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

Optimal. Leaf size=37 \[ \frac{2}{3} x^{3/2} (A c+b B)-\frac{2 A b}{\sqrt{x}}+\frac{2}{7} B c x^{7/2} \]

[Out]

(-2*A*b)/Sqrt[x] + (2*(b*B + A*c)*x^(3/2))/3 + (2*B*c*x^(7/2))/7

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Rubi [A]  time = 0.0205999, antiderivative size = 37, 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} \[ \frac{2}{3} x^{3/2} (A c+b B)-\frac{2 A b}{\sqrt{x}}+\frac{2}{7} B c x^{7/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A*b)/Sqrt[x] + (2*(b*B + A*c)*x^(3/2))/3 + (2*B*c*x^(7/2))/7

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

Mathematica [A]  time = 0.0098416, size = 35, normalized size = 0.95 \[ \frac{2 \left (-21 A b+7 A c x^2+7 b B x^2+3 B c x^4\right )}{21 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-21*A*b + 7*b*B*x^2 + 7*A*c*x^2 + 3*B*c*x^4))/(21*Sqrt[x])

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Maple [A]  time = 0.004, size = 32, normalized size = 0.9 \begin{align*} -{\frac{-6\,Bc{x}^{4}-14\,A{x}^{2}c-14\,B{x}^{2}b+42\,Ab}{21}{\frac{1}{\sqrt{x}}}} \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^(7/2),x)

[Out]

-2/21/x^(1/2)*(-3*B*c*x^4-7*A*c*x^2-7*B*b*x^2+21*A*b)

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Maxima [A]  time = 1.14998, size = 36, normalized size = 0.97 \begin{align*} \frac{2}{7} \, B c x^{\frac{7}{2}} + \frac{2}{3} \,{\left (B b + A c\right )} x^{\frac{3}{2}} - \frac{2 \, A b}{\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^(7/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.61498, size = 74, normalized size = 2. \begin{align*} \frac{2 \,{\left (3 \, B c x^{4} + 7 \,{\left (B b + A c\right )} x^{2} - 21 \, A b\right )}}{21 \, \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^(7/2),x, algorithm="fricas")

[Out]

2/21*(3*B*c*x^4 + 7*(B*b + A*c)*x^2 - 21*A*b)/sqrt(x)

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Sympy [A]  time = 7.5185, size = 44, normalized size = 1.19 \begin{align*} - \frac{2 A b}{\sqrt{x}} + \frac{2 A c x^{\frac{3}{2}}}{3} + \frac{2 B b x^{\frac{3}{2}}}{3} + \frac{2 B c x^{\frac{7}{2}}}{7} \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**(7/2),x)

[Out]

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

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Giac [A]  time = 1.17079, size = 39, normalized size = 1.05 \begin{align*} \frac{2}{7} \, B c x^{\frac{7}{2}} + \frac{2}{3} \, B b x^{\frac{3}{2}} + \frac{2}{3} \, A c x^{\frac{3}{2}} - \frac{2 \, A b}{\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^(7/2),x, algorithm="giac")

[Out]

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