Integrand size = 20, antiderivative size = 183 \[ \int \frac {x^6 \left (A+B x^3\right )}{a+b x^3} \, dx=-\frac {a (A b-a B) x}{b^3}+\frac {(A b-a B) x^4}{4 b^2}+\frac {B x^7}{7 b}-\frac {a^{4/3} (A b-a B) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{10/3}}+\frac {a^{4/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{10/3}}-\frac {a^{4/3} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{10/3}} \]
-a*(A*b-B*a)*x/b^3+1/4*(A*b-B*a)*x^4/b^2+1/7*B*x^7/b+1/3*a^(4/3)*(A*b-B*a) *ln(a^(1/3)+b^(1/3)*x)/b^(10/3)-1/6*a^(4/3)*(A*b-B*a)*ln(a^(2/3)-a^(1/3)*b ^(1/3)*x+b^(2/3)*x^2)/b^(10/3)-1/3*a^(4/3)*(A*b-B*a)*arctan(1/3*(a^(1/3)-2 *b^(1/3)*x)/a^(1/3)*3^(1/2))/b^(10/3)*3^(1/2)
Time = 0.11 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.93 \[ \int \frac {x^6 \left (A+B x^3\right )}{a+b x^3} \, dx=\frac {84 a \sqrt [3]{b} (-A b+a B) x+21 b^{4/3} (A b-a B) x^4+12 b^{7/3} B x^7+28 \sqrt {3} a^{4/3} (-A b+a B) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-28 a^{4/3} (-A b+a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+14 a^{4/3} (-A b+a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{84 b^{10/3}} \]
(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^(1 0/3))
Time = 0.31 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.85, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {959, 831, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^6 \left (A+B x^3\right )}{a+b x^3} \, dx\) |
\(\Big \downarrow \) 959 |
\(\displaystyle \frac {(A b-a B) \int \frac {x^6}{b x^3+a}dx}{b}+\frac {B x^7}{7 b}\) |
\(\Big \downarrow \) 831 |
\(\displaystyle \frac {(A b-a B) \int \left (\frac {x^3}{b}+\frac {a^2}{b^2 \left (b x^3+a\right )}-\frac {a}{b^2}\right )dx}{b}+\frac {B x^7}{7 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(A b-a B) \left (-\frac {a^{4/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{7/3}}-\frac {a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{7/3}}+\frac {a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{7/3}}-\frac {a x}{b^2}+\frac {x^4}{4 b}\right )}{b}+\frac {B x^7}{7 b}\) |
(B*x^7)/(7*b) + ((A*b - a*B)*(-((a*x)/b^2) + x^4/(4*b) - (a^(4/3)*ArcTan[( a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^(7/3)) + (a^(4/3)*Lo g[a^(1/3) + b^(1/3)*x])/(3*b^(7/3)) - (a^(4/3)*Log[a^(2/3) - a^(1/3)*b^(1/ 3)*x + b^(2/3)*x^2])/(6*b^(7/3))))/b
3.1.56.3.1 Defintions of rubi rules used
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] && Gt Q[m, 2*n - 1]
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] - Simp[(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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.01 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.46
method | result | size |
risch | \(\frac {B \,x^{7}}{7 b}+\frac {A \,x^{4}}{4 b}-\frac {B a \,x^{4}}{4 b^{2}}-\frac {a A x}{b^{2}}+\frac {a^{2} B x}{b^{3}}+\frac {a^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (A b -B a \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{3 b^{4}}\) | \(84\) |
default | \(-\frac {-\frac {1}{7} b^{2} B \,x^{7}-\frac {1}{4} A \,b^{2} x^{4}+\frac {1}{4} B a b \,x^{4}+a A b x -a^{2} B x}{b^{3}}+\frac {\left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) a^{2} \left (A b -B a \right )}{b^{3}}\) | \(151\) |
1/7*B*x^7/b+1/4/b*A*x^4-1/4/b^2*B*a*x^4-1/b^2*a*A*x+1/b^3*a^2*B*x+1/3/b^4* a^2*sum((A*b-B*a)/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+a))
