3.74 \(\int \frac{x^6 (A+B x^3)}{(a+b x^3)^2} \, dx\)

Optimal. Leaf size=213 \[ \frac{\sqrt [3]{a} (4 A b-7 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{10/3}}-\frac{x^4 (4 A b-7 a B)}{12 a b^2}+\frac{x (4 A b-7 a B)}{3 b^3}-\frac{\sqrt [3]{a} (4 A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{10/3}}+\frac{\sqrt [3]{a} (4 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} b^{10/3}}+\frac{x^7 (A b-a B)}{3 a b \left (a+b x^3\right )} \]

[Out]

((4*A*b - 7*a*B)*x)/(3*b^3) - ((4*A*b - 7*a*B)*x^4)/(12*a*b^2) + ((A*b - a*B)*x^7)/(3*a*b*(a + b*x^3)) + (a^(1
/3)*(4*A*b - 7*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*b^(10/3)) - (a^(1/3)*(4*A*b
- 7*a*B)*Log[a^(1/3) + b^(1/3)*x])/(9*b^(10/3)) + (a^(1/3)*(4*A*b - 7*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b
^(2/3)*x^2])/(18*b^(10/3))

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Rubi [A]  time = 0.130409, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {457, 302, 200, 31, 634, 617, 204, 628} \[ \frac{\sqrt [3]{a} (4 A b-7 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{10/3}}-\frac{x^4 (4 A b-7 a B)}{12 a b^2}+\frac{x (4 A b-7 a B)}{3 b^3}-\frac{\sqrt [3]{a} (4 A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{10/3}}+\frac{\sqrt [3]{a} (4 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} b^{10/3}}+\frac{x^7 (A b-a B)}{3 a b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*A*b - 7*a*B)*x)/(3*b^3) - ((4*A*b - 7*a*B)*x^4)/(12*a*b^2) + ((A*b - a*B)*x^7)/(3*a*b*(a + b*x^3)) + (a^(1
/3)*(4*A*b - 7*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*b^(10/3)) - (a^(1/3)*(4*A*b
- 7*a*B)*Log[a^(1/3) + b^(1/3)*x])/(9*b^(10/3)) + (a^(1/3)*(4*A*b - 7*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b
^(2/3)*x^2])/(18*b^(10/3))

Rule 457

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d
)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*b*e*n*(p + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b
*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] &&
LeQ[-1, m, -(n*(p + 1))]))

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*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{x^6 \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx &=\frac{(A b-a B) x^7}{3 a b \left (a+b x^3\right )}+\frac{(-4 A b+7 a B) \int \frac{x^6}{a+b x^3} \, dx}{3 a b}\\ &=\frac{(A b-a B) x^7}{3 a b \left (a+b x^3\right )}+\frac{(-4 A b+7 a B) \int \left (-\frac{a}{b^2}+\frac{x^3}{b}+\frac{a^2}{b^2 \left (a+b x^3\right )}\right ) \, dx}{3 a b}\\ &=\frac{(4 A b-7 a B) x}{3 b^3}-\frac{(4 A b-7 a B) x^4}{12 a b^2}+\frac{(A b-a B) x^7}{3 a b \left (a+b x^3\right )}-\frac{(a (4 A b-7 a B)) \int \frac{1}{a+b x^3} \, dx}{3 b^3}\\ &=\frac{(4 A b-7 a B) x}{3 b^3}-\frac{(4 A b-7 a B) x^4}{12 a b^2}+\frac{(A b-a B) x^7}{3 a b \left (a+b x^3\right )}-\frac{\left (\sqrt [3]{a} (4 A b-7 a B)\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 b^3}-\frac{\left (\sqrt [3]{a} (4 A b-7 a B)\right ) \int \frac{2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 b^3}\\ &=\frac{(4 A b-7 a B) x}{3 b^3}-\frac{(4 A b-7 a B) x^4}{12 a b^2}+\frac{(A b-a B) x^7}{3 a b \left (a+b x^3\right )}-\frac{\sqrt [3]{a} (4 A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{10/3}}+\frac{\left (\sqrt [3]{a} (4 A b-7 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}{18 b^{10/3}}-\frac{\left (a^{2/3} (4 A b-7 a B)\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 b^3}\\ &=\frac{(4 A b-7 a B) x}{3 b^3}-\frac{(4 A b-7 a B) x^4}{12 a b^2}+\frac{(A b-a B) x^7}{3 a b \left (a+b x^3\right )}-\frac{\sqrt [3]{a} (4 A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{10/3}}+\frac{\sqrt [3]{a} (4 A b-7 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{10/3}}-\frac{\left (\sqrt [3]{a} (4 A b-7 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 )}{3 b^{10/3}}\\ &=\frac{(4 A b-7 a B) x}{3 b^3}-\frac{(4 A b-7 a B) x^4}{12 a b^2}+\frac{(A b-a B) x^7}{3 a b \left (a+b x^3\right )}+\frac{\sqrt [3]{a} (4 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} b^{10/3}}-\frac{\sqrt [3]{a} (4 A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{10/3}}+\frac{\sqrt [3]{a} (4 A b-7 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{10/3}}\\ \end{align*}

