[next] [prev] [prev-tail] [tail] [up]

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

Optimal. Leaf size=318 \[ -\frac{(7 A b-a B) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{13/6} b^{5/6}}+\frac{(7 A b-a B) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{13/6} b^{5/6}}+\frac{(7 A b-a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{18 a^{13/6} b^{5/6}}-\frac{(7 A b-a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{18 a^{13/6} b^{5/6}}-\frac{(7 A b-a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{9 a^{13/6} b^{5/6}}-\frac{7 A b-a B}{3 a^2 b \sqrt{x}}+\frac{A b-a B}{3 a b \sqrt{x} \left (a+b x^3\right )} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 1.60311, antiderivative size = 318, 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 \[ -\frac{(7 A b-a B) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{13/6} b^{5/6}}+\frac{(7 A b-a B) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{13/6} b^{5/6}}+\frac{(7 A b-a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{18 a^{13/6} b^{5/6}}-\frac{(7 A b-a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{18 a^{13/6} b^{5/6}}-\frac{(7 A b-a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{9 a^{13/6} b^{5/6}}-\frac{7 A b-a B}{3 a^2 b \sqrt{x}}+\frac{A b-a B}{3 a b \sqrt{x} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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((B*x**3+A)/x**(3/2)/(b*x**3+a)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 0.728351, size = 274, normalized size = 0.86 \[ \frac{\frac{\sqrt{3} (a B-7 A b) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{5/6}}+\frac{\sqrt{3} (7 A b-a B) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{5/6}}+\frac{2 (7 A b-a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{b^{5/6}}-\frac{2 (7 A b-a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{b^{5/6}}+\frac{4 (a B-7 A b) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{b^{5/6}}+\frac{12 \sqrt [6]{a} x^{5/2} (a B-A b)}{a+b x^3}-\frac{72 \sqrt [6]{a} A}{\sqrt{x}}}{36 a^{13/6}} \]

Antiderivative was successfully verified.

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

[Out]

((-72*a^(1/6)*A)/Sqrt[x] + (12*a^(1/6)*(-(A*b) + a*B)*x^(5/2))/(a + b*x^3) + (2*
(7*A*b - a*B)*ArcTan[Sqrt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/b^(5/6) - (2*(7*A*b
 - a*B)*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a^(1/6)])/b^(5/6) + (4*(-7*A*b + a*
B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/b^(5/6) + (Sqrt[3]*(-7*A*b + a*B)*Log[a^(1
/3) - Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/b^(5/6) + (Sqrt[3]*(7*A*b -
a*B)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/b^(5/6))/(36*a^
(13/6))

