3.299 \(\int \frac{b x^2+c x^4}{x^{7/2}} \, dx\)

Optimal. Leaf size=19 \[ \frac{2}{3} c x^{3/2}-\frac{2 b}{\sqrt{x}} \]

[Out]

(-2*b)/Sqrt[x] + (2*c*x^(3/2))/3

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Rubi [A]  time = 0.0050894, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {14} \[ \frac{2}{3} c x^{3/2}-\frac{2 b}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)/x^(7/2),x]

[Out]

(-2*b)/Sqrt[x] + (2*c*x^(3/2))/3

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{b x^2+c x^4}{x^{7/2}} \, dx &=\int \left (\frac{b}{x^{3/2}}+c \sqrt{x}\right ) \, dx\\ &=-\frac{2 b}{\sqrt{x}}+\frac{2}{3} c x^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0060615, size = 19, normalized size = 1. \[ \frac{2}{3} c x^{3/2}-\frac{2 b}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)/x^(7/2),x]

[Out]

(-2*b)/Sqrt[x] + (2*c*x^(3/2))/3

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Maple [A]  time = 0.045, size = 16, normalized size = 0.8 \begin{align*} -{\frac{-2\,c{x}^{2}+6\,b}{3}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)/x^(7/2),x)

[Out]

-2/3/x^(1/2)*(-c*x^2+3*b)

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Maxima [A]  time = 0.980051, size = 18, normalized size = 0.95 \begin{align*} \frac{2}{3} \, c x^{\frac{3}{2}} - \frac{2 \, b}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)/x^(7/2),x, algorithm="maxima")

[Out]

2/3*c*x^(3/2) - 2*b/sqrt(x)

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Fricas [A]  time = 1.37157, size = 36, normalized size = 1.89 \begin{align*} \frac{2 \,{\left (c x^{2} - 3 \, b\right )}}{3 \, \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)/x^(7/2),x, algorithm="fricas")

[Out]

2/3*(c*x^2 - 3*b)/sqrt(x)

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Sympy [A]  time = 2.09811, size = 17, normalized size = 0.89 \begin{align*} - \frac{2 b}{\sqrt{x}} + \frac{2 c x^{\frac{3}{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)/x**(7/2),x)

[Out]

-2*b/sqrt(x) + 2*c*x**(3/2)/3

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Giac [A]  time = 1.18016, size = 18, normalized size = 0.95 \begin{align*} \frac{2}{3} \, c x^{\frac{3}{2}} - \frac{2 \, b}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)/x^(7/2),x, algorithm="giac")

[Out]

2/3*c*x^(3/2) - 2*b/sqrt(x)