Optimal. Leaf size=115 \[ \frac{b^2 (5 A b-6 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{7/2}}-\frac{b \sqrt{a+b x} (5 A b-6 a B)}{8 a^3 x}+\frac{\sqrt{a+b x} (5 A b-6 a B)}{12 a^2 x^2}-\frac{A \sqrt{a+b x}}{3 a x^3} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.151893, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{b^2 (5 A b-6 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{7/2}}-\frac{b \sqrt{a+b x} (5 A b-6 a B)}{8 a^3 x}+\frac{\sqrt{a+b x} (5 A b-6 a B)}{12 a^2 x^2}-\frac{A \sqrt{a+b x}}{3 a x^3} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x^4*Sqrt[a + b*x]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 12.7658, size = 105, normalized size = 0.91 \[ - \frac{A \sqrt{a + b x}}{3 a x^{3}} + \frac{\sqrt{a + b x} \left (5 A b - 6 B a\right )}{12 a^{2} x^{2}} - \frac{b \sqrt{a + b x} \left (5 A b - 6 B a\right )}{8 a^{3} x} + \frac{b^{2} \left (5 A b - 6 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{8 a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/x**4/(b*x+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.155461, size = 93, normalized size = 0.81 \[ \frac{b^2 (5 A b-6 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{7/2}}+\frac{\sqrt{a+b x} \left (-4 a^2 (2 A+3 B x)+2 a b x (5 A+9 B x)-15 A b^2 x^2\right )}{24 a^3 x^3} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x^4*Sqrt[a + b*x]),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.017, size = 104, normalized size = 0.9 \[ 2\,{b}^{2} \left ({\frac{1}{{x}^{3}{b}^{3}} \left ( -1/16\,{\frac{ \left ( 5\,Ab-6\,Ba \right ) \left ( bx+a \right ) ^{5/2}}{{a}^{3}}}+1/6\,{\frac{ \left ( 5\,Ab-6\,Ba \right ) \left ( bx+a \right ) ^{3/2}}{{a}^{2}}}-1/16\,{\frac{ \left ( 11\,Ab-10\,Ba \right ) \sqrt{bx+a}}{a}} \right ) }+1/16\,{\frac{5\,Ab-6\,Ba}{{a}^{7/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x^4/(b*x+a)^(1/2),x)
[Out]
_______________________________________________________________________________________
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 + A)/(sqrt(b*x + a)*x^4),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.229695, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (6 \, B a b^{2} - 5 \, A b^{3}\right )} x^{3} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (8 \, A a^{2} - 3 \,{\left (6 \, B a b - 5 \, A b^{2}\right )} x^{2} + 2 \,{\left (6 \, B a^{2} - 5 \, A a b\right )} x\right )} \sqrt{b x + a} \sqrt{a}}{48 \, a^{\frac{7}{2}} x^{3}}, \frac{3 \,{\left (6 \, B a b^{2} - 5 \, A b^{3}\right )} x^{3} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) -{\left (8 \, A a^{2} - 3 \,{\left (6 \, B a b - 5 \, A b^{2}\right )} x^{2} + 2 \,{\left (6 \, B a^{2} - 5 \, A a b\right )} x\right )} \sqrt{b x + a} \sqrt{-a}}{24 \, \sqrt{-a} a^{3} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(sqrt(b*x + a)*x^4),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 90.4866, size = 245, normalized size = 2.13 \[ - \frac{A}{3 \sqrt{b} x^{\frac{7}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{A \sqrt{b}}{12 a x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{5 A b^{\frac{3}{2}}}{24 a^{2} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{5 A b^{\frac{5}{2}}}{8 a^{3} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{5 A b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{8 a^{\frac{7}{2}}} - \frac{B}{2 \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{B \sqrt{b}}{4 a x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{3 B b^{\frac{3}{2}}}{4 a^{2} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} - \frac{3 B b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 a^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/x**4/(b*x+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.214179, size = 194, normalized size = 1.69 \[ \frac{\frac{3 \,{\left (6 \, B a b^{3} - 5 \, A b^{4}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{18 \,{\left (b x + a\right )}^{\frac{5}{2}} B a b^{3} - 48 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{2} b^{3} + 30 \, \sqrt{b x + a} B a^{3} b^{3} - 15 \,{\left (b x + a\right )}^{\frac{5}{2}} A b^{4} + 40 \,{\left (b x + a\right )}^{\frac{3}{2}} A a b^{4} - 33 \, \sqrt{b x + a} A a^{2} b^{4}}{a^{3} b^{3} x^{3}}}{24 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(sqrt(b*x + a)*x^4),x, algorithm="giac")
[Out]