3.389 \(\int \frac{A+B x^2}{x^{5/2} \left (a+b x^2\right )^3} \, dx\)

Optimal. Leaf size=322 \[ \frac{7 (11 A b-3 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{15/4} \sqrt [4]{b}}-\frac{7 (11 A b-3 a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{15/4} \sqrt [4]{b}}+\frac{7 (11 A b-3 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{15/4} \sqrt [4]{b}}-\frac{7 (11 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{15/4} \sqrt [4]{b}}-\frac{7 (11 A b-3 a B)}{48 a^3 b x^{3/2}}+\frac{11 A b-3 a B}{16 a^2 b x^{3/2} \left (a+b x^2\right )}+\frac{A b-a B}{4 a b x^{3/2} \left (a+b x^2\right )^2} \]

[Out]

(-7*(11*A*b - 3*a*B))/(48*a^3*b*x^(3/2)) + (A*b - a*B)/(4*a*b*x^(3/2)*(a + b*x^2
)^2) + (11*A*b - 3*a*B)/(16*a^2*b*x^(3/2)*(a + b*x^2)) + (7*(11*A*b - 3*a*B)*Arc
Tan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(15/4)*b^(1/4)) - (7*(
11*A*b - 3*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(15
/4)*b^(1/4)) + (7*(11*A*b - 3*a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x]
+ Sqrt[b]*x])/(64*Sqrt[2]*a^(15/4)*b^(1/4)) - (7*(11*A*b - 3*a*B)*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(15/4)*b^(1/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.537527, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ \frac{7 (11 A b-3 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{15/4} \sqrt [4]{b}}-\frac{7 (11 A b-3 a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{15/4} \sqrt [4]{b}}+\frac{7 (11 A b-3 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{15/4} \sqrt [4]{b}}-\frac{7 (11 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{15/4} \sqrt [4]{b}}-\frac{7 (11 A b-3 a B)}{48 a^3 b x^{3/2}}+\frac{11 A b-3 a B}{16 a^2 b x^{3/2} \left (a+b x^2\right )}+\frac{A b-a B}{4 a b x^{3/2} \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(x^(5/2)*(a + b*x^2)^3),x]

[Out]

(-7*(11*A*b - 3*a*B))/(48*a^3*b*x^(3/2)) + (A*b - a*B)/(4*a*b*x^(3/2)*(a + b*x^2
)^2) + (11*A*b - 3*a*B)/(16*a^2*b*x^(3/2)*(a + b*x^2)) + (7*(11*A*b - 3*a*B)*Arc
Tan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(15/4)*b^(1/4)) - (7*(
11*A*b - 3*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(15
/4)*b^(1/4)) + (7*(11*A*b - 3*a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x]
+ Sqrt[b]*x])/(64*Sqrt[2]*a^(15/4)*b^(1/4)) - (7*(11*A*b - 3*a*B)*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(15/4)*b^(1/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 86.9963, size = 306, normalized size = 0.95 \[ \frac{A b - B a}{4 a b x^{\frac{3}{2}} \left (a + b x^{2}\right )^{2}} + \frac{11 A b - 3 B a}{16 a^{2} b x^{\frac{3}{2}} \left (a + b x^{2}\right )} - \frac{7 \left (11 A b - 3 B a\right )}{48 a^{3} b x^{\frac{3}{2}}} + \frac{7 \sqrt{2} \left (11 A b - 3 B a\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{128 a^{\frac{15}{4}} \sqrt [4]{b}} - \frac{7 \sqrt{2} \left (11 A b - 3 B a\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{128 a^{\frac{15}{4}} \sqrt [4]{b}} + \frac{7 \sqrt{2} \left (11 A b - 3 B a\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{15}{4}} \sqrt [4]{b}} - \frac{7 \sqrt{2} \left (11 A b - 3 B a\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{15}{4}} \sqrt [4]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/x**(5/2)/(b*x**2+a)**3,x)

[Out]