_______________________________________________________________________________________

Maple [A]  time = 0.066, size = 395, normalized size = 1.2 \[ -2\,{\frac{A}{{a}^{2}\sqrt{x}}}-{\frac{Ab}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }{x}^{{\frac{5}{2}}}}+{\frac{B}{3\,a \left ( b{x}^{3}+a \right ) }{x}^{{\frac{5}{2}}}}-{\frac{7\,A}{9\,{a}^{2}}\arctan \left ({1\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-{\frac{7\,Ab\sqrt{3}}{36\,{a}^{3}} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ( x-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{7\,A}{18\,{a}^{2}}\arctan \left ( -\sqrt{3}+2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+{\frac{7\,Ab\sqrt{3}}{36\,{a}^{3}} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ( x+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{7\,A}{18\,{a}^{2}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+{\frac{B}{9\,ab}\arctan \left ({1\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+{\frac{B\sqrt{3}}{36\,{a}^{2}} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ( x-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{B}{18\,ab}\arctan \left ( -\sqrt{3}+2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-{\frac{B\sqrt{3}}{36\,{a}^{2}} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ( x+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{B}{18\,ab}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A/a^2/x^(1/2)-1/3/a^2*x^(5/2)/(b*x^3+a)*A*b+1/3/a*x^(5/2)/(b*x^3+a)*B-7/9/a^2
*A/(a/b)^(1/6)*arctan(x^(1/2)/(a/b)^(1/6))-7/36/a^3*A*b*(a/b)^(5/6)*3^(1/2)*ln(x
-3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))-7/18/a^2*A/(a/b)^(1/6)*arctan(-3^(1/2)
+2*x^(1/2)/(a/b)^(1/6))+7/36/a^3*A*b*3^(1/2)*(a/b)^(5/6)*ln(x+3^(1/2)*(a/b)^(1/6
)*x^(1/2)+(a/b)^(1/3))-7/18/a^2*A/(a/b)^(1/6)*arctan(2*x^(1/2)/(a/b)^(1/6)+3^(1/
2))+1/9/a*B/b/(a/b)^(1/6)*arctan(x^(1/2)/(a/b)^(1/6))+1/36/a^2*B*(a/b)^(5/6)*3^(
1/2)*ln(x-3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))+1/18/a*B/b/(a/b)^(1/6)*arctan
(-3^(1/2)+2*x^(1/2)/(a/b)^(1/6))-1/36/a^2*B*3^(1/2)*(a/b)^(5/6)*ln(x+3^(1/2)*(a/
b)^(1/6)*x^(1/2)+(a/b)^(1/3))+1/18/a*B/b/(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)/((b*x^3 + a)^2*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.303071, size = 4690, normalized size = 14.75 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/36*(12*(B*a - 7*A*b)*x^3 - 4*sqrt(3)*(a^2*b*x^3 + a^3)*sqrt(x)*(-(B^6*a^6 - 42
*A*B^5*a^5*b + 735*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 36015*A^4*B^2*a^2*b^
4 - 100842*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^5))^(1/6)*arctan(-sqrt(3)*a^11*
b^4*(-(B^6*a^6 - 42*A*B^5*a^5*b + 735*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 3
6015*A^4*B^2*a^2*b^4 - 100842*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^5))^(5/6)/(a
^11*b^4*(-(B^6*a^6 - 42*A*B^5*a^5*b + 735*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3
 + 36015*A^4*B^2*a^2*b^4 - 100842*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^5))^(5/6
) + 2*(B^5*a^5 - 35*A*B^4*a^4*b + 490*A^2*B^3*a^3*b^2 - 3430*A^3*B^2*a^2*b^3 + 1
2005*A^4*B*a*b^4 - 16807*A^5*b^5)*sqrt(x) - 2*sqrt((B^5*a^16*b^4 - 35*A*B^4*a^15
*b^5 + 490*A^2*B^3*a^14*b^6 - 3430*A^3*B^2*a^13*b^7 + 12005*A^4*B*a^12*b^8 - 168
07*A^5*a^11*b^9)*sqrt(x)*(-(B^6*a^6 - 42*A*B^5*a^5*b + 735*A^2*B^4*a^4*b^2 - 686
0*A^3*B^3*a^3*b^3 + 36015*A^4*B^2*a^2*b^4 - 100842*A^5*B*a*b^5 + 117649*A^6*b^6)
/(a^13*b^5))^(5/6) + (B^10*a^10 - 70*A*B^9*a^9*b + 2205*A^2*B^8*a^8*b^2 - 41160*
A^3*B^7*a^7*b^3 + 504210*A^4*B^6*a^6*b^4 - 4235364*A^5*B^5*a^5*b^5 + 24706290*A^
