3.366 \(\int \frac{a+b x^2}{x^4 \sqrt{-c+d x} \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=75 \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (2 a d^2+3 b c^2\right )}{3 c^4 x}+\frac{a \sqrt{d x-c} \sqrt{c+d x}}{3 c^2 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0693193, antiderivative size = 75, 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, 95} \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (2 a d^2+3 b c^2\right )}{3 c^4 x}+\frac{a \sqrt{d x-c} \sqrt{c+d x}}{3 c^2 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sqrt[-c + d*x]*Sqrt[c + d*x])/(3*c^2*x^3) + ((3*b*c^2 + 2*a*d^2)*Sqrt[-c + d*x]*Sqrt[c + d*x])/(3*c^4*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 95

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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]

Rubi steps

\begin{align*} \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{1}{3} \left (3 b+\frac{2 a d^2}{c^2}\right ) \int \frac{1}{x^2 \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}\\ \end{align*}

Mathematica [A]  time = 0.0257343, size = 66, normalized size = 0.88 \[ -\frac{\left (c^2-d^2 x^2\right ) \left (a \left (c^2+2 d^2 x^2\right )+3 b c^2 x^2\right )}{3 c^4 x^3 \sqrt{d x-c} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 49, normalized size = 0.7 \begin{align*}{\frac{2\,a{d}^{2}{x}^{2}+3\,b{c}^{2}{x}^{2}+a{c}^{2}}{3\,{x}^{3}{c}^{4}}\sqrt{dx+c}\sqrt{dx-c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(d*x+c)^(1/2)*(2*a*d^2*x^2+3*b*c^2*x^2+a*c^2)/x^3/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^4/(d*x-c)^(1/2)/(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.48573, size = 144, normalized size = 1.92 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)/x^4/(d*x-c)^(1/2)/(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

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)

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Sympy [C]  time = 44.3197, size = 170, normalized size = 2.27 \begin{align*} - \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)

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Giac [B]  time = 1.19709, size = 185, normalized size = 2.47 \begin{align*} \frac{8 \,{\left (3 \, b d^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{8} + 24 \, b c^{2} d^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 24 \, a d^{4}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 48 \, b c^{4} d^{2} + 32 \, a c^{2} d^{4}\right )}}{3 \,{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 4 \, c^{2}\right )}^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)/x^4/(d*x-c)^(1/2)/(d*x+c)^(1/2),x, algorithm="giac")

[Out]

8/3*(3*b*d^2*(sqrt(d*x + c) - sqrt(d*x - c))^8 + 24*b*c^2*d^2*(sqrt(d*x + c) - sqrt(d*x - c))^4 + 24*a*d^4*(sq
rt(d*x + c) - sqrt(d*x - c))^4 + 48*b*c^4*d^2 + 32*a*c^2*d^4)/(((sqrt(d*x + c) - sqrt(d*x - c))^4 + 4*c^2)^3*d
)