3.1.87 \(\int \frac {A+B x^3}{x^7 (a+b x^3)^2} \, dx\)

Optimal. Leaf size=97 \[ -\frac {b (3 A b-2 a B) \log \left (a+b x^3\right )}{3 a^4}+\frac {b \log (x) (3 A b-2 a B)}{a^4}+\frac {b (A b-a B)}{3 a^3 \left (a+b x^3\right )}+\frac {2 A b-a B}{3 a^3 x^3}-\frac {A}{6 a^2 x^6} \]

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Rubi [A]  time = 0.10, antiderivative size = 97, 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} \begin {gather*} \frac {b (A b-a B)}{3 a^3 \left (a+b x^3\right )}+\frac {2 A b-a B}{3 a^3 x^3}-\frac {b (3 A b-2 a B) \log \left (a+b x^3\right )}{3 a^4}+\frac {b \log (x) (3 A b-2 a B)}{a^4}-\frac {A}{6 a^2 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x^3)/(x^7*(a + b*x^3)^2),x]

[Out]

-A/(6*a^2*x^6) + (2*A*b - a*B)/(3*a^3*x^3) + (b*(A*b - a*B))/(3*a^3*(a + b*x^3)) + (b*(3*A*b - 2*a*B)*Log[x])/
a^4 - (b*(3*A*b - 2*a*B)*Log[a + b*x^3])/(3*a^4)

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 {A+B x^3}{x^7 \left (a+b x^3\right )^2} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {A+B x}{x^3 (a+b x)^2} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {A}{a^2 x^3}+\frac {-2 A b+a B}{a^3 x^2}-\frac {b (-3 A b+2 a B)}{a^4 x}+\frac {b^2 (-A b+a B)}{a^3 (a+b x)^2}+\frac {b^2 (-3 A b+2 a B)}{a^4 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {A}{6 a^2 x^6}+\frac {2 A b-a B}{3 a^3 x^3}+\frac {b (A b-a B)}{3 a^3 \left (a+b x^3\right )}+\frac {b (3 A b-2 a B) \log (x)}{a^4}-\frac {b (3 A b-2 a B) \log \left (a+b x^3\right )}{3 a^4}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 85, normalized size = 0.88 \begin {gather*} -\frac {\frac {a^2 A}{x^6}+\frac {2 a b (a B-A b)}{a+b x^3}+\frac {2 a (a B-2 A b)}{x^3}+2 b (3 A b-2 a B) \log \left (a+b x^3\right )-6 b \log (x) (3 A b-2 a B)}{6 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^3)/(x^7*(a + b*x^3)^2),x]

[Out]

-1/6*((a^2*A)/x^6 + (2*a*(-2*A*b + a*B))/x^3 + (2*a*b*(-(A*b) + a*B))/(a + b*x^3) - 6*b*(3*A*b - 2*a*B)*Log[x]
+ 2*b*(3*A*b - 2*a*B)*Log[a + b*x^3])/a^4

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x^3}{x^7 \left (a+b x^3\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(A + B*x^3)/(x^7*(a + b*x^3)^2),x]

[Out]

IntegrateAlgebraic[(A + B*x^3)/(x^7*(a + b*x^3)^2), x]

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fricas [A]  time = 0.46, size = 154, normalized size = 1.59 \begin {gather*} -\frac {2 \, {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{6} + A a^{3} + {\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} x^{3} - 2 \, {\left ({\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} x^{9} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{6}\right )} \log \left (b x^{3} + a\right ) + 6 \, {\left ({\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} x^{9} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{6}\right )} \log \relax (x)}{6 \, {\left (a^{4} b x^{9} + a^{5} x^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^3+A)/x^7/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

-1/6*(2*(2*B*a^2*b - 3*A*a*b^2)*x^6 + A*a^3 + (2*B*a^3 - 3*A*a^2*b)*x^3 - 2*((2*B*a*b^2 - 3*A*b^3)*x^9 + (2*B*
a^2*b - 3*A*a*b^2)*x^6)*log(b*x^3 + a) + 6*((2*B*a*b^2 - 3*A*b^3)*x^9 + (2*B*a^2*b - 3*A*a*b^2)*x^6)*log(x))/(
a^4*b*x^9 + a^5*x^6)

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giac [A]  time = 0.17, size = 149, normalized size = 1.54 \begin {gather*} -\frac {{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {{\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{4} b} - \frac {2 \, B a b^{2} x^{3} - 3 \, A b^{3} x^{3} + 3 \, B a^{2} b - 4 \, A a b^{2}}{3 \, {\left (b x^{3} + a\right )} a^{4}} + \frac {6 \, B a b x^{6} - 9 \, A b^{2} x^{6} - 2 \, B a^{2} x^{3} + 4 \, A a b x^{3} - A a^{2}}{6 \, a^{4} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^3+A)/x^7/(b*x^3+a)^2,x, algorithm="giac")

