3.159 \(\int \frac{A+B x^3}{\sqrt{x} \left (a+b x^3\right )} \, dx\)
Optimal. Leaf size=268 \[ -\frac{(A b-a B) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \sqrt{3} a^{5/6} b^{7/6}}+\frac{(A b-a B) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \sqrt{3} a^{5/6} b^{7/6}}-\frac{(A b-a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{3 a^{5/6} b^{7/6}}+\frac{(A b-a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{3 a^{5/6} b^{7/6}}+\frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{3 a^{5/6} b^{7/6}}+\frac{2 B \sqrt{x}}{b} \]
[Out]
(2*B*Sqrt[x])/b - ((A*b - a*B)*ArcTan[Sqrt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(3
*a^(5/6)*b^(7/6)) + ((A*b - a*B)*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a^(1/6)])/
(3*a^(5/6)*b^(7/6)) + (2*(A*b - a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(3*a^(5/
6)*b^(7/6)) - ((A*b - a*B)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/
3)*x])/(2*Sqrt[3]*a^(5/6)*b^(7/6)) + ((A*b - a*B)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*
b^(1/6)*Sqrt[x] + b^(1/3)*x])/(2*Sqrt[3]*a^(5/6)*b^(7/6))
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Rubi [A] time = 1.02544, antiderivative size = 268, normalized size of antiderivative = 1.,
number of steps used = 12, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364
\[ -\frac{(A b-a B) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \sqrt{3} a^{5/6} b^{7/6}}+\frac{(A b-a B) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \sqrt{3} a^{5/6} b^{7/6}}-\frac{(A b-a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{3 a^{5/6} b^{7/6}}+\frac{(A b-a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{3 a^{5/6} b^{7/6}}+\frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{3 a^{5/6} b^{7/6}}+\frac{2 B \sqrt{x}}{b} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^3)/(Sqrt[x]*(a + b*x^3)),x]
[Out]
(2*B*Sqrt[x])/b - ((A*b - a*B)*ArcTan[Sqrt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(3
*a^(5/6)*b^(7/6)) + ((A*b - a*B)*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a^(1/6)])/
(3*a^(5/6)*b^(7/6)) + (2*(A*b - a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(3*a^(5/
6)*b^(7/6)) - ((A*b - a*B)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/
3)*x])/(2*Sqrt[3]*a^(5/6)*b^(7/6)) + ((A*b - a*B)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*
b^(1/6)*Sqrt[x] + b^(1/3)*x])/(2*Sqrt[3]*a^(5/6)*b^(7/6))
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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)/(b*x**3+a)/x**(1/2),x)
[Out]
Timed out
