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

Optimal. Leaf size=312 \[ -\frac{(A b-7 a B) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{5/6} b^{13/6}}+\frac{(A b-7 a B) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{5/6} b^{13/6}}-\frac{(A b-7 a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{13/6}}+\frac{(A b-7 a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{18 a^{5/6} b^{13/6}}+\frac{(A b-7 a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{9 a^{5/6} b^{13/6}}-\frac{\sqrt{x} (A b-7 a B)}{3 a b^2}+\frac{x^{7/2} (A b-a B)}{3 a b \left (a+b x^3\right )} \]

[Out]

-((A*b - 7*a*B)*Sqrt[x])/(3*a*b^2) + ((A*b - a*B)*x^(7/2))/(3*a*b*(a + b*x^3)) -
((A*b - 7*a*B)*ArcTan[Sqrt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(18*a^(5/6)*b^(13
/6)) + ((A*b - 7*a*B)*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(18*a^(5/6)
*b^(13/6)) + ((A*b - 7*a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(9*a^(5/6)*b^(13/
6)) - ((A*b - 7*a*B)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])
/(12*Sqrt[3]*a^(5/6)*b^(13/6)) + ((A*b - 7*a*B)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^
(1/6)*Sqrt[x] + b^(1/3)*x])/(12*Sqrt[3]*a^(5/6)*b^(13/6))

_______________________________________________________________________________________

Rubi [A]  time = 1.17273, antiderivative size = 312, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ -\frac{(A b-7 a B) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{5/6} b^{13/6}}+\frac{(A b-7 a B) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{5/6} b^{13/6}}-\frac{(A b-7 a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{13/6}}+\frac{(A b-7 a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{18 a^{5/6} b^{13/6}}+\frac{(A b-7 a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{9 a^{5/6} b^{13/6}}-\frac{\sqrt{x} (A b-7 a B)}{3 a b^2}+\frac{x^{7/2} (A b-a B)}{3 a b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((A*b - 7*a*B)*Sqrt[x])/(3*a*b^2) + ((A*b - a*B)*x^(7/2))/(3*a*b*(a + b*x^3)) -
((A*b - 7*a*B)*ArcTan[Sqrt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(18*a^(5/6)*b^(13
/6)) + ((A*b - 7*a*B)*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(18*a^(5/6)
*b^(13/6)) + ((A*b - 7*a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(9*a^(5/6)*b^(13/
6)) - ((A*b - 7*a*B)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])
/(12*Sqrt[3]*a^(5/6)*b^(13/6)) + ((A*b - 7*a*B)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^
(1/6)*Sqrt[x] + b^(1/3)*x])/(12*Sqrt[3]*a^(5/6)*b^(13/6))

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.540781, size = 273, normalized size = 0.88 \[ \frac{\frac{\sqrt{3} (7 a B-A b) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/6}}+\frac{\sqrt{3} (A b-7 a B) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/6}}+\frac{2 (7 a B-A b) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{a^{5/6}}+\frac{2 (A b-7 a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{a^{5/6}}+\frac{4 (A b-7 a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{a^{5/6}}-\frac{12 \sqrt [6]{b} \sqrt{x} (A b-a B)}{a+b x^3}+72 \sqrt [6]{b} B \sqrt{x}}{36 b^{13/6}} \]

Antiderivative was successfully verified.

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

[Out]

(72*b^(1/6)*B*Sqrt[x] - (12*b^(1/6)*(A*b - a*B)*Sqrt[x])/(a + b*x^3) + (2*(-(A*b
) + 7*a*B)*ArcTan[Sqrt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/a^(5/6) + (2*(A*b - 7*
a*B)*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a^(1/6)])/a^(5/6) + (4*(A*b - 7*a*B)*A
rcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/a^(5/6) + (Sqrt[3]*(-(A*b) + 7*a*B)*Log[a^(1/3
) - Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/a^(5/6) + (Sqrt[3]*(A*b - 7*a*
B)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/a^(5/6))/(36*b^(1
3/6))

