3.323 \(\int \frac{A+B x}{x^{11/2} (a+b x)} \, dx\)

Optimal. Leaf size=136 \[ -\frac{2 b^{7/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{11/2}}-\frac{2 b^3 (A b-a B)}{a^5 \sqrt{x}}+\frac{2 b^2 (A b-a B)}{3 a^4 x^{3/2}}-\frac{2 b (A b-a B)}{5 a^3 x^{5/2}}+\frac{2 (A b-a B)}{7 a^2 x^{7/2}}-\frac{2 A}{9 a x^{9/2}} \]

[Out]

(-2*A)/(9*a*x^(9/2)) + (2*(A*b - a*B))/(7*a^2*x^(7/2)) - (2*b*(A*b - a*B))/(5*a^
3*x^(5/2)) + (2*b^2*(A*b - a*B))/(3*a^4*x^(3/2)) - (2*b^3*(A*b - a*B))/(a^5*Sqrt
[x]) - (2*b^(7/2)*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(11/2)

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Rubi [A]  time = 0.191084, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{2 b^{7/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{11/2}}-\frac{2 b^3 (A b-a B)}{a^5 \sqrt{x}}+\frac{2 b^2 (A b-a B)}{3 a^4 x^{3/2}}-\frac{2 b (A b-a B)}{5 a^3 x^{5/2}}+\frac{2 (A b-a B)}{7 a^2 x^{7/2}}-\frac{2 A}{9 a x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(11/2)*(a + b*x)),x]

[Out]

(-2*A)/(9*a*x^(9/2)) + (2*(A*b - a*B))/(7*a^2*x^(7/2)) - (2*b*(A*b - a*B))/(5*a^
3*x^(5/2)) + (2*b^2*(A*b - a*B))/(3*a^4*x^(3/2)) - (2*b^3*(A*b - a*B))/(a^5*Sqrt
[x]) - (2*b^(7/2)*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(11/2)

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Rubi in Sympy [A]  time = 24.4444, size = 129, normalized size = 0.95 \[ - \frac{2 A}{9 a x^{\frac{9}{2}}} + \frac{2 \left (A b - B a\right )}{7 a^{2} x^{\frac{7}{2}}} - \frac{2 b \left (A b - B a\right )}{5 a^{3} x^{\frac{5}{2}}} + \frac{2 b^{2} \left (A b - B a\right )}{3 a^{4} x^{\frac{3}{2}}} - \frac{2 b^{3} \left (A b - B a\right )}{a^{5} \sqrt{x}} - \frac{2 b^{\frac{7}{2}} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(11/2)/(b*x+a),x)

[Out]

-2*A/(9*a*x**(9/2)) + 2*(A*b - B*a)/(7*a**2*x**(7/2)) - 2*b*(A*b - B*a)/(5*a**3*
x**(5/2)) + 2*b**2*(A*b - B*a)/(3*a**4*x**(3/2)) - 2*b**3*(A*b - B*a)/(a**5*sqrt
(x)) - 2*b**(7/2)*(A*b - B*a)*atan(sqrt(b)*sqrt(x)/sqrt(a))/a**(11/2)

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Mathematica [A]  time = 0.175334, size = 122, normalized size = 0.9 \[ \frac{2 b^{7/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{11/2}}-\frac{2 \left (5 a^4 (7 A+9 B x)-9 a^3 b x (5 A+7 B x)+21 a^2 b^2 x^2 (3 A+5 B x)-105 a b^3 x^3 (A+3 B x)+315 A b^4 x^4\right )}{315 a^5 x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(11/2)*(a + b*x)),x]

[Out]

(-2*(315*A*b^4*x^4 - 105*a*b^3*x^3*(A + 3*B*x) + 21*a^2*b^2*x^2*(3*A + 5*B*x) -
9*a^3*b*x*(5*A + 7*B*x) + 5*a^4*(7*A + 9*B*x)))/(315*a^5*x^(9/2)) + (2*b^(7/2)*(
-(A*b) + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(11/2)

