3.155 \(\int \frac{x^{7/2} \left (A+B x^3\right )}{a+b x^3} \, dx\)

Optimal. Leaf size=73 \[ -\frac{2 \sqrt{a} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{3 b^{5/2}}+\frac{2 x^{3/2} (A b-a B)}{3 b^2}+\frac{2 B x^{9/2}}{9 b} \]

[Out]

(2*(A*b - a*B)*x^(3/2))/(3*b^2) + (2*B*x^(9/2))/(9*b) - (2*Sqrt[a]*(A*b - a*B)*A
rcTan[(Sqrt[b]*x^(3/2))/Sqrt[a]])/(3*b^(5/2))

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Rubi [A]  time = 0.148463, antiderivative size = 73, 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 \[ -\frac{2 \sqrt{a} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{3 b^{5/2}}+\frac{2 x^{3/2} (A b-a B)}{3 b^2}+\frac{2 B x^{9/2}}{9 b} \]

Antiderivative was successfully verified.

[In]  Int[(x^(7/2)*(A + B*x^3))/(a + b*x^3),x]

[Out]

(2*(A*b - a*B)*x^(3/2))/(3*b^2) + (2*B*x^(9/2))/(9*b) - (2*Sqrt[a]*(A*b - a*B)*A
rcTan[(Sqrt[b]*x^(3/2))/Sqrt[a]])/(3*b^(5/2))

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Rubi in Sympy [A]  time = 16.8753, size = 66, normalized size = 0.9 \[ \frac{2 B x^{\frac{9}{2}}}{9 b} - \frac{2 \sqrt{a} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a}} \right )}}{3 b^{\frac{5}{2}}} + \frac{2 x^{\frac{3}{2}} \left (A b - B a\right )}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)*(B*x**3+A)/(b*x**3+a),x)

[Out]

2*B*x**(9/2)/(9*b) - 2*sqrt(a)*(A*b - B*a)*atan(sqrt(b)*x**(3/2)/sqrt(a))/(3*b**
(5/2)) + 2*x**(3/2)*(A*b - B*a)/(3*b**2)

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Mathematica [B]  time = 0.175819, size = 180, normalized size = 2.47 \[ \frac{2 \sqrt{a} (a B-A b) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}-\sqrt{3} \sqrt [6]{a}}{\sqrt [6]{a}}\right )}{3 b^{5/2}}+\frac{2 \sqrt{a} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{3 b^{5/2}}-\frac{2 \sqrt{a} (a B-A b) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{3 b^{5/2}}+\frac{2 x^{3/2} (A b-a B)}{3 b^2}+\frac{2 B x^{9/2}}{9 b} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(7/2)*(A + B*x^3))/(a + b*x^3),x]

[Out]

(2*(A*b - a*B)*x^(3/2))/(3*b^2) + (2*B*x^(9/2))/(9*b) + (2*Sqrt[a]*(-(A*b) + a*B
)*ArcTan[(-(Sqrt[3]*a^(1/6)) + 2*b^(1/6)*Sqrt[x])/a^(1/6)])/(3*b^(5/2)) + (2*Sqr
t[a]*(-(A*b) + a*B)*ArcTan[(Sqrt[3]*a^(1/6) + 2*b^(1/6)*Sqrt[x])/a^(1/6)])/(3*b^
(5/2)) - (2*Sqrt[a]*(-(A*b) + a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(3*b^(5/2)
)

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Maple [A]  time = 0.013, size = 78, normalized size = 1.1 \[{\frac{2\,B}{9\,b}{x}^{{\frac{9}{2}}}}+{\frac{2\,A}{3\,b}{x}^{{\frac{3}{2}}}}-{\frac{2\,Ba}{3\,{b}^{2}}{x}^{{\frac{3}{2}}}}-{\frac{2\,Aa}{3\,b}\arctan \left ({b{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{2\,{a}^{2}B}{3\,{b}^{2}}\arctan \left ({b{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)*(B*x^3+A)/(b*x^3+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244268, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (B a - A b\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{3} - 2 \, b x^{\frac{3}{2}} \sqrt{-\frac{a}{b}} - a}{b x^{3} + a}\right ) - 2 \,{\left (B b x^{4} - 3 \,{\left (B a - A b\right )} x\right )} \sqrt{x}}{9 \, b^{2}}, \frac{2 \,{\left (3 \,{\left (B a - A b\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x^{\frac{3}{2}}}{\sqrt{\frac{a}{b}}}\right ) +{\left (B b x^{4} - 3 \,{\left (B a - A b\right )} x\right )} \sqrt{x}\right )}}{9 \, b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^(7/2)/(b*x^3 + a),x, algorithm="fricas")

[Out]

[-1/9*(3*(B*a - A*b)*sqrt(-a/b)*log((b*x^3 - 2*b*x^(3/2)*sqrt(-a/b) - a)/(b*x^3
+ a)) - 2*(B*b*x^4 - 3*(B*a - A*b)*x)*sqrt(x))/b^2, 2/9*(3*(B*a - A*b)*sqrt(a/b)
*arctan(x^(3/2)/sqrt(a/b)) + (B*b*x^4 - 3*(B*a - A*b)*x)*sqrt(x))/b^2]

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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(x**(7/2)*(B*x**3+A)/(b*x**3+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.21628, size = 86, normalized size = 1.18 \[ \frac{2 \,{\left (B a^{2} - A a b\right )} \arctan \left (\frac{b x^{\frac{3}{2}}}{\sqrt{a b}}\right )}{3 \, \sqrt{a b} b^{2}} + \frac{2 \,{\left (B b^{2} x^{\frac{9}{2}} - 3 \, B a b x^{\frac{3}{2}} + 3 \, A b^{2} x^{\frac{3}{2}}\right )}}{9 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*x^(7/2)/(b*x^3 + a),x, algorithm="giac")

[Out]

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