6*B^4*a^4*b^6 - 98825160*A^7*B^3*a^3*b^7 + 259416045*A^8*B^2*a^2*b^8 - 403536070
*A^9*B*a*b^9 + 282475249*A^10*b^10)*x - (B^6*a^15*b^3 - 42*A*B^5*a^14*b^4 + 735*
A^2*B^4*a^13*b^5 - 6860*A^3*B^3*a^12*b^6 + 36015*A^4*B^2*a^11*b^7 - 100842*A^5*B
*a^10*b^8 + 117649*A^6*a^9*b^9)*(-(B^6*a^6 - 42*A*B^5*a^5*b + 735*A^2*B^4*a^4*b^
2 - 6860*A^3*B^3*a^3*b^3 + 36015*A^4*B^2*a^2*b^4 - 100842*A^5*B*a*b^5 + 117649*A
^6*b^6)/(a^13*b^5))^(2/3)))) - 4*sqrt(3)*(a^2*b*x^3 + a^3)*sqrt(x)*(-(B^6*a^6 -
42*A*B^5*a^5*b + 735*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 36015*A^4*B^2*a^2*
b^4 - 100842*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^5))^(1/6)*arctan(sqrt(3)*a^11
*b^4*(-(B^6*a^6 - 42*A*B^5*a^5*b + 735*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 +
36015*A^4*B^2*a^2*b^4 - 100842*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^5))^(5/6)/(
a^11*b^4*(-(B^6*a^6 - 42*A*B^5*a^5*b + 735*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^
3 + 36015*A^4*B^2*a^2*b^4 - 100842*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^5))^(5/
6) - 2*(B^5*a^5 - 35*A*B^4*a^4*b + 490*A^2*B^3*a^3*b^2 - 3430*A^3*B^2*a^2*b^3 +
12005*A^4*B*a*b^4 - 16807*A^5*b^5)*sqrt(x) + 2*sqrt(-(B^5*a^16*b^4 - 35*A*B^4*a^
15*b^5 + 490*A^2*B^3*a^14*b^6 - 3430*A^3*B^2*a^13*b^7 + 12005*A^4*B*a^12*b^8 - 1
6807*A^5*a^11*b^9)*sqrt(x)*(-(B^6*a^6 - 42*A*B^5*a^5*b + 735*A^2*B^4*a^4*b^2 - 6
860*A^3*B^3*a^3*b^3 + 36015*A^4*B^2*a^2*b^4 - 100842*A^5*B*a*b^5 + 117649*A^6*b^
6)/(a^13*b^5))^(5/6) + (B^10*a^10 - 70*A*B^9*a^9*b + 2205*A^2*B^8*a^8*b^2 - 4116
0*A^3*B^7*a^7*b^3 + 504210*A^4*B^6*a^6*b^4 - 4235364*A^5*B^5*a^5*b^5 + 24706290*
A^6*B^4*a^4*b^6 - 98825160*A^7*B^3*a^3*b^7 + 259416045*A^8*B^2*a^2*b^8 - 4035360
70*A^9*B*a*b^9 + 282475249*A^10*b^10)*x - (B^6*a^15*b^3 - 42*A*B^5*a^14*b^4 + 73
5*A^2*B^4*a^13*b^5 - 6860*A^3*B^3*a^12*b^6 + 36015*A^4*B^2*a^11*b^7 - 100842*A^5
*B*a^10*b^8 + 117649*A^6*a^9*b^9)*(-(B^6*a^6 - 42*A*B^5*a^5*b + 735*A^2*B^4*a^4*
b^2 - 6860*A^3*B^3*a^3*b^3 + 36015*A^4*B^2*a^2*b^4 - 100842*A^5*B*a*b^5 + 117649
*A^6*b^6)/(a^13*b^5))^(2/3)))) - 2*(a^2*b*x^3 + a^3)*sqrt(x)*(-(B^6*a^6 - 42*A*B
^5*a^5*b + 735*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 36015*A^4*B^2*a^2*b^4 -
100842*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^5))^(1/6)*log(a^11*b^4*(-(B^6*a^6 -
 42*A*B^5*a^5*b + 735*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 36015*A^4*B^2*a^2
*b^4 - 100842*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^5))^(5/6) - (B^5*a^5 - 35*A*
B^4*a^4*b + 490*A^2*B^3*a^3*b^2 - 3430*A^3*B^2*a^2*b^3 + 12005*A^4*B*a*b^4 - 168
07*A^5*b^5)*sqrt(x)) + 2*(a^2*b*x^3 + a^3)*sqrt(x)*(-(B^6*a^6 - 42*A*B^5*a^5*b +
 735*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 36015*A^4*B^2*a^2*b^4 - 100842*A^5
*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^5))^(1/6)*log(-a^11*b^4*(-(B^6*a^6 - 42*A*B^5
*a^5*b + 735*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 36015*A^4*B^2*a^2*b^4 - 10
0842*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^5))^(5/6) - (B^5*a^5 - 35*A*B^4*a^4*b
 + 490*A^2*B^3*a^3*b^2 - 3430*A^3*B^2*a^2*b^3 + 12005*A^4*B*a*b^4 - 16807*A^5*b^
5)*sqrt(x)) + (a^2*b*x^3 + a^3)*sqrt(x)*(-(B^6*a^6 - 42*A*B^5*a^5*b + 735*A^2*B^
4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 36015*A^4*B^2*a^2*b^4 - 100842*A^5*B*a*b^5 +
117649*A^6*b^6)/(a^13*b^5))^(1/6)*log((B^5*a^16*b^4 - 35*A*B^4*a^15*b^5 + 490*A^
2*B^3*a^14*b^6 - 3430*A^3*B^2*a^13*b^7 + 12005*A^4*B*a^12*b^8 - 16807*A^5*a^11*b
