3.455 \(\int \frac{1}{x^{7/2} (a+b x)} \, dx\)

Optimal. Leaf size=68 \[ -\frac{2 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}-\frac{2 b^2}{a^3 \sqrt{x}}+\frac{2 b}{3 a^2 x^{3/2}}-\frac{2}{5 a x^{5/2}} \]

[Out]

-2/(5*a*x^(5/2)) + (2*b)/(3*a^2*x^(3/2)) - (2*b^2)/(a^3*Sqrt[x]) - (2*b^(5/2)*Ar
cTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(7/2)

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Rubi [A]  time = 0.0589313, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{2 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}-\frac{2 b^2}{a^3 \sqrt{x}}+\frac{2 b}{3 a^2 x^{3/2}}-\frac{2}{5 a x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(7/2)*(a + b*x)),x]

[Out]

-2/(5*a*x^(5/2)) + (2*b)/(3*a^2*x^(3/2)) - (2*b^2)/(a^3*Sqrt[x]) - (2*b^(5/2)*Ar
cTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(7/2)

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Rubi in Sympy [A]  time = 11.8811, size = 65, normalized size = 0.96 \[ - \frac{2}{5 a x^{\frac{5}{2}}} + \frac{2 b}{3 a^{2} x^{\frac{3}{2}}} - \frac{2 b^{2}}{a^{3} \sqrt{x}} - \frac{2 b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(7/2)/(b*x+a),x)

[Out]

-2/(5*a*x**(5/2)) + 2*b/(3*a**2*x**(3/2)) - 2*b**2/(a**3*sqrt(x)) - 2*b**(5/2)*a
tan(sqrt(b)*sqrt(x)/sqrt(a))/a**(7/2)

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Mathematica [A]  time = 0.0449019, size = 61, normalized size = 0.9 \[ -\frac{2 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{7/2}}-\frac{2 \left (3 a^2-5 a b x+15 b^2 x^2\right )}{15 a^3 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(7/2)*(a + b*x)),x]

[Out]

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

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Maple [A]  time = 0.014, size = 54, normalized size = 0.8 \[ -{\frac{2}{5\,a}{x}^{-{\frac{5}{2}}}}-2\,{\frac{{b}^{2}}{{a}^{3}\sqrt{x}}}+{\frac{2\,b}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}-2\,{\frac{{b}^{3}}{{a}^{3}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(7/2)/(b*x+a),x)

[Out]

-2/5/a/x^(5/2)-2*b^2/a^3/x^(1/2)+2/3*b/a^2/x^(3/2)-2/a^3*b^3/(a*b)^(1/2)*arctan(
x^(1/2)*b/(a*b)^(1/2))

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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(1/((b*x + a)*x^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227736, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{2} x^{\frac{5}{2}} \sqrt{-\frac{b}{a}} \log \left (\frac{b x - 2 \, a \sqrt{x} \sqrt{-\frac{b}{a}} - a}{b x + a}\right ) - 30 \, b^{2} x^{2} + 10 \, a b x - 6 \, a^{2}}{15 \, a^{3} x^{\frac{5}{2}}}, \frac{2 \,{\left (15 \, b^{2} x^{\frac{5}{2}} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) - 15 \, b^{2} x^{2} + 5 \, a b x - 3 \, a^{2}\right )}}{15 \, a^{3} x^{\frac{5}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*x^(7/2)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 12.2853, size = 65, normalized size = 0.96 \[ - \frac{2}{5 a x^{\frac{5}{2}}} + \frac{2 b}{3 a^{2} x^{\frac{3}{2}}} - \frac{2 b^{2}}{a^{3} \sqrt{x}} - \frac{2 b^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(7/2)/(b*x+a),x)

[Out]

-2/(5*a*x**(5/2)) + 2*b/(3*a**2*x**(3/2)) - 2*b**2/(a**3*sqrt(x)) - 2*b**(5/2)*a
tan(sqrt(b)*sqrt(x)/sqrt(a))/a**(7/2)

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GIAC/XCAS [A]  time = 0.203083, size = 70, normalized size = 1.03 \[ -\frac{2 \, b^{3} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{3}} - \frac{2 \,{\left (15 \, b^{2} x^{2} - 5 \, a b x + 3 \, a^{2}\right )}}{15 \, a^{3} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)*x^(7/2)),x, algorithm="giac")

[Out]

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