3.749 \(\int \frac{x^4 \left (c+d x^2\right )^{5/2}}{\left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=258 \[ \frac{x \sqrt{c+d x^2} \left (32 a^2 d^2-52 a b c d+19 b^2 c^2\right )}{16 b^4}+\frac{\left (-64 a^3 d^3+120 a^2 b c d^2-60 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 b^5 \sqrt{d}}-\frac{\sqrt{a} (3 b c-8 a d) (b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 b^5}+\frac{d x^3 \sqrt{c+d x^2} (7 b c-8 a d)}{8 b^3}-\frac{x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac{2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2} \]

[Out]

((19*b^2*c^2 - 52*a*b*c*d + 32*a^2*d^2)*x*Sqrt[c + d*x^2])/(16*b^4) + (d*(7*b*c
- 8*a*d)*x^3*Sqrt[c + d*x^2])/(8*b^3) + (2*d*x^3*(c + d*x^2)^(3/2))/(3*b^2) - (x
^3*(c + d*x^2)^(5/2))/(2*b*(a + b*x^2)) - (Sqrt[a]*(3*b*c - 8*a*d)*(b*c - a*d)^(
3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*b^5) + ((5*b^3*c^
3 - 60*a*b^2*c^2*d + 120*a^2*b*c*d^2 - 64*a^3*d^3)*ArcTanh[(Sqrt[d]*x)/Sqrt[c +
d*x^2]])/(16*b^5*Sqrt[d])

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Rubi [A]  time = 1.18009, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x \sqrt{c+d x^2} \left (32 a^2 d^2-52 a b c d+19 b^2 c^2\right )}{16 b^4}+\frac{\left (-64 a^3 d^3+120 a^2 b c d^2-60 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 b^5 \sqrt{d}}-\frac{\sqrt{a} (3 b c-8 a d) (b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 b^5}+\frac{d x^3 \sqrt{c+d x^2} (7 b c-8 a d)}{8 b^3}-\frac{x^3 \left (c+d x^2\right )^{5/2}}{2 b \left (a+b x^2\right )}+\frac{2 d x^3 \left (c+d x^2\right )^{3/2}}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(c + d*x^2)^(5/2))/(a + b*x^2)^2,x]

[Out]

((19*b^2*c^2 - 52*a*b*c*d + 32*a^2*d^2)*x*Sqrt[c + d*x^2])/(16*b^4) + (d*(7*b*c
- 8*a*d)*x^3*Sqrt[c + d*x^2])/(8*b^3) + (2*d*x^3*(c + d*x^2)^(3/2))/(3*b^2) - (x
^3*(c + d*x^2)^(5/2))/(2*b*(a + b*x^2)) - (Sqrt[a]*(3*b*c - 8*a*d)*(b*c - a*d)^(
3/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*b^5) + ((5*b^3*c^
3 - 60*a*b^2*c^2*d + 120*a^2*b*c*d^2 - 64*a^3*d^3)*ArcTanh[(Sqrt[d]*x)/Sqrt[c +
d*x^2]])/(16*b^5*Sqrt[d])

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Rubi in Sympy [A]  time = 148.015, size = 246, normalized size = 0.95 \[ \frac{\sqrt{a} \left (a d - b c\right )^{\frac{3}{2}} \left (8 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{2 b^{5}} - \frac{x^{3} \left (c + d x^{2}\right )^{\frac{5}{2}}}{2 b \left (a + b x^{2}\right )} + \frac{2 d x^{3} \left (c + d x^{2}\right )^{\frac{3}{2}}}{3 b^{2}} - \frac{d x^{3} \sqrt{c + d x^{2}} \left (8 a d - 7 b c\right )}{8 b^{3}} + \frac{x \sqrt{c + d x^{2}} \left (32 a^{2} d^{2} - 52 a b c d + 19 b^{2} c^{2}\right )}{16 b^{4}} - \frac{\left (64 a^{3} d^{3} - 120 a^{2} b c d^{2} + 60 a b^{2} c^{2} d - 5 b^{3} c^{3}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{16 b^{5} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(d*x**2+c)**(5/2)/(b*x**2+a)**2,x)

