3.52 \(\int \frac{\sqrt{a+b x^2}}{\left (c+d x^2\right )^4} \, dx\)

Optimal. Leaf size=208 \[ \frac{a \left (5 a^2 d^2-12 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{16 c^{7/2} (b c-a d)^{5/2}}+\frac{x \sqrt{a+b x^2} (2 b c-5 a d) (4 b c-3 a d)}{48 c^3 \left (c+d x^2\right ) (b c-a d)^2}+\frac{x \sqrt{a+b x^2} (4 b c-5 a d)}{24 c^2 \left (c+d x^2\right )^2 (b c-a d)}+\frac{x \sqrt{a+b x^2}}{6 c \left (c+d x^2\right )^3} \]

[Out]

(x*Sqrt[a + b*x^2])/(6*c*(c + d*x^2)^3) + ((4*b*c - 5*a*d)*x*Sqrt[a + b*x^2])/(2
4*c^2*(b*c - a*d)*(c + d*x^2)^2) + ((2*b*c - 5*a*d)*(4*b*c - 3*a*d)*x*Sqrt[a + b
*x^2])/(48*c^3*(b*c - a*d)^2*(c + d*x^2)) + (a*(8*b^2*c^2 - 12*a*b*c*d + 5*a^2*d
^2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(16*c^(7/2)*(b*c - a
*d)^(5/2))

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Rubi [A]  time = 0.554456, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{a \left (5 a^2 d^2-12 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{16 c^{7/2} (b c-a d)^{5/2}}+\frac{x \sqrt{a+b x^2} (2 b c-5 a d) (4 b c-3 a d)}{48 c^3 \left (c+d x^2\right ) (b c-a d)^2}+\frac{x \sqrt{a+b x^2} (4 b c-5 a d)}{24 c^2 \left (c+d x^2\right )^2 (b c-a d)}+\frac{x \sqrt{a+b x^2}}{6 c \left (c+d x^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[a + b*x^2])/(6*c*(c + d*x^2)^3) + ((4*b*c - 5*a*d)*x*Sqrt[a + b*x^2])/(2
4*c^2*(b*c - a*d)*(c + d*x^2)^2) + ((2*b*c - 5*a*d)*(4*b*c - 3*a*d)*x*Sqrt[a + b
*x^2])/(48*c^3*(b*c - a*d)^2*(c + d*x^2)) + (a*(8*b^2*c^2 - 12*a*b*c*d + 5*a^2*d
^2)*ArcTanh[(Sqrt[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(16*c^(7/2)*(b*c - a
*d)^(5/2))

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Rubi in Sympy [A]  time = 82.8335, size = 190, normalized size = 0.91 \[ \frac{a \left (5 a^{2} d^{2} - 12 a b c d + 8 b^{2} c^{2}\right ) \operatorname{atan}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{c} \sqrt{a + b x^{2}}} \right )}}{16 c^{\frac{7}{2}} \left (a d - b c\right )^{\frac{5}{2}}} + \frac{x \sqrt{a + b x^{2}}}{6 c \left (c + d x^{2}\right )^{3}} + \frac{x \sqrt{a + b x^{2}} \left (5 a d - 4 b c\right )}{24 c^{2} \left (c + d x^{2}\right )^{2} \left (a d - b c\right )} + \frac{x \sqrt{a + b x^{2}} \left (3 a d - 4 b c\right ) \left (5 a d - 2 b c\right )}{48 c^{3} \left (c + d x^{2}\right ) \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(1/2)/(d*x**2+c)**4,x)

[Out]

a*(5*a**2*d**2 - 12*a*b*c*d + 8*b**2*c**2)*atan(x*sqrt(a*d - b*c)/(sqrt(c)*sqrt(
a + b*x**2)))/(16*c**(7/2)*(a*d - b*c)**(5/2)) + x*sqrt(a + b*x**2)/(6*c*(c + d*
x**2)**3) + x*sqrt(a + b*x**2)*(5*a*d - 4*b*c)/(24*c**2*(c + d*x**2)**2*(a*d - b
*c)) + x*sqrt(a + b*x**2)*(3*a*d - 4*b*c)*(5*a*d - 2*b*c)/(48*c**3*(c + d*x**2)*
(a*d - b*c)**2)