[Out]

-(2*B*a*b - 3*A*b^2)*log(abs(x))/a^4 + 1/3*(2*B*a*b^2 - 3*A*b^3)*log(abs(b*x^3 + a))/(a^4*b) - 1/3*(2*B*a*b^2*
x^3 - 3*A*b^3*x^3 + 3*B*a^2*b - 4*A*a*b^2)/((b*x^3 + a)*a^4) + 1/6*(6*B*a*b*x^6 - 9*A*b^2*x^6 - 2*B*a^2*x^3 +
4*A*a*b*x^3 - A*a^2)/(a^4*x^6)

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maple [A]  time = 0.08, size = 116, normalized size = 1.20 \begin {gather*} \frac {A \,b^{2}}{3 \left (b \,x^{3}+a \right ) a^{3}}+\frac {3 A \,b^{2} \ln \relax (x )}{a^{4}}-\frac {A \,b^{2} \ln \left (b \,x^{3}+a \right )}{a^{4}}-\frac {B b}{3 \left (b \,x^{3}+a \right ) a^{2}}-\frac {2 B b \ln \relax (x )}{a^{3}}+\frac {2 B b \ln \left (b \,x^{3}+a \right )}{3 a^{3}}+\frac {2 A b}{3 a^{3} x^{3}}-\frac {B}{3 a^{2} x^{3}}-\frac {A}{6 a^{2} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^3+A)/x^7/(b*x^3+a)^2,x)

[Out]

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

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maxima [A]  time = 0.65, size = 106, normalized size = 1.09 \begin {gather*} -\frac {2 \, {\left (2 \, B a b - 3 \, A b^{2}\right )} x^{6} + {\left (2 \, B a^{2} - 3 \, A a b\right )} x^{3} + A a^{2}}{6 \, {\left (a^{3} b x^{9} + a^{4} x^{6}\right )}} + \frac {{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \left (b x^{3} + a\right )}{3 \, a^{4}} - \frac {{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \left (x^{3}\right )}{3 \, a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^3+A)/x^7/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

-1/6*(2*(2*B*a*b - 3*A*b^2)*x^6 + (2*B*a^2 - 3*A*a*b)*x^3 + A*a^2)/(a^3*b*x^9 + a^4*x^6) + 1/3*(2*B*a*b - 3*A*
b^2)*log(b*x^3 + a)/a^4 - 1/3*(2*B*a*b - 3*A*b^2)*log(x^3)/a^4

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mupad [B]  time = 0.14, size = 100, normalized size = 1.03 \begin {gather*} \frac {\frac {x^3\,\left (3\,A\,b-2\,B\,a\right )}{6\,a^2}-\frac {A}{6\,a}+\frac {b\,x^6\,\left (3\,A\,b-2\,B\,a\right )}{3\,a^3}}{b\,x^9+a\,x^6}-\frac {\ln \left (b\,x^3+a\right )\,\left (3\,A\,b^2-2\,B\,a\,b\right )}{3\,a^4}+\frac {\ln \relax (x)\,\left (3\,A\,b^2-2\,B\,a\,b\right )}{a^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^3)/(x^7*(a + b*x^3)^2),x)

[Out]

((x^3*(3*A*b - 2*B*a))/(6*a^2) - A/(6*a) + (b*x^6*(3*A*b - 2*B*a))/(3*a^3))/(a*x^6 + b*x^9) - (log(a + b*x^3)*
(3*A*b^2 - 2*B*a*b))/(3*a^4) + (log(x)*(3*A*b^2 - 2*B*a*b))/a^4

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sympy [A]  time = 2.40, size = 100, normalized size = 1.03 \begin {gather*} \frac {- A a^{2} + x^{6} \left (6 A b^{2} - 4 B a b\right ) + x^{3} \left (3 A a b - 2 B a^{2}\right )}{6 a^{4} x^{6} + 6 a^{3} b x^{9}} - \frac {b \left (- 3 A b + 2 B a\right ) \log {\relax (x )}}{a^{4}} + \frac {b \left (- 3 A b + 2 B a\right ) \log {\left (\frac {a}{b} + x^{3} \right )}}{3 a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**3+A)/x**7/(b*x**3+a)**2,x)

[Out]

(-A*a**2 + x**6*(6*A*b**2 - 4*B*a*b) + x**3*(3*A*a*b - 2*B*a**2))/(6*a**4*x**6 + 6*a**3*b*x**9) - b*(-3*A*b +
2*B*a)*log(x)/a**4 + b*(-3*A*b + 2*B*a)*log(a/b + x**3)/(3*a**4)

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