Optimal. Leaf size=243 \[ \frac{3 b \log (x) (a+b x) (2 A b-a B)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 b (a+b x) (2 A b-a B) \log (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b (3 A b-2 a B)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (3 A b-a B)}{a^4 x \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b (A b-a B)}{2 a^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A (a+b x)}{2 a^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.402413, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{3 b \log (x) (a+b x) (2 A b-a B)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 b (a+b x) (2 A b-a B) \log (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b (3 A b-2 a B)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(a+b x) (3 A b-a B)}{a^4 x \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b (A b-a B)}{2 a^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{A (a+b x)}{2 a^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x^3*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 36.7465, size = 245, normalized size = 1.01 \[ - \frac{A \left (2 a + 2 b x\right )}{4 a x^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{\left (2 a + 2 b x\right ) \left (2 A b - B a\right )}{4 a^{2} x \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{3 \left (2 A b - B a\right )}{2 a^{3} x \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{3 b \left (2 A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a^{5} \left (a + b x\right )} - \frac{3 b \left (2 A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{a^{5} \left (a + b x\right )} + \frac{3 \left (2 A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{a^{5} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/x**3/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.15029, size = 133, normalized size = 0.55 \[ \frac{-a \left (a^3 (A+2 B x)+a^2 b x (9 B x-4 A)+6 a b^2 x^2 (B x-3 A)-12 A b^3 x^3\right )+6 b x^2 \log (x) (a+b x)^2 (2 A b-a B)+6 b x^2 (a+b x)^2 (a B-2 A b) \log (a+b x)}{2 a^5 x^2 (a+b x) \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x^3*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.024, size = 263, normalized size = 1.1 \[{\frac{ \left ( 12\,A\ln \left ( x \right ){x}^{4}{b}^{4}-12\,A\ln \left ( bx+a \right ){x}^{4}{b}^{4}-6\,B\ln \left ( x \right ){x}^{4}a{b}^{3}+6\,B\ln \left ( bx+a \right ){x}^{4}a{b}^{3}+24\,A\ln \left ( x \right ){x}^{3}a{b}^{3}-24\,A\ln \left ( bx+a \right ){x}^{3}a{b}^{3}-12\,B\ln \left ( x \right ){x}^{3}{a}^{2}{b}^{2}+12\,B\ln \left ( bx+a \right ){x}^{3}{a}^{2}{b}^{2}+12\,A\ln \left ( x \right ){x}^{2}{a}^{2}{b}^{2}-12\,A\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{2}+12\,aA{b}^{3}{x}^{3}-6\,B\ln \left ( x \right ){x}^{2}{a}^{3}b+6\,B\ln \left ( bx+a \right ){x}^{2}{a}^{3}b-6\,B{x}^{3}{a}^{2}{b}^{2}+18\,{a}^{2}A{b}^{2}{x}^{2}-9\,B{x}^{2}{a}^{3}b+4\,{a}^{3}Abx-2\,{a}^{4}Bx-A{a}^{4} \right ) \left ( bx+a \right ) }{2\,{a}^{5}{x}^{2}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x^3/(b^2*x^2+2*a*b*x+a^2)^(3/2),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)/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*x^3),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.291352, size = 304, normalized size = 1.25 \[ -\frac{A a^{4} + 6 \,{\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} + 9 \,{\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2} + 2 \,{\left (B a^{4} - 2 \, A a^{3} b\right )} x - 6 \,{\left ({\left (B a b^{3} - 2 \, A b^{4}\right )} x^{4} + 2 \,{\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} +{\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) + 6 \,{\left ({\left (B a b^{3} - 2 \, A b^{4}\right )} x^{4} + 2 \,{\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} +{\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{5} b^{2} x^{4} + 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*x^3),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{x^{3} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/x**3/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.593798, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*x^3),x, algorithm="giac")
[Out]