3.376 \(\int \frac{x^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=289 \[ \frac{(3 A b-7 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} b^{11/4}}-\frac{(3 A b-7 a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} b^{11/4}}-\frac{(3 A b-7 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} \sqrt [4]{a} b^{11/4}}+\frac{(3 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} \sqrt [4]{a} b^{11/4}}-\frac{x^{3/2} (3 A b-7 a B)}{6 a b^2}+\frac{x^{7/2} (A b-a B)}{2 a b \left (a+b x^2\right )} \]

[Out]

-((3*A*b - 7*a*B)*x^(3/2))/(6*a*b^2) + ((A*b - a*B)*x^(7/2))/(2*a*b*(a + b*x^2))
- ((3*A*b - 7*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^
(1/4)*b^(11/4)) + ((3*A*b - 7*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]
)/(4*Sqrt[2]*a^(1/4)*b^(11/4)) + ((3*A*b - 7*a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*
b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(1/4)*b^(11/4)) - ((3*A*b - 7*a*B)*Lo
g[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(1/4)*b^(
11/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.469584, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ \frac{(3 A b-7 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} b^{11/4}}-\frac{(3 A b-7 a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} b^{11/4}}-\frac{(3 A b-7 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} \sqrt [4]{a} b^{11/4}}+\frac{(3 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} \sqrt [4]{a} b^{11/4}}-\frac{x^{3/2} (3 A b-7 a B)}{6 a b^2}+\frac{x^{7/2} (A b-a B)}{2 a b \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((3*A*b - 7*a*B)*x^(3/2))/(6*a*b^2) + ((A*b - a*B)*x^(7/2))/(2*a*b*(a + b*x^2))
- ((3*A*b - 7*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^
(1/4)*b^(11/4)) + ((3*A*b - 7*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]
)/(4*Sqrt[2]*a^(1/4)*b^(11/4)) + ((3*A*b - 7*a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*
b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(1/4)*b^(11/4)) - ((3*A*b - 7*a*B)*Lo
g[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(1/4)*b^(
11/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 80.0589, size = 269, normalized size = 0.93 \[ \frac{x^{\frac{7}{2}} \left (A b - B a\right )}{2 a b \left (a + b x^{2}\right )} - \frac{x^{\frac{3}{2}} \left (3 A b - 7 B a\right )}{6 a b^{2}} + \frac{\sqrt{2} \left (3 A b - 7 B a\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 \sqrt [4]{a} b^{\frac{11}{4}}} - \frac{\sqrt{2} \left (3 A b - 7 B a\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 \sqrt [4]{a} b^{\frac{11}{4}}} - \frac{\sqrt{2} \left (3 A b - 7 B a\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 \sqrt [4]{a} b^{\frac{11}{4}}} + \frac{\sqrt{2} \left (3 A b - 7 B a\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 \sqrt [4]{a} b^{\frac{11}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(B*x**2+A)/(b*x**2+a)**2,x)

[Out]

x**(7/2)*(A*b - B*a)/(2*a*b*(a + b*x**2)) - x**(3/2)*(3*A*b - 7*B*a)/(6*a*b**2)
+ sqrt(2)*(3*A*b - 7*B*a)*log(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqr
t(b)*x)/(16*a**(1/4)*b**(11/4)) - sqrt(2)*(3*A*b - 7*B*a)*log(sqrt(2)*a**(1/4)*b
**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(16*a**(1/4)*b**(11/4)) - sqrt(2)*(3*A*b
- 7*B*a)*atan(1 - sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(8*a**(1/4)*b**(11/4)) + sq
rt(2)*(3*A*b - 7*B*a)*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(8*a**(1/4)*b*
*(11/4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.449962, size = 256, normalized size = 0.89 \[ \frac{-\frac{24 b^{3/4} x^{3/2} (A b-a B)}{a+b x^2}+\frac{3 \sqrt{2} (3 A b-7 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{a}}+\frac{3 \sqrt{2} (7 a B-3 A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{a}}+\frac{6 \sqrt{2} (7 a B-3 A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a}}+\frac{6 \sqrt{2} (3 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{a}}+32 b^{3/4} B x^{3/2}}{48 b^{11/4}} \]

Antiderivative was successfully verified.

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

[Out]

(32*b^(3/4)*B*x^(3/2) - (24*b^(3/4)*(A*b - a*B)*x^(3/2))/(a + b*x^2) + (6*Sqrt[2
]*(-3*A*b + 7*a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(1/4) + (6*S
qrt[2]*(3*A*b - 7*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(1/4) +
(3*Sqrt[2]*(3*A*b - 7*a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[
b]*x])/a^(1/4) + (3*Sqrt[2]*(-3*A*b + 7*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/
4)*Sqrt[x] + Sqrt[b]*x])/a^(1/4))/(48*b^(11/4))

