3.57 \(\int \frac{x^5 (A+B x^3)}{a+b x^3} \, dx\)

Optimal. Leaf size=54 \[ \frac{x^3 (A b-a B)}{3 b^2}-\frac{a (A b-a B) \log \left (a+b x^3\right )}{3 b^3}+\frac{B x^6}{6 b} \]

[Out]

((A*b - a*B)*x^3)/(3*b^2) + (B*x^6)/(6*b) - (a*(A*b - a*B)*Log[a + b*x^3])/(3*b^3)

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Rubi [A]  time = 0.0572361, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {446, 77} \[ \frac{x^3 (A b-a B)}{3 b^2}-\frac{a (A b-a B) \log \left (a+b x^3\right )}{3 b^3}+\frac{B x^6}{6 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*(A + B*x^3))/(a + b*x^3),x]

[Out]

((A*b - a*B)*x^3)/(3*b^2) + (B*x^6)/(6*b) - (a*(A*b - a*B)*Log[a + b*x^3])/(3*b^3)

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{x^5 \left (A+B x^3\right )}{a+b x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x (A+B x)}{a+b x} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{A b-a B}{b^2}+\frac{B x}{b}+\frac{a (-A b+a B)}{b^2 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac{(A b-a B) x^3}{3 b^2}+\frac{B x^6}{6 b}-\frac{a (A b-a B) \log \left (a+b x^3\right )}{3 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0204856, size = 47, normalized size = 0.87 \[ \frac{b x^3 \left (-2 a B+2 A b+b B x^3\right )+2 a (a B-A b) \log \left (a+b x^3\right )}{6 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(A + B*x^3))/(a + b*x^3),x]

[Out]

(b*x^3*(2*A*b - 2*a*B + b*B*x^3) + 2*a*(-(A*b) + a*B)*Log[a + b*x^3])/(6*b^3)

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Maple [A]  time = 0.004, size = 62, normalized size = 1.2 \begin{align*}{\frac{B{x}^{6}}{6\,b}}+{\frac{A{x}^{3}}{3\,b}}-{\frac{B{x}^{3}a}{3\,{b}^{2}}}-{\frac{a\ln \left ( b{x}^{3}+a \right ) A}{3\,{b}^{2}}}+{\frac{{a}^{2}\ln \left ( b{x}^{3}+a \right ) B}{3\,{b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(B*x^3+A)/(b*x^3+a),x)

[Out]

1/6*B*x^6/b+1/3/b*A*x^3-1/3/b^2*B*x^3*a-1/3*a/b^2*ln(b*x^3+a)*A+1/3*a^2/b^3*ln(b*x^3+a)*B

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Maxima [A]  time = 1.14137, size = 68, normalized size = 1.26 \begin{align*} \frac{B b x^{6} - 2 \,{\left (B a - A b\right )} x^{3}}{6 \, b^{2}} + \frac{{\left (B a^{2} - A a b\right )} \log \left (b x^{3} + a\right )}{3 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x^3+A)/(b*x^3+a),x, algorithm="maxima")

[Out]

1/6*(B*b*x^6 - 2*(B*a - A*b)*x^3)/b^2 + 1/3*(B*a^2 - A*a*b)*log(b*x^3 + a)/b^3

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Fricas [A]  time = 1.44918, size = 108, normalized size = 2. \begin{align*} \frac{B b^{2} x^{6} - 2 \,{\left (B a b - A b^{2}\right )} x^{3} + 2 \,{\left (B a^{2} - A a b\right )} \log \left (b x^{3} + a\right )}{6 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x^3+A)/(b*x^3+a),x, algorithm="fricas")

[Out]

1/6*(B*b^2*x^6 - 2*(B*a*b - A*b^2)*x^3 + 2*(B*a^2 - A*a*b)*log(b*x^3 + a))/b^3

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Sympy [A]  time = 1.12372, size = 44, normalized size = 0.81 \begin{align*} \frac{B x^{6}}{6 b} + \frac{a \left (- A b + B a\right ) \log{\left (a + b x^{3} \right )}}{3 b^{3}} - \frac{x^{3} \left (- A b + B a\right )}{3 b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(B*x**3+A)/(b*x**3+a),x)

[Out]

B*x**6/(6*b) + a*(-A*b + B*a)*log(a + b*x**3)/(3*b**3) - x**3*(-A*b + B*a)/(3*b**2)

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Giac [A]  time = 1.17429, size = 70, normalized size = 1.3 \begin{align*} \frac{B b x^{6} - 2 \, B a x^{3} + 2 \, A b x^{3}}{6 \, b^{2}} + \frac{{\left (B a^{2} - A a b\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(B*x^3+A)/(b*x^3+a),x, algorithm="giac")

[Out]

1/6*(B*b*x^6 - 2*B*a*x^3 + 2*A*b*x^3)/b^2 + 1/3*(B*a^2 - A*a*b)*log(abs(b*x^3 + a))/b^3