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Mathematica [A] time = 0.248682, size = 228, normalized size = 0.85 \[ \frac{12 a^{5/6} \sqrt [6]{b} B \sqrt{x}-\sqrt{3} (A b-a B) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )+\sqrt{3} (A b-a B) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 (A b-a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )+2 (A b-a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )+4 (A b-a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{6 a^{5/6} b^{7/6}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^3)/(Sqrt[x]*(a + b*x^3)),x]
[Out]
(12*a^(5/6)*b^(1/6)*B*Sqrt[x] - 2*(A*b - a*B)*ArcTan[Sqrt[3] - (2*b^(1/6)*Sqrt[x
])/a^(1/6)] + 2*(A*b - a*B)*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a^(1/6)] + 4*(A
*b - a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)] - Sqrt[3]*(A*b - a*B)*Log[a^(1/3) -
Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x] + Sqrt[3]*(A*b - a*B)*Log[a^(1/3) +
Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(6*a^(5/6)*b^(7/6))
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Maple [A] time = 0.048, size = 347, normalized size = 1.3 \[ 2\,{\frac{B\sqrt{x}}{b}}+{\frac{2\,A}{3\,a}\sqrt [6]{{\frac{a}{b}}}\arctan \left ({1\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }-{\frac{2\,B}{3\,b}\sqrt [6]{{\frac{a}{b}}}\arctan \left ({1\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }-{\frac{\sqrt{3}A}{6\,a}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{\sqrt{3}B}{6\,b}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{A}{3\,a}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( -\sqrt{3}+2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }-{\frac{B}{3\,b}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( -\sqrt{3}+2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }+{\frac{\sqrt{3}A}{6\,a}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{\sqrt{3}B}{6\,b}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{A}{3\,a}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ) }-{\frac{B}{3\,b}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^3+A)/(b*x^3+a)/x^(1/2),x)
[Out]
2*B*x^(1/2)/b+2/3/a*(a/b)^(1/6)*arctan(x^(1/2)/(a/b)^(1/6))*A-2/3/b*(a/b)^(1/6)*
arctan(x^(1/2)/(a/b)^(1/6))*B-1/6/a*3^(1/2)*(a/b)^(1/6)*ln(x-3^(1/2)*(a/b)^(1/6)
*x^(1/2)+(a/b)^(1/3))*A+1/6/b*3^(1/2)*(a/b)^(1/6)*ln(x-3^(1/2)*(a/b)^(1/6)*x^(1/
2)+(a/b)^(1/3))*B+1/3/a*(a/b)^(1/6)*arctan(-3^(1/2)+2*x^(1/2)/(a/b)^(1/6))*A-1/3
/b*(a/b)^(1/6)*arctan(-3^(1/2)+2*x^(1/2)/(a/b)^(1/6))*B+1/6/a*3^(1/2)*(a/b)^(1/6
)*ln(x+3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))*A-1/6/b*3^(1/2)*(a/b)^(1/6)*ln(x