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Mathematica [A]  time = 0.436169, size = 184, normalized size = 0.88 \[ \frac{\frac{\sqrt{c} x \sqrt{a+b x^2} \left (\frac{\left (c+d x^2\right )^2 \left (15 a^2 d^2-26 a b c d+8 b^2 c^2\right )}{(b c-a d)^2}+\frac{2 c \left (c+d x^2\right ) (4 b c-5 a d)}{b c-a d}+8 c^2\right )}{\left (c+d x^2\right )^3}+\frac{3 a \left (5 a^2 d^2-12 a b c d+8 b^2 c^2\right ) \tan ^{-1}\left (\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{(a d-b c)^{5/2}}}{48 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[c]*x*Sqrt[a + b*x^2]*(8*c^2 + (2*c*(4*b*c - 5*a*d)*(c + d*x^2))/(b*c - a*
d) + ((8*b^2*c^2 - 26*a*b*c*d + 15*a^2*d^2)*(c + d*x^2)^2)/(b*c - a*d)^2))/(c +
d*x^2)^3 + (3*a*(8*b^2*c^2 - 12*a*b*c*d + 5*a^2*d^2)*ArcTan[(Sqrt[-(b*c) + a*d]*
x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(-(b*c) + a*d)^(5/2))/(48*c^(7/2))

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Maple [B]  time = 0.047, size = 7922, normalized size = 38.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{2} + a}}{{\left (d x^{2} + c\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 1.25559, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/192*(4*((8*b^2*c^2*d^2 - 26*a*b*c*d^3 + 15*a^2*d^4)*x^5 + 2*(12*b^2*c^3*d - 3
5*a*b*c^2*d^2 + 20*a^2*c*d^3)*x^3 + 3*(8*b^2*c^4 - 20*a*b*c^3*d + 11*a^2*c^2*d^2
)*x)*sqrt(b*c^2 - a*c*d)*sqrt(b*x^2 + a) + 3*(8*a*b^2*c^5 - 12*a^2*b*c^4*d + 5*a
^3*c^3*d^2 + (8*a*b^2*c^2*d^3 - 12*a^2*b*c*d^4 + 5*a^3*d^5)*x^6 + 3*(8*a*b^2*c^3
*d^2 - 12*a^2*b*c^2*d^3 + 5*a^3*c*d^4)*x^4 + 3*(8*a*b^2*c^4*d - 12*a^2*b*c^3*d^2
+ 5*a^3*c^2*d^3)*x^2)*log((((8*b^2*c^2 - 8*a*b*c*d + a^2*d^2)*x^4 + a^2*c^2 + 2
*(4*a*b*c^2 - 3*a^2*c*d)*x^2)*sqrt(b*c^2 - a*c*d) + 4*((2*b^2*c^3 - 3*a*b*c^2*d
+ a^2*c*d^2)*x^3 + (a*b*c^3 - a^2*c^2*d)*x)*sqrt(b*x^2 + a))/(d^2*x^4 + 2*c*d*x^
2 + c^2)))/((b^2*c^8 - 2*a*b*c^7*d + a^2*c^6*d^2 + (b^2*c^5*d^3 - 2*a*b*c^4*d^4
+ a^2*c^3*d^5)*x^6 + 3*(b^2*c^6*d^2 - 2*a*b*c^5*d^3 + a^2*c^4*d^4)*x^4 + 3*(b^2*
c^7*d - 2*a*b*c^6*d^2 + a^2*c^5*d^3)*x^2)*sqrt(b*c^2 - a*c*d)), 1/96*(2*((8*b^2*
c^2*d^2 - 26*a*b*c*d^3 + 15*a^2*d^4)*x^5 + 2*(12*b^2*c^3*d - 35*a*b*c^2*d^2 + 20
*a^2*c*d^3)*x^3 + 3*(8*b^2*c^4 - 20*a*b*c^3*d + 11*a^2*c^2*d^2)*x)*sqrt(-b*c^2 +
a*c*d)*sqrt(b*x^2 + a) + 3*(8*a*b^2*c^5 - 12*a^2*b*c^4*d + 5*a^3*c^3*d^2 + (8*a
*b^2*c^2*d^3 - 12*a^2*b*c*d^4 + 5*a^3*d^5)*x^6 + 3*(8*a*b^2*c^3*d^2 - 12*a^2*b*c
^2*d^3 + 5*a^3*c*d^4)*x^4 + 3*(8*a*b^2*c^4*d - 12*a^2*b*c^3*d^2 + 5*a^3*c^2*d^3)
*x^2)*arctan(1/2*sqrt(-b*c^2 + a*c*d)*((2*b*c - a*d)*x^2 + a*c)/((b*c^2 - a*c*d)
*sqrt(b*x^2 + a)*x)))/((b^2*c^8 - 2*a*b*c^7*d + a^2*c^6*d^2 + (b^2*c^5*d^3 - 2*a
*b*c^4*d^4 + a^2*c^3*d^5)*x^6 + 3*(b^2*c^6*d^2 - 2*a*b*c^5*d^3 + a^2*c^4*d^4)*x^
4 + 3*(b^2*c^7*d - 2*a*b*c^6*d^2 + a^2*c^5*d^3)*x^2)*sqrt(-b*c^2 + a*c*d))]

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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)**(1/2)/(d*x**2+c)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 32.1371, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x