3.764 \(\int \frac{A+B x}{x^{7/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx\)

Optimal. Leaf size=131 \[ -\frac{b^{3/2} (7 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{9/2}}-\frac{b (7 A b-5 a B)}{a^4 \sqrt{x}}+\frac{7 A b-5 a B}{3 a^3 x^{3/2}}-\frac{7 A b-5 a B}{5 a^2 b x^{5/2}}+\frac{A b-a B}{a b x^{5/2} (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.168621, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ -\frac{b^{3/2} (7 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{9/2}}-\frac{b (7 A b-5 a B)}{a^4 \sqrt{x}}+\frac{7 A b-5 a B}{3 a^3 x^{3/2}}-\frac{7 A b-5 a B}{5 a^2 b x^{5/2}}+\frac{A b-a B}{a b x^{5/2} (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 38.2437, size = 119, normalized size = 0.91 \[ \frac{A b - B a}{a b x^{\frac{5}{2}} \left (a + b x\right )} - \frac{7 A b - 5 B a}{5 a^{2} b x^{\frac{5}{2}}} + \frac{7 A b - 5 B a}{3 a^{3} x^{\frac{3}{2}}} - \frac{b \left (7 A b - 5 B a\right )}{a^{4} \sqrt{x}} - \frac{b^{\frac{3}{2}} \left (7 A b - 5 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(7/2)/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

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

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Mathematica [A]  time = 0.192387, size = 112, normalized size = 0.85 \[ \frac{b^{3/2} (5 a B-7 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{9/2}}+\frac{-2 a^3 (3 A+5 B x)+2 a^2 b x (7 A+25 B x)+5 a b^2 x^2 (15 B x-14 A)-105 A b^3 x^3}{15 a^4 x^{5/2} (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)),x]

[Out]

(-105*A*b^3*x^3 - 2*a^3*(3*A + 5*B*x) + 5*a*b^2*x^2*(-14*A + 15*B*x) + 2*a^2*b*x
*(7*A + 25*B*x))/(15*a^4*x^(5/2)*(a + b*x)) + (b^(3/2)*(-7*A*b + 5*a*B)*ArcTan[(
Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(9/2)

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Maple [A]  time = 0.029, size = 139, normalized size = 1.1 \[ -{\frac{2\,A}{5\,{a}^{2}}{x}^{-{\frac{5}{2}}}}+{\frac{4\,Ab}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}}-{\frac{2\,B}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}-6\,{\frac{{b}^{2}A}{{a}^{4}\sqrt{x}}}+4\,{\frac{Bb}{{a}^{3}\sqrt{x}}}-{\frac{A{b}^{3}}{{a}^{4} \left ( bx+a \right ) }\sqrt{x}}+{\frac{{b}^{2}B}{{a}^{3} \left ( bx+a \right ) }\sqrt{x}}-7\,{\frac{A{b}^{3}}{{a}^{4}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) }+5\,{\frac{{b}^{2}B}{{a}^{3}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(7/2)/(b^2*x^2+2*a*b*x+a^2),x)

[Out]

-2/5*A/a^2/x^(5/2)+4/3/x^(3/2)/a^3*A*b-2/3/x^(3/2)/a^2*B-6*b^2/a^4/x^(1/2)*A+4*b
/a^3/x^(1/2)*B-b^3/a^4*x^(1/2)/(b*x+a)*A+b^2/a^3*x^(1/2)/(b*x+a)*B-7*b^3/a^4/(a*
b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*A+5*b^2/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^2*x^2 + 2*a*b*x + a^2)*x^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.319414, size = 1, normalized size = 0.01 \[ \left [-\frac{12 \, A a^{3} - 30 \,{\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} - 20 \,{\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 15 \,{\left ({\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} +{\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2}\right )} \sqrt{x} \sqrt{-\frac{b}{a}} \log \left (\frac{b x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) + 4 \,{\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x}{30 \,{\left (a^{4} b x^{3} + a^{5} x^{2}\right )} \sqrt{x}}, -\frac{6 \, A a^{3} - 15 \,{\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} - 10 \,{\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 15 \,{\left ({\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} +{\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2}\right )} \sqrt{x} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) + 2 \,{\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x}{15 \,{\left (a^{4} b x^{3} + a^{5} x^{2}\right )} \sqrt{x}}\right ] \]

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)*x^(7/2)),x, algorithm="fricas")

[Out]

[-1/30*(12*A*a^3 - 30*(5*B*a*b^2 - 7*A*b^3)*x^3 - 20*(5*B*a^2*b - 7*A*a*b^2)*x^2
+ 15*((5*B*a*b^2 - 7*A*b^3)*x^3 + (5*B*a^2*b - 7*A*a*b^2)*x^2)*sqrt(x)*sqrt(-b/
a)*log((b*x - 2*a*sqrt(x)*sqrt(-b/a) - a)/(b*x + a)) + 4*(5*B*a^3 - 7*A*a^2*b)*x
)/((a^4*b*x^3 + a^5*x^2)*sqrt(x)), -1/15*(6*A*a^3 - 15*(5*B*a*b^2 - 7*A*b^3)*x^3
- 10*(5*B*a^2*b - 7*A*a*b^2)*x^2 + 15*((5*B*a*b^2 - 7*A*b^3)*x^3 + (5*B*a^2*b -
7*A*a*b^2)*x^2)*sqrt(x)*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x))) + 2*(5*B*a^3
- 7*A*a^2*b)*x)/((a^4*b*x^3 + a^5*x^2)*sqrt(x))]

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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**(7/2)/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.272901, size = 149, normalized size = 1.14 \[ \frac{{\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{4}} + \frac{B a b^{2} \sqrt{x} - A b^{3} \sqrt{x}}{{\left (b x + a\right )} a^{4}} + \frac{2 \,{\left (30 \, B a b x^{2} - 45 \, A b^{2} x^{2} - 5 \, B a^{2} x + 10 \, A a b x - 3 \, A a^{2}\right )}}{15 \, a^{4} x^{\frac{5}{2}}} \]

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)*x^(7/2)),x, algorithm="giac")

[Out]

(5*B*a*b^2 - 7*A*b^3)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4) + (B*a*b^2*sqr
t(x) - A*b^3*sqrt(x))/((b*x + a)*a^4) + 2/15*(30*B*a*b*x^2 - 45*A*b^2*x^2 - 5*B*
a^2*x + 10*A*a*b*x - 3*A*a^2)/(a^4*x^(5/2))