+3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))*B+1/3/a*(a/b)^(1/6)*arctan(2*x^(1/2)/(
a/b)^(1/6)+3^(1/2))*A-1/3/b*(a/b)^(1/6)*arctan(2*x^(1/2)/(a/b)^(1/6)+3^(1/2))*B
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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)*sqrt(x)),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.2741, size = 2805, normalized size = 10.47 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)/((b*x^3 + a)*sqrt(x)),x, algorithm="fricas")
[Out]
-1/6*(4*sqrt(3)*b*(-(B^6*a^6 - 6*A*B^5*a^5*b + 15*A^2*B^4*a^4*b^2 - 20*A^3*B^3*a
^3*b^3 + 15*A^4*B^2*a^2*b^4 - 6*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^7))^(1/6)*arctan(-
sqrt(3)*a*b*(-(B^6*a^6 - 6*A*B^5*a^5*b + 15*A^2*B^4*a^4*b^2 - 20*A^3*B^3*a^3*b^3
+ 15*A^4*B^2*a^2*b^4 - 6*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^7))^(1/6)/(a*b*(-(B^6*a^
6 - 6*A*B^5*a^5*b + 15*A^2*B^4*a^4*b^2 - 20*A^3*B^3*a^3*b^3 + 15*A^4*B^2*a^2*b^4
- 6*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^7))^(1/6) + 2*(B*a - A*b)*sqrt(x) - 2*sqrt(a^
2*b^2*(-(B^6*a^6 - 6*A*B^5*a^5*b + 15*A^2*B^4*a^4*b^2 - 20*A^3*B^3*a^3*b^3 + 15*
A^4*B^2*a^2*b^4 - 6*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^7))^(1/3) + (B^2*a^2 - 2*A*B*a
*b + A^2*b^2)*x + (B*a^2*b - A*a*b^2)*sqrt(x)*(-(B^6*a^6 - 6*A*B^5*a^5*b + 15*A^
2*B^4*a^4*b^2 - 20*A^3*B^3*a^3*b^3 + 15*A^4*B^2*a^2*b^4 - 6*A^5*B*a*b^5 + A^6*b^
6)/(a^5*b^7))^(1/6)))) + 4*sqrt(3)*b*(-(B^6*a^6 - 6*A*B^5*a^5*b + 15*A^2*B^4*a^4
*b^2 - 20*A^3*B^3*a^3*b^3 + 15*A^4*B^2*a^2*b^4 - 6*A^5*B*a*b^5 + A^6*b^6)/(a^5*b
^7))^(1/6)*arctan(sqrt(3)*a*b*(-(B^6*a^6 - 6*A*B^5*a^5*b + 15*A^2*B^4*a^4*b^2 -
20*A^3*B^3*a^3*b^3 + 15*A^4*B^2*a^2*b^4 - 6*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^7))^(1
/6)/(a*b*(-(B^6*a^6 - 6*A*B^5*a^5*b + 15*A^2*B^4*a^4*b^2 - 20*A^3*B^3*a^3*b^3 +
15*A^4*B^2*a^2*b^4 - 6*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^7))^(1/6) - 2*(B*a - A*b)*s
qrt(x) + 2*sqrt(a^2*b^2*(-(B^6*a^6 - 6*A*B^5*a^5*b + 15*A^2*B^4*a^4*b^2 - 20*A^3
*B^3*a^3*b^3 + 15*A^4*B^2*a^2*b^4 - 6*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^7))^(1/3) +
(B^2*a^2 - 2*A*B*a*b + A^2*b^2)*x - (B*a^2*b - A*a*b^2)*sqrt(x)*(-(B^6*a^6 - 6*A
*B^5*a^5*b + 15*A^2*B^4*a^4*b^2 - 20*A^3*B^3*a^3*b^3 + 15*A^4*B^2*a^2*b^4 - 6*A^
5*B*a*b^5 + A^6*b^6)/(a^5*b^7))^(1/6)))) + b*(-(B^6*a^6 - 6*A*B^5*a^5*b + 15*A^2
*B^4*a^4*b^2 - 20*A^3*B^3*a^3*b^3 + 15*A^4*B^2*a^2*b^4 - 6*A^5*B*a*b^5 + A^6*b^6
)/(a^5*b^7))^(1/6)*log(4*a^2*b^2*(-(B^6*a^6 - 6*A*B^5*a^5*b + 15*A^2*B^4*a^4*b^2
- 20*A^3*B^3*a^3*b^3 + 15*A^4*B^2*a^2*b^4 - 6*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^7))
