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

Optimal. Leaf size=36 \[ \frac {2}{13} b^2 x^{13/2}+\frac {4}{17} b c x^{17/2}+\frac {2}{21} c^2 x^{21/2} \]

[Out]

2/13*b^2*x^(13/2)+4/17*b*c*x^(17/2)+2/21*c^2*x^(21/2)

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Rubi [A]  time = 0.01, antiderivative size = 36, 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} \[ \frac {2}{13} b^2 x^{13/2}+\frac {4}{17} b c x^{17/2}+\frac {2}{21} c^2 x^{21/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(b*x^2 + c*x^4)^2,x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 30, normalized size = 0.83 \[ \frac {2 x^{13/2} \left (357 b^2+546 b c x^2+221 c^2 x^4\right )}{4641} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(13/2)*(357*b^2 + 546*b*c*x^2 + 221*c^2*x^4))/4641

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fricas [A]  time = 0.56, size = 29, normalized size = 0.81 \[ \frac {2}{4641} \, {\left (221 \, c^{2} x^{10} + 546 \, b c x^{8} + 357 \, b^{2} x^{6}\right )} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/4641*(221*c^2*x^10 + 546*b*c*x^8 + 357*b^2*x^6)*sqrt(x)

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giac [A]  time = 0.16, size = 24, normalized size = 0.67 \[ \frac {2}{21} \, c^{2} x^{\frac {21}{2}} + \frac {4}{17} \, b c x^{\frac {17}{2}} + \frac {2}{13} \, b^{2} x^{\frac {13}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*c^2*x^(21/2) + 4/17*b*c*x^(17/2) + 2/13*b^2*x^(13/2)

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maple [A]  time = 0.01, size = 27, normalized size = 0.75 \[ \frac {2 \left (221 c^{2} x^{4}+546 b c \,x^{2}+357 b^{2}\right ) x^{\frac {13}{2}}}{4641} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/4641*x^(13/2)*(221*c^2*x^4+546*b*c*x^2+357*b^2)

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maxima [A]  time = 1.23, size = 24, normalized size = 0.67 \[ \frac {2}{21} \, c^{2} x^{\frac {21}{2}} + \frac {4}{17} \, b c x^{\frac {17}{2}} + \frac {2}{13} \, b^{2} x^{\frac {13}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*c^2*x^(21/2) + 4/17*b*c*x^(17/2) + 2/13*b^2*x^(13/2)

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mupad [B]  time = 4.27, size = 25, normalized size = 0.69 \[ x^{13/2}\,\left (\frac {2\,b^2}{13}+\frac {4\,b\,c\,x^2}{17}+\frac {2\,c^2\,x^4}{21}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(b*x^2 + c*x^4)^2,x)

[Out]

x^(13/2)*((2*b^2)/13 + (2*c^2*x^4)/21 + (4*b*c*x^2)/17)

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sympy [A]  time = 11.38, size = 34, normalized size = 0.94 \[ \frac {2 b^{2} x^{\frac {13}{2}}}{13} + \frac {4 b c x^{\frac {17}{2}}}{17} + \frac {2 c^{2} x^{\frac {21}{2}}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b**2*x**(13/2)/13 + 4*b*c*x**(17/2)/17 + 2*c**2*x**(21/2)/21

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