c}\, \operatorname {csgn}\left (d \right )^{2}}{c^{4} \left (-d x +c \right ) x \sqrt {d x +c}}\)	\(60\)
risch	\(\frac {a \left (-d x +c \right ) \sqrt {d x +c}}{c^{4} x \sqrt {d x -c}}-\frac {\left (a \,d^{2}+b \,c^{2}\right ) x \sqrt {\left (d x -c \right ) \left (d x +c \right )}}{\sqrt {-\left (d x +c \right ) \left (-d x +c \right )}\, c^{4} \sqrt {d x -c}\, \sqrt {d x +c}}\)	\(95\)

[In]

int((b*x^2+a)/x^2/(d*x-c)^(3/2)/(d*x+c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.54 \[ \int \frac {a+b x^2}{x^2 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=-\frac {{\left (b c^{2} d^{2} + 2 \, a d^{4}\right )} x^{3} - {\left (a c^{2} d - {\left (b c^{2} d + 2 \, a d^{3}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {d x - c} - {\left (b c^{4} + 2 \, a c^{2} d^{2}\right )} x}{c^{4} d^{3} x^{3} - c^{6} d x} \]

[In]

integrate((b*x^2+a)/x^2/(d*x-c)^(3/2)/(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

-((b*c^2*d^2 + 2*a*d^4)*x^3 - (a*c^2*d - (b*c^2*d + 2*a*d^3)*x^2)*sqrt(d*x + c)*sqrt(d*x - c) - (b*c^4 + 2*a*c
^2*d^2)*x)/(c^4*d^3*x^3 - c^6*d*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b x^2}{x^2 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate((b*x**2+a)/x**2/(d*x-c)**(3/2)/(d*x+c)**(3/2),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.06 \[ \int \frac {a+b x^2}{x^2 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=-\frac {b x}{\sqrt {d^{2} x^{2} - c^{2}} c^{2}} - \frac {2 \, a d^{2} x}{\sqrt {d^{2} x^{2} - c^{2}} c^{4}} + \frac {a}{\sqrt {d^{2} x^{2} - c^{2}} c^{2} x} \]

[In]

integrate((b*x^2+a)/x^2/(d*x-c)^(3/2)/(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

-b*x/(sqrt(d^2*x^2 - c^2)*c^2) - 2*a*d^2*x/(sqrt(d^2*x^2 - c^2)*c^4) + a/(sqrt(d^2*x^2 - c^2)*c^2*x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (59) = 118\).

Time = 0.34 (sec) , antiderivative size = 219, normalized size of antiderivative = 3.27 \[ \int \frac {a+b x^2}{x^2 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=-\frac {{\left (b c^{2} + a d^{2}\right )} \sqrt {d x + c}}{2 \, \sqrt {d x - c} c^{4} d} - \frac {2 \, {\left (b c^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{4} + a d^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{4} + 4 \, a c d^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2} + 4 \, b c^{4} + 12 \, a c^{2} d^{2}\right )}}{{\left ({\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{6} + 2 \, c {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{4} + 4 \, c^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2} + 8 \, c^{3}\right )} c^{3} d} \]

[In]

integrate((b*x^2+a)/x^2/(d*x-c)^(3/2)/(d*x+c)^(3/2),x, algorithm="giac")

[Out]

-1/2*(b*c^2 + a*d^2)*sqrt(d*x + c)/(sqrt(d*x - c)*c^4*d) - 2*(b*c^2*(sqrt(d*x + c) - sqrt(d*x - c))^4 + a*d^2*
(sqrt(d*x + c) - sqrt(d*x - c))^4 + 4*a*c*d^2*(sqrt(d*x + c) - sqrt(d*x - c))^2 + 4*b*c^4 + 12*a*c^2*d^2)/(((s
qrt(d*x + c) - sqrt(d*x - c))^6 + 2*c*(sqrt(d*x + c) - sqrt(d*x - c))^4 + 4*c^2*(sqrt(d*x + c) - sqrt(d*x - c)
)^2 + 8*c^3)*c^3*d)

Mupad [B] (verification not implemented)

Time = 7.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.09 \[ \int \frac {a+b x^2}{x^2 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {2\,a\,d^2\,x^2\,\sqrt {d\,x-c}-a\,c^2\,\sqrt {d\,x-c}+b\,c^2\,x^2\,\sqrt {d\,x-c}}{c^4\,x\,\sqrt {c+d\,x}\,\left (c-d\,x\right )} \]

[In]

int((a + b*x^2)/(x^2*(c + d*x)^(3/2)*(d*x - c)^(3/2)),x)

[Out]

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