^(1/3) + 4*(B^2*a^2 - 2*A*B*a*b + A^2*b^2)*x + 4*(B*a^2*b - A*a*b^2)*sqrt(x)*(-(
B^6*a^6 - 6*A*B^5*a^5*b + 15*A^2*B^4*a^4*b^2 - 20*A^3*B^3*a^3*b^3 + 15*A^4*B^2*a
^2*b^4 - 6*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^7))^(1/6)) - b*(-(B^6*a^6 - 6*A*B^5*a^5
*b + 15*A^2*B^4*a^4*b^2 - 20*A^3*B^3*a^3*b^3 + 15*A^4*B^2*a^2*b^4 - 6*A^5*B*a*b^
5 + A^6*b^6)/(a^5*b^7))^(1/6)*log(4*a^2*b^2*(-(B^6*a^6 - 6*A*B^5*a^5*b + 15*A^2*
B^4*a^4*b^2 - 20*A^3*B^3*a^3*b^3 + 15*A^4*B^2*a^2*b^4 - 6*A^5*B*a*b^5 + A^6*b^6)
/(a^5*b^7))^(1/3) + 4*(B^2*a^2 - 2*A*B*a*b + A^2*b^2)*x - 4*(B*a^2*b - A*a*b^2)*
sqrt(x)*(-(B^6*a^6 - 6*A*B^5*a^5*b + 15*A^2*B^4*a^4*b^2 - 20*A^3*B^3*a^3*b^3 + 1
5*A^4*B^2*a^2*b^4 - 6*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^7))^(1/6)) - 2*b*(-(B^6*a^6
- 6*A*B^5*a^5*b + 15*A^2*B^4*a^4*b^2 - 20*A^3*B^3*a^3*b^3 + 15*A^4*B^2*a^2*b^4 -
6*A^5*B*a*b^5 + A^6*b^6)/(a^5*b^7))^(1/6)*log(a*b*(-(B^6*a^6 - 6*A*B^5*a^5*b +
15*A^2*B^4*a^4*b^2 - 20*A^3*B^3*a^3*b^3 + 15*A^4*B^2*a^2*b^4 - 6*A^5*B*a*b^5 + A
^6*b^6)/(a^5*b^7))^(1/6) - (B*a - A*b)*sqrt(x)) + 2*b*(-(B^6*a^6 - 6*A*B^5*a^5*b
+ 15*A^2*B^4*a^4*b^2 - 20*A^3*B^3*a^3*b^3 + 15*A^4*B^2*a^2*b^4 - 6*A^5*B*a*b^5
+ A^6*b^6)/(a^5*b^7))^(1/6)*log(-a*b*(-(B^6*a^6 - 6*A*B^5*a^5*b + 15*A^2*B^4*a^4
*b^2 - 20*A^3*B^3*a^3*b^3 + 15*A^4*B^2*a^2*b^4 - 6*A^5*B*a*b^5 + A^6*b^6)/(a^5*b
^7))^(1/6) - (B*a - A*b)*sqrt(x)) - 12*B*sqrt(x))/b
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Sympy [A] time = 100.154, size = 864, normalized size = 3.22 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**3+A)/(b*x**3+a)/x**(1/2),x)
[Out]
Piecewise((zoo*(-2*A/(5*x**(5/2)) + 2*B*sqrt(x)), Eq(a, 0) & Eq(b, 0)), ((2*A*sq
rt(x) + 2*B*x**(7/2)/7)/a, Eq(b, 0)), ((-2*A/(5*x**(5/2)) + 2*B*sqrt(x))/b, Eq(a
, 0)), (-(-1)**(1/6)*A*log(-(-1)**(1/6)*a**(1/6)*(1/b)**(1/6) + sqrt(x))/(3*a**(
5/6)*b**15*(1/b)**(89/6)) + (-1)**(1/6)*A*log((-1)**(1/6)*a**(1/6)*(1/b)**(1/6)
+ sqrt(x))/(3*a**(5/6)*b**15*(1/b)**(89/6)) - (-1)**(1/6)*A*log(-4*(-1)**(1/6)*a
**(1/6)*sqrt(x)*(1/b)**(1/6) + 4*(-1)**(1/3)*a**(1/3)*(1/b)**(1/3) + 4*x)/(6*a**
(5/6)*b**15*(1/b)**(89/6)) + (-1)**(1/6)*A*log(4*(-1)**(1/6)*a**(1/6)*sqrt(x)*(1
/b)**(1/6) + 4*(-1)**(1/3)*a**(1/3)*(1/b)**(1/3) + 4*x)/(6*a**(5/6)*b**15*(1/b)*
*(89/6)) + (-1)**(1/6)*sqrt(3)*A*atan(sqrt(3)/3 - 2*(-1)**(5/6)*sqrt(3)*sqrt(x)/
(3*a**(1/6)*(1/b)**(1/6)))/(3*a**(5/6)*b**15*(1/b)**(89/6)) - (-1)**(1/6)*sqrt(3
)*A*atan(sqrt(3)/3 + 2*(-1)**(5/6)*sqrt(3)*sqrt(x)/(3*a**(1/6)*(1/b)**(1/6)))/(3
*a**(5/6)*b**15*(1/b)**(89/6)) + (-1)**(1/6)*B*a**(1/6)*log(-(-1)**(1/6)*a**(1/6
)*(1/b)**(1/6) + sqrt(x))/(3*b**16*(1/b)**(89/6)) - (-1)**(1/6)*B*a**(1/6)*log((