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 317, normalized size = 1.1 \[{\frac{2\,B}{3\,{b}^{2}}{x}^{{\frac{3}{2}}}}-{\frac{A}{2\,b \left ( b{x}^{2}+a \right ) }{x}^{{\frac{3}{2}}}}+{\frac{Ba}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }{x}^{{\frac{3}{2}}}}-{\frac{7\,\sqrt{2}Ba}{16\,{b}^{3}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{7\,\sqrt{2}Ba}{8\,{b}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{7\,\sqrt{2}Ba}{8\,{b}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{3\,\sqrt{2}A}{16\,{b}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{3\,\sqrt{2}A}{8\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{3\,\sqrt{2}A}{8\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*B*x^(3/2)/b^2-1/2/b*x^(3/2)/(b*x^2+a)*A+1/2/b^2*x^(3/2)/(b*x^2+a)*B*a-7/16/b
^3/(a/b)^(1/4)*2^(1/2)*B*a*ln((x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x+(a/
b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))-7/8/b^3/(a/b)^(1/4)*2^(1/2)*B*a*arctan(2^
(1/2)/(a/b)^(1/4)*x^(1/2)+1)-7/8/b^3/(a/b)^(1/4)*2^(1/2)*B*a*arctan(2^(1/2)/(a/b
)^(1/4)*x^(1/2)-1)+3/16/b^2/(a/b)^(1/4)*2^(1/2)*A*ln((x-(a/b)^(1/4)*x^(1/2)*2^(1
/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))+3/8/b^2/(a/b)^(1/4
)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+3/8/b^2/(a/b)^(1/4)*2^(1/2)*A*
arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.254998, size = 1095, normalized size = 3.79 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(12*(b^3*x^2 + a*b^2)*(-(2401*B^4*a^4 - 4116*A*B^3*a^3*b + 2646*A^2*B^2*a^2
*b^2 - 756*A^3*B*a*b^3 + 81*A^4*b^4)/(a*b^11))^(1/4)*arctan(-a*b^8*(-(2401*B^4*a
^4 - 4116*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 756*A^3*B*a*b^3 + 81*A^4*b^4)/(a*
b^11))^(3/4)/((343*B^3*a^3 - 441*A*B^2*a^2*b + 189*A^2*B*a*b^2 - 27*A^3*b^3)*sqr
t(x) - sqrt((117649*B^6*a^6 - 302526*A*B^5*a^5*b + 324135*A^2*B^4*a^4*b^2 - 1852
20*A^3*B^3*a^3*b^3 + 59535*A^4*B^2*a^2*b^4 - 10206*A^5*B*a*b^5 + 729*A^6*b^6)*x
- (2401*B^4*a^5*b^5 - 4116*A*B^3*a^4*b^6 + 2646*A^2*B^2*a^3*b^7 - 756*A^3*B*a^2*
b^8 + 81*A^4*a*b^9)*sqrt(-(2401*B^4*a^4 - 4116*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^
2 - 756*A^3*B*a*b^3 + 81*A^4*b^4)/(a*b^11))))) + 3*(b^3*x^2 + a*b^2)*(-(2401*B^4
*a^4 - 4116*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 756*A^3*B*a*b^3 + 81*A^4*b^4)/(
a*b^11))^(1/4)*log(a*b^8*(-(2401*B^4*a^4 - 4116*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b
^2 - 756*A^3*B*a*b^3 + 81*A^4*b^4)/(a*b^11))^(3/4) - (343*B^3*a^3 - 441*A*B^2*a^
2*b + 189*A^2*B*a*b^2 - 27*A^3*b^3)*sqrt(x)) - 3*(b^3*x^2 + a*b^2)*(-(2401*B^4*a
^4 - 4116*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^2 - 756*A^3*B*a*b^3 + 81*A^4*b^4)/(a*
b^11))^(1/4)*log(-a*b^8*(-(2401*B^4*a^4 - 4116*A*B^3*a^3*b + 2646*A^2*B^2*a^2*b^
2 - 756*A^3*B*a*b^3 + 81*A^4*b^4)/(a*b^11))^(3/4) - (343*B^3*a^3 - 441*A*B^2*a^2
*b + 189*A^2*B*a*b^2 - 27*A^3*b^3)*sqrt(x)) + 4*(4*B*b*x^3 + (7*B*a - 3*A*b)*x)*
sqrt(x))/(b^3*x^2 + a*b^2)

_______________________________________________________________________________________

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**(5/2)*(B*x**2+A)/(b*x**2+a)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.254201, size = 382, normalized size = 1.32 \[ \frac{2 \, B x^{\frac{3}{2}}}{3 \, b^{2}} + \frac{B a x^{\frac{3}{2}} - A b x^{\frac{3}{2}}}{2 \,{\left (b x^{2} + a\right )} b^{2}} - \frac{\sqrt{2}{\left (7 \, \left (a b^{3}\right )^{\frac{3}{4}} B a - 3 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a b^{5}} - \frac{\sqrt{2}{\left (7 \, \left (a b^{3}\right )^{\frac{3}{4}} B a - 3 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a b^{5}} + \frac{\sqrt{2}{\left (7 \, \left (a b^{3}\right )^{\frac{3}{4}} B a - 3 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a b^{5}} - \frac{\sqrt{2}{\left (7 \, \left (a b^{3}\right )^{\frac{3}{4}} B a - 3 \, \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*B*x^(3/2)/b^2 + 1/2*(B*a*x^(3/2) - A*b*x^(3/2))/((b*x^2 + a)*b^2) - 1/8*sqrt
(2)*(7*(a*b^3)^(3/4)*B*a - 3*(a*b^3)^(3/4)*A*b)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b
)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a*b^5) - 1/8*sqrt(2)*(7*(a*b^3)^(3/4)*B*a - 3
*(a*b^3)^(3/4)*A*b)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^
(1/4))/(a*b^5) + 1/16*sqrt(2)*(7*(a*b^3)^(3/4)*B*a - 3*(a*b^3)^(3/4)*A*b)*ln(sqr
t(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a*b^5) - 1/16*sqrt(2)*(7*(a*b^3)^(3/4
)*B*a - 3*(a*b^3)^(3/4)*A*b)*ln(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a
*b^5)