3.2.96 \(\int \frac {(b x^2+c x^4)^3}{x^{5/2}} \, dx\)

Optimal. Leaf size=51 \[ \frac {2}{9} b^3 x^{9/2}+\frac {6}{13} b^2 c x^{13/2}+\frac {6}{17} b c^2 x^{17/2}+\frac {2}{21} c^3 x^{21/2} \]

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Rubi [A]  time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1584, 270} \begin {gather*} \frac {6}{13} b^2 c x^{13/2}+\frac {2}{9} b^3 x^{9/2}+\frac {6}{17} b c^2 x^{17/2}+\frac {2}{21} c^3 x^{21/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^3/x^(5/2),x]

[Out]

(2*b^3*x^(9/2))/9 + (6*b^2*c*x^(13/2))/13 + (6*b*c^2*x^(17/2))/17 + (2*c^3*x^(21/2))/21

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin {align*} \int \frac {\left (b x^2+c x^4\right )^3}{x^{5/2}} \, dx &=\int x^{7/2} \left (b+c x^2\right )^3 \, dx\\ &=\int \left (b^3 x^{7/2}+3 b^2 c x^{11/2}+3 b c^2 x^{15/2}+c^3 x^{19/2}\right ) \, dx\\ &=\frac {2}{9} b^3 x^{9/2}+\frac {6}{13} b^2 c x^{13/2}+\frac {6}{17} b c^2 x^{17/2}+\frac {2}{21} c^3 x^{21/2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 41, normalized size = 0.80 \begin {gather*} \frac {2 x^{9/2} \left (1547 b^3+3213 b^2 c x^2+2457 b c^2 x^4+663 c^3 x^6\right )}{13923} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^3/x^(5/2),x]

[Out]

(2*x^(9/2)*(1547*b^3 + 3213*b^2*c*x^2 + 2457*b*c^2*x^4 + 663*c^3*x^6))/13923

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IntegrateAlgebraic [A]  time = 0.03, size = 47, normalized size = 0.92 \begin {gather*} \frac {2 \left (1547 b^3 x^{9/2}+3213 b^2 c x^{13/2}+2457 b c^2 x^{17/2}+663 c^3 x^{21/2}\right )}{13923} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b*x^2 + c*x^4)^3/x^(5/2),x]

[Out]

(2*(1547*b^3*x^(9/2) + 3213*b^2*c*x^(13/2) + 2457*b*c^2*x^(17/2) + 663*c^3*x^(21/2)))/13923

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fricas [A]  time = 0.86, size = 40, normalized size = 0.78 \begin {gather*} \frac {2}{13923} \, {\left (663 \, c^{3} x^{10} + 2457 \, b c^{2} x^{8} + 3213 \, b^{2} c x^{6} + 1547 \, b^{3} x^{4}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^(5/2),x, algorithm="fricas")

[Out]

2/13923*(663*c^3*x^10 + 2457*b*c^2*x^8 + 3213*b^2*c*x^6 + 1547*b^3*x^4)*sqrt(x)

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giac [A]  time = 0.15, size = 35, normalized size = 0.69 \begin {gather*} \frac {2}{21} \, c^{3} x^{\frac {21}{2}} + \frac {6}{17} \, b c^{2} x^{\frac {17}{2}} + \frac {6}{13} \, b^{2} c x^{\frac {13}{2}} + \frac {2}{9} \, b^{3} x^{\frac {9}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^(5/2),x, algorithm="giac")

[Out]

2/21*c^3*x^(21/2) + 6/17*b*c^2*x^(17/2) + 6/13*b^2*c*x^(13/2) + 2/9*b^3*x^(9/2)

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maple [A]  time = 0.01, size = 38, normalized size = 0.75 \begin {gather*} \frac {2 \left (663 c^{3} x^{6}+2457 b \,c^{2} x^{4}+3213 b^{2} c \,x^{2}+1547 b^{3}\right ) x^{\frac {9}{2}}}{13923} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^3/x^(5/2),x)

[Out]

2/13923*x^(9/2)*(663*c^3*x^6+2457*b*c^2*x^4+3213*b^2*c*x^2+1547*b^3)

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maxima [A]  time = 1.35, size = 35, normalized size = 0.69 \begin {gather*} \frac {2}{21} \, c^{3} x^{\frac {21}{2}} + \frac {6}{17} \, b c^{2} x^{\frac {17}{2}} + \frac {6}{13} \, b^{2} c x^{\frac {13}{2}} + \frac {2}{9} \, b^{3} x^{\frac {9}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^(5/2),x, algorithm="maxima")

[Out]

2/21*c^3*x^(21/2) + 6/17*b*c^2*x^(17/2) + 6/13*b^2*c*x^(13/2) + 2/9*b^3*x^(9/2)

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mupad [B]  time = 0.05, size = 35, normalized size = 0.69 \begin {gather*} \frac {2\,b^3\,x^{9/2}}{9}+\frac {2\,c^3\,x^{21/2}}{21}+\frac {6\,b^2\,c\,x^{13/2}}{13}+\frac {6\,b\,c^2\,x^{17/2}}{17} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2 + c*x^4)^3/x^(5/2),x)

[Out]

(2*b^3*x^(9/2))/9 + (2*c^3*x^(21/2))/21 + (6*b^2*c*x^(13/2))/13 + (6*b*c^2*x^(17/2))/17

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sympy [A]  time = 22.20, size = 49, normalized size = 0.96 \begin {gather*} \frac {2 b^{3} x^{\frac {9}{2}}}{9} + \frac {6 b^{2} c x^{\frac {13}{2}}}{13} + \frac {6 b c^{2} x^{\frac {17}{2}}}{17} + \frac {2 c^{3} x^{\frac {21}{2}}}{21} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**3/x**(5/2),x)

[Out]

2*b**3*x**(9/2)/9 + 6*b**2*c*x**(13/2)/13 + 6*b*c**2*x**(17/2)/17 + 2*c**3*x**(21/2)/21

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