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

Optimal. Leaf size=90 \[ -\frac{2 b^{3/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}-\frac{2 b (A b-a B)}{a^3 \sqrt{x}}+\frac{2 (A b-a B)}{3 a^2 x^{3/2}}-\frac{2 A}{5 a x^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.127212, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{2 b^{3/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}-\frac{2 b (A b-a B)}{a^3 \sqrt{x}}+\frac{2 (A b-a B)}{3 a^2 x^{3/2}}-\frac{2 A}{5 a x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.0477, size = 85, normalized size = 0.94 \[ - \frac{2 A}{5 a x^{\frac{5}{2}}} + \frac{2 \left (A b - B a\right )}{3 a^{2} x^{\frac{3}{2}}} - \frac{2 b \left (A b - B a\right )}{a^{3} \sqrt{x}} - \frac{2 b^{\frac{3}{2}} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.112648, size = 83, normalized size = 0.92 \[ \frac{2 b^{3/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}-\frac{2 \left (a^2 (3 A+5 B x)-5 a b x (A+3 B x)+15 A b^2 x^2\right )}{15 a^3 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 102, normalized size = 1.1 \[ -{\frac{2\,A}{5\,a}{x}^{-{\frac{5}{2}}}}+{\frac{2\,Ab}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}-{\frac{2\,B}{3\,a}{x}^{-{\frac{3}{2}}}}-2\,{\frac{{b}^{2}A}{{a}^{3}\sqrt{x}}}+2\,{\frac{Bb}{{a}^{2}\sqrt{x}}}-2\,{\frac{A{b}^{3}}{{a}^{3}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) }+2\,{\frac{{b}^{2}B}{{a}^{2}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(7/2)/(b*x+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218835, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (B a b - A b^{2}\right )} x^{\frac{5}{2}} \sqrt{-\frac{b}{a}} \log \left (\frac{b x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) + 6 \, A a^{2} - 30 \,{\left (B a b - A b^{2}\right )} x^{2} + 10 \,{\left (B a^{2} - A a b\right )} x}{15 \, a^{3} x^{\frac{5}{2}}}, -\frac{2 \,{\left (15 \,{\left (B a b - A b^{2}\right )} x^{\frac{5}{2}} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) + 3 \, A a^{2} - 15 \,{\left (B a b - A b^{2}\right )} x^{2} + 5 \,{\left (B a^{2} - A a b\right )} x\right )}}{15 \, a^{3} x^{\frac{5}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)*x^(7/2)),x, algorithm="fricas")

[Out]

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

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.214657, size = 108, normalized size = 1.2 \[ \frac{2 \,{\left (B a b^{2} - A b^{3}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{3}} + \frac{2 \,{\left (15 \, B a b x^{2} - 15 \, A b^{2} x^{2} - 5 \, B a^{2} x + 5 \, A a b x - 3 \, A a^{2}\right )}}{15 \, a^{3} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)*x^(7/2)),x, algorithm="giac")

[Out]

2*(B*a*b^2 - A*b^3)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^3) + 2/15*(15*B*a*b
*x^2 - 15*A*b^2*x^2 - 5*B*a^2*x + 5*A*a*b*x - 3*A*a^2)/(a^3*x^(5/2))