[Out]

sqrt(a)*(a*d - b*c)**(3/2)*(8*a*d - 3*b*c)*atanh(x*sqrt(a*d - b*c)/(sqrt(a)*sqrt
(c + d*x**2)))/(2*b**5) - x**3*(c + d*x**2)**(5/2)/(2*b*(a + b*x**2)) + 2*d*x**3
*(c + d*x**2)**(3/2)/(3*b**2) - d*x**3*sqrt(c + d*x**2)*(8*a*d - 7*b*c)/(8*b**3)
+ x*sqrt(c + d*x**2)*(32*a**2*d**2 - 52*a*b*c*d + 19*b**2*c**2)/(16*b**4) - (64
*a**3*d**3 - 120*a**2*b*c*d**2 + 60*a*b**2*c**2*d - 5*b**3*c**3)*atanh(sqrt(d)*x
/sqrt(c + d*x**2))/(16*b**5*sqrt(d))

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Mathematica [A]  time = 0.576895, size = 219, normalized size = 0.85 \[ \frac{b x \sqrt{c+d x^2} \left (72 a^2 d^2+2 b d x^2 (13 b c-12 a d)+\frac{24 a (b c-a d)^2}{a+b x^2}-108 a b c d+33 b^2 c^2+8 b^2 d^2 x^4\right )+\frac{3 \left (-64 a^3 d^3+120 a^2 b c d^2-60 a b^2 c^2 d+5 b^3 c^3\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{\sqrt{d}}+24 \sqrt{a} (8 a d-3 b c) (b c-a d)^{3/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{48 b^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(c + d*x^2)^(5/2))/(a + b*x^2)^2,x]

[Out]

(b*x*Sqrt[c + d*x^2]*(33*b^2*c^2 - 108*a*b*c*d + 72*a^2*d^2 + 2*b*d*(13*b*c - 12
*a*d)*x^2 + 8*b^2*d^2*x^4 + (24*a*(b*c - a*d)^2)/(a + b*x^2)) + 24*Sqrt[a]*(b*c
- a*d)^(3/2)*(-3*b*c + 8*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2
])] + (3*(5*b^3*c^3 - 60*a*b^2*c^2*d + 120*a^2*b*c*d^2 - 64*a^3*d^3)*Log[d*x + S
qrt[d]*Sqrt[c + d*x^2]])/Sqrt[d])/(48*b^5)

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Maple [B]  time = 0.041, size = 7611, normalized size = 29.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(d*x^2+c)^(5/2)/(b*x^2+a)^2,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )}^{\frac{5}{2}} x^{4}}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^(5/2)*x^4/(b*x^2 + a)^2,x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)^(5/2)*x^4/(b*x^2 + a)^2, x)

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Fricas [A]  time = 3.03263, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^(5/2)*x^4/(b*x^2 + a)^2,x, algorithm="fricas")

[Out]

