Integrand size = 22, antiderivative size = 71 \[ \int \frac {\sqrt {x} \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx=\frac {(A b-a B) x^{3/2}}{3 a b \left (a+b x^3\right )}+\frac {(A b+a B) \arctan \left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )}{3 a^{3/2} b^{3/2}} \]
1/3*(A*b-B*a)*x^(3/2)/a/b/(b*x^3+a)+1/3*(A*b+B*a)*arctan(x^(3/2)*b^(1/2)/a ^(1/2))/a^(3/2)/b^(3/2)
Time = 0.14 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {x} \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx=-\frac {(-A b+a B) x^{3/2}}{3 a b \left (a+b x^3\right )}+\frac {(A b+a B) \arctan \left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )}{3 a^{3/2} b^{3/2}} \]
-1/3*((-(A*b) + a*B)*x^(3/2))/(a*b*(a + b*x^3)) + ((A*b + a*B)*ArcTan[(Sqr t[b]*x^(3/2))/Sqrt[a]])/(3*a^(3/2)*b^(3/2))
Time = 0.22 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {957, 851, 807, 218}
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.
\(\displaystyle \int \frac {\sqrt {x} \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx\) |
\(\Big \downarrow \) 957 |
\(\displaystyle \frac {(a B+A b) \int \frac {\sqrt {x}}{b x^3+a}dx}{2 a b}+\frac {x^{3/2} (A b-a B)}{3 a b \left (a+b x^3\right )}\) |
\(\Big \downarrow \) 851 |
\(\displaystyle \frac {(a B+A b) \int \frac {x}{b x^3+a}d\sqrt {x}}{a b}+\frac {x^{3/2} (A b-a B)}{3 a b \left (a+b x^3\right )}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {(a B+A b) \int \frac {1}{a+b x}dx^{3/2}}{3 a b}+\frac {x^{3/2} (A b-a B)}{3 a b \left (a+b x^3\right )}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {(a B+A b) \arctan \left (\frac {\sqrt {b} x^{3/2}}{\sqrt {a}}\right )}{3 a^{3/2} b^{3/2}}+\frac {x^{3/2} (A b-a B)}{3 a b \left (a+b x^3\right )}\) |
((A*b - a*B)*x^(3/2))/(3*a*b*(a + b*x^3)) + ((A*b + a*B)*ArcTan[(Sqrt[b]*x ^(3/2))/Sqrt[a]])/(3*a^(3/2)*b^(3/2))
3.2.66.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
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] - Simp[(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] && N eQ[p, -5/4]) || !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1 , m, (-n)*(p + 1)]))
Time = 4.14 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {\left (A b -B a \right ) x^{\frac {3}{2}}}{3 a b \left (b \,x^{3}+a \right )}+\frac {\left (A b +B a \right ) \arctan \left (\frac {b \,x^{\frac {3}{2}}}{\sqrt {a b}}\right )}{3 a b \sqrt {a b}}\) | \(61\) |
default | \(\frac {\left (A b -B a \right ) x^{\frac {3}{2}}}{3 a b \left (b \,x^{3}+a \right )}+\frac {\left (A b +B a \right ) \arctan \left (\frac {b \,x^{\frac {3}{2}}}{\sqrt {a b}}\right )}{3 a b \sqrt {a b}}\) | \(61\) |
1/3*(A*b-B*a)*x^(3/2)/a/b/(b*x^3+a)+1/3*(A*b+B*a)/a/b/(a*b)^(1/2)*arctan(b *x^(3/2)/(a*b)^(1/2))
Time = 0.27 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.68 \[ \int \frac {\sqrt {x} \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx=\left [-\frac {2 \, {\left (B a^{2} b - A a b^{2}\right )} x^{\frac {3}{2}} + {\left ({\left (B a b + A b^{2}\right )} x^{3} + B a^{2} + A a b\right )} \sqrt {-a b} \log \left (\frac {b x^{3} - 2 \, \sqrt {-a b} x^{\frac {3}{2}} - a}{b x^{3} + a}\right )}{6 \, {\left (a^{2} b^{3} x^{3} + a^{3} b^{2}\right )}}, -\frac {{\left (B a^{2} b - A a b^{2}\right )} x^{\frac {3}{2}} - {\left ({\left (B a b + A b^{2}\right )} x^{3} + B a^{2} + A a b\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x^{\frac {3}{2}}}{a}\right )}{3 \, {\left (a^{2} b^{3} x^{3} + a^{3} b^{2}\right )}}\right ] \]
[-1/6*(2*(B*a^2*b - A*a*b^2)*x^(3/2) + ((B*a*b + A*b^2)*x^3 + B*a^2 + A*a* b)*sqrt(-a*b)*log((b*x^3 - 2*sqrt(-a*b)*x^(3/2) - a)/(b*x^3 + a)))/(a^2*b^ 3*x^3 + a^3*b^2), -1/3*((B*a^2*b - A*a*b^2)*x^(3/2) - ((B*a*b + A*b^2)*x^3 + B*a^2 + A*a*b)*sqrt(a*b)*arctan(sqrt(a*b)*x^(3/2)/a))/(a^2*b^3*x^3 + a^ 3*b^2)]
Leaf count of result is larger than twice the leaf count of optimal. 1042 vs. \(2 (61) = 122\).