Mathematica [A]  time = 0.1235, size = 181, normalized size = 0.85 \[ \frac{-2 \sqrt [3]{a} (7 a B-4 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+\frac{12 a \sqrt [3]{b} x (A b-a B)}{a+b x^3}+36 \sqrt [3]{b} x (A b-2 a B)+4 \sqrt [3]{a} (7 a B-4 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-4 \sqrt{3} \sqrt [3]{a} (7 a B-4 A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+9 b^{4/3} B x^4}{36 b^{10/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(36*b^(1/3)*(A*b - 2*a*B)*x + 9*b^(4/3)*B*x^4 + (12*a*b^(1/3)*(A*b - a*B)*x)/(a + b*x^3) - 4*Sqrt[3]*a^(1/3)*(
-4*A*b + 7*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 4*a^(1/3)*(-4*A*b + 7*a*B)*Log[a^(1/3) + b^(1/3)
*x] - 2*a^(1/3)*(-4*A*b + 7*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(36*b^(10/3))

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Maple [A]  time = 0.009, size = 257, normalized size = 1.2 \begin{align*}{\frac{B{x}^{4}}{4\,{b}^{2}}}+{\frac{Ax}{{b}^{2}}}-2\,{\frac{Bax}{{b}^{3}}}+{\frac{aAx}{3\,{b}^{2} \left ( b{x}^{3}+a \right ) }}-{\frac{{a}^{2}Bx}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}-{\frac{4\,Aa}{9\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,Aa}{9\,{b}^{3}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{4\,Aa\sqrt{3}}{9\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{7\,{a}^{2}B}{9\,{b}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{7\,{a}^{2}B}{18\,{b}^{4}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{7\,{a}^{2}B\sqrt{3}}{9\,{b}^{4}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(B*x^3+A)/(b*x^3+a)^2,x)

[Out]

1/4/b^2*B*x^4+1/b^2*A*x-2/b^3*B*a*x+1/3*a/b^2*x/(b*x^3+a)*A-1/3*a^2/b^3*x/(b*x^3+a)*B-4/9*a/b^3*A/(a/b)^(2/3)*
ln(x+(a/b)^(1/3))+2/9*a/b^3*A/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-4/9*a/b^3*A/(a/b)^(2/3)*3^(1/2)*ar
ctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+7/9*a^2/b^4*B/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-7/18*a^2/b^4*B/(a/b)^(2/3)*l
n(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+7/9*a^2/b^4*B/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))

