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 = 0.0053334, 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, Rules used = {45, 37} \[ \frac{4 b (a+b x)^{3/2}}{15 a^2 x^{3/2}}-\frac{2 (a+b x)^{3/2}}{5 a x^{5/2}} \]

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))

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{\sqrt{a+b x}}{x^{7/2}} \, dx &=-\frac{2 (a+b x)^{3/2}}{5 a x^{5/2}}-\frac{(2 b) \int \frac{\sqrt{a+b x}}{x^{5/2}} \, dx}{5 a}\\ &=-\frac{2 (a+b x)^{3/2}}{5 a x^{5/2}}+\frac{4 b (a+b x)^{3/2}}{15 a^2 x^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0086478, size = 29, normalized size = 0.66 \[ -\frac{2 (3 a-2 b x) (a+b x)^{3/2}}{15 a^2 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*x]/x^(7/2),x]

[Out]

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

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Maple [A]  time = 0.003, size = 24, normalized size = 0.6 \begin{align*} -{\frac{-4\,bx+6\,a}{15\,{a}^{2}} \left ( bx+a \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{5}{2}}}} \end{align*}

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 = 1.09814, size = 42, normalized size = 0.95 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/2)/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 = 1.57641, size = 84, normalized size = 1.91 \begin{align*} \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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/2)/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 = 20.8099, size = 65, normalized size = 1.48 \begin{align*} - \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}} \end{align*}

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 [A]  time = 1.30399, size = 68, normalized size = 1.55 \begin{align*} -\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 |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(1/2)/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))