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

Optimal. Leaf size=39 \[ \frac{2}{7} x^{7/2} (A c+b B)+\frac{2}{3} A b x^{3/2}+\frac{2}{11} B c x^{11/2} \]

[Out]

(2*A*b*x^(3/2))/3 + (2*(b*B + A*c)*x^(7/2))/7 + (2*B*c*x^(11/2))/11

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Rubi [A]  time = 0.0212394, 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} \[ \frac{2}{7} x^{7/2} (A c+b B)+\frac{2}{3} A b x^{3/2}+\frac{2}{11} B c x^{11/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*A*b*x^(3/2))/3 + (2*(b*B + A*c)*x^(7/2))/7 + (2*B*c*x^(11/2))/11

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 \frac{\left (A+B x^2\right ) \left (b x^2+c x^4\right )}{x^{3/2}} \, dx &=\int \sqrt{x} \left (A+B x^2\right ) \left (b+c x^2\right ) \, dx\\ &=\int \left (A b \sqrt{x}+(b B+A c) x^{5/2}+B c x^{9/2}\right ) \, dx\\ &=\frac{2}{3} A b x^{3/2}+\frac{2}{7} (b B+A c) x^{7/2}+\frac{2}{11} B c x^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.0139968, size = 33, normalized size = 0.85 \[ \frac{2}{231} x^{3/2} \left (33 x^2 (A c+b B)+77 A b+21 B c x^4\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(3/2)*(77*A*b + 33*(b*B + A*c)*x^2 + 21*B*c*x^4))/231

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Maple [A]  time = 0.004, size = 32, normalized size = 0.8 \begin{align*}{\frac{42\,Bc{x}^{4}+66\,A{x}^{2}c+66\,B{x}^{2}b+154\,Ab}{231}{x}^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/231*x^(3/2)*(21*B*c*x^4+33*A*c*x^2+33*B*b*x^2+77*A*b)

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Maxima [A]  time = 1.19645, size = 36, normalized size = 0.92 \begin{align*} \frac{2}{11} \, B c x^{\frac{11}{2}} + \frac{2}{7} \,{\left (B b + A c\right )} x^{\frac{7}{2}} + \frac{2}{3} \, A b x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*B*c*x^(11/2) + 2/7*(B*b + A*c)*x^(7/2) + 2/3*A*b*x^(3/2)

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Fricas [A]  time = 1.65898, size = 81, normalized size = 2.08 \begin{align*} \frac{2}{231} \,{\left (21 \, B c x^{5} + 33 \,{\left (B b + A c\right )} x^{3} + 77 \, A b x\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/231*(21*B*c*x^5 + 33*(B*b + A*c)*x^3 + 77*A*b*x)*sqrt(x)

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Sympy [A]  time = 2.66208, size = 46, normalized size = 1.18 \begin{align*} \frac{2 A b x^{\frac{3}{2}}}{3} + \frac{2 A c x^{\frac{7}{2}}}{7} + \frac{2 B b x^{\frac{7}{2}}}{7} + \frac{2 B c x^{\frac{11}{2}}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*b*x**(3/2)/3 + 2*A*c*x**(7/2)/7 + 2*B*b*x**(7/2)/7 + 2*B*c*x**(11/2)/11

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Giac [A]  time = 1.14787, size = 39, normalized size = 1. \begin{align*} \frac{2}{11} \, B c x^{\frac{11}{2}} + \frac{2}{7} \, B b x^{\frac{7}{2}} + \frac{2}{7} \, A c x^{\frac{7}{2}} + \frac{2}{3} \, A b x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*B*c*x^(11/2) + 2/7*B*b*x^(7/2) + 2/7*A*c*x^(7/2) + 2/3*A*b*x^(3/2)