-1)**(1/6)*a**(1/6)*(1/b)**(1/6) + sqrt(x))/(3*b**16*(1/b)**(89/6)) + (-1)**(1/6
)*B*a**(1/6)*log(-4*(-1)**(1/6)*a**(1/6)*sqrt(x)*(1/b)**(1/6) + 4*(-1)**(1/3)*a*
*(1/3)*(1/b)**(1/3) + 4*x)/(6*b**16*(1/b)**(89/6)) - (-1)**(1/6)*B*a**(1/6)*log(
4*(-1)**(1/6)*a**(1/6)*sqrt(x)*(1/b)**(1/6) + 4*(-1)**(1/3)*a**(1/3)*(1/b)**(1/3
) + 4*x)/(6*b**16*(1/b)**(89/6)) - (-1)**(1/6)*sqrt(3)*B*a**(1/6)*atan(sqrt(3)/3
- 2*(-1)**(5/6)*sqrt(3)*sqrt(x)/(3*a**(1/6)*(1/b)**(1/6)))/(3*b**16*(1/b)**(89/
6)) + (-1)**(1/6)*sqrt(3)*B*a**(1/6)*atan(sqrt(3)/3 + 2*(-1)**(5/6)*sqrt(3)*sqrt
(x)/(3*a**(1/6)*(1/b)**(1/6)))/(3*b**16*(1/b)**(89/6)) + 2*B*sqrt(x)/b, True))
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GIAC/XCAS [A] time = 0.225896, size = 378, normalized size = 1.41 \[ \frac{2 \, B \sqrt{x}}{b} - \frac{\sqrt{3}{\left (\left (a b^{5}\right )^{\frac{1}{6}} B a - \left (a b^{5}\right )^{\frac{1}{6}} A b\right )}{\rm ln}\left (\sqrt{3} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{6}} + x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{6 \, a b^{2}} + \frac{\sqrt{3}{\left (\left (a b^{5}\right )^{\frac{1}{6}} B a - \left (a b^{5}\right )^{\frac{1}{6}} A b\right )}{\rm ln}\left (-\sqrt{3} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{6}} + x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{6 \, a b^{2}} - \frac{{\left (\left (a b^{5}\right )^{\frac{1}{6}} B a - \left (a b^{5}\right )^{\frac{1}{6}} A b\right )} \arctan \left (\frac{\sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}} + 2 \, \sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{3 \, a b^{2}} - \frac{{\left (\left (a b^{5}\right )^{\frac{1}{6}} B a - \left (a b^{5}\right )^{\frac{1}{6}} A b\right )} \arctan \left (-\frac{\sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}} - 2 \, \sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{3 \, a b^{2}} - \frac{2 \,{\left (\left (a b^{5}\right )^{\frac{1}{6}} B a - \left (a b^{5}\right )^{\frac{1}{6}} A b\right )} \arctan \left (\frac{\sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{3 \, a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)/((b*x^3 + a)*sqrt(x)),x, algorithm="giac")
[Out]
2*B*sqrt(x)/b - 1/6*sqrt(3)*((a*b^5)^(1/6)*B*a - (a*b^5)^(1/6)*A*b)*ln(sqrt(3)*s
qrt(x)*(a/b)^(1/6) + x + (a/b)^(1/3))/(a*b^2) + 1/6*sqrt(3)*((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^2) -
1/3*((a*b^5)^(1/6)*B*a - (a*b^5)^(1/6)*A*b)*arctan((sqrt(3)*(a/b)^(1/6) + 2*sqr
t(x))/(a/b)^(1/6))/(a*b^2) - 1/3*((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^2) - 2/3*((a*b^5)^(1/6)*B*a
- (a*b^5)^(1/6)*A*b)*arctan(sqrt(x)/(a/b)^(1/6))/(a*b^2)