[1/96*(12*(3*a*b^2*c^2 - 11*a^2*b*c*d + 8*a^3*d^2 + (3*b^3*c^2 - 11*a*b^2*c*d +
8*a^2*b*d^2)*x^2)*sqrt(-a*b*c + a^2*d)*sqrt(d)*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 - 4*((b*c - 2*a*d)*x^3 - a*c
*x)*sqrt(-a*b*c + a^2*d)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 2*(8*b^
4*d^2*x^7 + 2*(13*b^4*c*d - 8*a*b^3*d^2)*x^5 + (33*b^4*c^2 - 82*a*b^3*c*d + 48*a
^2*b^2*d^2)*x^3 + 3*(19*a*b^3*c^2 - 52*a^2*b^2*c*d + 32*a^3*b*d^2)*x)*sqrt(d*x^2
+ c)*sqrt(d) - 3*(5*a*b^3*c^3 - 60*a^2*b^2*c^2*d + 120*a^3*b*c*d^2 - 64*a^4*d^3
+ (5*b^4*c^3 - 60*a*b^3*c^2*d + 120*a^2*b^2*c*d^2 - 64*a^3*b*d^3)*x^2)*log(2*sq
rt(d*x^2 + c)*d*x - (2*d*x^2 + c)*sqrt(d)))/((b^6*x^2 + a*b^5)*sqrt(d)), 1/48*(6
*(3*a*b^2*c^2 - 11*a^2*b*c*d + 8*a^3*d^2 + (3*b^3*c^2 - 11*a*b^2*c*d + 8*a^2*b*d
^2)*x^2)*sqrt(-a*b*c + a^2*d)*sqrt(-d)*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 - 4*((b*c - 2*a*d)*x^3 - a*c*x)*sqrt
(-a*b*c + a^2*d)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + (8*b^4*d^2*x^7
+ 2*(13*b^4*c*d - 8*a*b^3*d^2)*x^5 + (33*b^4*c^2 - 82*a*b^3*c*d + 48*a^2*b^2*d^2
)*x^3 + 3*(19*a*b^3*c^2 - 52*a^2*b^2*c*d + 32*a^3*b*d^2)*x)*sqrt(d*x^2 + c)*sqrt
(-d) + 3*(5*a*b^3*c^3 - 60*a^2*b^2*c^2*d + 120*a^3*b*c*d^2 - 64*a^4*d^3 + (5*b^4
*c^3 - 60*a*b^3*c^2*d + 120*a^2*b^2*c*d^2 - 64*a^3*b*d^3)*x^2)*arctan(sqrt(-d)*x
/sqrt(d*x^2 + c)))/((b^6*x^2 + a*b^5)*sqrt(-d)), 1/96*(24*(3*a*b^2*c^2 - 11*a^2*
b*c*d + 8*a^3*d^2 + (3*b^3*c^2 - 11*a*b^2*c*d + 8*a^2*b*d^2)*x^2)*sqrt(a*b*c - a
^2*d)*sqrt(d)*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)) + 2*(8*b^4*d^2*x^7 + 2*(13*b^4*c*d - 8*a*b^3*d^2)*x^5 + (33*b^4*c^2
- 82*a*b^3*c*d + 48*a^2*b^2*d^2)*x^3 + 3*(19*a*b^3*c^2 - 52*a^2*b^2*c*d + 32*a^
3*b*d^2)*x)*sqrt(d*x^2 + c)*sqrt(d) - 3*(5*a*b^3*c^3 - 60*a^2*b^2*c^2*d + 120*a^
3*b*c*d^2 - 64*a^4*d^3 + (5*b^4*c^3 - 60*a*b^3*c^2*d + 120*a^2*b^2*c*d^2 - 64*a^
3*b*d^3)*x^2)*log(2*sqrt(d*x^2 + c)*d*x - (2*d*x^2 + c)*sqrt(d)))/((b^6*x^2 + a*
b^5)*sqrt(d)), 1/48*(12*(3*a*b^2*c^2 - 11*a^2*b*c*d + 8*a^3*d^2 + (3*b^3*c^2 - 1
1*a*b^2*c*d + 8*a^2*b*d^2)*x^2)*sqrt(a*b*c - a^2*d)*sqrt(-d)*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)) + (8*b^4*d^2*x^7 + 2
*(13*b^4*c*d - 8*a*b^3*d^2)*x^5 + (33*b^4*c^2 - 82*a*b^3*c*d + 48*a^2*b^2*d^2)*x
^3 + 3*(19*a*b^3*c^2 - 52*a^2*b^2*c*d + 32*a^3*b*d^2)*x)*sqrt(d*x^2 + c)*sqrt(-d
) + 3*(5*a*b^3*c^3 - 60*a^2*b^2*c^2*d + 120*a^3*b*c*d^2 - 64*a^4*d^3 + (5*b^4*c^
3 - 60*a*b^3*c^2*d + 120*a^2*b^2*c*d^2 - 64*a^3*b*d^3)*x^2)*arctan(sqrt(-d)*x/sq
rt(d*x^2 + c)))/((b^6*x^2 + a*b^5)*sqrt(-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(x**4*(d*x**2+c)**(5/2)/(b*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.581435, 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)^(5/2)*x^4/(b*x^2 + a)^2,x, algorithm="giac")

[Out]

sage0*x