3.277 \(\int \frac{a+b x^2}{x^4 (-c+d x)^{3/2} (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=119 \[ -\frac{2 d^2 x \left (4 a d^2+3 b c^2\right )}{3 c^6 \sqrt{d x-c} \sqrt{c+d x}}+\frac{4 a d^2+3 b c^2}{3 c^4 x \sqrt{d x-c} \sqrt{c+d x}}+\frac{a}{3 c^2 x^3 \sqrt{d x-c} \sqrt{c+d x}} \]

[Out]

a/(3*c^2*x^3*Sqrt[-c + d*x]*Sqrt[c + d*x]) + (3*b*c^2 + 4*a*d^2)/(3*c^4*x*Sqrt[-
c + d*x]*Sqrt[c + d*x]) - (2*d^2*(3*b*c^2 + 4*a*d^2)*x)/(3*c^6*Sqrt[-c + d*x]*Sq
rt[c + d*x])

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Rubi [A]  time = 0.323861, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ -\frac{2 d^2 x \left (4 a d^2+3 b c^2\right )}{3 c^6 \sqrt{d x-c} \sqrt{c+d x}}+\frac{4 a d^2+3 b c^2}{3 c^4 x \sqrt{d x-c} \sqrt{c+d x}}+\frac{a}{3 c^2 x^3 \sqrt{d x-c} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

a/(3*c^2*x^3*Sqrt[-c + d*x]*Sqrt[c + d*x]) + (3*b*c^2 + 4*a*d^2)/(3*c^4*x*Sqrt[-
c + d*x]*Sqrt[c + d*x]) - (2*d^2*(3*b*c^2 + 4*a*d^2)*x)/(3*c^6*Sqrt[-c + d*x]*Sq
rt[c + d*x])

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Rubi in Sympy [A]  time = 17.9077, size = 105, normalized size = 0.88 \[ \frac{a}{3 c^{2} x^{3} \sqrt{- c + d x} \sqrt{c + d x}} + \frac{4 a d^{2} + 3 b c^{2}}{3 c^{4} x \sqrt{- c + d x} \sqrt{c + d x}} - \frac{2 d^{2} x \left (4 a d^{2} + 3 b c^{2}\right )}{3 c^{6} \sqrt{- c + d x} \sqrt{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)/x**4/(d*x-c)**(3/2)/(d*x+c)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.107597, size = 77, normalized size = 0.65 \[ \frac{a \left (c^4+4 c^2 d^2 x^2-8 d^4 x^4\right )+3 b c^2 x^2 \left (c^2-2 d^2 x^2\right )}{3 c^6 x^3 \sqrt{d x-c} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

(3*b*c^2*x^2*(c^2 - 2*d^2*x^2) + a*(c^4 + 4*c^2*d^2*x^2 - 8*d^4*x^4))/(3*c^6*x^3
*Sqrt[-c + d*x]*Sqrt[c + d*x])

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Maple [A]  time = 0.008, size = 73, normalized size = 0.6 \[{\frac{-8\,a{d}^{4}{x}^{4}-6\,b{c}^{2}{d}^{2}{x}^{4}+4\,a{c}^{2}{d}^{2}{x}^{2}+3\,b{c}^{4}{x}^{2}+a{c}^{4}}{3\,{x}^{3}{c}^{6}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{dx-c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235783, size = 193, normalized size = 1.62 \[ \frac{6 \, b d^{2} x^{4} - a c^{2} -{\left (3 \, b c^{2} - 4 \, a d^{2}\right )} x^{2} - 2 \,{\left (3 \, b d x^{3} + 2 \, a d x\right )} \sqrt{d x + c} \sqrt{d x - c}}{3 \,{\left (8 \, d^{5} x^{8} - 12 \, c^{2} d^{3} x^{6} + 4 \, c^{4} d x^{4} -{\left (8 \, d^{4} x^{7} - 8 \, c^{2} d^{2} x^{5} + c^{4} x^{3}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)/((d*x + c)^(3/2)*(d*x - c)^(3/2)*x^4),x, algorithm="fricas")

[Out]

1/3*(6*b*d^2*x^4 - a*c^2 - (3*b*c^2 - 4*a*d^2)*x^2 - 2*(3*b*d*x^3 + 2*a*d*x)*sqr
t(d*x + c)*sqrt(d*x - c))/(8*d^5*x^8 - 12*c^2*d^3*x^6 + 4*c^4*d*x^4 - (8*d^4*x^7
- 8*c^2*d^2*x^5 + c^4*x^3)*sqrt(d*x + c)*sqrt(d*x - c))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)/x**4/(d*x-c)**(3/2)/(d*x+c)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.313585, size = 327, normalized size = 2.75 \[ -\frac{{\left (b c^{2} d + a d^{3}\right )} \sqrt{d x + c}}{2 \, \sqrt{d x - c} c^{6}} - \frac{2 \,{\left (b c^{2} d + a d^{3}\right )}}{{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2} + 2 \, c\right )} c^{5}} - \frac{8 \,{\left (3 \, b c^{2} d{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{8} + 3 \, a d^{3}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{8} + 24 \, b c^{4} d{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 48 \, a c^{2} d^{3}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 48 \, b c^{6} d + 80 \, a c^{4} d^{3}\right )}}{3 \,{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 4 \, c^{2}\right )}^{3} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)/((d*x + c)^(3/2)*(d*x - c)^(3/2)*x^4),x, algorithm="giac")

[Out]

-1/2*(b*c^2*d + a*d^3)*sqrt(d*x + c)/(sqrt(d*x - c)*c^6) - 2*(b*c^2*d + a*d^3)/(
((sqrt(d*x + c) - sqrt(d*x - c))^2 + 2*c)*c^5) - 8/3*(3*b*c^2*d*(sqrt(d*x + c) -
sqrt(d*x - c))^8 + 3*a*d^3*(sqrt(d*x + c) - sqrt(d*x - c))^8 + 24*b*c^4*d*(sqrt
(d*x + c) - sqrt(d*x - c))^4 + 48*a*c^2*d^3*(sqrt(d*x + c) - sqrt(d*x - c))^4 +
48*b*c^6*d + 80*a*c^4*d^3)/(((sqrt(d*x + c) - sqrt(d*x - c))^4 + 4*c^2)^3*c^4)