(A*b - B*a)/(4*a*b*x**(3/2)*(a + b*x**2)**2) + (11*A*b - 3*B*a)/(16*a**2*b*x**(3
/2)*(a + b*x**2)) - 7*(11*A*b - 3*B*a)/(48*a**3*b*x**(3/2)) + 7*sqrt(2)*(11*A*b
- 3*B*a)*log(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(128*a**(
15/4)*b**(1/4)) - 7*sqrt(2)*(11*A*b - 3*B*a)*log(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(
x) + sqrt(a) + sqrt(b)*x)/(128*a**(15/4)*b**(1/4)) + 7*sqrt(2)*(11*A*b - 3*B*a)*
atan(1 - sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(64*a**(15/4)*b**(1/4)) - 7*sqrt(2)*
(11*A*b - 3*B*a)*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(64*a**(15/4)*b**(1
/4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.497686, size = 286, normalized size = 0.89 \[ \frac{\frac{96 a^{7/4} \sqrt{x} (a B-A b)}{\left (a+b x^2\right )^2}+\frac{24 a^{3/4} \sqrt{x} (7 a B-15 A b)}{a+b x^2}-\frac{256 a^{3/4} A}{x^{3/2}}+\frac{21 \sqrt{2} (11 A b-3 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{b}}+\frac{21 \sqrt{2} (3 a B-11 A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{b}}+\frac{42 \sqrt{2} (11 A b-3 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}-\frac{42 \sqrt{2} (11 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{b}}}{384 a^{15/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(x^(5/2)*(a + b*x^2)^3),x]

[Out]

((-256*a^(3/4)*A)/x^(3/2) + (96*a^(7/4)*(-(A*b) + a*B)*Sqrt[x])/(a + b*x^2)^2 +
(24*a^(3/4)*(-15*A*b + 7*a*B)*Sqrt[x])/(a + b*x^2) + (42*Sqrt[2]*(11*A*b - 3*a*B
)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) - (42*Sqrt[2]*(11*A*b -
3*a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) + (21*Sqrt[2]*(11
*A*b - 3*a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(1/4
) + (21*Sqrt[2]*(-11*A*b + 3*a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x]
+ Sqrt[b]*x])/b^(1/4))/(384*a^(15/4))

_______________________________________________________________________________________

Maple [A]  time = 0.026, size = 357, normalized size = 1.1 \[ -{\frac{2\,A}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}}-{\frac{15\,{b}^{2}A}{16\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{5}{2}}}}+{\frac{7\,Bb}{16\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}{x}^{{\frac{5}{2}}}}-{\frac{19\,Ab}{16\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}\sqrt{x}}+{\frac{11\,B}{16\,a \left ( b{x}^{2}+a \right ) ^{2}}\sqrt{x}}-{\frac{77\,\sqrt{2}Ab}{64\,{a}^{4}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }-{\frac{77\,\sqrt{2}Ab}{128\,{a}^{4}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{77\,\sqrt{2}Ab}{64\,{a}^{4}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{21\,\sqrt{2}B}{64\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{21\,\sqrt{2}B}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{21\,\sqrt{2}B}{64\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^(5/2)/(b*x^2+a)^3,x)

[Out]

