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

Optimal. Leaf size=183 \[ -\frac{a^{4/3} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{10/3}}+\frac{a^{4/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{10/3}}-\frac{a^{4/3} (A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{10/3}}+\frac{x^4 (A b-a B)}{4 b^2}-\frac{a x (A b-a B)}{b^3}+\frac{B x^7}{7 b} \]

[Out]

-((a*(A*b - a*B)*x)/b^3) + ((A*b - a*B)*x^4)/(4*b^2) + (B*x^7)/(7*b) - (a^(4/3)*(A*b - a*B)*ArcTan[(a^(1/3) -
2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^(10/3)) + (a^(4/3)*(A*b - a*B)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(10/
3)) - (a^(4/3)*(A*b - a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(10/3))

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Rubi [A]  time = 0.149132, antiderivative size = 183, 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 = {459, 302, 200, 31, 634, 617, 204, 628} \[ -\frac{a^{4/3} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{10/3}}+\frac{a^{4/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{10/3}}-\frac{a^{4/3} (A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{10/3}}+\frac{x^4 (A b-a B)}{4 b^2}-\frac{a x (A b-a B)}{b^3}+\frac{B x^7}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^6*(A + B*x^3))/(a + b*x^3),x]

[Out]

-((a*(A*b - a*B)*x)/b^3) + ((A*b - a*B)*x^4)/(4*b^2) + (B*x^7)/(7*b) - (a^(4/3)*(A*b - a*B)*ArcTan[(a^(1/3) -
2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^(10/3)) + (a^(4/3)*(A*b - a*B)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(10/
3)) - (a^(4/3)*(A*b - a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(10/3))

Rule 459

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

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 )}{a+b x^3} \, dx &=\frac{B x^7}{7 b}-\frac{(-7 A b+7 a B) \int \frac{x^6}{a+b x^3} \, dx}{7 b}\\ &=\frac{B x^7}{7 b}-\frac{(-7 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}{7 b}\\ &=-\frac{a (A b-a B) x}{b^3}+\frac{(A b-a B) x^4}{4 b^2}+\frac{B x^7}{7 b}+\frac{\left (a^2 (A b-a B)\right ) \int \frac{1}{a+b x^3} \, dx}{b^3}\\ &=-\frac{a (A b-a B) x}{b^3}+\frac{(A b-a B) x^4}{4 b^2}+\frac{B x^7}{7 b}+\frac{\left (a^{4/3} (A b-a B)\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^3}+\frac{\left (a^{4/3} (A b-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}{3 b^3}\\ &=-\frac{a (A b-a B) x}{b^3}+\frac{(A b-a B) x^4}{4 b^2}+\frac{B x^7}{7 b}+\frac{a^{4/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{10/3}}-\frac{\left (a^{4/3} (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 b^{10/3}}+\frac{\left (a^{5/3} (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 b^3}\\ &=-\frac{a (A b-a B) x}{b^3}+\frac{(A b-a B) x^4}{4 b^2}+\frac{B x^7}{7 b}+\frac{a^{4/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{10/3}}-\frac{a^{4/3} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{10/3}}+\frac{\left (a^{4/3} (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 )}{b^{10/3}}\\ &=-\frac{a (A b-a B) x}{b^3}+\frac{(A b-a B) x^4}{4 b^2}+\frac{B x^7}{7 b}-\frac{a^{4/3} (A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{10/3}}+\frac{a^{4/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{10/3}}-\frac{a^{4/3} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{10/3}}\\ \end{align*}

Mathematica [A]  time = 0.106341, size = 171, normalized size = 0.93 \[ \frac{14 a^{4/3} (a B-A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-28 a^{4/3} (a B-A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+28 \sqrt{3} a^{4/3} (a B-A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+21 b^{4/3} x^4 (A b-a B)+84 a \sqrt [3]{b} x (a B-A b)+12 b^{7/3} B x^7}{84 b^{10/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(84*a*b^(1/3)*(-(A*b) + a*B)*x + 21*b^(4/3)*(A*b - a*B)*x^4 + 12*b^(7/3)*B*x^7 + 28*Sqrt[3]*a^(4/3)*(-(A*b) +
a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 28*a^(4/3)*(-(A*b) + a*B)*Log[a^(1/3) + b^(1/3)*x] + 14*a^(
4/3)*(-(A*b) + a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(84*b^(10/3))

