3.3.97 \(\int \frac {b x^2+c x^4}{x^{3/2}} \, dx\) [297]

Optimal. Leaf size=21 \[ \frac {2}{3} b x^{3/2}+\frac {2}{7} c x^{7/2} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 21, 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} b x^{3/2}+\frac {2}{7} c x^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)/x^(3/2),x]

[Out]

(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 {b x^2+c x^4}{x^{3/2}} \, dx &=\int \left (b \sqrt {x}+c x^{5/2}\right ) \, dx\\ &=\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 = 21, normalized size = 1.00 \begin {gather*} \frac {2}{21} \left (7 b x^{3/2}+3 c x^{7/2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)/x^(3/2),x]

[Out]

(2*(7*b*x^(3/2) + 3*c*x^(7/2)))/21

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Maple [A]
time = 0.02, size = 14, normalized size = 0.67

method	result	size
derivativedivides	\(\frac {2 b \,x^{\frac {3}{2}}}{3}+\frac {2 c \,x^{\frac {7}{2}}}{7}\)	\(14\)
default	\(\frac {2 b \,x^{\frac {3}{2}}}{3}+\frac {2 c \,x^{\frac {7}{2}}}{7}\)	\(14\)
gosper	\(\frac {2 x^{\frac {3}{2}} \left (3 c \,x^{2}+7 b \right )}{21}\)	\(16\)
trager	\(\frac {2 x^{\frac {3}{2}} \left (3 c \,x^{2}+7 b \right )}{21}\)	\(16\)
risch	\(\frac {2 x^{\frac {3}{2}} \left (3 c \,x^{2}+7 b \right )}{21}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)/x^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 13, normalized size = 0.62 \begin {gather*} \frac {2}{7} \, c x^{\frac {7}{2}} + \frac {2}{3} \, b x^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*c*x^(7/2) + 2/3*b*x^(3/2)

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Fricas [A]
time = 0.34, size = 16, normalized size = 0.76 \begin {gather*} \frac {2}{21} \, {\left (3 \, c x^{3} + 7 \, b x\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(3*c*x^3 + 7*b*x)*sqrt(x)

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Sympy [A]
time = 0.18, size = 19, normalized size = 0.90 \begin {gather*} \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)/x**(3/2),x)

[Out]

2*b*x**(3/2)/3 + 2*c*x**(7/2)/7

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Giac [A]
time = 3.72, size = 13, normalized size = 0.62 \begin {gather*} \frac {2}{7} \, c x^{\frac {7}{2}} + \frac {2}{3} \, b x^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*c*x^(7/2) + 2/3*b*x^(3/2)

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Mupad [B]
time = 0.03, size = 15, normalized size = 0.71 \begin {gather*} \frac {2\,x^{3/2}\,\left (3\,c\,x^2+7\,b\right )}{21} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2 + c*x^4)/x^(3/2),x)

[Out]

(2*x^(3/2)*(7*b + 3*c*x^2))/21

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