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

Optimal. Leaf size=39 \[ \frac {2}{13} x^{13/2} (A c+b B)+\frac {2}{9} A b x^{9/2}+\frac {2}{17} B c x^{17/2} \]

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Rubi [A]  time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1584, 448} \begin {gather*} \frac {2}{13} x^{13/2} (A c+b B)+\frac {2}{9} A b x^{9/2}+\frac {2}{17} B c x^{17/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(A + B*x^2)*(b*x^2 + c*x^4),x]

[Out]

(2*A*b*x^(9/2))/9 + (2*(b*B + A*c)*x^(13/2))/13 + (2*B*c*x^(17/2))/17

Rule 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 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 (A+B x^2\right ) \left (b x^2+c x^4\right ) \, dx &=\int x^{7/2} \left (A+B x^2\right ) \left (b+c x^2\right ) \, dx\\ &=\int \left (A b x^{7/2}+(b B+A c) x^{11/2}+B c x^{15/2}\right ) \, dx\\ &=\frac {2}{9} A b x^{9/2}+\frac {2}{13} (b B+A c) x^{13/2}+\frac {2}{17} B c x^{17/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 33, normalized size = 0.85 \begin {gather*} \frac {2 x^{9/2} \left (153 x^2 (A c+b B)+221 A b+117 B c x^4\right )}{1989} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(9/2)*(221*A*b + 153*(b*B + A*c)*x^2 + 117*B*c*x^4))/1989

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IntegrateAlgebraic [A]  time = 0.03, size = 41, normalized size = 1.05 \begin {gather*} \frac {2 \left (221 A b x^{9/2}+153 A c x^{13/2}+153 b B x^{13/2}+117 B c x^{17/2}\right )}{1989} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(221*A*b*x^(9/2) + 153*b*B*x^(13/2) + 153*A*c*x^(13/2) + 117*B*c*x^(17/2)))/1989

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fricas [A]  time = 0.40, size = 32, normalized size = 0.82 \begin {gather*} \frac {2}{1989} \, {\left (117 \, B c x^{8} + 153 \, {\left (B b + A c\right )} x^{6} + 221 \, A b x^{4}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1989*(117*B*c*x^8 + 153*(B*b + A*c)*x^6 + 221*A*b*x^4)*sqrt(x)

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giac [A]  time = 0.15, size = 29, normalized size = 0.74 \begin {gather*} \frac {2}{17} \, B c x^{\frac {17}{2}} + \frac {2}{13} \, B b x^{\frac {13}{2}} + \frac {2}{13} \, A c x^{\frac {13}{2}} + \frac {2}{9} \, A b x^{\frac {9}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/17*B*c*x^(17/2) + 2/13*B*b*x^(13/2) + 2/13*A*c*x^(13/2) + 2/9*A*b*x^(9/2)

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maple [A]  time = 0.05, size = 32, normalized size = 0.82 \begin {gather*} \frac {2 \left (117 B c \,x^{4}+153 A c \,x^{2}+153 B b \,x^{2}+221 A b \right ) x^{\frac {9}{2}}}{1989} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1989*x^(9/2)*(117*B*c*x^4+153*A*c*x^2+153*B*b*x^2+221*A*b)

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maxima [A]  time = 1.33, size = 27, normalized size = 0.69 \begin {gather*} \frac {2}{17} \, B c x^{\frac {17}{2}} + \frac {2}{13} \, {\left (B b + A c\right )} x^{\frac {13}{2}} + \frac {2}{9} \, A b x^{\frac {9}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/17*B*c*x^(17/2) + 2/13*(B*b + A*c)*x^(13/2) + 2/9*A*b*x^(9/2)

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mupad [B]  time = 0.04, size = 31, normalized size = 0.79 \begin {gather*} \frac {2\,x^{9/2}\,\left (221\,A\,b+153\,A\,c\,x^2+153\,B\,b\,x^2+117\,B\,c\,x^4\right )}{1989} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*x^(9/2)*(221*A*b + 153*A*c*x^2 + 153*B*b*x^2 + 117*B*c*x^4))/1989

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sympy [A]  time = 5.59, size = 46, normalized size = 1.18 \begin {gather*} \frac {2 A b x^{\frac {9}{2}}}{9} + \frac {2 A c x^{\frac {13}{2}}}{13} + \frac {2 B b x^{\frac {13}{2}}}{13} + \frac {2 B c x^{\frac {17}{2}}}{17} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*b*x**(9/2)/9 + 2*A*c*x**(13/2)/13 + 2*B*b*x**(13/2)/13 + 2*B*c*x**(17/2)/17

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