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

Optimal. Leaf size=122 \[ \frac{b (3 A b-2 a B)}{3 a^4 \left (a+b x^3\right )}+\frac{b (A b-a B)}{6 a^3 \left (a+b x^3\right )^2}+\frac{3 A b-a B}{3 a^4 x^3}-\frac{b (2 A b-a B) \log \left (a+b x^3\right )}{a^5}+\frac{3 b \log (x) (2 A b-a B)}{a^5}-\frac{A}{6 a^3 x^6} \]

[Out]

-A/(6*a^3*x^6) + (3*A*b - a*B)/(3*a^4*x^3) + (b*(A*b - a*B))/(6*a^3*(a + b*x^3)^2) + (b*(3*A*b - 2*a*B))/(3*a^
4*(a + b*x^3)) + (3*b*(2*A*b - a*B)*Log[x])/a^5 - (b*(2*A*b - a*B)*Log[a + b*x^3])/a^5

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Rubi [A]  time = 0.129556, antiderivative size = 122, 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} \[ \frac{b (3 A b-2 a B)}{3 a^4 \left (a+b x^3\right )}+\frac{b (A b-a B)}{6 a^3 \left (a+b x^3\right )^2}+\frac{3 A b-a B}{3 a^4 x^3}-\frac{b (2 A b-a B) \log \left (a+b x^3\right )}{a^5}+\frac{3 b \log (x) (2 A b-a B)}{a^5}-\frac{A}{6 a^3 x^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-A/(6*a^3*x^6) + (3*A*b - a*B)/(3*a^4*x^3) + (b*(A*b - a*B))/(6*a^3*(a + b*x^3)^2) + (b*(3*A*b - 2*a*B))/(3*a^
4*(a + b*x^3)) + (3*b*(2*A*b - a*B)*Log[x])/a^5 - (b*(2*A*b - a*B)*Log[a + b*x^3])/a^5

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{A+B x^3}{x^7 \left (a+b x^3\right )^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{A+B x}{x^3 (a+b x)^3} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{A}{a^3 x^3}+\frac{-3 A b+a B}{a^4 x^2}-\frac{3 b (-2 A b+a B)}{a^5 x}+\frac{b^2 (-A b+a B)}{a^3 (a+b x)^3}+\frac{b^2 (-3 A b+2 a B)}{a^4 (a+b x)^2}+\frac{3 b^2 (-2 A b+a B)}{a^5 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac{A}{6 a^3 x^6}+\frac{3 A b-a B}{3 a^4 x^3}+\frac{b (A b-a B)}{6 a^3 \left (a+b x^3\right )^2}+\frac{b (3 A b-2 a B)}{3 a^4 \left (a+b x^3\right )}+\frac{3 b (2 A b-a B) \log (x)}{a^5}-\frac{b (2 A b-a B) \log \left (a+b x^3\right )}{a^5}\\ \end{align*}

Mathematica [A]  time = 0.0767724, size = 108, normalized size = 0.89 \[ \frac{\frac{a^2 b (A b-a B)}{\left (a+b x^3\right )^2}-\frac{a^2 A}{x^6}+\frac{2 a b (3 A b-2 a B)}{a+b x^3}-\frac{2 a (a B-3 A b)}{x^3}+6 b (a B-2 A b) \log \left (a+b x^3\right )+18 b \log (x) (2 A b-a B)}{6 a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((a^2*A)/x^6) - (2*a*(-3*A*b + a*B))/x^3 + (a^2*b*(A*b - a*B))/(a + b*x^3)^2 + (2*a*b*(3*A*b - 2*a*B))/(a +
b*x^3) + 18*b*(2*A*b - a*B)*Log[x] + 6*b*(-2*A*b + a*B)*Log[a + b*x^3])/(6*a^5)

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Maple [A]  time = 0.014, size = 147, normalized size = 1.2 \begin{align*}{\frac{A{b}^{2}}{{a}^{4} \left ( b{x}^{3}+a \right ) }}-{\frac{2\,Bb}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) }}-2\,{\frac{{b}^{2}\ln \left ( b{x}^{3}+a \right ) A}{{a}^{5}}}+{\frac{b\ln \left ( b{x}^{3}+a \right ) B}{{a}^{4}}}+{\frac{A{b}^{2}}{6\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{Bb}{6\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{A}{6\,{a}^{3}{x}^{6}}}+{\frac{Ab}{{a}^{4}{x}^{3}}}-{\frac{B}{3\,{a}^{3}{x}^{3}}}+6\,{\frac{A\ln \left ( x \right ){b}^{2}}{{a}^{5}}}-3\,{\frac{bB\ln \left ( x \right ) }{{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.957756, size = 184, normalized size = 1.51 \begin{align*} -\frac{6 \,{\left (B a b^{2} - 2 \, A b^{3}\right )} x^{9} + 9 \,{\left (B a^{2} b - 2 \, A a b^{2}\right )} x^{6} + A a^{3} + 2 \,{\left (B a^{3} - 2 \, A a^{2} b\right )} x^{3}}{6 \,{\left (a^{4} b^{2} x^{12} + 2 \, a^{5} b x^{9} + a^{6} x^{6}\right )}} + \frac{{\left (B a b - 2 \, A b^{2}\right )} \log \left (b x^{3} + a\right )}{a^{5}} - \frac{{\left (B a b - 2 \, A b^{2}\right )} \log \left (x^{3}\right )}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(6*(B*a*b^2 - 2*A*b^3)*x^9 + 9*(B*a^2*b - 2*A*a*b^2)*x^6 + A*a^3 + 2*(B*a^3 - 2*A*a^2*b)*x^3)/(a^4*b^2*x^
12 + 2*a^5*b*x^9 + a^6*x^6) + (B*a*b - 2*A*b^2)*log(b*x^3 + a)/a^5 - (B*a*b - 2*A*b^2)*log(x^3)/a^5

