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

Optimal. Leaf size=120 \[ -\frac{b (5 A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{4 a^{7/2}}+\frac{\sqrt{a+b x^3} (5 A b-4 a B)}{4 a^3 x^3}-\frac{5 A b-4 a B}{6 a^2 x^3 \sqrt{a+b x^3}}-\frac{A}{6 a x^6 \sqrt{a+b x^3}} \]

[Out]

-A/(6*a*x^6*Sqrt[a + b*x^3]) - (5*A*b - 4*a*B)/(6*a^2*x^3*Sqrt[a + b*x^3]) + ((5
*A*b - 4*a*B)*Sqrt[a + b*x^3])/(4*a^3*x^3) - (b*(5*A*b - 4*a*B)*ArcTanh[Sqrt[a +
b*x^3]/Sqrt[a]])/(4*a^(7/2))

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Rubi [A]  time = 0.273545, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{b (5 A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{4 a^{7/2}}+\frac{\sqrt{a+b x^3} (5 A b-4 a B)}{4 a^3 x^3}-\frac{5 A b-4 a B}{6 a^2 x^3 \sqrt{a+b x^3}}-\frac{A}{6 a x^6 \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

-A/(6*a*x^6*Sqrt[a + b*x^3]) - (5*A*b - 4*a*B)/(6*a^2*x^3*Sqrt[a + b*x^3]) + ((5
*A*b - 4*a*B)*Sqrt[a + b*x^3])/(4*a^3*x^3) - (b*(5*A*b - 4*a*B)*ArcTanh[Sqrt[a +
b*x^3]/Sqrt[a]])/(4*a^(7/2))

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Rubi in Sympy [A]  time = 18.1814, size = 110, normalized size = 0.92 \[ - \frac{A}{6 a x^{6} \sqrt{a + b x^{3}}} - \frac{5 A b - 4 B a}{6 a^{2} x^{3} \sqrt{a + b x^{3}}} + \frac{\sqrt{a + b x^{3}} \left (5 A b - 4 B a\right )}{4 a^{3} x^{3}} - \frac{b \left (5 A b - 4 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{4 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(6*a*x**6*sqrt(a + b*x**3)) - (5*A*b - 4*B*a)/(6*a**2*x**3*sqrt(a + b*x**3))
+ sqrt(a + b*x**3)*(5*A*b - 4*B*a)/(4*a**3*x**3) - b*(5*A*b - 4*B*a)*atanh(sqrt(
a + b*x**3)/sqrt(a))/(4*a**(7/2))

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Mathematica [A]  time = 0.597257, size = 95, normalized size = 0.79 \[ \frac{-\frac{2 a^2 \left (A+2 B x^3\right )}{x^6}+a b \left (\frac{5 A}{x^3}-12 B\right )+3 b \sqrt{\frac{b x^3}{a}+1} (4 a B-5 A b) \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )+15 A b^2}{12 a^3 \sqrt{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(15*A*b^2 + a*b*(-12*B + (5*A)/x^3) - (2*a^2*(A + 2*B*x^3))/x^6 + 3*b*(-5*A*b +
4*a*B)*Sqrt[1 + (b*x^3)/a]*ArcTanh[Sqrt[1 + (b*x^3)/a]])/(12*a^3*Sqrt[a + b*x^3]
)

