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*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])

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Rubi [A]  time = 0.076683, antiderivative size = 67, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [A]  time = 0.0241339, 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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Maple [A]  time = 0.004, size = 48, normalized size = 0.7 \begin{align*}{\frac{-2\,a{d}^{2}{x}^{2}-b{c}^{2}{x}^{2}+a{c}^{2}}{x{c}^{4}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{dx-c}}}} \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 1.49661, size = 197, normalized size = 2.94 \begin{align*} -\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} \end{align*}

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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Sympy [C]  time = 58.0321, size = 165, normalized size = 2.46 \begin{align*} 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 ) \end{align*}

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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Giac [B]  time = 1.36761, size = 296, normalized size = 4.42 \begin{align*} -\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} \end{align*}

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)