Integrand size = 33, antiderivative size = 82 \[ \int \frac {(-b+a x) \sqrt [4]{b x^3+a x^4}}{x (b+a x)} \, dx=\sqrt [4]{b x^3+a x^4}+\frac {7 b \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{2 a^{3/4}}-\frac {7 b \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{2 a^{3/4}} \]
(a*x^4+b*x^3)^(1/4)+7/2*b*arctan(a^(1/4)*x/(a*x^4+b*x^3)^(1/4))/a^(3/4)-7/ 2*b*arctanh(a^(1/4)*x/(a*x^4+b*x^3)^(1/4))/a^(3/4)
Time = 0.38 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.35 \[ \int \frac {(-b+a x) \sqrt [4]{b x^3+a x^4}}{x (b+a x)} \, dx=\frac {x^{9/4} \left (2 a^{3/4} x^{3/4} (b+a x)+7 b (b+a x)^{3/4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )-7 b (b+a x)^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )\right )}{2 a^{3/4} \left (x^3 (b+a x)\right )^{3/4}} \]
(x^(9/4)*(2*a^(3/4)*x^(3/4)*(b + a*x) + 7*b*(b + a*x)^(3/4)*ArcTan[(a^(1/4 )*x^(1/4))/(b + a*x)^(1/4)] - 7*b*(b + a*x)^(3/4)*ArcTanh[(a^(1/4)*x^(1/4) )/(b + a*x)^(1/4)]))/(2*a^(3/4)*(x^3*(b + a*x))^(3/4))
Time = 0.42 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.37, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {2467, 25, 90, 73, 854, 827, 216, 219}
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.
| \(\displaystyle \int \frac {(a x-b) \sqrt [4]{a x^4+b x^3}}{x (a x+b)} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{a x^4+b x^3} \int -\frac {b-a x}{\sqrt [4]{x} (b+a x)^{3/4}}dx}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \int \frac {b-a x}{\sqrt [4]{x} (b+a x)^{3/4}}dx}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (\frac {7}{4} b \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}}dx-x^{3/4} \sqrt [4]{a x+b}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (7 b \int \frac {\sqrt {x}}{(b+a x)^{3/4}}d\sqrt [4]{x}-x^{3/4} \sqrt [4]{a x+b}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (7 b \int \frac {\sqrt {x}}{1-a x}d\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}-x^{3/4} \sqrt [4]{a x+b}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (7 b \left (\frac {\int \frac {1}{1-\sqrt {a} \sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}}{2 \sqrt {a}}-\frac {\int \frac {1}{\sqrt {a} \sqrt {x}+1}d\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}}{2 \sqrt {a}}\right )-x^{3/4} \sqrt [4]{a x+b}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (7 b \left (\frac {\int \frac {1}{1-\sqrt {a} \sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}}{2 \sqrt {a}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{3/4}}\right )-x^{3/4} \sqrt [4]{a x+b}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (7 b \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{3/4}}\right )-x^{3/4} \sqrt [4]{a x+b}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
-(((b*x^3 + a*x^4)^(1/4)*(-(x^(3/4)*(b + a*x)^(1/4)) + 7*b*(-1/2*ArcTan[(a ^(1/4)*x^(1/4))/(b + a*x)^(1/4)]/a^(3/4) + ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)]/(2*a^(3/4)))))/(x^(3/4)*(b + a*x)^(1/4)))
3.12.3.3.1 Defintions of rubi rules used
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*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n )^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 2^(-1)] && IntegersQ[m, p + (m + 1)/n]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Time = 1.15 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.05
| method | result | size |
