3.174 \(\int \frac{\sqrt{x} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx\)

Optimal. Leaf size=104 \[ \frac{(a B+3 A b) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{12 a^{5/2} b^{3/2}}+\frac{x^{3/2} (a B+3 A b)}{12 a^2 b \left (a+b x^3\right )}+\frac{x^{3/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

[Out]

((A*b - a*B)*x^(3/2))/(6*a*b*(a + b*x^3)^2) + ((3*A*b + a*B)*x^(3/2))/(12*a^2*b*
(a + b*x^3)) + ((3*A*b + a*B)*ArcTan[(Sqrt[b]*x^(3/2))/Sqrt[a]])/(12*a^(5/2)*b^(
3/2))

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Rubi [A]  time = 0.169202, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(a B+3 A b) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{12 a^{5/2} b^{3/2}}+\frac{x^{3/2} (a B+3 A b)}{12 a^2 b \left (a+b x^3\right )}+\frac{x^{3/2} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - a*B)*x^(3/2))/(6*a*b*(a + b*x^3)^2) + ((3*A*b + a*B)*x^(3/2))/(12*a^2*b*
(a + b*x^3)) + ((3*A*b + a*B)*ArcTan[(Sqrt[b]*x^(3/2))/Sqrt[a]])/(12*a^(5/2)*b^(
3/2))

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Rubi in Sympy [A]  time = 19.3505, size = 88, normalized size = 0.85 \[ \frac{x^{\frac{3}{2}} \left (A b - B a\right )}{6 a b \left (a + b x^{3}\right )^{2}} + \frac{x^{\frac{3}{2}} \left (3 A b + B a\right )}{12 a^{2} b \left (a + b x^{3}\right )} + \frac{\left (3 A b + B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a}} \right )}}{12 a^{\frac{5}{2}} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**3+A)*x**(1/2)/(b*x**3+a)**3,x)

[Out]

x**(3/2)*(A*b - B*a)/(6*a*b*(a + b*x**3)**2) + x**(3/2)*(3*A*b + B*a)/(12*a**2*b
*(a + b*x**3)) + (3*A*b + B*a)*atan(sqrt(b)*x**(3/2)/sqrt(a))/(12*a**(5/2)*b**(3
/2))

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Mathematica [A]  time = 0.256254, size = 176, normalized size = 1.69 \[ \frac{-\frac{2 a^{3/2} \sqrt{b} x^{3/2} (a B-A b)}{\left (a+b x^3\right )^2}+\frac{\sqrt{a} \sqrt{b} x^{3/2} (a B+3 A b)}{a+b x^3}-(a B+3 A b) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )+(a B+3 A b) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )-(a B+3 A b) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{12 a^{5/2} b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((-2*a^(3/2)*Sqrt[b]*(-(A*b) + a*B)*x^(3/2))/(a + b*x^3)^2 + (Sqrt[a]*Sqrt[b]*(3
*A*b + a*B)*x^(3/2))/(a + b*x^3) - (3*A*b + a*B)*ArcTan[Sqrt[3] - (2*b^(1/6)*Sqr
t[x])/a^(1/6)] + (3*A*b + a*B)*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a^(1/6)] - (
3*A*b + a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(12*a^(5/2)*b^(3/2))

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Maple [A]  time = 0.013, size = 97, normalized size = 0.9 \[{\frac{2}{3\, \left ( b{x}^{3}+a \right ) ^{2}} \left ({\frac{3\,Ab+Ba}{8\,{a}^{2}}{x}^{{\frac{9}{2}}}}+{\frac{5\,Ab-Ba}{8\,ab}{x}^{{\frac{3}{2}}}} \right ) }+{\frac{A}{4\,{a}^{2}}\arctan \left ({b{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{B}{12\,ab}\arctan \left ({b{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^3+A)*x^(1/2)/(b*x^3+a)^3,x)

[Out]

