3.9.12 \(\int \frac {a+b x^2+c x^4}{x^{3/2}} \, dx\)

Optimal. Leaf size=29 \[ -\frac {2 a}{\sqrt {x}}+\frac {2}{3} b x^{3/2}+\frac {2}{7} c x^{7/2} \]

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Rubi [A]  time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {14} \begin {gather*} -\frac {2 a}{\sqrt {x}}+\frac {2}{3} b x^{3/2}+\frac {2}{7} c x^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/x^(3/2),x]

[Out]

(-2*a)/Sqrt[x] + (2*b*x^(3/2))/3 + (2*c*x^(7/2))/7

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

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Mathematica [A]  time = 0.01, size = 25, normalized size = 0.86 \begin {gather*} \frac {2 \left (-21 a+7 b x^2+3 c x^4\right )}{21 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/x^(3/2),x]

[Out]

(2*(-21*a + 7*b*x^2 + 3*c*x^4))/(21*Sqrt[x])

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IntegrateAlgebraic [A]  time = 0.02, size = 25, normalized size = 0.86 \begin {gather*} \frac {2 \left (-21 a+7 b x^2+3 c x^4\right )}{21 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x^2 + c*x^4)/x^(3/2),x]

[Out]

(2*(-21*a + 7*b*x^2 + 3*c*x^4))/(21*Sqrt[x])

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fricas [A]  time = 1.00, size = 21, normalized size = 0.72 \begin {gather*} \frac {2 \, {\left (3 \, c x^{4} + 7 \, b x^{2} - 21 \, a\right )}}{21 \, \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^(3/2),x, algorithm="fricas")

[Out]

2/21*(3*c*x^4 + 7*b*x^2 - 21*a)/sqrt(x)

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giac [A]  time = 0.15, size = 19, normalized size = 0.66 \begin {gather*} \frac {2}{7} \, c x^{\frac {7}{2}} + \frac {2}{3} \, b x^{\frac {3}{2}} - \frac {2 \, a}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^(3/2),x, algorithm="giac")

[Out]

2/7*c*x^(7/2) + 2/3*b*x^(3/2) - 2*a/sqrt(x)

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maple [A]  time = 0.00, size = 22, normalized size = 0.76 \begin {gather*} -\frac {2 \left (-3 c \,x^{4}-7 b \,x^{2}+21 a \right )}{21 \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)/x^(3/2),x)

[Out]

-2/21*(-3*c*x^4-7*b*x^2+21*a)/x^(1/2)

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maxima [A]  time = 1.09, size = 19, normalized size = 0.66 \begin {gather*} \frac {2}{7} \, c x^{\frac {7}{2}} + \frac {2}{3} \, b x^{\frac {3}{2}} - \frac {2 \, a}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^(3/2),x, algorithm="maxima")

[Out]

2/7*c*x^(7/2) + 2/3*b*x^(3/2) - 2*a/sqrt(x)

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mupad [B]  time = 0.04, size = 21, normalized size = 0.72 \begin {gather*} \frac {6\,c\,x^4+14\,b\,x^2-42\,a}{21\,\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)/x^(3/2),x)

[Out]

(14*b*x^2 - 42*a + 6*c*x^4)/(21*x^(1/2))

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sympy [A]  time = 1.04, size = 27, normalized size = 0.93 \begin {gather*} - \frac {2 a}{\sqrt {x}} + \frac {2 b x^{\frac {3}{2}}}{3} + \frac {2 c x^{\frac {7}{2}}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)/x**(3/2),x)

[Out]

-2*a/sqrt(x) + 2*b*x**(3/2)/3 + 2*c*x**(7/2)/7

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