_______________________________________________________________________________________

Maple [A]  time = 0.06, size = 399, normalized size = 1.3 \[ 2\,{\frac{B\sqrt{x}}{{b}^{2}}}-{\frac{A}{3\,b \left ( b{x}^{3}+a \right ) }\sqrt{x}}+{\frac{Ba}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }\sqrt{x}}-{\frac{7\,B}{9\,{b}^{2}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ({1\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }+{\frac{7\,B\sqrt{3}}{36\,{b}^{2}}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{7\,B}{18\,{b}^{2}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( -\sqrt{3}+2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }-{\frac{7\,B\sqrt{3}}{36\,{b}^{2}}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{7\,B}{18\,{b}^{2}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ) }+{\frac{A}{9\,ab}\sqrt [6]{{\frac{a}{b}}}\arctan \left ({1\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }-{\frac{A\sqrt{3}}{36\,ab}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{A}{18\,ab}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( -\sqrt{3}+2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }+{\frac{A\sqrt{3}}{36\,ab}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{A}{18\,ab}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*x^(1/2)/b^2-1/3/b*x^(1/2)/(b*x^3+a)*A+1/3/b^2*x^(1/2)/(b*x^3+a)*B*a-7/9/b^2*
B*(a/b)^(1/6)*arctan(x^(1/2)/(a/b)^(1/6))+7/36/b^2*B*3^(1/2)*(a/b)^(1/6)*ln(x-3^
(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))-7/18/b^2*B*(a/b)^(1/6)*arctan(-3^(1/2)+2*
x^(1/2)/(a/b)^(1/6))-7/36/b^2*B*3^(1/2)*(a/b)^(1/6)*ln(x+3^(1/2)*(a/b)^(1/6)*x^(
1/2)+(a/b)^(1/3))-7/18/b^2*B*(a/b)^(1/6)*arctan(2*x^(1/2)/(a/b)^(1/6)+3^(1/2))+1
/9/b*A/a*(a/b)^(1/6)*arctan(x^(1/2)/(a/b)^(1/6))-1/36/b*A/a*3^(1/2)*(a/b)^(1/6)*
ln(x-3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))+1/18/b*A/a*(a/b)^(1/6)*arctan(-3^(
1/2)+2*x^(1/2)/(a/b)^(1/6))+1/36/b*A/a*3^(1/2)*(a/b)^(1/6)*ln(x+3^(1/2)*(a/b)^(1
/6)*x^(1/2)+(a/b)^(1/3))+1/18/b*A/a*(a/b)^(1/6)*arctan(2*x^(1/2)/(a/b)^(1/6)+3^(
1/2))