| pseudoelliptic | \(-\frac {7 \left (\ln \left (\frac {a^{\frac {1}{4}} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}\right ) b +2 \arctan \left (\frac {\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) b -\frac {4 \left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}} a^{\frac {3}{4}}}{7}\right )}{4 a^{\frac {3}{4}}}\) | \(86\) |
-7/4/a^(3/4)*(ln((a^(1/4)*x+(x^3*(a*x+b))^(1/4))/(-a^(1/4)*x+(x^3*(a*x+b)) ^(1/4)))*b+2*arctan(1/a^(1/4)/x*(x^3*(a*x+b))^(1/4))*b-4/7*(x^3*(a*x+b))^( 1/4)*a^(3/4))
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.43 \[ \int \frac {(-b+a x) \sqrt [4]{b x^3+a x^4}}{x (b+a x)} \, dx=-\frac {7}{4} \, \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} \log \left (\frac {7 \, {\left (a \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b\right )}}{x}\right ) + \frac {7}{4} \, \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} \log \left (-\frac {7 \, {\left (a \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b\right )}}{x}\right ) + \frac {7}{4} i \, \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} \log \left (-\frac {7 \, {\left (i \, a \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b\right )}}{x}\right ) - \frac {7}{4} i \, \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} \log \left (-\frac {7 \, {\left (-i \, a \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b\right )}}{x}\right ) + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} \]
-7/4*(b^4/a^3)^(1/4)*log(7*(a*(b^4/a^3)^(1/4)*x + (a*x^4 + b*x^3)^(1/4)*b) /x) + 7/4*(b^4/a^3)^(1/4)*log(-7*(a*(b^4/a^3)^(1/4)*x - (a*x^4 + b*x^3)^(1 /4)*b)/x) + 7/4*I*(b^4/a^3)^(1/4)*log(-7*(I*a*(b^4/a^3)^(1/4)*x - (a*x^4 + b*x^3)^(1/4)*b)/x) - 7/4*I*(b^4/a^3)^(1/4)*log(-7*(-I*a*(b^4/a^3)^(1/4)*x - (a*x^4 + b*x^3)^(1/4)*b)/x) + (a*x^4 + b*x^3)^(1/4)
\[ \int \frac {(-b+a x) \sqrt [4]{b x^3+a x^4}}{x (b+a x)} \, dx=\int \frac {\sqrt [4]{x^{3} \left (a x + b\right )} \left (a x - b\right )}{x \left (a x + b\right )}\, dx \]
\[ \int \frac {(-b+a x) \sqrt [4]{b x^3+a x^4}}{x (b+a x)} \, dx=\int { \frac {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (a x - b\right )}}{{\left (a x + b\right )} x} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (64) = 128\).
Time = 0.29 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.52 \[ \int \frac {(-b+a x) \sqrt [4]{b x^3+a x^4}}{x (b+a x)} \, dx=\frac {\frac {14 \, \sqrt {2} b^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}}} + \frac {14 \, \sqrt {2} b^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}}} + \frac {7 \, \sqrt {2} b^{2} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{\left (-a\right )^{\frac {3}{4}}} + \frac {7 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{a} + 8 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} b x}{8 \, b} \]
1/8*(14*sqrt(2)*b^2*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a + b/x)^( 1/4))/(-a)^(1/4))/(-a)^(3/4) + 14*sqrt(2)*b^2*arctan(-1/2*sqrt(2)*(sqrt(2) *(-a)^(1/4) - 2*(a + b/x)^(1/4))/(-a)^(1/4))/(-a)^(3/4) + 7*sqrt(2)*b^2*lo g(sqrt(2)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x))/(-a)^(3/4 ) + 7*sqrt(2)*(-a)^(1/4)*b^2*log(-sqrt(2)*(-a)^(1/4)*(a + b/x)^(1/4) + sqr t(-a) + sqrt(a + b/x))/a + 8*(a + b/x)^(1/4)*b*x)/b
Timed out. \[ \int \frac {(-b+a x) \sqrt [4]{b x^3+a x^4}}{x (b+a x)} \, dx=\int -\frac {{\left (a\,x^4+b\,x^3\right )}^{1/4}\,\left (b-a\,x\right )}{x\,\left (b+a\,x\right )} \,d x \]