Integrand size = 22, antiderivative size = 85 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^7} \, dx=-\frac {A \sqrt {a+b x^3}}{6 x^6}-\frac {(A b+4 a B) \sqrt {a+b x^3}}{12 a x^3}+\frac {b (A b-4 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{12 a^{3/2}} \] Output:
-1/6*A*(b*x^3+a)^(1/2)/x^6-1/12*(A*b+4*B*a)*(b*x^3+a)^(1/2)/a/x^3+1/12*b*( A*b-4*B*a)*arctanh((b*x^3+a)^(1/2)/a^(1/2))/a^(3/2)
Time = 0.21 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^7} \, dx=\frac {\sqrt {a+b x^3} \left (-2 a A-A b x^3-4 a B x^3\right )}{12 a x^6}-\frac {b (-A b+4 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{12 a^{3/2}} \] Input:
Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/x^7,x]
Output:
(Sqrt[a + b*x^3]*(-2*a*A - A*b*x^3 - 4*a*B*x^3))/(12*a*x^6) - (b*(-(A*b) + 4*a*B)*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(12*a^(3/2))
Time = 0.34 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {948, 87, 51, 73, 221}
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.
\(\displaystyle \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^7} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{3} \int \frac {\sqrt {b x^3+a} \left (B x^3+A\right )}{x^9}dx^3\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {1}{3} \left (-\frac {(A b-4 a B) \int \frac {\sqrt {b x^3+a}}{x^6}dx^3}{4 a}-\frac {A \left (a+b x^3\right )^{3/2}}{2 a x^6}\right )\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{3} \left (-\frac {(A b-4 a B) \left (\frac {1}{2} b \int \frac {1}{x^3 \sqrt {b x^3+a}}dx^3-\frac {\sqrt {a+b x^3}}{x^3}\right )}{4 a}-\frac {A \left (a+b x^3\right )^{3/2}}{2 a x^6}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} \left (-\frac {(A b-4 a B) \left (\int \frac {1}{\frac {x^6}{b}-\frac {a}{b}}d\sqrt {b x^3+a}-\frac {\sqrt {a+b x^3}}{x^3}\right )}{4 a}-\frac {A \left (a+b x^3\right )^{3/2}}{2 a x^6}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{3} \left (-\frac {(A b-4 a B) \left (-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {a+b x^3}}{x^3}\right )}{4 a}-\frac {A \left (a+b x^3\right )^{3/2}}{2 a x^6}\right )\) |
Input:
Int[(Sqrt[a + b*x^3]*(A + B*x^3))/x^7,x]
Output:
(-1/2*(A*(a + b*x^3)^(3/2))/(a*x^6) - ((A*b - 4*a*B)*(-(Sqrt[a + b*x^3]/x^ 3) - (b*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/Sqrt[a]))/(4*a))/3
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[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]]
Time = 0.87 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.76
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{3}+a}\, \left (A b \,x^{3}+4 B a \,x^{3}+2 A a \right )}{12 x^{6} a}+\frac {b \left (A b -4 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{12 a^{\frac {3}{2}}}\) | \(65\) |
elliptic | \(-\frac {A \sqrt {b \,x^{3}+a}}{6 x^{6}}-\frac {\left (A b +4 B a \right ) \sqrt {b \,x^{3}+a}}{12 a \,x^{3}}+\frac {b \left (A b -4 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{12 a^{\frac {3}{2}}}\) | \(70\) |