-2/3*A/a^3/x^(3/2)-15/16/a^3/(b*x^2+a)^2*x^(5/2)*b^2*A+7/16/a^2/(b*x^2+a)^2*x^(5
/2)*b*B-19/16/a^2/(b*x^2+a)^2*A*x^(1/2)*b+11/16/a/(b*x^2+a)^2*B*x^(1/2)-77/64/a^
4*(a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*b-77/128/a^4*(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)))*b-77/64/a^4*(a/b)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(a
/b)^(1/4)*x^(1/2)+1)*b+21/64/a^3*(a/b)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(a/b)^(1/4
)*x^(1/2)-1)+21/128/a^3*(a/b)^(1/4)*2^(1/2)*B*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)))+21/64/a^3*(a/b)^(1/4)*
2^(1/2)*B*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)/((b*x^2 + a)^3*x^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.256562, size = 929, normalized size = 2.89 \[ \frac{28 \,{\left (3 \, B a b - 11 \, A b^{2}\right )} x^{4} - 128 \, A a^{2} + 44 \,{\left (3 \, B a^{2} - 11 \, A a b\right )} x^{2} + 84 \,{\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )} \sqrt{x} \left (-\frac{81 \, B^{4} a^{4} - 1188 \, A B^{3} a^{3} b + 6534 \, A^{2} B^{2} a^{2} b^{2} - 15972 \, A^{3} B a b^{3} + 14641 \, A^{4} b^{4}}{a^{15} b}\right )^{\frac{1}{4}} \arctan \left (-\frac{a^{4} \left (-\frac{81 \, B^{4} a^{4} - 1188 \, A B^{3} a^{3} b + 6534 \, A^{2} B^{2} a^{2} b^{2} - 15972 \, A^{3} B a b^{3} + 14641 \, A^{4} b^{4}}{a^{15} b}\right )^{\frac{1}{4}}}{{\left (3 \, B a - 11 \, A b\right )} \sqrt{x} - \sqrt{a^{8} \sqrt{-\frac{81 \, B^{4} a^{4} - 1188 \, A B^{3} a^{3} b + 6534 \, A^{2} B^{2} a^{2} b^{2} - 15972 \, A^{3} B a b^{3} + 14641 \, A^{4} b^{4}}{a^{15} b}} +{\left (9 \, B^{2} a^{2} - 66 \, A B a b + 121 \, A^{2} b^{2}\right )} x}}\right ) - 21 \,{\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )} \sqrt{x} \left (-\frac{81 \, B^{4} a^{4} - 1188 \, A B^{3} a^{3} b + 6534 \, A^{2} B^{2} a^{2} b^{2} - 15972 \, A^{3} B a b^{3} + 14641 \, A^{4} b^{4}}{a^{15} b}\right )^{\frac{1}{4}} \log \left (7 \, a^{4} \left (-\frac{81 \, B^{4} a^{4} - 1188 \, A B^{3} a^{3} b + 6534 \, A^{2} B^{2} a^{2} b^{2} - 15972 \, A^{3} B a b^{3} + 14641 \, A^{4} b^{4}}{a^{15} b}\right )^{\frac{1}{4}} - 7 \,{\left (3 \, B a - 11 \, A b\right )} \sqrt{x}\right ) + 21 \,{\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )} \sqrt{x} \left (-\frac{81 \, B^{4} a^{4} - 1188 \, A B^{3} a^{3} b + 6534 \, A^{2} B^{2} a^{2} b^{2} - 15972 \, A^{3} B a b^{3} + 14641 \, A^{4} b^{4}}{a^{15} b}\right )^{\frac{1}{4}} \log \left (-7 \, a^{4} \left (-\frac{81 \, B^{4} a^{4} - 1188 \, A B^{3} a^{3} b + 6534 \, A^{2} B^{2} a^{2} b^{2} - 15972 \, A^{3} B a b^{3} + 14641 \, A^{4} b^{4}}{a^{15} b}\right )^{\frac{1}{4}} - 7 \,{\left (3 \, B a - 11 \, A b\right )} \sqrt{x}\right )}{192 \,{\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^3*x^(5/2)),x, algorithm="fricas")

[Out]

