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

Optimal. Leaf size=44 \[ \frac{4 b (a+b x)^{3/2}}{15 a^2 x^{3/2}}-\frac{2 (a+b x)^{3/2}}{5 a x^{5/2}} \]

[Out]

(-2*(a + b*x)^(3/2))/(5*a*x^(5/2)) + (4*b*(a + b*x)^(3/2))/(15*a^2*x^(3/2))

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Rubi [A]  time = 0.0265544, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{4 b (a+b x)^{3/2}}{15 a^2 x^{3/2}}-\frac{2 (a+b x)^{3/2}}{5 a x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x]/x^(7/2),x]

[Out]

(-2*(a + b*x)^(3/2))/(5*a*x^(5/2)) + (4*b*(a + b*x)^(3/2))/(15*a^2*x^(3/2))

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Rubi in Sympy [A]  time = 3.54827, size = 39, normalized size = 0.89 \[ - \frac{2 \left (a + b x\right )^{\frac{3}{2}}}{5 a x^{\frac{5}{2}}} + \frac{4 b \left (a + b x\right )^{\frac{3}{2}}}{15 a^{2} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(a + b*x)**(3/2)/(5*a*x**(5/2)) + 4*b*(a + b*x)**(3/2)/(15*a**2*x**(3/2))

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Mathematica [A]  time = 0.01915, size = 39, normalized size = 0.89 \[ -\frac{2 \sqrt{a+b x} \left (3 a^2+a b x-2 b^2 x^2\right )}{15 a^2 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 24, normalized size = 0.6 \[ -{\frac{-4\,bx+6\,a}{15\,{a}^{2}} \left ( bx+a \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/2)/x^(7/2),x)

[Out]

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

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Maxima [A]  time = 1.34821, size = 42, normalized size = 0.95 \[ \frac{2 \,{\left (\frac{5 \,{\left (b x + a\right )}^{\frac{3}{2}} b}{x^{\frac{3}{2}}} - \frac{3 \,{\left (b x + a\right )}^{\frac{5}{2}}}{x^{\frac{5}{2}}}\right )}}{15 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.208085, size = 46, normalized size = 1.05 \[ \frac{2 \,{\left (2 \, b^{2} x^{2} - a b x - 3 \, a^{2}\right )} \sqrt{b x + a}}{15 \, a^{2} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 140.216, size = 65, normalized size = 1.48 \[ - \frac{2 \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{5 x^{2}} - \frac{2 b^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}}{15 a x} + \frac{4 b^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}}{15 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(b)*sqrt(a/(b*x) + 1)/(5*x**2) - 2*b**(3/2)*sqrt(a/(b*x) + 1)/(15*a*x) +
4*b**(5/2)*sqrt(a/(b*x) + 1)/(15*a**2)

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GIAC/XCAS [A]  time = 0.208339, size = 68, normalized size = 1.55 \[ -\frac{{\left (b x + a\right )}^{\frac{3}{2}} b{\left (\frac{2 \,{\left (b x + a\right )}}{a^{3} b^{4}} - \frac{5}{a^{2} b^{4}}\right )}}{480 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{5}{2}}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/480*(b*x + a)^(3/2)*b*(2*(b*x + a)/(a^3*b^4) - 5/(a^2*b^4))/(((b*x + a)*b - a
*b)^(5/2)*abs(b))