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

Optimal. Leaf size=165 \[ \frac{\sqrt [3]{b} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{7/3}}+\frac{A b-a B}{a^2 x}-\frac{\sqrt [3]{b} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{7/3}}-\frac{\sqrt [3]{b} (A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{7/3}}-\frac{A}{4 a x^4} \]

[Out]

-A/(4*a*x^4) + (A*b - a*B)/(a^2*x) - (b^(1/3)*(A*b - a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(
Sqrt[3]*a^(7/3)) - (b^(1/3)*(A*b - a*B)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(7/3)) + (b^(1/3)*(A*b - a*B)*Log[a^(2/
3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(7/3))

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Rubi [A]  time = 0.114788, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {453, 325, 292, 31, 634, 617, 204, 628} \[ \frac{\sqrt [3]{b} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{7/3}}+\frac{A b-a B}{a^2 x}-\frac{\sqrt [3]{b} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{7/3}}-\frac{\sqrt [3]{b} (A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{7/3}}-\frac{A}{4 a x^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^3)/(x^5*(a + b*x^3)),x]

[Out]

-A/(4*a*x^4) + (A*b - a*B)/(a^2*x) - (b^(1/3)*(A*b - a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(
Sqrt[3]*a^(7/3)) - (b^(1/3)*(A*b - a*B)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(7/3)) + (b^(1/3)*(A*b - a*B)*Log[a^(2/
3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(7/3))

Rule 453

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

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

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

Mathematica [A]  time = 0.115591, size = 154, normalized size = 0.93 \[ \frac{2 \sqrt [3]{b} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac{3 a^{4/3} A}{x^4}+\frac{12 \sqrt [3]{a} (A b-a B)}{x}+4 \sqrt [3]{b} (a B-A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-4 \sqrt{3} \sqrt [3]{b} (A b-a B) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{12 a^{7/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-3*a^(4/3)*A)/x^4 + (12*a^(1/3)*(A*b - a*B))/x - 4*Sqrt[3]*b^(1/3)*(A*b - a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(
1/3))/Sqrt[3]] + 4*b^(1/3)*(-(A*b) + a*B)*Log[a^(1/3) + b^(1/3)*x] + 2*b^(1/3)*(A*b - a*B)*Log[a^(2/3) - a^(1/
3)*b^(1/3)*x + b^(2/3)*x^2])/(12*a^(7/3))

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Maple [A]  time = 0.008, size = 216, normalized size = 1.3 \begin{align*} -{\frac{Ab}{3\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{B}{3\,a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{Ab}{6\,{a}^{2}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{B}{6\,a}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{b\sqrt{3}A}{3\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{\sqrt{3}B}{3\,a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{A}{4\,a{x}^{4}}}+{\frac{Ab}{{a}^{2}x}}-{\frac{B}{ax}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.48209, size = 371, normalized size = 2.25 \begin{align*} -\frac{4 \, \sqrt{3}{\left (B a - A b\right )} x^{4} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{2}{3} \, \sqrt{3} x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) - 2 \,{\left (B a - A b\right )} x^{4} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (-\frac{b}{a}\right )^{\frac{2}{3}} - a \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right ) + 4 \,{\left (B a - A b\right )} x^{4} \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right ) + 12 \,{\left (B a - A b\right )} x^{3} + 3 \, A a}{12 \, a^{2} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(4*sqrt(3)*(B*a - A*b)*x^4*(-b/a)^(1/3)*arctan(2/3*sqrt(3)*x*(-b/a)^(1/3) + 1/3*sqrt(3)) - 2*(B*a - A*b)
*x^4*(-b/a)^(1/3)*log(b*x^2 - a*x*(-b/a)^(2/3) - a*(-b/a)^(1/3)) + 4*(B*a - A*b)*x^4*(-b/a)^(1/3)*log(b*x + a*
(-b/a)^(2/3)) + 12*(B*a - A*b)*x^3 + 3*A*a)/(a^2*x^4)

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Sympy [A]  time = 1.40904, size = 112, normalized size = 0.68 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} a^{7} + A^{3} b^{4} - 3 A^{2} B a b^{3} + 3 A B^{2} a^{2} b^{2} - B^{3} a^{3} b, \left ( t \mapsto t \log{\left (\frac{9 t^{2} a^{5}}{A^{2} b^{3} - 2 A B a b^{2} + B^{2} a^{2} b} + x \right )} \right )\right )} - \frac{A a + x^{3} \left (- 4 A b + 4 B a\right )}{4 a^{2} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(27*_t**3*a**7 + A**3*b**4 - 3*A**2*B*a*b**3 + 3*A*B**2*a**2*b**2 - B**3*a**3*b, Lambda(_t, _t*log(9*_t
**2*a**5/(A**2*b**3 - 2*A*B*a*b**2 + B**2*a**2*b) + x))) - (A*a + x**3*(-4*A*b + 4*B*a))/(4*a**2*x**4)

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Giac [A]  time = 1.14394, size = 266, normalized size = 1.61 \begin{align*} \frac{{\left (B a b \left (-\frac{a}{b}\right )^{\frac{1}{3}} - A b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{3}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{2}{3}} B a - \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{3} b} - \frac{{\left (\left (-a b^{2}\right )^{\frac{2}{3}} B a - \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{3} b} - \frac{4 \, B a x^{3} - 4 \, A b x^{3} + A a}{4 \, a^{2} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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