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Maple [A]  time = 0.013, size = 141, normalized size = 1.2 \[ A \left ( -{\frac{1}{6\,{a}^{2}{x}^{6}}\sqrt{b{x}^{3}+a}}+{\frac{7\,b}{12\,{a}^{3}{x}^{3}}\sqrt{b{x}^{3}+a}}+{\frac{2\,{b}^{2}}{3\,{a}^{3}}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{a}{b}} \right ) b}}}}-{\frac{5\,{b}^{2}}{4}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{7}{2}}}} \right ) +B \left ( -{\frac{1}{3\,{a}^{2}{x}^{3}}\sqrt{b{x}^{3}+a}}-{\frac{2\,b}{3\,{a}^{2}}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{a}{b}} \right ) b}}}}+{b{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{5}{2}}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(-1/6/a^2*(b*x^3+a)^(1/2)/x^6+7/12/a^3*b*(b*x^3+a)^(1/2)/x^3+2/3*b^2/a^3/((x^3
+a/b)*b)^(1/2)-5/4/a^(7/2)*b^2*arctanh((b*x^3+a)^(1/2)/a^(1/2)))+B*(-1/3/a^2*(b*
x^3+a)^(1/2)/x^3-2/3*b/a^2/((x^3+a/b)*b)^(1/2)+b/a^(5/2)*arctanh((b*x^3+a)^(1/2)
/a^(1/2)))

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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)^(3/2)*x^7),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.270753, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \, \sqrt{b x^{3} + a}{\left (4 \, B a b - 5 \, A b^{2}\right )} x^{6} \log \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{3} + a} a}{x^{3}}\right ) + 2 \,{\left (3 \,{\left (4 \, B a b - 5 \, A b^{2}\right )} x^{6} +{\left (4 \, B a^{2} - 5 \, A a b\right )} x^{3} + 2 \, A a^{2}\right )} \sqrt{a}}{24 \, \sqrt{b x^{3} + a} a^{\frac{7}{2}} x^{6}}, -\frac{3 \, \sqrt{b x^{3} + a}{\left (4 \, B a b - 5 \, A b^{2}\right )} x^{6} \arctan \left (\frac{a}{\sqrt{b x^{3} + a} \sqrt{-a}}\right ) +{\left (3 \,{\left (4 \, B a b - 5 \, A b^{2}\right )} x^{6} +{\left (4 \, B a^{2} - 5 \, A a b\right )} x^{3} + 2 \, A a^{2}\right )} \sqrt{-a}}{12 \, \sqrt{b x^{3} + a} \sqrt{-a} a^{3} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/24*(3*sqrt(b*x^3 + a)*(4*B*a*b - 5*A*b^2)*x^6*log(((b*x^3 + 2*a)*sqrt(a) - 2
*sqrt(b*x^3 + a)*a)/x^3) + 2*(3*(4*B*a*b - 5*A*b^2)*x^6 + (4*B*a^2 - 5*A*a*b)*x^
3 + 2*A*a^2)*sqrt(a))/(sqrt(b*x^3 + a)*a^(7/2)*x^6), -1/12*(3*sqrt(b*x^3 + a)*(4
*B*a*b - 5*A*b^2)*x^6*arctan(a/(sqrt(b*x^3 + a)*sqrt(-a))) + (3*(4*B*a*b - 5*A*b
^2)*x^6 + (4*B*a^2 - 5*A*a*b)*x^3 + 2*A*a^2)*sqrt(-a))/(sqrt(b*x^3 + a)*sqrt(-a)
*a^3*x^6)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221717, size = 185, normalized size = 1.54 \[ -\frac{{\left (4 \, B a b - 5 \, A b^{2}\right )} \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a} a^{3}} - \frac{2 \,{\left (B a b - A b^{2}\right )}}{3 \, \sqrt{b x^{3} + a} a^{3}} - \frac{4 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} B a b - 4 \, \sqrt{b x^{3} + a} B a^{2} b - 7 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} A b^{2} + 9 \, \sqrt{b x^{3} + a} A a b^{2}}{12 \, a^{3} b^{2} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(4*B*a*b - 5*A*b^2)*arctan(sqrt(b*x^3 + a)/sqrt(-a))/(sqrt(-a)*a^3) - 2/3*(
B*a*b - A*b^2)/(sqrt(b*x^3 + a)*a^3) - 1/12*(4*(b*x^3 + a)^(3/2)*B*a*b - 4*sqrt(
b*x^3 + a)*B*a^2*b - 7*(b*x^3 + a)^(3/2)*A*b^2 + 9*sqrt(b*x^3 + a)*A*a*b^2)/(a^3
*b^2*x^6)