3.320 \(\int \frac{A+B x}{x^{5/2} (a+b x)} \, dx\)

Optimal. Leaf size=69 \[ \frac{2 \sqrt{b} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{2 (A b-a B)}{a^2 \sqrt{x}}-\frac{2 A}{3 a x^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.102135, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{2 \sqrt{b} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{2 (A b-a B)}{a^2 \sqrt{x}}-\frac{2 A}{3 a x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^(5/2)*(a + b*x)),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.5715, size = 65, normalized size = 0.94 \[ - \frac{2 A}{3 a x^{\frac{3}{2}}} + \frac{2 \left (A b - B a\right )}{a^{2} \sqrt{x}} + \frac{2 \sqrt{b} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**(5/2)/(b*x+a),x)

[Out]

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

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Mathematica [A]  time = 0.0808479, size = 64, normalized size = 0.93 \[ \frac{2 \sqrt{b} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{2 (a (A+3 B x)-3 A b x)}{3 a^2 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^(5/2)*(a + b*x)),x]

[Out]

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

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Maple [A]  time = 0.016, size = 78, normalized size = 1.1 \[ -{\frac{2\,A}{3\,a}{x}^{-{\frac{3}{2}}}}+2\,{\frac{Ab}{\sqrt{x}{a}^{2}}}-2\,{\frac{B}{\sqrt{x}a}}+2\,{\frac{A{b}^{2}}{{a}^{2}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) }-2\,{\frac{Bb}{a\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^(5/2)/(b*x+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221126, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (B a - A b\right )} x^{\frac{3}{2}} \sqrt{-\frac{b}{a}} \log \left (\frac{b x + 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) + 2 \, A a + 6 \,{\left (B a - A b\right )} x}{3 \, a^{2} x^{\frac{3}{2}}}, \frac{2 \,{\left (3 \,{\left (B a - A b\right )} x^{\frac{3}{2}} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) - A a - 3 \,{\left (B a - A b\right )} x\right )}}{3 \, a^{2} x^{\frac{3}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)*x^(5/2)),x, algorithm="fricas")

[Out]

[-1/3*(3*(B*a - A*b)*x^(3/2)*sqrt(-b/a)*log((b*x + 2*a*sqrt(x)*sqrt(-b/a) - a)/(
b*x + a)) + 2*A*a + 6*(B*a - A*b)*x)/(a^2*x^(3/2)), 2/3*(3*(B*a - A*b)*x^(3/2)*s
qrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x))) - A*a - 3*(B*a - A*b)*x)/(a^2*x^(3/2))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{x^{\frac{5}{2}} \left (a + b x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/x**(5/2)/(b*x+a),x)

[Out]

Integral((A + B*x)/(x**(5/2)*(a + b*x)), x)

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GIAC/XCAS [A]  time = 0.214453, size = 74, normalized size = 1.07 \[ -\frac{2 \,{\left (B a b - A b^{2}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2}} - \frac{2 \,{\left (3 \, B a x - 3 \, A b x + A a\right )}}{3 \, a^{2} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((b*x + a)*x^(5/2)),x, algorithm="giac")

[Out]

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