pseudoelliptic | \(-\frac {-b \,x^{6} \left (A b -4 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )+\sqrt {b \,x^{3}+a}\, \left (\left (4 B \,x^{3}+2 A \right ) a^{\frac {3}{2}}+A \sqrt {a}\, b \,x^{3}\right )}{12 a^{\frac {3}{2}} x^{6}}\) | \(72\) |
default | \(A \left (-\frac {\sqrt {b \,x^{3}+a}}{6 x^{6}}-\frac {b \sqrt {b \,x^{3}+a}}{12 a \,x^{3}}+\frac {b^{2} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{12 a^{\frac {3}{2}}}\right )+B \left (-\frac {\sqrt {b \,x^{3}+a}}{3 x^{3}}-\frac {b \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 \sqrt {a}}\right )\) | \(96\) |
Input:
int((b*x^3+a)^(1/2)*(B*x^3+A)/x^7,x,method=_RETURNVERBOSE)
Output:
-1/12*(b*x^3+a)^(1/2)*(A*b*x^3+4*B*a*x^3+2*A*a)/x^6/a+1/12*b*(A*b-4*B*a)*a rctanh((b*x^3+a)^(1/2)/a^(1/2))/a^(3/2)
Time = 0.11 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.99 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^7} \, dx=\left [-\frac {{\left (4 \, B a b - A b^{2}\right )} \sqrt {a} x^{6} \log \left (\frac {b x^{3} + 2 \, \sqrt {b x^{3} + a} \sqrt {a} + 2 \, a}{x^{3}}\right ) + 2 \, {\left ({\left (4 \, B a^{2} + A a b\right )} x^{3} + 2 \, A a^{2}\right )} \sqrt {b x^{3} + a}}{24 \, a^{2} x^{6}}, \frac {{\left (4 \, B a b - A b^{2}\right )} \sqrt {-a} x^{6} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{3} + a}}\right ) - {\left ({\left (4 \, B a^{2} + A a b\right )} x^{3} + 2 \, A a^{2}\right )} \sqrt {b x^{3} + a}}{12 \, a^{2} x^{6}}\right ] \] Input:
integrate((b*x^3+a)^(1/2)*(B*x^3+A)/x^7,x, algorithm="fricas")
Output:
[-1/24*((4*B*a*b - A*b^2)*sqrt(a)*x^6*log((b*x^3 + 2*sqrt(b*x^3 + a)*sqrt( a) + 2*a)/x^3) + 2*((4*B*a^2 + A*a*b)*x^3 + 2*A*a^2)*sqrt(b*x^3 + a))/(a^2 *x^6), 1/12*((4*B*a*b - A*b^2)*sqrt(-a)*x^6*arctan(sqrt(-a)/sqrt(b*x^3 + a )) - ((4*B*a^2 + A*a*b)*x^3 + 2*A*a^2)*sqrt(b*x^3 + a))/(a^2*x^6)]
Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (75) = 150\).
Time = 39.33 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.88 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^7} \, dx=- \frac {A a}{6 \sqrt {b} x^{\frac {15}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {A \sqrt {b}}{4 x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {A b^{\frac {3}{2}}}{12 a x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {A b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{12 a^{\frac {3}{2}}} - \frac {B \sqrt {b} \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} - \frac {B b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{3 \sqrt {a}} \] Input:
integrate((b*x**3+a)**(1/2)*(B*x**3+A)/x**7,x)
Output:
-A*a/(6*sqrt(b)*x**(15/2)*sqrt(a/(b*x**3) + 1)) - A*sqrt(b)/(4*x**(9/2)*sq rt(a/(b*x**3) + 1)) - A*b**(3/2)/(12*a*x**(3/2)*sqrt(a/(b*x**3) + 1)) + A* b**2*asinh(sqrt(a)/(sqrt(b)*x**(3/2)))/(12*a**(3/2)) - B*sqrt(b)*sqrt(a/(b *x**3) + 1)/(3*x**(3/2)) - B*b*asinh(sqrt(a)/(sqrt(b)*x**(3/2)))/(3*sqrt(a ))
Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (70) = 140\).