_______________________________________________________________________________________

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((B*x^3 + A)*x^(5/2)/(b*x^3 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.287061, size = 3002, normalized size = 9.62 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/36*(4*sqrt(3)*(b^3*x^3 + a*b^2)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 3601
5*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5
+ A^6*b^6)/(a^5*b^13))^(1/6)*arctan(-sqrt(3)*a*b^2*(-(117649*B^6*a^6 - 100842*A*
B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 -
42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/6)/(a*b^2*(-(117649*B^6*a^6 - 100842*A
*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4
- 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/6) + 2*(7*B*a - A*b)*sqrt(x) - 2*sqrt
(a^2*b^4*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A
^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/
3) + (49*B^2*a^2 - 14*A*B*a*b + A^2*b^2)*x + (7*B*a^2*b^2 - A*a*b^3)*sqrt(x)*(-(
117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b
^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/6)))) + 4*sq
rt(3)*(b^3*x^3 + a*b^2)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a
^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/
(a^5*b^13))^(1/6)*arctan(sqrt(3)*a*b^2*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b +
36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*
b^5 + A^6*b^6)/(a^5*b^13))^(1/6)/(a*b^2*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b +
36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a
*b^5 + A^6*b^6)/(a^5*b^13))^(1/6) - 2*(7*B*a - A*b)*sqrt(x) + 2*sqrt(a^2*b^4*(-(
117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b
^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/3) + (49*B^2
*a^2 - 14*A*B*a*b + A^2*b^2)*x - (7*B*a^2*b^2 - A*a*b^3)*sqrt(x)*(-(117649*B^6*a
^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4
*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/6)))) + (b^3*x^3 + a*b^2
)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*
a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/6)*log(
a^2*b^4*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^
3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/3
) + (49*B^2*a^2 - 14*A*B*a*b + A^2*b^2)*x + (7*B*a^2*b^2 - A*a*b^3)*sqrt(x)*(-(1
17649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^
3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/6)) - (b^3*x^
3 + a*b^2)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860
*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(
1/6)*log(a^2*b^4*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2
- 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^
13))^(1/3) + (49*B^2*a^2 - 14*A*B*a*b + A^2*b^2)*x - (7*B*a^2*b^2 - A*a*b^3)*sqr
t(x)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B
^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/6))
- 2*(b^3*x^3 + a*b^2)*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4
*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a
^5*b^13))^(1/6)*log(a*b^2*(-(117649*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4
*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6
)/(a^5*b^13))^(1/6) - (7*B*a - A*b)*sqrt(x)) + 2*(b^3*x^3 + a*b^2)*(-(117649*B^6
*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 735*A
^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/6)*log(-a*b^2*(-(11764
9*B^6*a^6 - 100842*A*B^5*a^5*b + 36015*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 +
735*A^4*B^2*a^2*b^4 - 42*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^13))^(1/6) - (7*B*a - A*b
)*sqrt(x)) - 12*(6*B*b*x^3 + 7*B*a - A*b)*sqrt(x))/(b^3*x^3 + 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**3+A)/(b*x**3+a)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.228286, size = 423, normalized size = 1.36 \[ \frac{2 \, B \sqrt{x}}{b^{2}} - \frac{\sqrt{3}{\left (7 \, \left (a b^{5}\right )^{\frac{1}{6}} B a - \left (a b^{5}\right )^{\frac{1}{6}} A b\right )}{\rm ln}\left (\sqrt{3} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{6}} + x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{36 \, a b^{3}} + \frac{\sqrt{3}{\left (7 \, \left (a b^{5}\right )^{\frac{1}{6}} B a - \left (a b^{5}\right )^{\frac{1}{6}} A b\right )}{\rm ln}\left (-\sqrt{3} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{6}} + x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{36 \, a b^{3}} + \frac{B a \sqrt{x} - A b \sqrt{x}}{3 \,{\left (b x^{3} + a\right )} b^{2}} - \frac{{\left (7 \, \left (a b^{5}\right )^{\frac{1}{6}} B a - \left (a b^{5}\right )^{\frac{1}{6}} A b\right )} \arctan \left (\frac{\sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}} + 2 \, \sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{18 \, a b^{3}} - \frac{{\left (7 \, \left (a b^{5}\right )^{\frac{1}{6}} B a - \left (a b^{5}\right )^{\frac{1}{6}} A b\right )} \arctan \left (-\frac{\sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}} - 2 \, \sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{18 \, a b^{3}} - \frac{{\left (7 \, \left (a b^{5}\right )^{\frac{1}{6}} B a - \left (a b^{5}\right )^{\frac{1}{6}} A b\right )} \arctan \left (\frac{\sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{9 \, a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*sqrt(x)/b^2 - 1/36*sqrt(3)*(7*(a*b^5)^(1/6)*B*a - (a*b^5)^(1/6)*A*b)*ln(sqrt
(3)*sqrt(x)*(a/b)^(1/6) + x + (a/b)^(1/3))/(a*b^3) + 1/36*sqrt(3)*(7*(a*b^5)^(1/
6)*B*a - (a*b^5)^(1/6)*A*b)*ln(-sqrt(3)*sqrt(x)*(a/b)^(1/6) + x + (a/b)^(1/3))/(
a*b^3) + 1/3*(B*a*sqrt(x) - A*b*sqrt(x))/((b*x^3 + a)*b^2) - 1/18*(7*(a*b^5)^(1/
6)*B*a - (a*b^5)^(1/6)*A*b)*arctan((sqrt(3)*(a/b)^(1/6) + 2*sqrt(x))/(a/b)^(1/6)
)/(a*b^3) - 1/18*(7*(a*b^5)^(1/6)*B*a - (a*b^5)^(1/6)*A*b)*arctan(-(sqrt(3)*(a/b
)^(1/6) - 2*sqrt(x))/(a/b)^(1/6))/(a*b^3) - 1/9*(7*(a*b^5)^(1/6)*B*a - (a*b^5)^(
1/6)*A*b)*arctan(sqrt(x)/(a/b)^(1/6))/(a*b^3)