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

Optimal. Leaf size=217 \[ \frac{x \sqrt{c+d x^2} \left (8 a^2 d^2-18 a b c d+11 b^2 c^2\right )}{16 b^3}+\frac{\left (-16 a^3 d^3+40 a^2 b c d^2-30 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 b^4 \sqrt{d}}-\frac{\sqrt{a} (b c-a d)^{5/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{b^4}+\frac{d x^3 \sqrt{c+d x^2} (3 b c-2 a d)}{8 b^2}+\frac{d x^3 \left (c+d x^2\right )^{3/2}}{6 b} \]

[Out]

((11*b^2*c^2 - 18*a*b*c*d + 8*a^2*d^2)*x*Sqrt[c + d*x^2])/(16*b^3) + (d*(3*b*c -
2*a*d)*x^3*Sqrt[c + d*x^2])/(8*b^2) + (d*x^3*(c + d*x^2)^(3/2))/(6*b) - (Sqrt[a
]*(b*c - a*d)^(5/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/b^4 +
((5*b^3*c^3 - 30*a*b^2*c^2*d + 40*a^2*b*c*d^2 - 16*a^3*d^3)*ArcTanh[(Sqrt[d]*x)
/Sqrt[c + d*x^2]])/(16*b^4*Sqrt[d])

_______________________________________________________________________________________

Rubi [A]  time = 1.00248, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x \sqrt{c+d x^2} \left (8 a^2 d^2-18 a b c d+11 b^2 c^2\right )}{16 b^3}+\frac{\left (-16 a^3 d^3+40 a^2 b c d^2-30 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{16 b^4 \sqrt{d}}-\frac{\sqrt{a} (b c-a d)^{5/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{b^4}+\frac{d x^3 \sqrt{c+d x^2} (3 b c-2 a d)}{8 b^2}+\frac{d x^3 \left (c+d x^2\right )^{3/2}}{6 b} \]

Antiderivative was successfully verified.

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

[Out]

((11*b^2*c^2 - 18*a*b*c*d + 8*a^2*d^2)*x*Sqrt[c + d*x^2])/(16*b^3) + (d*(3*b*c -
2*a*d)*x^3*Sqrt[c + d*x^2])/(8*b^2) + (d*x^3*(c + d*x^2)^(3/2))/(6*b) - (Sqrt[a
]*(b*c - a*d)^(5/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/b^4 +
((5*b^3*c^3 - 30*a*b^2*c^2*d + 40*a^2*b*c*d^2 - 16*a^3*d^3)*ArcTanh[(Sqrt[d]*x)
/Sqrt[c + d*x^2]])/(16*b^4*Sqrt[d])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 126.215, size = 207, normalized size = 0.95 \[ \frac{\sqrt{a} \left (a d - b c\right )^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{a} \sqrt{c + d x^{2}}} \right )}}{b^{4}} + \frac{d x^{3} \left (c + d x^{2}\right )^{\frac{3}{2}}}{6 b} - \frac{d x^{3} \sqrt{c + d x^{2}} \left (2 a d - 3 b c\right )}{8 b^{2}} + \frac{x \sqrt{c + d x^{2}} \left (8 a^{2} d^{2} - 18 a b c d + 11 b^{2} c^{2}\right )}{16 b^{3}} - \frac{\left (16 a^{3} d^{3} - 40 a^{2} b c d^{2} + 30 a b^{2} c^{2} d - 5 b^{3} c^{3}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{16 b^{4} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a)*(a*d - b*c)**(5/2)*atanh(x*sqrt(a*d - b*c)/(sqrt(a)*sqrt(c + d*x**2)))/b
**4 + d*x**3*(c + d*x**2)**(3/2)/(6*b) - d*x**3*sqrt(c + d*x**2)*(2*a*d - 3*b*c)
/(8*b**2) + x*sqrt(c + d*x**2)*(8*a**2*d**2 - 18*a*b*c*d + 11*b**2*c**2)/(16*b**
3) - (16*a**3*d**3 - 40*a**2*b*c*d**2 + 30*a*b**2*c**2*d - 5*b**3*c**3)*atanh(sq
rt(d)*x/sqrt(c + d*x**2))/(16*b**4*sqrt(d))

_______________________________________________________________________________________

Mathematica [A]  time = 0.244738, size = 187, normalized size = 0.86 \[ \frac{b x \sqrt{c+d x^2} \left (24 a^2 d^2-6 a b d \left (9 c+2 d x^2\right )+b^2 \left (33 c^2+26 c d x^2+8 d^2 x^4\right )\right )+\frac{3 \left (-16 a^3 d^3+40 a^2 b c d^2-30 a b^2 c^2 d+5 b^3 c^3\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{\sqrt{d}}-48 \sqrt{a} (b c-a d)^{5/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{48 b^4} \]

Antiderivative was successfully verified.