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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(x^6*(B*x^3+A)/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.77681, size = 551, normalized size = 2.59 \begin{align*} \frac{9 \, B b^{2} x^{7} - 9 \,{\left (7 \, B a b - 4 \, A b^{2}\right )} x^{4} - 4 \, \sqrt{3}{\left ({\left (7 \, B a b - 4 \, A b^{2}\right )} x^{3} + 7 \, B a^{2} - 4 \, A a b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x \left (-\frac{a}{b}\right )^{\frac{2}{3}} - \sqrt{3} a}{3 \, a}\right ) + 2 \,{\left ({\left (7 \, B a b - 4 \, A b^{2}\right )} x^{3} + 7 \, B a^{2} - 4 \, A a b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right ) - 4 \,{\left ({\left (7 \, B a b - 4 \, A b^{2}\right )} x^{3} + 7 \, B a^{2} - 4 \, A a b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x - \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right ) - 12 \,{\left (7 \, B a^{2} - 4 \, A a b\right )} x}{36 \,{\left (b^{4} x^{3} + a b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(B*x^3+A)/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

1/36*(9*B*b^2*x^7 - 9*(7*B*a*b - 4*A*b^2)*x^4 - 4*sqrt(3)*((7*B*a*b - 4*A*b^2)*x^3 + 7*B*a^2 - 4*A*a*b)*(-a/b)
^(1/3)*arctan(1/3*(2*sqrt(3)*b*x*(-a/b)^(2/3) - sqrt(3)*a)/a) + 2*((7*B*a*b - 4*A*b^2)*x^3 + 7*B*a^2 - 4*A*a*b
)*(-a/b)^(1/3)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3)) - 4*((7*B*a*b - 4*A*b^2)*x^3 + 7*B*a^2 - 4*A*a*b)*(-a/
b)^(1/3)*log(x - (-a/b)^(1/3)) - 12*(7*B*a^2 - 4*A*a*b)*x)/(b^4*x^3 + a*b^3)

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Sympy [A]  time = 1.43998, size = 124, normalized size = 0.58 \begin{align*} \frac{B x^{4}}{4 b^{2}} - \frac{x \left (- A a b + B a^{2}\right )}{3 a b^{3} + 3 b^{4} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} b^{10} + 64 A^{3} a b^{3} - 336 A^{2} B a^{2} b^{2} + 588 A B^{2} a^{3} b - 343 B^{3} a^{4}, \left ( t \mapsto t \log{\left (\frac{9 t b^{3}}{- 4 A b + 7 B a} + x \right )} \right )\right )} - \frac{x \left (- A b + 2 B a\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(B*x**3+A)/(b*x**3+a)**2,x)

[Out]

B*x**4/(4*b**2) - x*(-A*a*b + B*a**2)/(3*a*b**3 + 3*b**4*x**3) + RootSum(729*_t**3*b**10 + 64*A**3*a*b**3 - 33
6*A**2*B*a**2*b**2 + 588*A*B**2*a**3*b - 343*B**3*a**4, Lambda(_t, _t*log(9*_t*b**3/(-4*A*b + 7*B*a) + x))) -
x*(-A*b + 2*B*a)/b**3

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Giac [A]  time = 1.12706, size = 285, normalized size = 1.34 \begin{align*} \frac{\sqrt{3}{\left (7 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - 4 \, \left (-a b^{2}\right )^{\frac{1}{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 )}{9 \, b^{4}} - \frac{{\left (7 \, B a^{2} - 4 \, A a b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b^{3}} + \frac{{\left (7 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - 4 \, \left (-a b^{2}\right )^{\frac{1}{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 )}{18 \, b^{4}} - \frac{B a^{2} x - A a b x}{3 \,{\left (b x^{3} + a\right )} b^{3}} + \frac{B b^{6} x^{4} - 8 \, B a b^{5} x + 4 \, A b^{6} x}{4 \, b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(B*x^3+A)/(b*x^3+a)^2,x, algorithm="giac")

[Out]

1/9*sqrt(3)*(7*(-a*b^2)^(1/3)*B*a - 4*(-a*b^2)^(1/3)*A*b)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3)
)/b^4 - 1/9*(7*B*a^2 - 4*A*a*b)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^3) + 1/18*(7*(-a*b^2)^(1/3)*B*a -
4*(-a*b^2)^(1/3)*A*b)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^4 - 1/3*(B*a^2*x - A*a*b*x)/((b*x^3 + a)*b^3
) + 1/4*(B*b^6*x^4 - 8*B*a*b^5*x + 4*A*b^6*x)/b^8