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

Optimal. Leaf size=67 \[ \frac {a}{c^2 x \sqrt {d x-c} \sqrt {c+d x}}-\frac {x \left (2 a d^2+b c^2\right )}{c^4 \sqrt {d x-c} \sqrt {c+d x}} \]

[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)

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Rubi [A]  time = 0.08, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {454, 39} \[ \frac {a}{c^2 x \sqrt {d x-c} \sqrt {c+d x}}-\frac {x \left (2 a d^2+b c^2\right )}{c^4 \sqrt {d x-c} \sqrt {c+d x}} \]

Antiderivative was successfully verified.

[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 454

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*} \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}}+\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*}

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Mathematica [A]  time = 0.03, size = 51, normalized size = 0.76 \[ \frac {a \left (c^2-2 d^2 x^2\right )-b c^2 x^2}{c^4 x \sqrt {d x-c} \sqrt {c+d x}} \]

Antiderivative was successfully verified.

[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])

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fricas [A]  time = 1.00, size = 103, normalized size = 1.54 \[ -\frac {{\left (b c^{2} d^{2} + 2 \, a d^{4}\right )} x^{3} - {\left (a c^{2} d - {\left (b c^{2} d + 2 \, a d^{3}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {d x - c} - {\left (b c^{4} + 2 \, a c^{2} d^{2}\right )} x}{c^{4} d^{3} x^{3} - c^{6} d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [B]  time = 0.46, size = 219, normalized size = 3.27 \[ -\frac {{\left (b c^{2} + a d^{2}\right )} \sqrt {d x + c}}{2 \, \sqrt {d x - c} c^{4} d} - \frac {2 \, {\left (b c^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{4} + a d^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{4} + 4 \, a c d^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2} + 4 \, b c^{4} + 12 \, a c^{2} d^{2}\right )}}{{\left ({\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{6} + 2 \, c {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{4} + 4 \, c^{2} {\left (\sqrt {d x + c} - \sqrt {d x - c}\right )}^{2} + 8 \, c^{3}\right )} c^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.05, size = 48, normalized size = 0.72 \[ \frac {-2 a \,d^{2} x^{2}-b \,c^{2} x^{2}+a \,c^{2}}{\sqrt {d x +c}\, \sqrt {d x -c}\, c^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.36, size = 71, normalized size = 1.06 \[ -\frac {b x}{\sqrt {d^{2} x^{2} - c^{2}} c^{2}} - \frac {2 \, a d^{2} x}{\sqrt {d^{2} x^{2} - c^{2}} c^{4}} + \frac {a}{\sqrt {d^{2} x^{2} - c^{2}} c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 2.87, size = 73, normalized size = 1.09 \[ \frac {2\,a\,d^2\,x^2\,\sqrt {d\,x-c}-a\,c^2\,\sqrt {d\,x-c}+b\,c^2\,x^2\,\sqrt {d\,x-c}}{c^4\,x\,\sqrt {c+d\,x}\,\left (c-d\,x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [C]  time = 136.13, size = 165, normalized size = 2.46 \[ a \left (- \frac {d {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {7}{4}, \frac {9}{4}, 1 & \frac {3}{2}, \frac {5}{2}, 3 \\\frac {7}{4}, 2, \frac {9}{4}, \frac {5}{2}, 3 & 0 \end {matrix} \middle | {\frac {c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} c^{4}} + \frac {i d {G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {1}{2}, 1, \frac {5}{4}, \frac {3}{2}, \frac {7}{4}, 1 & \\\frac {5}{4}, \frac {7}{4} & \frac {1}{2}, 1, 2, 0 \end {matrix} \middle | {\frac {c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} c^{4}}\right ) + b \left (- \frac {{G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4}, 1 & \frac {1}{2}, \frac {3}{2}, 2 \\\frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2}, 2 & 0 \end {matrix} \middle | {\frac {c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} c^{2} d} + \frac {i {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1 & \\\frac {1}{4}, \frac {3}{4} & - \frac {1}{2}, 0, 1, 0 \end {matrix} \middle | {\frac {c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}} c^{2} d}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*(-d*meijerg(((7/4, 9/4, 1), (3/2, 5/2, 3)), ((7/4, 2, 9/4, 5/2, 3), (0,)), c**2/(d**2*x**2))/(2*pi**(3/2)*c*
*4) + I*d*meijerg(((1/2, 1, 5/4, 3/2, 7/4, 1), ()), ((5/4, 7/4), (1/2, 1, 2, 0)), c**2*exp_polar(2*I*pi)/(d**2
*x**2))/(2*pi**(3/2)*c**4)) + b*(-meijerg(((3/4, 5/4, 1), (1/2, 3/2, 2)), ((3/4, 1, 5/4, 3/2, 2), (0,)), c**2/
(d**2*x**2))/(2*pi**(3/2)*c**2*d) + I*meijerg(((-1/2, 0, 1/4, 1/2, 3/4, 1), ()), ((1/4, 3/4), (-1/2, 0, 1, 0))
, c**2*exp_polar(2*I*pi)/(d**2*x**2))/(2*pi**(3/2)*c**2*d))

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