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

[Out]

(b*x*Sqrt[c + d*x^2]*(24*a^2*d^2 - 6*a*b*d*(9*c + 2*d*x^2) + b^2*(33*c^2 + 26*c*
d*x^2 + 8*d^2*x^4)) - 48*Sqrt[a]*(b*c - a*d)^(5/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(S
qrt[a]*Sqrt[c + d*x^2])] + (3*(5*b^3*c^3 - 30*a*b^2*c^2*d + 40*a^2*b*c*d^2 - 16*
a^3*d^3)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/Sqrt[d])/(48*b^4)

_______________________________________________________________________________________

Maple [B]  time = 0.021, size = 3235, normalized size = 14.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*a/(-a*b)^(1/2)/b*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)
^(1/2))-(a*d-b*c)/b)^(1/2)*c^2-15/16*a/b^2*d^(1/2)*ln((d*(-a*b)^(1/2)/b+(x-1/b*(
-a*b)^(1/2))*d)/d^(1/2)+((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*
b)^(1/2))-(a*d-b*c)/b)^(1/2))*c^2+1/6*a^2/(-a*b)^(1/2)/b^2*((x-1/b*(-a*b)^(1/2))
^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*d-1/6*a/(-a*b)^(
1/2)/b*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*
c)/b)^(3/2)*c+1/2*a^3/(-a*b)^(1/2)/b^3*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2
)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*d^2+1/2*a/(-a*b)^(1/2)/b*((x+1/b*(-a
*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*c^2-1/
8*a/b^2*d*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d
-b*c)/b)^(3/2)*x-15/16*a/b^2*d^(1/2)*ln((-d*(-a*b)^(1/2)/b+(x+1/b*(-a*b)^(1/2))*
d)/d^(1/2)+((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*
d-b*c)/b)^(1/2))*c^2-1/6*a^2/(-a*b)^(1/2)/b^2*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*
b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*d+1/6*a/(-a*b)^(1/2)/b*((x+1/
b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*c
+1/4*a^2/b^3*d^2*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2
))-(a*d-b*c)/b)^(1/2)*x+5/4*a^2/b^3*d^(3/2)*ln((-d*(-a*b)^(1/2)/b+(x+1/b*(-a*b)^
(1/2))*d)/d^(1/2)+((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/
2))-(a*d-b*c)/b)^(1/2))*c-1/8*a/b^2*d*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)
/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*x+1/6/b*x*(d*x^2+c)^(5/2)-a^2/(-a*b)^
(1/2)/b^2*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d
-b*c)/b)^(1/2)*d*c+1/2*a^4/(-a*b)^(1/2)/b^4/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c
)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x+1/b*(-a*b
)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+1/b*
(-a*b)^(1/2)))*d^3-1/2*a/(-a*b)^(1/2)/b/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-
2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x+1/b*(-a*b)^(1
/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*
b)^(1/2)))*c^3-1/2*a^3/(-a*b)^(1/2)/b^3*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/
2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*d^2-1/10*a/(-a*b)^(1/2)/b*((x-1/b*(
-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(5/2)-1/2*
a^3/b^4*d^(5/2)*ln((d*(-a*b)^(1/2)/b+(x-1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x-1/b*(-a
*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))+5/24/
b*c*x*(d*x^2+c)^(3/2)+5/16/b*c^2*x*(d*x^2+c)^(1/2)+5/16/b*c^3/d^(1/2)*ln(x*d^(1/
2)+(d*x^2+c)^(1/2))+1/10*a/(-a*b)^(1/2)/b*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(
1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(5/2)-1/2*a^3/b^4*d^(5/2)*ln((-d*(-a*b)
^(1/2)/b+(x+1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1
/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))+1/4*a^2/b^3*d^2*((x-1/b*(-a*b)^(1
/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x+5/4*a^2/b^
3*d^(3/2)*ln((d*(-a*b)^(1/2)/b+(x-1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x-1/b*(-a*b)^(1
/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))*c-7/16*a/b^
2*d*c*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c
)/b)^(1/2)*x-7/16*a/b^2*d*c*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*
(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x+a^2/(-a*b)^(1/2)/b^2*((x-1/b*(-a*b)^(1/2))^2*
d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*d*c-1/2*a^4/(-a*b)^
(1/2)/b^4/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*
b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x
-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))*d^3+1/2*a/(-a*b)^(1
/2)/b/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(
1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b
*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))*c^3-3/2*a^3/(-a*b)^(1/2
)/b^3/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(
1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b
*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))*d^2*c+3/2*a^2/(-a*b)^(1
/2)/b^2/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)
^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1
/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))*d*c^2+3/2*a^3/(-a*b)^
(1/2)/b^3/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*
b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x
-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))*d^2*c-3/2*a^2/(-a*b
)^(1/2)/b^2/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-
a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*
(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))*d*c^2

_______________________________________________________________________________________

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((d*x^2 + c)^(5/2)*x^2/(b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 3.01344, 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^2/(b*x^2 + a),x, algorithm="fricas")

[Out]