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Fricas [B]  time = 1.543, size = 478, normalized size = 3.92 \begin{align*} -\frac{6 \,{\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{9} + 9 \,{\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{6} + A a^{4} + 2 \,{\left (B a^{4} - 2 \, A a^{3} b\right )} x^{3} - 6 \,{\left ({\left (B a b^{3} - 2 \, A b^{4}\right )} x^{12} + 2 \,{\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{9} +{\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{6}\right )} \log \left (b x^{3} + a\right ) + 18 \,{\left ({\left (B a b^{3} - 2 \, A b^{4}\right )} x^{12} + 2 \,{\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{9} +{\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{6}\right )} \log \left (x\right )}{6 \,{\left (a^{5} b^{2} x^{12} + 2 \, a^{6} b x^{9} + a^{7} x^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(6*(B*a^2*b^2 - 2*A*a*b^3)*x^9 + 9*(B*a^3*b - 2*A*a^2*b^2)*x^6 + A*a^4 + 2*(B*a^4 - 2*A*a^3*b)*x^3 - 6*((
B*a*b^3 - 2*A*b^4)*x^12 + 2*(B*a^2*b^2 - 2*A*a*b^3)*x^9 + (B*a^3*b - 2*A*a^2*b^2)*x^6)*log(b*x^3 + a) + 18*((B
*a*b^3 - 2*A*b^4)*x^12 + 2*(B*a^2*b^2 - 2*A*a*b^3)*x^9 + (B*a^3*b - 2*A*a^2*b^2)*x^6)*log(x))/(a^5*b^2*x^12 +
2*a^6*b*x^9 + a^7*x^6)

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Sympy [A]  time = 9.36646, size = 133, normalized size = 1.09 \begin{align*} - \frac{A a^{3} + x^{9} \left (- 12 A b^{3} + 6 B a b^{2}\right ) + x^{6} \left (- 18 A a b^{2} + 9 B a^{2} b\right ) + x^{3} \left (- 4 A a^{2} b + 2 B a^{3}\right )}{6 a^{6} x^{6} + 12 a^{5} b x^{9} + 6 a^{4} b^{2} x^{12}} - \frac{3 b \left (- 2 A b + B a\right ) \log{\left (x \right )}}{a^{5}} + \frac{b \left (- 2 A b + B a\right ) \log{\left (\frac{a}{b} + x^{3} \right )}}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(A*a**3 + x**9*(-12*A*b**3 + 6*B*a*b**2) + x**6*(-18*A*a*b**2 + 9*B*a**2*b) + x**3*(-4*A*a**2*b + 2*B*a**3))/
(6*a**6*x**6 + 12*a**5*b*x**9 + 6*a**4*b**2*x**12) - 3*b*(-2*A*b + B*a)*log(x)/a**5 + b*(-2*A*b + B*a)*log(a/b
+ x**3)/a**5

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Giac [A]  time = 1.12672, size = 177, normalized size = 1.45 \begin{align*} -\frac{3 \,{\left (B a b - 2 \, A b^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac{{\left (B a b^{2} - 2 \, A b^{3}\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{a^{5} b} - \frac{6 \, B a b^{2} x^{9} - 12 \, A b^{3} x^{9} + 9 \, B a^{2} b x^{6} - 18 \, A a b^{2} x^{6} + 2 \, B a^{3} x^{3} - 4 \, A a^{2} b x^{3} + A a^{3}}{6 \,{\left (b x^{6} + a x^{3}\right )}^{2} a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*(B*a*b - 2*A*b^2)*log(abs(x))/a^5 + (B*a*b^2 - 2*A*b^3)*log(abs(b*x^3 + a))/(a^5*b) - 1/6*(6*B*a*b^2*x^9 -
12*A*b^3*x^9 + 9*B*a^2*b*x^6 - 18*A*a*b^2*x^6 + 2*B*a^3*x^3 - 4*A*a^2*b*x^3 + A*a^3)/((b*x^6 + a*x^3)^2*a^4)