1/192*(28*(3*B*a*b - 11*A*b^2)*x^4 - 128*A*a^2 + 44*(3*B*a^2 - 11*A*a*b)*x^2 + 8
4*(a^3*b^2*x^5 + 2*a^4*b*x^3 + a^5*x)*sqrt(x)*(-(81*B^4*a^4 - 1188*A*B^3*a^3*b +
6534*A^2*B^2*a^2*b^2 - 15972*A^3*B*a*b^3 + 14641*A^4*b^4)/(a^15*b))^(1/4)*arcta
n(-a^4*(-(81*B^4*a^4 - 1188*A*B^3*a^3*b + 6534*A^2*B^2*a^2*b^2 - 15972*A^3*B*a*b
^3 + 14641*A^4*b^4)/(a^15*b))^(1/4)/((3*B*a - 11*A*b)*sqrt(x) - sqrt(a^8*sqrt(-(
81*B^4*a^4 - 1188*A*B^3*a^3*b + 6534*A^2*B^2*a^2*b^2 - 15972*A^3*B*a*b^3 + 14641
*A^4*b^4)/(a^15*b)) + (9*B^2*a^2 - 66*A*B*a*b + 121*A^2*b^2)*x))) - 21*(a^3*b^2*
x^5 + 2*a^4*b*x^3 + a^5*x)*sqrt(x)*(-(81*B^4*a^4 - 1188*A*B^3*a^3*b + 6534*A^2*B
^2*a^2*b^2 - 15972*A^3*B*a*b^3 + 14641*A^4*b^4)/(a^15*b))^(1/4)*log(7*a^4*(-(81*
B^4*a^4 - 1188*A*B^3*a^3*b + 6534*A^2*B^2*a^2*b^2 - 15972*A^3*B*a*b^3 + 14641*A^
4*b^4)/(a^15*b))^(1/4) - 7*(3*B*a - 11*A*b)*sqrt(x)) + 21*(a^3*b^2*x^5 + 2*a^4*b
*x^3 + a^5*x)*sqrt(x)*(-(81*B^4*a^4 - 1188*A*B^3*a^3*b + 6534*A^2*B^2*a^2*b^2 -
15972*A^3*B*a*b^3 + 14641*A^4*b^4)/(a^15*b))^(1/4)*log(-7*a^4*(-(81*B^4*a^4 - 11
88*A*B^3*a^3*b + 6534*A^2*B^2*a^2*b^2 - 15972*A^3*B*a*b^3 + 14641*A^4*b^4)/(a^15
*b))^(1/4) - 7*(3*B*a - 11*A*b)*sqrt(x)))/((a^3*b^2*x^5 + 2*a^4*b*x^3 + a^5*x)*s
qrt(x))

_______________________________________________________________________________________

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)/x**(5/2)/(b*x**2+a)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.26433, size = 410, normalized size = 1.27 \[ \frac{7 \, \sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - 11 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{4} b} + \frac{7 \, \sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - 11 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{4} b} + \frac{7 \, \sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - 11 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{4} b} - \frac{7 \, \sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - 11 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{4} b} - \frac{2 \, A}{3 \, a^{3} x^{\frac{3}{2}}} + \frac{7 \, B a b x^{\frac{5}{2}} - 15 \, A b^{2} x^{\frac{5}{2}} + 11 \, B a^{2} \sqrt{x} - 19 \, A a b \sqrt{x}}{16 \,{\left (b x^{2} + a\right )}^{2} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)^3*x^(5/2)),x, algorithm="giac")

[Out]

7/64*sqrt(2)*(3*(a*b^3)^(1/4)*B*a - 11*(a*b^3)^(1/4)*A*b)*arctan(1/2*sqrt(2)*(sq
rt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a^4*b) + 7/64*sqrt(2)*(3*(a*b^3)^(1
/4)*B*a - 11*(a*b^3)^(1/4)*A*b)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqr
t(x))/(a/b)^(1/4))/(a^4*b) + 7/128*sqrt(2)*(3*(a*b^3)^(1/4)*B*a - 11*(a*b^3)^(1/
4)*A*b)*ln(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^4*b) - 7/128*sqrt(2)*
(3*(a*b^3)^(1/4)*B*a - 11*(a*b^3)^(1/4)*A*b)*ln(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x
+ sqrt(a/b))/(a^4*b) - 2/3*A/(a^3*x^(3/2)) + 1/16*(7*B*a*b*x^(5/2) - 15*A*b^2*x
^(5/2) + 11*B*a^2*sqrt(x) - 19*A*a*b*sqrt(x))/((b*x^2 + a)^2*a^3)