3.218 \(\int \frac{A+B x^3}{x^7 \sqrt{a+b x^3}} \, dx\)

Optimal. Leaf size=90 \[ -\frac{b (3 A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{12 a^{5/2}}+\frac{\sqrt{a+b x^3} (3 A b-4 a B)}{12 a^2 x^3}-\frac{A \sqrt{a+b x^3}}{6 a x^6} \]

[Out]

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

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Rubi [A]  time = 0.216429, antiderivative size = 90, 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 \[ -\frac{b (3 A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{12 a^{5/2}}+\frac{\sqrt{a+b x^3} (3 A b-4 a B)}{12 a^2 x^3}-\frac{A \sqrt{a+b x^3}}{6 a x^6} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.5486, size = 82, normalized size = 0.91 \[ - \frac{A \sqrt{a + b x^{3}}}{6 a x^{6}} + \frac{\sqrt{a + b x^{3}} \left (3 A b - 4 B a\right )}{12 a^{2} x^{3}} - \frac{b \left (3 A b - 4 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{12 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**3+A)/x**7/(b*x**3+a)**(1/2),x)

[Out]

-A*sqrt(a + b*x**3)/(6*a*x**6) + sqrt(a + b*x**3)*(3*A*b - 4*B*a)/(12*a**2*x**3)
- b*(3*A*b - 4*B*a)*atanh(sqrt(a + b*x**3)/sqrt(a))/(12*a**(5/2))

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Mathematica [A]  time = 0.170723, size = 101, normalized size = 1.12 \[ \frac{b \sqrt{a+b x^3} (4 a B-3 A b) \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{12 a^3 \sqrt{\frac{b x^3}{a}+1}}+\sqrt{a+b x^3} \left (\frac{3 A b-4 a B}{12 a^2 x^3}-\frac{A}{6 a x^6}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 102, normalized size = 1.1 \[ A \left ( -{\frac{1}{6\,a{x}^{6}}\sqrt{b{x}^{3}+a}}+{\frac{b}{4\,{a}^{2}{x}^{3}}\sqrt{b{x}^{3}+a}}-{\frac{{b}^{2}}{4}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{5}{2}}}} \right ) +B \left ( -{\frac{1}{3\,a{x}^{3}}\sqrt{b{x}^{3}+a}}+{\frac{b}{3}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^3+A)/x^7/(b*x^3+a)^(1/2),x)

[Out]

A*(-1/6/a*(b*x^3+a)^(1/2)/x^6+1/4*b/a^2*(b*x^3+a)^(1/2)/x^3-1/4/a^(5/2)*b^2*arct
anh((b*x^3+a)^(1/2)/a^(1/2)))+B*(-1/3/a*(b*x^3+a)^(1/2)/x^3+1/3*b/a^(3/2)*arctan
h((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)/(sqrt(b*x^3 + a)*x^7),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.274893, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (4 \, B a b - 3 \, A b^{2}\right )} x^{6} \log \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{3} + a} a}{x^{3}}\right ) + 2 \,{\left ({\left (4 \, B a - 3 \, A b\right )} x^{3} + 2 \, A a\right )} \sqrt{b x^{3} + a} \sqrt{a}}{24 \, a^{\frac{5}{2}} x^{6}}, -\frac{{\left (4 \, B a b - 3 \, A b^{2}\right )} x^{6} \arctan \left (\frac{a}{\sqrt{b x^{3} + a} \sqrt{-a}}\right ) +{\left ({\left (4 \, B a - 3 \, A b\right )} x^{3} + 2 \, A a\right )} \sqrt{b x^{3} + a} \sqrt{-a}}{12 \, \sqrt{-a} a^{2} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 50.3168, size = 163, normalized size = 1.81 \[ - \frac{A}{6 \sqrt{b} x^{\frac{15}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{A \sqrt{b}}{12 a x^{\frac{9}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{A b^{\frac{3}{2}}}{4 a^{2} x^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{A b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{4 a^{\frac{5}{2}}} - \frac{B \sqrt{b} \sqrt{\frac{a}{b x^{3}} + 1}}{3 a x^{\frac{3}{2}}} + \frac{B b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{3 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**3+A)/x**7/(b*x**3+a)**(1/2),x)

[Out]

-A/(6*sqrt(b)*x**(15/2)*sqrt(a/(b*x**3) + 1)) + A*sqrt(b)/(12*a*x**(9/2)*sqrt(a/
(b*x**3) + 1)) + A*b**(3/2)/(4*a**2*x**(3/2)*sqrt(a/(b*x**3) + 1)) - A*b**2*asin
h(sqrt(a)/(sqrt(b)*x**(3/2)))/(4*a**(5/2)) - B*sqrt(b)*sqrt(a/(b*x**3) + 1)/(3*a
*x**(3/2)) + B*b*asinh(sqrt(a)/(sqrt(b)*x**(3/2)))/(3*a**(3/2))

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GIAC/XCAS [A]  time = 0.219377, size = 163, normalized size = 1.81 \[ -\frac{\frac{{\left (4 \, B a b^{2} - 3 \, A b^{3}\right )} \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{4 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} B a b^{2} - 4 \, \sqrt{b x^{3} + a} B a^{2} b^{2} - 3 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} A b^{3} + 5 \, \sqrt{b x^{3} + a} A a b^{3}}{a^{2} b^{2} x^{6}}}{12 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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