Time = 67.82 (sec) , antiderivative size = 1042, normalized size of antiderivative = 14.68 \[ \int \frac {\sqrt {x} \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 A}{9 x^{\frac {9}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {2 A x^{\frac {3}{2}}}{3} + \frac {2 B x^{\frac {9}{2}}}{9}}{a^{2}} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{9 x^{\frac {9}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}}{b^{2}} & \text {for}\: a = 0 \\\frac {2 A a b x^{\frac {3}{2}}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} - \frac {A a b \sqrt {- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [6]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} + \frac {A a b \sqrt {- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [6]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} + \frac {A a b \sqrt {- \frac {a}{b}} \log {\left (- 4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} - \frac {A a b \sqrt {- \frac {a}{b}} \log {\left (4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} - \frac {A b^{2} x^{3} \sqrt {- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [6]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} + \frac {A b^{2} x^{3} \sqrt {- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [6]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} + \frac {A b^{2} x^{3} \sqrt {- \frac {a}{b}} \log {\left (- 4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} - \frac {A b^{2} x^{3} \sqrt {- \frac {a}{b}} \log {\left (4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} - \frac {2 B a^{2} x^{\frac {3}{2}}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} - \frac {B a^{2} \sqrt {- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [6]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} + \frac {B a^{2} \sqrt {- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [6]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} + \frac {B a^{2} \sqrt {- \frac {a}{b}} \log {\left (- 4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} - \frac {B a^{2} \sqrt {- \frac {a}{b}} \log {\left (4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} - \frac {B a b x^{3} \sqrt {- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [6]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} + \frac {B a b x^{3} \sqrt {- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [6]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} + \frac {B a b x^{3} \sqrt {- \frac {a}{b}} \log {\left (- 4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} - \frac {B a b x^{3} \sqrt {- \frac {a}{b}} \log {\left (4 \sqrt {x} \sqrt [6]{- \frac {a}{b}} + 4 x + 4 \sqrt [3]{- \frac {a}{b}} \right )}}{6 a^{3} b + 6 a^{2} b^{2} x^{3}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*(-2*A/(9*x**(9/2)) - 2*B/(3*x**(3/2))), Eq(a, 0) & Eq(b, 0) ), ((2*A*x**(3/2)/3 + 2*B*x**(9/2)/9)/a**2, Eq(b, 