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Maple [A]  time = 0.007, size = 249, normalized size = 1.4 \begin{align*}{\frac{B{x}^{7}}{7\,b}}+{\frac{A{x}^{4}}{4\,b}}-{\frac{B{x}^{4}a}{4\,{b}^{2}}}-{\frac{aAx}{{b}^{2}}}+{\frac{{a}^{2}Bx}{{b}^{3}}}+{\frac{{a}^{2}A}{3\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B{a}^{3}}{3\,{b}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{a}^{2}A}{6\,{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{B{a}^{3}}{6\,{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{{a}^{2}\sqrt{3}A}{3\,{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{{a}^{3}\sqrt{3}B}{3\,{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),x)

[Out]

1/7*B*x^7/b+1/4/b*A*x^4-1/4/b^2*B*x^4*a-1/b^2*a*A*x+1/b^3*a^2*B*x+1/3*a^2/b^3/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*A-
1/3*a^3/b^4/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*B-1/6*a^2/b^3/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*A+1/6*a^
3/b^4/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*B+1/3*a^2/b^3/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a
/b)^(1/3)*x-1))*A-1/3*a^3/b^4/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*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(x^6*(B*x^3+A)/(b*x^3+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.52742, size = 397, normalized size = 2.17 \begin{align*} \frac{12 \, B b^{2} x^{7} - 21 \,{\left (B a b - A b^{2}\right )} x^{4} - 28 \, \sqrt{3}{\left (B a^{2} - 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 ) + 14 \,{\left (B a^{2} - 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 ) - 28 \,{\left (B a^{2} - A a b\right )} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right ) + 84 \,{\left (B a^{2} - A a b\right )} x}{84 \, 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),x, algorithm="fricas")

[Out]

1/84*(12*B*b^2*x^7 - 21*(B*a*b - A*b^2)*x^4 - 28*sqrt(3)*(B*a^2 - A*a*b)*(a/b)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x
*(a/b)^(2/3) - sqrt(3)*a)/a) + 14*(B*a^2 - A*a*b)*(a/b)^(1/3)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3)) - 28*(B*a
^2 - A*a*b)*(a/b)^(1/3)*log(x + (a/b)^(1/3)) + 84*(B*a^2 - A*a*b)*x)/b^3

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Sympy [A]  time = 1.22068, size = 110, normalized size = 0.6 \begin{align*} \frac{B x^{7}}{7 b} + \operatorname{RootSum}{\left (27 t^{3} b^{10} - A^{3} a^{4} b^{3} + 3 A^{2} B a^{5} b^{2} - 3 A B^{2} a^{6} b + B^{3} a^{7}, \left ( t \mapsto t \log{\left (- \frac{3 t b^{3}}{- A a b + B a^{2}} + x \right )} \right )\right )} - \frac{x^{4} \left (- A b + B a\right )}{4 b^{2}} + \frac{x \left (- A a b + B a^{2}\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),x)

[Out]

B*x**7/(7*b) + RootSum(27*_t**3*b**10 - A**3*a**4*b**3 + 3*A**2*B*a**5*b**2 - 3*A*B**2*a**6*b + B**3*a**7, Lam
bda(_t, _t*log(-3*_t*b**3/(-A*a*b + B*a**2) + x))) - x**4*(-A*b + B*a)/(4*b**2) + x*(-A*a*b + B*a**2)/b**3

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Giac [A]  time = 1.1233, size = 293, normalized size = 1.6 \begin{align*} -\frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} B a^{2} - \left (-a b^{2}\right )^{\frac{1}{3}} A 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 \, b^{4}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} B a^{2} - \left (-a b^{2}\right )^{\frac{1}{3}} A 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 \, b^{4}} + \frac{{\left (B a^{3} b^{4} - A a^{2} b^{5}\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 b^{7}} + \frac{4 \, B b^{6} x^{7} - 7 \, B a b^{5} x^{4} + 7 \, A b^{6} x^{4} + 28 \, B a^{2} b^{4} x - 28 \, A a b^{5} x}{28 \, b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(3)*((-a*b^2)^(1/3)*B*a^2 - (-a*b^2)^(1/3)*A*a*b)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3
))/b^4 - 1/6*((-a*b^2)^(1/3)*B*a^2 - (-a*b^2)^(1/3)*A*a*b)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^4 + 1/3*
(B*a^3*b^4 - A*a^2*b^5)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^7) + 1/28*(4*B*b^6*x^7 - 7*B*a*b^5*x^4 +
7*A*b^6*x^4 + 28*B*a^2*b^4*x - 28*A*a*b^5*x)/b^7