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

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

[Out]

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

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Rubi [A]  time = 0.298351, 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 \[ \frac{c^2 (5 b c-6 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{16 a^{7/2} (b c-a d)^{3/2}}+\frac{c x \sqrt{c+d x^2} (5 b c-6 a d)}{16 a^3 \left (a+b x^2\right ) (b c-a d)}+\frac{x \left (c+d x^2\right )^{3/2} (5 b c-6 a d)}{24 a^2 \left (a+b x^2\right )^2 (b c-a d)}+\frac{b x \left (c+d x^2\right )^{5/2}}{6 a \left (a+b x^2\right )^3 (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 44.8537, size = 175, normalized size = 0.88 \[ - \frac{b x \left (c + d x^{2}\right )^{\frac{5}{2}}}{6 a \left (a + b x^{2}\right )^{3} \left (a d - b c\right )} + \frac{x \left (c + d x^{2}\right )^{\frac{3}{2}} \left (6 a d - 5 b c\right )}{24 a^{2} \left (a + b x^{2}\right )^{2} \left (a d - b c\right )} + \frac{c x \sqrt{c + d x^{2}} \left (6 a d - 5 b c\right )}{16 a^{3} \left (a + b x^{2}\right ) \left (a d - b c\right )} + \frac{c^{2} \left (6 a d - 5 b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{16 a^{\frac{7}{2}} \left (a d - b c\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.413399, size = 179, normalized size = 0.9 \[ \frac{\frac{3 c^2 (5 b c-6 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{(b c-a d)^{3/2}}-\frac{\sqrt{a} x \sqrt{c+d x^2} \left (-6 a^3 d \left (5 c+2 d x^2\right )+a^2 b \left (33 c^2-22 c d x^2-4 d^2 x^4\right )+8 a b^2 c x^2 \left (5 c-d x^2\right )+15 b^3 c^2 x^4\right )}{\left (a+b x^2\right )^3 (a d-b c)}}{48 a^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-((Sqrt[a]*x*Sqrt[c + d*x^2]*(15*b^3*c^2*x^4 + 8*a*b^2*c*x^2*(5*c - d*x^2) - 6*
a^3*d*(5*c + 2*d*x^2) + a^2*b*(33*c^2 - 22*c*d*x^2 - 4*d^2*x^4)))/((-(b*c) + a*d
)*(a + b*x^2)^3)) + (3*c^2*(5*b*c - 6*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*S
qrt[c + d*x^2])])/(b*c - a*d)^(3/2))/(48*a^(7/2))

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Maple [B]  time = 0.074, size = 13964, normalized size = 70.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )}^{\frac{3}{2}}}{{\left (b x^{2} + a\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.620245, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 19.5275, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x