[1/96*(24*(b^2*c^2 - 2*a*b*c*d + a^2*d^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^3*d^2*x^5 + 2*(13*b^3*c*d - 6*a*b^2*d^2)*x^3 + 3*(11*b^3
*c^2 - 18*a*b^2*c*d + 8*a^2*b*d^2)*x)*sqrt(d*x^2 + c)*sqrt(d) - 3*(5*b^3*c^3 - 3
0*a*b^2*c^2*d + 40*a^2*b*c*d^2 - 16*a^3*d^3)*log(2*sqrt(d*x^2 + c)*d*x - (2*d*x^
2 + c)*sqrt(d)))/(b^4*sqrt(d)), 1/48*(12*(b^2*c^2 - 2*a*b*c*d + a^2*d^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^3*d^2*x^5 + 2*(13*b^3*c*d
- 6*a*b^2*d^2)*x^3 + 3*(11*b^3*c^2 - 18*a*b^2*c*d + 8*a^2*b*d^2)*x)*sqrt(d*x^2 +
c)*sqrt(-d) + 3*(5*b^3*c^3 - 30*a*b^2*c^2*d + 40*a^2*b*c*d^2 - 16*a^3*d^3)*arct
an(sqrt(-d)*x/sqrt(d*x^2 + c)))/(b^4*sqrt(-d)), 1/96*(48*(b^2*c^2 - 2*a*b*c*d +
a^2*d^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^3*d^2*x^5 + 2*(13*b^3*c*d - 6*a*b^2
*d^2)*x^3 + 3*(11*b^3*c^2 - 18*a*b^2*c*d + 8*a^2*b*d^2)*x)*sqrt(d*x^2 + c)*sqrt(
d) - 3*(5*b^3*c^3 - 30*a*b^2*c^2*d + 40*a^2*b*c*d^2 - 16*a^3*d^3)*log(2*sqrt(d*x
^2 + c)*d*x - (2*d*x^2 + c)*sqrt(d)))/(b^4*sqrt(d)), 1/48*(24*(b^2*c^2 - 2*a*b*c
*d + a^2*d^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^3*d^2*x^5 + 2*(13*b^3*c*d - 6*a
*b^2*d^2)*x^3 + 3*(11*b^3*c^2 - 18*a*b^2*c*d + 8*a^2*b*d^2)*x)*sqrt(d*x^2 + c)*s
qrt(-d) + 3*(5*b^3*c^3 - 30*a*b^2*c^2*d + 40*a^2*b*c*d^2 - 16*a^3*d^3)*arctan(sq
rt(-d)*x/sqrt(d*x^2 + c)))/(b^4*sqrt(-d))]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2*(c + d*x**2)**(5/2)/(a + b*x**2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.263947, size = 373, normalized size = 1.72 \[ \frac{1}{48} \,{\left (2 \,{\left (\frac{4 \, d^{2} x^{2}}{b} + \frac{13 \, b^{9} c d^{5} - 6 \, a b^{8} d^{6}}{b^{10} d^{4}}\right )} x^{2} + \frac{3 \,{\left (11 \, b^{9} c^{2} d^{4} - 18 \, a b^{8} c d^{5} + 8 \, a^{2} b^{7} d^{6}\right )}}{b^{10} d^{4}}\right )} \sqrt{d x^{2} + c} x + \frac{{\left (a b^{3} c^{3} \sqrt{d} - 3 \, a^{2} b^{2} c^{2} d^{\frac{3}{2}} + 3 \, a^{3} b c d^{\frac{5}{2}} - a^{4} d^{\frac{7}{2}}\right )} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{\sqrt{a b c d - a^{2} d^{2}} b^{4}} - \frac{{\left (5 \, b^{3} c^{3} - 30 \, a b^{2} c^{2} d + 40 \, a^{2} b c d^{2} - 16 \, a^{3} d^{3}\right )}{\rm ln}\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{32 \, b^{4} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(2*(4*d^2*x^2/b + (13*b^9*c*d^5 - 6*a*b^8*d^6)/(b^10*d^4))*x^2 + 3*(11*b^9*
c^2*d^4 - 18*a*b^8*c*d^5 + 8*a^2*b^7*d^6)/(b^10*d^4))*sqrt(d*x^2 + c)*x + (a*b^3
*c^3*sqrt(d) - 3*a^2*b^2*c^2*d^(3/2) + 3*a^3*b*c*d^(5/2) - a^4*d^(7/2))*arctan(1
/2*((sqrt(d)*x - sqrt(d*x^2 + c))^2*b - b*c + 2*a*d)/sqrt(a*b*c*d - a^2*d^2))/(s
qrt(a*b*c*d - a^2*d^2)*b^4) - 1/32*(5*b^3*c^3 - 30*a*b^2*c^2*d + 40*a^2*b*c*d^2
- 16*a^3*d^3)*ln((sqrt(d)*x - sqrt(d*x^2 + c))^2)/(b^4*sqrt(d))