3.322 \(\int \frac{A+B x}{x^{9/2} (a+b x)} \, dx\)

Optimal. Leaf size=113 \[ \frac{2 b^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{9/2}}+\frac{2 b^2 (A b-a B)}{a^4 \sqrt{x}}-\frac{2 b (A b-a B)}{3 a^3 x^{3/2}}+\frac{2 (A b-a B)}{5 a^2 x^{5/2}}-\frac{2 A}{7 a x^{7/2}} \]

[Out]

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

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Rubi [A]  time = 0.158595, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{2 b^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{9/2}}+\frac{2 b^2 (A b-a B)}{a^4 \sqrt{x}}-\frac{2 b (A b-a B)}{3 a^3 x^{3/2}}+\frac{2 (A b-a B)}{5 a^2 x^{5/2}}-\frac{2 A}{7 a x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(9/2)*(a + b*x)),x]

[Out]

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

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Rubi in Sympy [A]  time = 19.5921, size = 107, normalized size = 0.95 \[ - \frac{2 A}{7 a x^{\frac{7}{2}}} + \frac{2 \left (A b - B a\right )}{5 a^{2} x^{\frac{5}{2}}} - \frac{2 b \left (A b - B a\right )}{3 a^{3} x^{\frac{3}{2}}} + \frac{2 b^{2} \left (A b - B a\right )}{a^{4} \sqrt{x}} + \frac{2 b^{\frac{5}{2}} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(9/2)/(b*x+a),x)

[Out]

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

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Mathematica [A]  time = 0.141294, size = 103, normalized size = 0.91 \[ \frac{2 b^{5/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{9/2}}+\frac{-6 a^3 (5 A+7 B x)+14 a^2 b x (3 A+5 B x)-70 a b^2 x^2 (A+3 B x)+210 A b^3 x^3}{105 a^4 x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(9/2)*(a + b*x)),x]

[Out]

(210*A*b^3*x^3 - 70*a*b^2*x^2*(A + 3*B*x) + 14*a^2*b*x*(3*A + 5*B*x) - 6*a^3*(5*
A + 7*B*x))/(105*a^4*x^(7/2)) + (2*b^(5/2)*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/
Sqrt[a]])/a^(9/2)

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Maple [A]  time = 0.017, size = 126, normalized size = 1.1 \[ -{\frac{2\,A}{7\,a}{x}^{-{\frac{7}{2}}}}+{\frac{2\,Ab}{5\,{a}^{2}}{x}^{-{\frac{5}{2}}}}-{\frac{2\,B}{5\,a}{x}^{-{\frac{5}{2}}}}-{\frac{2\,{b}^{2}A}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}}+{\frac{2\,Bb}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}+2\,{\frac{{b}^{3}A}{{a}^{4}\sqrt{x}}}-2\,{\frac{{b}^{2}B}{{a}^{3}\sqrt{x}}}+2\,{\frac{A{b}^{4}}{{a}^{4}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) }-2\,{\frac{{b}^{3}B}{{a}^{3}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(9/2)/(b*x+a),x)

[Out]

-2/7*A/a/x^(7/2)+2/5/a^2/x^(5/2)*A*b-2/5/a/x^(5/2)*B-2/3/a^3*b^2/x^(3/2)*A+2/3/a
^2*b/x^(3/2)*B+2/a^4*b^3/x^(1/2)*A-2/a^3*b^2/x^(1/2)*B+2*b^4/a^4/(a*b)^(1/2)*arc
tan(x^(1/2)*b/(a*b)^(1/2))*A-2*b^3/a^3/(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^(9/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223307, size = 1, normalized size = 0.01 \[ \left [-\frac{105 \,{\left (B a b^{2} - A b^{3}\right )} x^{\frac{7}{2}} \sqrt{-\frac{b}{a}} \log \left (\frac{b x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) + 30 \, A a^{3} + 210 \,{\left (B a b^{2} - A b^{3}\right )} x^{3} - 70 \,{\left (B a^{2} b - A a b^{2}\right )} x^{2} + 42 \,{\left (B a^{3} - A a^{2} b\right )} x}{105 \, a^{4} x^{\frac{7}{2}}}, \frac{2 \,{\left (105 \,{\left (B a b^{2} - A b^{3}\right )} x^{\frac{7}{2}} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) - 15 \, A a^{3} - 105 \,{\left (B a b^{2} - A b^{3}\right )} x^{3} + 35 \,{\left (B a^{2} b - A a b^{2}\right )} x^{2} - 21 \,{\left (B a^{3} - A a^{2} b\right )} x\right )}}{105 \, a^{4} x^{\frac{7}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)*x^(9/2)),x, algorithm="fricas")

[Out]

[-1/105*(105*(B*a*b^2 - A*b^3)*x^(7/2)*sqrt(-b/a)*log((b*x + 2*a*sqrt(x)*sqrt(-b
/a) - a)/(b*x + a)) + 30*A*a^3 + 210*(B*a*b^2 - A*b^3)*x^3 - 70*(B*a^2*b - A*a*b
^2)*x^2 + 42*(B*a^3 - A*a^2*b)*x)/(a^4*x^(7/2)), 2/105*(105*(B*a*b^2 - A*b^3)*x^
(7/2)*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x))) - 15*A*a^3 - 105*(B*a*b^2 - A*b^
3)*x^3 + 35*(B*a^2*b - A*a*b^2)*x^2 - 21*(B*a^3 - A*a^2*b)*x)/(a^4*x^(7/2))]

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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**(9/2)/(b*x+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213955, size = 140, normalized size = 1.24 \[ -\frac{2 \,{\left (B a b^{3} - A b^{4}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{4}} - \frac{2 \,{\left (105 \, B a b^{2} x^{3} - 105 \, A b^{3} x^{3} - 35 \, B a^{2} b x^{2} + 35 \, A a b^{2} x^{2} + 21 \, B a^{3} x - 21 \, A a^{2} b x + 15 \, A a^{3}\right )}}{105 \, a^{4} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)*x^(9/2)),x, algorithm="giac")

[Out]

-2*(B*a*b^3 - A*b^4)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4) - 2/105*(105*B*
a*b^2*x^3 - 105*A*b^3*x^3 - 35*B*a^2*b*x^2 + 35*A*a*b^2*x^2 + 21*B*a^3*x - 21*A*
a^2*b*x + 15*A*a^3)/(a^4*x^(7/2))