3.154 \(\int \frac{x^6 (A+B x^2)}{(b x^2+c x^4)^{3/2}} \, dx\)

Optimal. Leaf size=104 \[ \frac{x \sqrt{b x^2+c x^4} (4 b B-3 A c)}{3 b c^2}-\frac{2 \sqrt{b x^2+c x^4} (4 b B-3 A c)}{3 c^3 x}-\frac{x^5 (b B-A c)}{b c \sqrt{b x^2+c x^4}} \]

[Out]

-(((b*B - A*c)*x^5)/(b*c*Sqrt[b*x^2 + c*x^4])) - (2*(4*b*B - 3*A*c)*Sqrt[b*x^2 + c*x^4])/(3*c^3*x) + ((4*b*B -
3*A*c)*x*Sqrt[b*x^2 + c*x^4])/(3*b*c^2)

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Rubi [A]  time = 0.199874, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {2037, 2016, 1588} \[ \frac{x \sqrt{b x^2+c x^4} (4 b B-3 A c)}{3 b c^2}-\frac{2 \sqrt{b x^2+c x^4} (4 b B-3 A c)}{3 c^3 x}-\frac{x^5 (b B-A c)}{b c \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]

Int[(x^6*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]

[Out]

-(((b*B - A*c)*x^5)/(b*c*Sqrt[b*x^2 + c*x^4])) - (2*(4*b*B - 3*A*c)*Sqrt[b*x^2 + c*x^4])/(3*c^3*x) + ((4*b*B -
3*A*c)*x*Sqrt[b*x^2 + c*x^4])/(3*b*c^2)

Rule 2037

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> -Si
mp[(e^(j - 1)*(b*c - a*d)*(e*x)^(m - j + 1)*(a*x^j + b*x^(j + n))^(p + 1))/(a*b*n*(p + 1)), x] - Dist[(e^j*(a*
d*(m + j*p + 1) - b*c*(m + n + p*(j + n) + 1)))/(a*b*n*(p + 1)), Int[(e*x)^(m - j)*(a*x^j + b*x^(j + n))^(p +
1), x], x] /; FreeQ[{a, b, c, d, e, j, m, n}, x] && EqQ[jn, j + n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] && Lt
Q[p, -1] && GtQ[j, 0] && LeQ[j, m] && (GtQ[e, 0] || IntegerQ[j])

Rule 2016

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c^(j - 1)*(c*x)^(m - j +
1)*(a*x^j + b*x^n)^(p + 1))/(a*(m + j*p + 1)), x] - Dist[(b*(m + n*p + n - j + 1))/(a*c^(n - j)*(m + j*p + 1)
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[
n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c,
0])

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{x^6 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=-\frac{(b B-A c) x^5}{b c \sqrt{b x^2+c x^4}}+\frac{(4 b B-3 A c) \int \frac{x^4}{\sqrt{b x^2+c x^4}} \, dx}{b c}\\ &=-\frac{(b B-A c) x^5}{b c \sqrt{b x^2+c x^4}}+\frac{(4 b B-3 A c) x \sqrt{b x^2+c x^4}}{3 b c^2}-\frac{(2 (4 b B-3 A c)) \int \frac{x^2}{\sqrt{b x^2+c x^4}} \, dx}{3 c^2}\\ &=-\frac{(b B-A c) x^5}{b c \sqrt{b x^2+c x^4}}-\frac{2 (4 b B-3 A c) \sqrt{b x^2+c x^4}}{3 c^3 x}+\frac{(4 b B-3 A c) x \sqrt{b x^2+c x^4}}{3 b c^2}\\ \end{align*}

Mathematica [A]  time = 0.0381198, size = 60, normalized size = 0.58 \[ \frac{x \left (b \left (6 A c-4 B c x^2\right )+c^2 x^2 \left (3 A+B x^2\right )-8 b^2 B\right )}{3 c^3 \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^6*(A + B*x^2))/(b*x^2 + c*x^4)^(3/2),x]

[Out]

(x*(-8*b^2*B + c^2*x^2*(3*A + B*x^2) + b*(6*A*c - 4*B*c*x^2)))/(3*c^3*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.005, size = 66, normalized size = 0.6 \begin{align*}{\frac{ \left ( c{x}^{2}+b \right ) \left ( B{c}^{2}{x}^{4}+3\,A{x}^{2}{c}^{2}-4\,B{x}^{2}bc+6\,Abc-8\,B{b}^{2} \right ){x}^{3}}{3\,{c}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(B*x^2+A)/(c*x^4+b*x^2)^(3/2),x)

[Out]

1/3*(c*x^2+b)*(B*c^2*x^4+3*A*c^2*x^2-4*B*b*c*x^2+6*A*b*c-8*B*b^2)*x^3/c^3/(c*x^4+b*x^2)^(3/2)

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Maxima [A]  time = 1.1939, size = 80, normalized size = 0.77 \begin{align*} \frac{{\left (c x^{2} + 2 \, b\right )} A}{\sqrt{c x^{2} + b} c^{2}} + \frac{{\left (c^{2} x^{4} - 4 \, b c x^{2} - 8 \, b^{2}\right )} B}{3 \, \sqrt{c x^{2} + b} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(B*x^2+A)/(c*x^4+b*x^2)^(3/2),x, algorithm="maxima")

[Out]

(c*x^2 + 2*b)*A/(sqrt(c*x^2 + b)*c^2) + 1/3*(c^2*x^4 - 4*b*c*x^2 - 8*b^2)*B/(sqrt(c*x^2 + b)*c^3)

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Fricas [A]  time = 1.48939, size = 139, normalized size = 1.34 \begin{align*} \frac{{\left (B c^{2} x^{4} - 8 \, B b^{2} + 6 \, A b c -{\left (4 \, B b c - 3 \, A c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{3 \,{\left (c^{4} x^{3} + b c^{3} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(B*x^2+A)/(c*x^4+b*x^2)^(3/2),x, algorithm="fricas")

[Out]

1/3*(B*c^2*x^4 - 8*B*b^2 + 6*A*b*c - (4*B*b*c - 3*A*c^2)*x^2)*sqrt(c*x^4 + b*x^2)/(c^4*x^3 + b*c^3*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6} \left (A + B x^{2}\right )}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(B*x**2+A)/(c*x**4+b*x**2)**(3/2),x)

[Out]

Integral(x**6*(A + B*x**2)/(x**2*(b + c*x**2))**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{2} + A\right )} x^{6}}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(B*x^2+A)/(c*x^4+b*x^2)^(3/2),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)*x^6/(c*x^4 + b*x^2)^(3/2), x)