Time = 0.25 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.91 \[ \int \frac {x^6 \left (A+B x^3\right )}{a+b x^3} \, dx=\frac {12 \, B b^{2} x^{7} - 21 \, {\left (B a b - A b^{2}\right )} x^{4} - 28 \, \sqrt {3} {\left (B a^{2} - A a b\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) + 14 \, {\left (B a^{2} - A a b\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) - 28 \, {\left (B a^{2} - A a b\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) + 84 \, {\left (B a^{2} - A a b\right )} x}{84 \, b^{3}} \]
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
Time = 0.31 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.62 \[ \int \frac {x^6 \left (A+B x^3\right )}{a+b x^3} \, dx=\frac {B x^{7}}{7 b} + x^{4} \left (\frac {A}{4 b} - \frac {B a}{4 b^{2}}\right ) + x \left (- \frac {A a}{b^{2}} + \frac {B a^{2}}{b^{3}}\right ) + \operatorname {RootSum} {\left (27 t^{3} b^{10} - A^{3} a^{4} b^{3} + 3 A^{2} B a^{5} b^{2} - 3 A B^{2} a^{6} b + B^{3} a^{7}, \left ( t \mapsto t \log {\left (- \frac {3 t b^{3}}{- A a b + B a^{2}} + x \right )} \right )\right )} \]
B*x**7/(7*b) + x**4*(A/(4*b) - B*a/(4*b**2)) + x*(-A*a/b**2 + B*a**2/b**3) + 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, Lambda(_t, _t*log(-3*_t*b**3/(-A*a*b + B*a**2) + x)))
Time = 0.30 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.99 \[ \int \frac {x^6 \left (A+B x^3\right )}{a+b x^3} \, dx=\frac {4 \, B b^{2} x^{7} - 7 \, {\left (B a b - A b^{2}\right )} x^{4} + 28 \, {\left (B a^{2} - A a b\right )} x}{28 \, b^{3}} - \frac {\sqrt {3} {\left (B a^{3} - A a^{2} b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (B a^{3} - A a^{2} b\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (B a^{3} - A a^{2} b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
1/28*(4*B*b^2*x^7 - 7*(B*a*b - A*b^2)*x^4 + 28*(B*a^2 - A*a*b)*x)/b^3 - 1/ 3*sqrt(3)*(B*a^3 - A*a^2*b)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^( 1/3))/(b^4*(a/b)^(2/3)) + 1/6*(B*a^3 - A*a^2*b)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^4*(a/b)^(2/3)) - 1/3*(B*a^3 - A*a^2*b)*log(x + (a/b)^(1/3) )/(b^4*(a/b)^(2/3))
Time = 0.28 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.19 \[ \int \frac {x^6 \left (A+B x^3\right )}{a+b x^3} \, dx=-\frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {1}{3}} B a^{2} - \left (-a b^{2}\right )^{\frac {1}{3}} A a b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{4}} - \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} B a^{2} - \left (-a b^{2}\right )^{\frac {1}{3}} A a b\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{4}} + \frac {{\left (B a^{3} b^{4} - A a^{2} b^{5}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{7}} + \frac {4 \, B b^{6} x^{7} - 7 \, B a b^{5} x^{4} + 7 \, A b^{6} x^{4} + 28 \, B a^{2} b^{4} x - 28 \, A a b^{5} x}{28 \, b^{7}} \]
-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
Time = 0.28 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.90 \[ \int \frac {x^6 \left (A+B x^3\right )}{a+b x^3} \, dx=x^4\,\left (\frac {A}{4\,b}-\frac {B\,a}{4\,b^2}\right )+\frac {B\,x^7}{7\,b}+\frac {a^{4/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (A\,b-B\,a\right )}{3\,b^{10/3}}-\frac {a\,x\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{b}-\frac {a^{4/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (A\,b-B\,a\right )}{3\,b^{10/3}}+\frac {a^{4/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (A\,b-B\,a\right )}{3\,b^{10/3}} \]
x^4*(A/(4*b) - (B*a)/(4*b^2)) + (B*x^7)/(7*b) + (a^(4/3)*log(b^(1/3)*x + a ^(1/3))*(A*b - B*a))/(3*b^(10/3)) - (a*x*(A/b - (B*a)/b^2))/b - (a^(4/3)*l og(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(A*b - B*a))/(3*b^(10/3)) + (a^(4/3)*log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^ (1/3))*((3^(1/2)*1i)/2 - 1/2)*(A*b - B*a))/(3*b^(10/3))