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

Optimal. Leaf size=199 \[ \frac{a^2 (6 b c-5 a d) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{16 c^{7/2} (b c-a d)^{3/2}}+\frac{a x \sqrt{a+b x^2} (6 b c-5 a d)}{16 c^3 \left (c+d x^2\right ) (b c-a d)}+\frac{x \left (a+b x^2\right )^{3/2} (6 b c-5 a d)}{24 c^2 \left (c+d x^2\right )^2 (b c-a d)}-\frac{d x \left (a+b x^2\right )^{5/2}}{6 c \left (c+d x^2\right )^3 (b c-a d)} \]

[Out]

-(d*x*(a + b*x^2)^(5/2))/(6*c*(b*c - a*d)*(c + d*x^2)^3) + ((6*b*c - 5*a*d)*x*(a
+ b*x^2)^(3/2))/(24*c^2*(b*c - a*d)*(c + d*x^2)^2) + (a*(6*b*c - 5*a*d)*x*Sqrt[
a + b*x^2])/(16*c^3*(b*c - a*d)*(c + d*x^2)) + (a^2*(6*b*c - 5*a*d)*ArcTanh[(Sqr
t[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(16*c^(7/2)*(b*c - a*d)^(3/2))

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Rubi [A]  time = 0.306788, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{a^2 (6 b c-5 a d) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{16 c^{7/2} (b c-a d)^{3/2}}+\frac{a x \sqrt{a+b x^2} (6 b c-5 a d)}{16 c^3 \left (c+d x^2\right ) (b c-a d)}+\frac{x \left (a+b x^2\right )^{3/2} (6 b c-5 a d)}{24 c^2 \left (c+d x^2\right )^2 (b c-a d)}-\frac{d x \left (a+b x^2\right )^{5/2}}{6 c \left (c+d x^2\right )^3 (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

-(d*x*(a + b*x^2)^(5/2))/(6*c*(b*c - a*d)*(c + d*x^2)^3) + ((6*b*c - 5*a*d)*x*(a
+ b*x^2)^(3/2))/(24*c^2*(b*c - a*d)*(c + d*x^2)^2) + (a*(6*b*c - 5*a*d)*x*Sqrt[
a + b*x^2])/(16*c^3*(b*c - a*d)*(c + d*x^2)) + (a^2*(6*b*c - 5*a*d)*ArcTanh[(Sqr
t[b*c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(16*c^(7/2)*(b*c - a*d)^(3/2))

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Rubi in Sympy [A]  time = 43.4536, size = 175, normalized size = 0.88 \[ \frac{a^{2} \left (5 a d - 6 b c\right ) \operatorname{atan}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{c} \sqrt{a + b x^{2}}} \right )}}{16 c^{\frac{7}{2}} \left (a d - b c\right )^{\frac{3}{2}}} + \frac{a x \sqrt{a + b x^{2}} \left (5 a d - 6 b c\right )}{16 c^{3} \left (c + d x^{2}\right ) \left (a d - b c\right )} + \frac{d x \left (a + b x^{2}\right )^{\frac{5}{2}}}{6 c \left (c + d x^{2}\right )^{3} \left (a d - b c\right )} + \frac{x \left (a + b x^{2}\right )^{\frac{3}{2}} \left (5 a d - 6 b c\right )}{24 c^{2} \left (c + d x^{2}\right )^{2} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.371867, size = 178, normalized size = 0.89 \[ \frac{\frac{\sqrt{c} x \sqrt{a+b x^2} \left (a^2 (-d) \left (33 c^2+40 c d x^2+15 d^2 x^4\right )+2 a b c \left (15 c^2+11 c d x^2+4 d^2 x^4\right )+4 b^2 c^2 x^2 \left (3 c+d x^2\right )\right )}{\left (c+d x^2\right )^3 (b c-a d)}-\frac{3 a^2 (6 b c-5 a d) \tan ^{-1}\left (\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{(a d-b c)^{3/2}}}{48 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[c]*x*Sqrt[a + b*x^2]*(4*b^2*c^2*x^2*(3*c + d*x^2) + 2*a*b*c*(15*c^2 + 11*
c*d*x^2 + 4*d^2*x^4) - a^2*d*(33*c^2 + 40*c*d*x^2 + 15*d^2*x^4)))/((b*c - a*d)*(
c + d*x^2)^3) - (3*a^2*(6*b*c - 5*a*d)*ArcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sq
rt[a + b*x^2])])/(-(b*c) + a*d)^(3/2))/(48*c^(7/2))

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Maple [B]  time = 0.059, size = 13766, normalized size = 69.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(3/2)/(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. \[ \int \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}{{\left (d x^{2} + c\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^(3/2)/(d*x^2 + c)^4, x)

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Fricas [A]  time = 0.564068, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/192*(4*((4*b^2*c^2*d + 8*a*b*c*d^2 - 15*a^2*d^3)*x^5 + 2*(6*b^2*c^3 + 11*a*b*
c^2*d - 20*a^2*c*d^2)*x^3 + 3*(10*a*b*c^3 - 11*a^2*c^2*d)*x)*sqrt(b*c^2 - a*c*d)
*sqrt(b*x^2 + a) + 3*(6*a^2*b*c^4 - 5*a^3*c^3*d + (6*a^2*b*c*d^3 - 5*a^3*d^4)*x^
6 + 3*(6*a^2*b*c^2*d^2 - 5*a^3*c*d^3)*x^4 + 3*(6*a^2*b*c^3*d - 5*a^3*c^2*d^2)*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*c^7 -
a*c^6*d + (b*c^4*d^3 - a*c^3*d^4)*x^6 + 3*(b*c^5*d^2 - a*c^4*d^3)*x^4 + 3*(b*c^
6*d - a*c^5*d^2)*x^2)*sqrt(b*c^2 - a*c*d)), 1/96*(2*((4*b^2*c^2*d + 8*a*b*c*d^2
- 15*a^2*d^3)*x^5 + 2*(6*b^2*c^3 + 11*a*b*c^2*d - 20*a^2*c*d^2)*x^3 + 3*(10*a*b*
c^3 - 11*a^2*c^2*d)*x)*sqrt(-b*c^2 + a*c*d)*sqrt(b*x^2 + a) + 3*(6*a^2*b*c^4 - 5
*a^3*c^3*d + (6*a^2*b*c*d^3 - 5*a^3*d^4)*x^6 + 3*(6*a^2*b*c^2*d^2 - 5*a^3*c*d^3)
*x^4 + 3*(6*a^2*b*c^3*d - 5*a^3*c^2*d^2)*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*c^7 - a*c^6*d
+ (b*c^4*d^3 - a*c^3*d^4)*x^6 + 3*(b*c^5*d^2 - a*c^4*d^3)*x^4 + 3*(b*c^6*d - a*c
^5*d^2)*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)**(3/2)/(d*x**2+c)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 28.9007, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x