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

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

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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 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]))))

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]]

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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.96, size = 51, normalized size = 0.94 \[ \frac {B b^{2} x^{6} - 2 \, {\left (B a b - A b^{2}\right )} x^{3} + 2 \, {\left (B a^{2} - A a b\right )} \log \left (b x^{3} + a\right )}{6 \, b^{3}} \]

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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giac [A]  time = 0.17, size = 52, normalized size = 0.96 \[ \frac {B b x^{6} - 2 \, B a x^{3} + 2 \, A b x^{3}}{6 \, b^{2}} + \frac {{\left (B a^{2} - A a b\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} \]

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

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maple [A]  time = 0.04, size = 62, normalized size = 1.15 \[ \frac {B \,x^{6}}{6 b}+\frac {A \,x^{3}}{3 b}-\frac {B a \,x^{3}}{3 b^{2}}-\frac {A a \ln \left (b \,x^{3}+a \right )}{3 b^{2}}+\frac {B \,a^{2} \ln \left (b \,x^{3}+a \right )}{3 b^{3}} \]

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*a*x^3-1/3*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 = 0.50, size = 50, normalized size = 0.93 \[ \frac {B b x^{6} - 2 \, {\left (B a - A b\right )} x^{3}}{6 \, b^{2}} + \frac {{\left (B a^{2} - A a b\right )} \log \left (b x^{3} + a\right )}{3 \, b^{3}} \]

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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mupad [B]  time = 0.08, size = 52, normalized size = 0.96 \[ x^3\,\left (\frac {A}{3\,b}-\frac {B\,a}{3\,b^2}\right )+\frac {\ln \left (b\,x^3+a\right )\,\left (B\,a^2-A\,a\,b\right )}{3\,b^3}+\frac {B\,x^6}{6\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5*(A + B*x^3))/(a + b*x^3),x)

[Out]

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

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sympy [A]  time = 0.96, size = 46, normalized size = 0.85 \[ \frac {B x^{6}}{6 b} + \frac {a \left (- A b + B a\right ) \log {\left (a + b x^{3} \right )}}{3 b^{3}} + x^{3} \left (\frac {A}{3 b} - \frac {B a}{3 b^{2}}\right ) \]

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/(3*b) - B*a/(3*b**2))

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