3.159 \(\int x^{7/2} (A+B x^2) (b x^2+c x^4) \, dx\)

Optimal. Leaf size=39 \[ \frac{2}{17} x^{17/2} (A c+b B)+\frac{2}{13} A b x^{13/2}+\frac{2}{21} B c x^{21/2} \]

[Out]

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

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Rubi [A]  time = 0.0226728, antiderivative size = 39, normalized size of antiderivative = 1., 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} \[ \frac{2}{17} x^{17/2} (A c+b B)+\frac{2}{13} A b x^{13/2}+\frac{2}{21} B c x^{21/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(7/2)*(A + B*x^2)*(b*x^2 + c*x^4),x]

[Out]

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

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]

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]

Rubi steps

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

Mathematica [A]  time = 0.0157103, size = 33, normalized size = 0.85 \[ \frac{2 x^{13/2} \left (273 x^2 (A c+b B)+357 A b+221 B c x^4\right )}{4641} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)*(A + B*x^2)*(b*x^2 + c*x^4),x]

[Out]

(2*x^(13/2)*(357*A*b + 273*(b*B + A*c)*x^2 + 221*B*c*x^4))/4641

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Maple [A]  time = 0.004, size = 32, normalized size = 0.8 \begin{align*}{\frac{442\,Bc{x}^{4}+546\,A{x}^{2}c+546\,B{x}^{2}b+714\,Ab}{4641}{x}^{{\frac{13}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(B*x^2+A)*(c*x^4+b*x^2),x)

[Out]

2/4641*x^(13/2)*(221*B*c*x^4+273*A*c*x^2+273*B*b*x^2+357*A*b)

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Maxima [A]  time = 1.15532, size = 36, normalized size = 0.92 \begin{align*} \frac{2}{21} \, B c x^{\frac{21}{2}} + \frac{2}{17} \,{\left (B b + A c\right )} x^{\frac{17}{2}} + \frac{2}{13} \, A b x^{\frac{13}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x^2+A)*(c*x^4+b*x^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.299, size = 90, normalized size = 2.31 \begin{align*} \frac{2}{4641} \,{\left (221 \, B c x^{10} + 273 \,{\left (B b + A c\right )} x^{8} + 357 \, A b x^{6}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x^2+A)*(c*x^4+b*x^2),x, algorithm="fricas")

[Out]

2/4641*(221*B*c*x^10 + 273*(B*b + A*c)*x^8 + 357*A*b*x^6)*sqrt(x)

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Sympy [A]  time = 27.7593, size = 46, normalized size = 1.18 \begin{align*} \frac{2 A b x^{\frac{13}{2}}}{13} + \frac{2 A c x^{\frac{17}{2}}}{17} + \frac{2 B b x^{\frac{17}{2}}}{17} + \frac{2 B c x^{\frac{21}{2}}}{21} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)*(B*x**2+A)*(c*x**4+b*x**2),x)

[Out]

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

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Giac [A]  time = 1.15707, size = 39, normalized size = 1. \begin{align*} \frac{2}{21} \, B c x^{\frac{21}{2}} + \frac{2}{17} \, B b x^{\frac{17}{2}} + \frac{2}{17} \, A c x^{\frac{17}{2}} + \frac{2}{13} \, A b x^{\frac{13}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x^2+A)*(c*x^4+b*x^2),x, algorithm="giac")

[Out]

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