0)), ((-2*A/(9*x**(9/2)) - 2*B/(3*x**(3/2)))/b**2, Eq(a, 0)), (2*A*a*b*x**(3/2)/(6*a**3*b + 6*a**2 *b**2*x**3) - A*a*b*sqrt(-a/b)*log(sqrt(x) - (-a/b)**(1/6))/(6*a**3*b + 6* a**2*b**2*x**3) + A*a*b*sqrt(-a/b)*log(sqrt(x) + (-a/b)**(1/6))/(6*a**3*b + 6*a**2*b**2*x**3) + A*a*b*sqrt(-a/b)*log(-4*sqrt(x)*(-a/b)**(1/6) + 4*x + 4*(-a/b)**(1/3))/(6*a**3*b + 6*a**2*b**2*x**3) - A*a*b*sqrt(-a/b)*log(4* sqrt(x)*(-a/b)**(1/6) + 4*x + 4*(-a/b)**(1/3))/(6*a**3*b + 6*a**2*b**2*x** 3) - A*b**2*x**3*sqrt(-a/b)*log(sqrt(x) - (-a/b)**(1/6))/(6*a**3*b + 6*a** 2*b**2*x**3) + A*b**2*x**3*sqrt(-a/b)*log(sqrt(x) + (-a/b)**(1/6))/(6*a**3 *b + 6*a**2*b**2*x**3) + A*b**2*x**3*sqrt(-a/b)*log(-4*sqrt(x)*(-a/b)**(1/ 6) + 4*x + 4*(-a/b)**(1/3))/(6*a**3*b + 6*a**2*b**2*x**3) - A*b**2*x**3*sq rt(-a/b)*log(4*sqrt(x)*(-a/b)**(1/6) + 4*x + 4*(-a/b)**(1/3))/(6*a**3*b + 6*a**2*b**2*x**3) - 2*B*a**2*x**(3/2)/(6*a**3*b + 6*a**2*b**2*x**3) - B*a* *2*sqrt(-a/b)*log(sqrt(x) - (-a/b)**(1/6))/(6*a**3*b + 6*a**2*b**2*x**3) + B*a**2*sqrt(-a/b)*log(sqrt(x) + (-a/b)**(1/6))/(6*a**3*b + 6*a**2*b**2*x* *3) + B*a**2*sqrt(-a/b)*log(-4*sqrt(x)*(-a/b)**(1/6) + 4*x + 4*(-a/b)**(1/ 3))/(6*a**3*b + 6*a**2*b**2*x**3) - B*a**2*sqrt(-a/b)*log(4*sqrt(x)*(-a/b) **(1/6) + 4*x + 4*(-a/b)**(1/3))/(6*a**3*b + 6*a**2*b**2*x**3) - B*a*b*x** 3*sqrt(-a/b)*log(sqrt(x) - (-a/b)**(1/6))/(6*a**3*b + 6*a**2*b**2*x**3)...
Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {x} \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx=-\frac {{\left (B a - A b\right )} x^{\frac {3}{2}}}{3 \, {\left (a b^{2} x^{3} + a^{2} b\right )}} + \frac {{\left (B a + A b\right )} \arctan \left (\frac {b x^{\frac {3}{2}}}{\sqrt {a b}}\right )}{3 \, \sqrt {a b} a b} \]
-1/3*(B*a - A*b)*x^(3/2)/(a*b^2*x^3 + a^2*b) + 1/3*(B*a + A*b)*arctan(b*x^ (3/2)/sqrt(a*b))/(sqrt(a*b)*a*b)
Time = 0.29 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {x} \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx=\frac {{\left (B a + A b\right )} \arctan \left (\frac {b x^{\frac {3}{2}}}{\sqrt {a b}}\right )}{3 \, \sqrt {a b} a b} - \frac {B a x^{\frac {3}{2}} - A b x^{\frac {3}{2}}}{3 \, {\left (b x^{3} + a\right )} a b} \]
1/3*(B*a + A*b)*arctan(b*x^(3/2)/sqrt(a*b))/(sqrt(a*b)*a*b) - 1/3*(B*a*x^( 3/2) - A*b*x^(3/2))/((b*x^3 + a)*a*b)
Time = 7.02 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.62 \[ \int \frac {\sqrt {x} \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx=\frac {B\,a^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,x^{3/2}}{\sqrt {a}}\right )+A\,b^2\,x^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,x^{3/2}}{\sqrt {a}}\right )+A\,a\,b\,\mathrm {atan}\left (\frac {\sqrt {b}\,x^{3/2}}{\sqrt {a}}\right )+A\,\sqrt {a}\,b^{3/2}\,x^{3/2}-B\,a^{3/2}\,\sqrt {b}\,x^{3/2}+B\,a\,b\,x^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,x^{3/2}}{\sqrt {a}}\right )}{3\,a^{5/2}\,b^{3/2}+3\,a^{3/2}\,b^{5/2}\,x^3} \]