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

Optimal. Leaf size=88 \[ \frac{b (A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{12 a^{3/2}}+\frac{\sqrt{a+b x^3} (A b-4 a B)}{12 a x^3}-\frac{A \left (a+b x^3\right )^{3/2}}{6 a x^6} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.211303, antiderivative size = 88, 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 \[ \frac{b (A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{12 a^{3/2}}+\frac{\sqrt{a+b x^3} (A b-4 a B)}{12 a x^3}-\frac{A \left (a+b x^3\right )^{3/2}}{6 a x^6} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.329, size = 76, normalized size = 0.86 \[ - \frac{A \left (a + b x^{3}\right )^{\frac{3}{2}}}{6 a x^{6}} + \frac{\sqrt{a + b x^{3}} \left (A b - 4 B a\right )}{12 a x^{3}} + \frac{b \left (A b - 4 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{3}}}{\sqrt{a}} \right )}}{12 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.332693, size = 82, normalized size = 0.93 \[ \frac{\sqrt{a+b x^3} \left (\frac{b (A b-4 a B) \tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{\sqrt{\frac{b x^3}{a}+1}}-\frac{a \left (2 a \left (A+2 B x^3\right )+A b x^3\right )}{x^6}\right )}{12 a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 96, normalized size = 1.1 \[ A \left ( -{\frac{1}{6\,{x}^{6}}\sqrt{b{x}^{3}+a}}-{\frac{b}{12\,a{x}^{3}}\sqrt{b{x}^{3}+a}}+{\frac{{b}^{2}}{12}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}} \right ) +B \left ( -{\frac{1}{3\,{x}^{3}}\sqrt{b{x}^{3}+a}}-{\frac{b}{3}{\it Artanh} \left ({1\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(-1/6*(b*x^3+a)^(1/2)/x^6-1/12*b/a*(b*x^3+a)^(1/2)/x^3+1/12/a^(3/2)*b^2*arctan
h((b*x^3+a)^(1/2)/a^(1/2)))+B*(-1/3*(b*x^3+a)^(1/2)/x^3-1/3*b*arctanh((b*x^3+a)^
(1/2)/a^(1/2))/a^(1/2))

_______________________________________________________________________________________

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

_______________________________________________________________________________________

Fricas [A]  time = 0.27319, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (4 \, B a b - A b^{2}\right )} x^{6} \log \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{3} + a} a}{x^{3}}\right ) + 2 \,{\left ({\left (4 \, B a + A b\right )} x^{3} + 2 \, A a\right )} \sqrt{b x^{3} + a} \sqrt{a}}{24 \, a^{\frac{3}{2}} x^{6}}, \frac{{\left (4 \, B a b - A b^{2}\right )} x^{6} \arctan \left (\frac{a}{\sqrt{b x^{3} + a} \sqrt{-a}}\right ) -{\left ({\left (4 \, B a + A b\right )} x^{3} + 2 \, A a\right )} \sqrt{b x^{3} + a} \sqrt{-a}}{12 \, \sqrt{-a} a x^{6}}\right ] \]

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

_______________________________________________________________________________________

Sympy [A]  time = 68.8284, size = 160, normalized size = 1.82 \[ - \frac{A a}{6 \sqrt{b} x^{\frac{15}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{A \sqrt{b}}{4 x^{\frac{9}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{A b^{\frac{3}{2}}}{12 a x^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{A b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{12 a^{\frac{3}{2}}} - \frac{B \sqrt{b} \sqrt{\frac{a}{b x^{3}} + 1}}{3 x^{\frac{3}{2}}} - \frac{B b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{3 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.221622, size = 162, normalized size = 1.84 \[ \frac{\frac{{\left (4 \, B a b^{2} - A b^{3}\right )} \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} - \frac{4 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} B a b^{2} - 4 \, \sqrt{b x^{3} + a} B a^{2} b^{2} +{\left (b x^{3} + a\right )}^{\frac{3}{2}} A b^{3} + \sqrt{b x^{3} + a} A a b^{3}}{a b^{2} x^{6}}}{12 \, b} \]

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