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Maple [A]  time = 0.017, size = 150, normalized size = 1.1 \[ -{\frac{2\,A}{9\,a}{x}^{-{\frac{9}{2}}}}+{\frac{2\,Ab}{7\,{a}^{2}}{x}^{-{\frac{7}{2}}}}-{\frac{2\,B}{7\,a}{x}^{-{\frac{7}{2}}}}-2\,{\frac{{b}^{4}A}{{a}^{5}\sqrt{x}}}+2\,{\frac{{b}^{3}B}{{a}^{4}\sqrt{x}}}-{\frac{2\,{b}^{2}A}{5\,{a}^{3}}{x}^{-{\frac{5}{2}}}}+{\frac{2\,Bb}{5\,{a}^{2}}{x}^{-{\frac{5}{2}}}}+{\frac{2\,{b}^{3}A}{3\,{a}^{4}}{x}^{-{\frac{3}{2}}}}-{\frac{2\,{b}^{2}B}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}}-2\,{\frac{A{b}^{5}}{{a}^{5}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) }+2\,{\frac{{b}^{4}B}{{a}^{4}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(11/2)/(b*x+a),x)

[Out]

-2/9*A/a/x^(9/2)+2/7/a^2/x^(7/2)*A*b-2/7/a/x^(7/2)*B-2/a^5*b^4/x^(1/2)*A+2/a^4*b
^3/x^(1/2)*B-2/5/a^3*b^2/x^(5/2)*A+2/5/a^2*b/x^(5/2)*B+2/3/a^4*b^3/x^(3/2)*A-2/3
/a^3*b^2/x^(3/2)*B-2*b^5/a^5/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*A+2*b^4/a
^4/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*B

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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 + A)/((b*x + a)*x^(11/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224919, size = 1, normalized size = 0.01 \[ \left [-\frac{315 \,{\left (B a b^{3} - A b^{4}\right )} x^{\frac{9}{2}} \sqrt{-\frac{b}{a}} \log \left (\frac{b x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) + 70 \, A a^{4} - 630 \,{\left (B a b^{3} - A b^{4}\right )} x^{4} + 210 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{3} - 126 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x^{2} + 90 \,{\left (B a^{4} - A a^{3} b\right )} x}{315 \, a^{5} x^{\frac{9}{2}}}, -\frac{2 \,{\left (315 \,{\left (B a b^{3} - A b^{4}\right )} x^{\frac{9}{2}} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) + 35 \, A a^{4} - 315 \,{\left (B a b^{3} - A b^{4}\right )} x^{4} + 105 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{3} - 63 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x^{2} + 45 \,{\left (B a^{4} - A a^{3} b\right )} x\right )}}{315 \, a^{5} x^{\frac{9}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)*x^(11/2)),x, algorithm="fricas")

[Out]

[-1/315*(315*(B*a*b^3 - A*b^4)*x^(9/2)*sqrt(-b/a)*log((b*x - 2*a*sqrt(x)*sqrt(-b
/a) - a)/(b*x + a)) + 70*A*a^4 - 630*(B*a*b^3 - A*b^4)*x^4 + 210*(B*a^2*b^2 - A*
a*b^3)*x^3 - 126*(B*a^3*b - A*a^2*b^2)*x^2 + 90*(B*a^4 - A*a^3*b)*x)/(a^5*x^(9/2
)), -2/315*(315*(B*a*b^3 - A*b^4)*x^(9/2)*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x
))) + 35*A*a^4 - 315*(B*a*b^3 - A*b^4)*x^4 + 105*(B*a^2*b^2 - A*a*b^3)*x^3 - 63*
(B*a^3*b - A*a^2*b^2)*x^2 + 45*(B*a^4 - A*a^3*b)*x)/(a^5*x^(9/2))]

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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((B*x+A)/x**(11/2)/(b*x+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219479, size = 173, normalized size = 1.27 \[ \frac{2 \,{\left (B a b^{4} - A b^{5}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{5}} + \frac{2 \,{\left (315 \, B a b^{3} x^{4} - 315 \, A b^{4} x^{4} - 105 \, B a^{2} b^{2} x^{3} + 105 \, A a b^{3} x^{3} + 63 \, B a^{3} b x^{2} - 63 \, A a^{2} b^{2} x^{2} - 45 \, B a^{4} x + 45 \, A a^{3} b x - 35 \, A a^{4}\right )}}{315 \, a^{5} x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)*x^(11/2)),x, algorithm="giac")

[Out]

2*(B*a*b^4 - A*b^5)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^5) + 2/315*(315*B*a
*b^3*x^4 - 315*A*b^4*x^4 - 105*B*a^2*b^2*x^3 + 105*A*a*b^3*x^3 + 63*B*a^3*b*x^2
- 63*A*a^2*b^2*x^2 - 45*B*a^4*x + 45*A*a^3*b*x - 35*A*a^4)/(a^5*x^(9/2))