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

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

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {2}{3} c x^{3/2}-\frac {2 b}{\sqrt {x}} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} \frac {2}{3} c x^{3/2}-\frac {2 b}{\sqrt {x}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.02, size = 18, normalized size = 0.95 \begin {gather*} \frac {2 \left (c x^2-3 b\right )}{3 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b*x^2 + c*x^4)/x^(7/2),x]

[Out]

(2*(-3*b + c*x^2))/(3*Sqrt[x])

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fricas [A]  time = 0.91, size = 14, normalized size = 0.74 \begin {gather*} \frac {2 \, {\left (c x^{2} - 3 \, b\right )}}{3 \, \sqrt {x}} \end {gather*}

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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giac [A]  time = 0.16, size = 13, normalized size = 0.68 \begin {gather*} \frac {2}{3} \, c x^{\frac {3}{2}} - \frac {2 \, b}{\sqrt {x}} \end {gather*}

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)

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maple [A]  time = 0.00, size = 16, normalized size = 0.84 \begin {gather*} -\frac {2 \left (-c \,x^{2}+3 b \right )}{3 \sqrt {x}} \end {gather*}

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 = 1.34, size = 13, normalized size = 0.68 \begin {gather*} \frac {2}{3} \, c x^{\frac {3}{2}} - \frac {2 \, b}{\sqrt {x}} \end {gather*}

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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mupad [B]  time = 0.03, size = 15, normalized size = 0.79 \begin {gather*} -\frac {6\,b-2\,c\,x^2}{3\,\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2 + c*x^4)/x^(7/2),x)

[Out]

-(6*b - 2*c*x^2)/(3*x^(1/2))

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sympy [A]  time = 1.80, size = 17, normalized size = 0.89 \begin {gather*} - \frac {2 b}{\sqrt {x}} + \frac {2 c x^{\frac {3}{2}}}{3} \end {gather*}

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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