Time = 0.11 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.86 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^7} \, dx=-\frac {1}{24} \, {\left (\frac {b^{2} \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2 \, {\left ({\left (b x^{3} + a\right )}^{\frac {3}{2}} b^{2} + \sqrt {b x^{3} + a} a b^{2}\right )}}{{\left (b x^{3} + a\right )}^{2} a - 2 \, {\left (b x^{3} + a\right )} a^{2} + a^{3}}\right )} A + \frac {1}{6} \, {\left (\frac {b \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{\sqrt {a}} - \frac {2 \, \sqrt {b x^{3} + a}}{x^{3}}\right )} B \] Input:
integrate((b*x^3+a)^(1/2)*(B*x^3+A)/x^7,x, algorithm="maxima")
Output:
-1/24*(b^2*log((sqrt(b*x^3 + a) - sqrt(a))/(sqrt(b*x^3 + a) + sqrt(a)))/a^ (3/2) + 2*((b*x^3 + a)^(3/2)*b^2 + sqrt(b*x^3 + a)*a*b^2)/((b*x^3 + a)^2*a - 2*(b*x^3 + a)*a^2 + a^3))*A + 1/6*(b*log((sqrt(b*x^3 + a) - sqrt(a))/(s qrt(b*x^3 + a) + sqrt(a)))/sqrt(a) - 2*sqrt(b*x^3 + a)/x^3)*B
Time = 0.13 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^7} \, dx=\frac {\frac {{\left (4 \, B a b^{2} - A b^{3}\right )} \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} - \frac {4 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} B a b^{2} - 4 \, \sqrt {b x^{3} + a} B a^{2} b^{2} + {\left (b x^{3} + a\right )}^{\frac {3}{2}} A b^{3} + \sqrt {b x^{3} + a} A a b^{3}}{a b^{2} x^{6}}}{12 \, b} \] Input:
integrate((b*x^3+a)^(1/2)*(B*x^3+A)/x^7,x, algorithm="giac")
Output:
1/12*((4*B*a*b^2 - A*b^3)*arctan(sqrt(b*x^3 + a)/sqrt(-a))/(sqrt(-a)*a) - (4*(b*x^3 + a)^(3/2)*B*a*b^2 - 4*sqrt(b*x^3 + a)*B*a^2*b^2 + (b*x^3 + a)^( 3/2)*A*b^3 + sqrt(b*x^3 + a)*A*a*b^3)/(a*b^2*x^6))/b
Time = 1.39 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^7} \, dx=\frac {b\,\ln \left (\frac {\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )\,{\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}^3}{x^6}\right )\,\left (A\,b-4\,B\,a\right )}{24\,a^{3/2}}-\frac {\left (4\,B\,a^2+A\,b\,a\right )\,\sqrt {b\,x^3+a}}{12\,a^2\,x^3}-\frac {A\,\sqrt {b\,x^3+a}}{6\,x^6} \] Input:
int(((A + B*x^3)*(a + b*x^3)^(1/2))/x^7,x)
Output:
(b*log((((a + b*x^3)^(1/2) - a^(1/2))*((a + b*x^3)^(1/2) + a^(1/2))^3)/x^6 )*(A*b - 4*B*a))/(24*a^(3/2)) - ((4*B*a^2 + A*a*b)*(a + b*x^3)^(1/2))/(12* a^2*x^3) - (A*(a + b*x^3)^(1/2))/(6*x^6)
Time = 0.22 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {a+b x^3} \left (A+B x^3\right )}{x^7} \, dx=\frac {-4 \sqrt {b \,x^{3}+a}\, a^{2}-10 \sqrt {b \,x^{3}+a}\, a b \,x^{3}+3 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}-\sqrt {a}\right ) b^{2} x^{6}-3 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}+\sqrt {a}\right ) b^{2} x^{6}}{24 a \,x^{6}} \] Input:
int((b*x^3+a)^(1/2)*(B*x^3+A)/x^7,x)
Output:
( - 4*sqrt(a + b*x**3)*a**2 - 10*sqrt(a + b*x**3)*a*b*x**3 + 3*sqrt(a)*log (sqrt(a + b*x**3) - sqrt(a))*b**2*x**6 - 3*sqrt(a)*log(sqrt(a + b*x**3) + sqrt(a))*b**2*x**6)/(24*a*x**6)