2/3*(1/8*(3*A*b+B*a)/a^2*x^(9/2)+1/8*(5*A*b-B*a)/a/b*x^(3/2))/(b*x^3+a)^2+1/4/a^
2/(a*b)^(1/2)*arctan(x^(3/2)*b/(a*b)^(1/2))*A+1/12/a/b/(a*b)^(1/2)*arctan(x^(3/2
)*b/(a*b)^(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)*sqrt(x)/(b*x^3 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.253216, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left ({\left (B a b + 3 \, A b^{2}\right )} x^{4} -{\left (B a^{2} - 5 \, A a b\right )} x\right )} \sqrt{-a b} \sqrt{x} +{\left ({\left (B a b^{2} + 3 \, A b^{3}\right )} x^{6} + B a^{3} + 3 \, A a^{2} b + 2 \,{\left (B a^{2} b + 3 \, A a b^{2}\right )} x^{3}\right )} \log \left (\frac{2 \, a b x^{\frac{3}{2}} +{\left (b x^{3} - a\right )} \sqrt{-a b}}{b x^{3} + a}\right )}{24 \,{\left (a^{2} b^{3} x^{6} + 2 \, a^{3} b^{2} x^{3} + a^{4} b\right )} \sqrt{-a b}}, \frac{{\left ({\left (B a b + 3 \, A b^{2}\right )} x^{4} -{\left (B a^{2} - 5 \, A a b\right )} x\right )} \sqrt{a b} \sqrt{x} +{\left ({\left (B a b^{2} + 3 \, A b^{3}\right )} x^{6} + B a^{3} + 3 \, A a^{2} b + 2 \,{\left (B a^{2} b + 3 \, A a b^{2}\right )} x^{3}\right )} \arctan \left (\frac{\sqrt{a b} x^{\frac{3}{2}}}{a}\right )}{12 \,{\left (a^{2} b^{3} x^{6} + 2 \, a^{3} b^{2} x^{3} + a^{4} b\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/24*(2*((B*a*b + 3*A*b^2)*x^4 - (B*a^2 - 5*A*a*b)*x)*sqrt(-a*b)*sqrt(x) + ((B*
a*b^2 + 3*A*b^3)*x^6 + B*a^3 + 3*A*a^2*b + 2*(B*a^2*b + 3*A*a*b^2)*x^3)*log((2*a
*b*x^(3/2) + (b*x^3 - a)*sqrt(-a*b))/(b*x^3 + a)))/((a^2*b^3*x^6 + 2*a^3*b^2*x^3
+ a^4*b)*sqrt(-a*b)), 1/12*(((B*a*b + 3*A*b^2)*x^4 - (B*a^2 - 5*A*a*b)*x)*sqrt(
a*b)*sqrt(x) + ((B*a*b^2 + 3*A*b^3)*x^6 + B*a^3 + 3*A*a^2*b + 2*(B*a^2*b + 3*A*a
*b^2)*x^3)*arctan(sqrt(a*b)*x^(3/2)/a))/((a^2*b^3*x^6 + 2*a^3*b^2*x^3 + a^4*b)*s
qrt(a*b))]

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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**(1/2)/(b*x**3+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.223794, size = 113, normalized size = 1.09 \[ \frac{{\left (B a + 3 \, A b\right )} \arctan \left (\frac{b x^{\frac{3}{2}}}{\sqrt{a b}}\right )}{12 \, \sqrt{a b} a^{2} b} + \frac{B a b x^{\frac{9}{2}} + 3 \, A b^{2} x^{\frac{9}{2}} - B a^{2} x^{\frac{3}{2}} + 5 \, A a b x^{\frac{3}{2}}}{12 \,{\left (b x^{3} + a\right )}^{2} a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(B*a + 3*A*b)*arctan(b*x^(3/2)/sqrt(a*b))/(sqrt(a*b)*a^2*b) + 1/12*(B*a*b*x
^(9/2) + 3*A*b^2*x^(9/2) - B*a^2*x^(3/2) + 5*A*a*b*x^(3/2))/((b*x^3 + a)^2*a^2*b
)