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

Optimal. Leaf size=19 \[ -\frac{2 a}{5 x^{5/2}}-\frac{2 b}{\sqrt{x}} \]

[Out]

(-2*a)/(5*x^(5/2)) - (2*b)/Sqrt[x]

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Rubi [A]  time = 0.0039203, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {14} \[ -\frac{2 a}{5 x^{5/2}}-\frac{2 b}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)/x^(7/2),x]

[Out]

(-2*a)/(5*x^(5/2)) - (2*b)/Sqrt[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0048721, size = 19, normalized size = 1. \[ -\frac{2 a}{5 x^{5/2}}-\frac{2 b}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)/x^(7/2),x]

[Out]

(-2*a)/(5*x^(5/2)) - (2*b)/Sqrt[x]

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Maple [A]  time = 0.002, size = 14, normalized size = 0.7 \begin{align*} -{\frac{10\,b{x}^{2}+2\,a}{5}{x}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 2.62221, size = 18, normalized size = 0.95 \begin{align*} -\frac{2 \,{\left (5 \, b x^{2} + a\right )}}{5 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*(5*b*x^2 + a)/x^(5/2)

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Fricas [A]  time = 1.48795, size = 38, normalized size = 2. \begin{align*} -\frac{2 \,{\left (5 \, b x^{2} + a\right )}}{5 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*(5*b*x^2 + a)/x^(5/2)

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Sympy [A]  time = 1.29671, size = 19, normalized size = 1. \begin{align*} - \frac{2 a}{5 x^{\frac{5}{2}}} - \frac{2 b}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a/(5*x**(5/2)) - 2*b/sqrt(x)

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Giac [A]  time = 2.76476, size = 18, normalized size = 0.95 \begin{align*} -\frac{2 \,{\left (5 \, b x^{2} + a\right )}}{5 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*(5*b*x^2 + a)/x^(5/2)