^9)*sqrt(x)*(-(B^6*a^6 - 42*A*B^5*a^5*b + 735*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*a^3
*b^3 + 36015*A^4*B^2*a^2*b^4 - 100842*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^5))^
(5/6) + (B^10*a^10 - 70*A*B^9*a^9*b + 2205*A^2*B^8*a^8*b^2 - 41160*A^3*B^7*a^7*b
^3 + 504210*A^4*B^6*a^6*b^4 - 4235364*A^5*B^5*a^5*b^5 + 24706290*A^6*B^4*a^4*b^6
 - 98825160*A^7*B^3*a^3*b^7 + 259416045*A^8*B^2*a^2*b^8 - 403536070*A^9*B*a*b^9
+ 282475249*A^10*b^10)*x - (B^6*a^15*b^3 - 42*A*B^5*a^14*b^4 + 735*A^2*B^4*a^13*
b^5 - 6860*A^3*B^3*a^12*b^6 + 36015*A^4*B^2*a^11*b^7 - 100842*A^5*B*a^10*b^8 + 1
17649*A^6*a^9*b^9)*(-(B^6*a^6 - 42*A*B^5*a^5*b + 735*A^2*B^4*a^4*b^2 - 6860*A^3*
B^3*a^3*b^3 + 36015*A^4*B^2*a^2*b^4 - 100842*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13
*b^5))^(2/3)) - (a^2*b*x^3 + a^3)*sqrt(x)*(-(B^6*a^6 - 42*A*B^5*a^5*b + 735*A^2*
B^4*a^4*b^2 - 6860*A^3*B^3*a^3*b^3 + 36015*A^4*B^2*a^2*b^4 - 100842*A^5*B*a*b^5
+ 117649*A^6*b^6)/(a^13*b^5))^(1/6)*log(-(B^5*a^16*b^4 - 35*A*B^4*a^15*b^5 + 490
*A^2*B^3*a^14*b^6 - 3430*A^3*B^2*a^13*b^7 + 12005*A^4*B*a^12*b^8 - 16807*A^5*a^1
1*b^9)*sqrt(x)*(-(B^6*a^6 - 42*A*B^5*a^5*b + 735*A^2*B^4*a^4*b^2 - 6860*A^3*B^3*
a^3*b^3 + 36015*A^4*B^2*a^2*b^4 - 100842*A^5*B*a*b^5 + 117649*A^6*b^6)/(a^13*b^5
))^(5/6) + (B^10*a^10 - 70*A*B^9*a^9*b + 2205*A^2*B^8*a^8*b^2 - 41160*A^3*B^7*a^
7*b^3 + 504210*A^4*B^6*a^6*b^4 - 4235364*A^5*B^5*a^5*b^5 + 24706290*A^6*B^4*a^4*
b^6 - 98825160*A^7*B^3*a^3*b^7 + 259416045*A^8*B^2*a^2*b^8 - 403536070*A^9*B*a*b
^9 + 282475249*A^10*b^10)*x - (B^6*a^15*b^3 - 42*A*B^5*a^14*b^4 + 735*A^2*B^4*a^
13*b^5 - 6860*A^3*B^3*a^12*b^6 + 36015*A^4*B^2*a^11*b^7 - 100842*A^5*B*a^10*b^8
+ 117649*A^6*a^9*b^9)*(-(B^6*a^6 - 42*A*B^5*a^5*b + 735*A^2*B^4*a^4*b^2 - 6860*A
^3*B^3*a^3*b^3 + 36015*A^4*B^2*a^2*b^4 - 100842*A^5*B*a*b^5 + 117649*A^6*b^6)/(a
^13*b^5))^(2/3)) - 72*A*a)/((a^2*b*x^3 + a^3)*sqrt(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**3+A)/x**(3/2)/(b*x**3+a)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.308136, size = 414, normalized size = 1.3 \[ \frac{B a x^{3} - 7 \, A b x^{3} - 6 \, A a}{3 \,{\left (b x^{\frac{7}{2}} + a \sqrt{x}\right )} a^{2}} - \frac{\sqrt{3}{\left (\left (a b^{5}\right )^{\frac{5}{6}} B a - 7 \, \left (a b^{5}\right )^{\frac{5}{6}} A b\right )}{\rm ln}\left (\sqrt{3} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{6}} + x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{36 \, a^{3} b^{5}} + \frac{\sqrt{3}{\left (\left (a b^{5}\right )^{\frac{5}{6}} B a - 7 \, \left (a b^{5}\right )^{\frac{5}{6}} A b\right )}{\rm ln}\left (-\sqrt{3} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{6}} + x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{36 \, a^{3} b^{5}} + \frac{{\left (\left (a b^{5}\right )^{\frac{5}{6}} B a - 7 \, \left (a b^{5}\right )^{\frac{5}{6}} A b\right )} \arctan \left (\frac{\sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}} + 2 \, \sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{18 \, a^{3} b^{5}} + \frac{{\left (\left (a b^{5}\right )^{\frac{5}{6}} B a - 7 \, \left (a b^{5}\right )^{\frac{5}{6}} A b\right )} \arctan \left (-\frac{\sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}} - 2 \, \sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{18 \, a^{3} b^{5}} + \frac{{\left (\left (a b^{5}\right )^{\frac{5}{6}} B a - 7 \, \left (a b^{5}\right )^{\frac{5}{6}} A b\right )} \arctan \left (\frac{\sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{9 \, a^{3} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[next] [prev] [prev-tail] [front] [up]