Optimal. Leaf size=357 \[ \frac{x^{11/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{9/2} (3 A b-11 a B)}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{105 \sqrt{a} (a+b x) (3 A b-11 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 b^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{105 \sqrt{x} (a+b x) (3 A b-11 a B)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 x^{3/2} (a+b x) (3 A b-11 a B)}{64 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{21 x^{5/2} (3 A b-11 a B)}{64 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 x^{7/2} (3 A b-11 a B)}{32 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.468065, antiderivative size = 357, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194 \[ \frac{x^{11/2} (A b-a B)}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^{9/2} (3 A b-11 a B)}{24 a b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{105 \sqrt{a} (a+b x) (3 A b-11 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 b^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{105 \sqrt{x} (a+b x) (3 A b-11 a B)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 x^{3/2} (a+b x) (3 A b-11 a B)}{64 a b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{21 x^{5/2} (3 A b-11 a B)}{64 a b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 x^{7/2} (3 A b-11 a B)}{32 a b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
[In] Int[(x^(9/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(9/2)*(B*x+A)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.226315, size = 167, normalized size = 0.47 \[ \frac{\sqrt{b} \sqrt{x} \left (-3465 a^5 B+105 a^4 b (9 A-121 B x)+231 a^3 b^2 x (15 A-73 B x)+9 a^2 b^3 x^2 (511 A-1023 B x)+a b^4 x^3 (2511 A-1408 B x)+128 b^5 x^4 (3 A+B x)\right )+315 \sqrt{a} (a+b x)^4 (11 a B-3 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{192 b^{13/2} (a+b x)^3 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^(9/2)*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.031, size = 407, normalized size = 1.1 \[{\frac{bx+a}{192\,{b}^{6}} \left ( -1408\,B\sqrt{ab}{x}^{9/2}a{b}^{4}+384\,A\sqrt{ab}{x}^{9/2}{b}^{5}-945\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{4}a{b}^{5}-16863\,B\sqrt{ab}{x}^{5/2}{a}^{3}{b}^{2}+3465\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{4}{a}^{2}{b}^{4}-3780\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{3}{a}^{2}{b}^{4}+13860\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{3}{a}^{3}{b}^{3}+3465\,A\sqrt{ab}{x}^{3/2}{a}^{3}{b}^{2}-5670\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{2}{a}^{3}{b}^{3}-12705\,B\sqrt{ab}{x}^{3/2}{a}^{4}b+20790\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){x}^{2}{a}^{4}{b}^{2}-3780\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) x{a}^{4}{b}^{2}+13860\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) x{a}^{5}b+945\,A\sqrt{ab}\sqrt{x}{a}^{4}b+3465\,B\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{6}+2511\,A\sqrt{ab}{x}^{7/2}a{b}^{4}-9207\,B\sqrt{ab}{x}^{7/2}{a}^{2}{b}^{3}+128\,B\sqrt{ab}{x}^{11/2}{b}^{5}+4599\,A\sqrt{ab}{x}^{5/2}{a}^{2}{b}^{3}-945\,A\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ){a}^{5}b-3465\,B\sqrt{ab}\sqrt{x}{a}^{5} \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(9/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^(5/2),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)*x^(9/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.338997, size = 1, normalized size = 0. \[ \left [-\frac{315 \,{\left (11 \, B a^{5} - 3 \, A a^{4} b +{\left (11 \, B a b^{4} - 3 \, A b^{5}\right )} x^{4} + 4 \,{\left (11 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{3} + 6 \,{\left (11 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} + 4 \,{\left (11 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (128 \, B b^{5} x^{5} - 3465 \, B a^{5} + 945 \, A a^{4} b - 128 \,{\left (11 \, B a b^{4} - 3 \, A b^{5}\right )} x^{4} - 837 \,{\left (11 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{3} - 1533 \,{\left (11 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} - 1155 \,{\left (11 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x\right )} \sqrt{x}}{384 \,{\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}}, \frac{315 \,{\left (11 \, B a^{5} - 3 \, A a^{4} b +{\left (11 \, B a b^{4} - 3 \, A b^{5}\right )} x^{4} + 4 \,{\left (11 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{3} + 6 \,{\left (11 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} + 4 \,{\left (11 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) +{\left (128 \, B b^{5} x^{5} - 3465 \, B a^{5} + 945 \, A a^{4} b - 128 \,{\left (11 \, B a b^{4} - 3 \, A b^{5}\right )} x^{4} - 837 \,{\left (11 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{3} - 1533 \,{\left (11 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} - 1155 \,{\left (11 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x\right )} \sqrt{x}}{192 \,{\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^(9/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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(x**(9/2)*(B*x+A)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.286569, size = 258, normalized size = 0.72 \[ \frac{105 \,{\left (11 \, B a^{2} - 3 \, A a b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{64 \, \sqrt{a b} b^{6}{\rm sign}\left (b x + a\right )} - \frac{2295 \, B a^{2} b^{3} x^{\frac{7}{2}} - 975 \, A a b^{4} x^{\frac{7}{2}} + 5855 \, B a^{3} b^{2} x^{\frac{5}{2}} - 2295 \, A a^{2} b^{3} x^{\frac{5}{2}} + 5153 \, B a^{4} b x^{\frac{3}{2}} - 1929 \, A a^{3} b^{2} x^{\frac{3}{2}} + 1545 \, B a^{5} \sqrt{x} - 561 \, A a^{4} b \sqrt{x}}{192 \,{\left (b x + a\right )}^{4} b^{6}{\rm sign}\left (b x + a\right )} + \frac{2 \,{\left (B b^{10} x^{\frac{3}{2}} - 15 \, B a b^{9} \sqrt{x} + 3 \, A b^{10} \sqrt{x}\right )}}{3 \, b^{15}{\rm sign}\left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^(9/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="giac")
[Out]