Integrand size = 17, antiderivative size = 112 \[ \int x \sqrt [4]{b x^3+a x^4} \, dx=\frac {\left (-7 b^2+4 a b x+32 a^2 x^2\right ) \sqrt [4]{b x^3+a x^4}}{96 a^2}-\frac {7 b^3 \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{64 a^{11/4}}+\frac {7 b^3 \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{64 a^{11/4}} \]
1/96*(32*a^2*x^2+4*a*b*x-7*b^2)*(a*x^4+b*x^3)^(1/4)/a^2-7/64*b^3*arctan(a^ (1/4)*x/(a*x^4+b*x^3)^(1/4))/a^(11/4)+7/64*b^3*arctanh(a^(1/4)*x/(a*x^4+b* x^3)^(1/4))/a^(11/4)
Time = 0.47 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.15 \[ \int x \sqrt [4]{b x^3+a x^4} \, dx=\frac {x^{9/4} (b+a x)^{3/4} \left (2 a^{3/4} x^{3/4} \sqrt [4]{b+a x} \left (-7 b^2+4 a b x+32 a^2 x^2\right )-21 b^3 \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )+21 b^3 \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )\right )}{192 a^{11/4} \left (x^3 (b+a x)\right )^{3/4}} \]
(x^(9/4)*(b + a*x)^(3/4)*(2*a^(3/4)*x^(3/4)*(b + a*x)^(1/4)*(-7*b^2 + 4*a* b*x + 32*a^2*x^2) - 21*b^3*ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)] + 21* b^3*ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)]))/(192*a^(11/4)*(x^3*(b + a *x))^(3/4))
Time = 0.37 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.57, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {1927, 1930, 1930, 1938, 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.
| \(\displaystyle \int x \sqrt [4]{a x^4+b x^3} \, dx\) |
| \(\Big \downarrow \) 1927 |
| \(\displaystyle \frac {1}{12} b \int \frac {x^4}{\left (a x^4+b x^3\right )^{3/4}}dx+\frac {1}{3} x^2 \sqrt [4]{a x^4+b x^3}\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle \frac {1}{12} b \left (\frac {x \sqrt [4]{a x^4+b x^3}}{2 a}-\frac {7 b \int \frac {x^3}{\left (a x^4+b x^3\right )^{3/4}}dx}{8 a}\right )+\frac {1}{3} x^2 \sqrt [4]{a x^4+b x^3}\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle \frac {1}{12} b \left (\frac {x \sqrt [4]{a x^4+b x^3}}{2 a}-\frac {7 b \left (\frac {\sqrt [4]{a x^4+b x^3}}{a}-\frac {3 b \int \frac {x^2}{\left (a x^4+b x^3\right )^{3/4}}dx}{4 a}\right )}{8 a}\right )+\frac {1}{3} x^2 \sqrt [4]{a x^4+b x^3}\) |
| \(\Big \downarrow \) 1938 |
| \(\displaystyle \frac {1}{12} b \left (\frac {x \sqrt [4]{a x^4+b x^3}}{2 a}-\frac {7 b \left (\frac {\sqrt [4]{a x^4+b x^3}}{a}-\frac {3 b x^{9/4} (a x+b)^{3/4} \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}}dx}{4 a \left (a x^4+b x^3\right )^{3/4}}\right )}{8 a}\right )+\frac {1}{3} x^2 \sqrt [4]{a x^4+b x^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{12} b \left (\frac {x \sqrt [4]{a x^4+b x^3}}{2 a}-\frac {7 b \left (\frac {\sqrt [4]{a x^4+b x^3}}{a}-\frac {3 b x^{9/4} (a x+b)^{3/4} \int \frac {\sqrt {x}}{(b+a x)^{3/4}}d\sqrt [4]{x}}{a \left (a x^4+b x^3\right )^{3/4}}\right )}{8 a}\right )+\frac {1}{3} x^2 \sqrt [4]{a x^4+b x^3}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle \frac {1}{12} b \left (\frac {x \sqrt [4]{a x^4+b x^3}}{2 a}-\frac {7 b \left (\frac {\sqrt [4]{a x^4+b x^3}}{a}-\frac {3 b x^{9/4} (a x+b)^{3/4} \int \frac {\sqrt {x}}{1-a x}d\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}}{a \left (a x^4+b x^3\right )^{3/4}}\right )}{8 a}\right )+\frac {1}{3} x^2 \sqrt [4]{a x^4+b x^3}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {1}{12} b \left (\frac {x \sqrt [4]{a x^4+b x^3}}{2 a}-\frac {7 b \left (\frac {\sqrt [4]{a x^4+b x^3}}{a}-\frac {3 b x^{9/4} (a x+b)^{3/4} \left (\frac {\int \frac {1}{1-\sqrt {a} \sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}}{2 \sqrt {a}}-\frac {\int \frac {1}{\sqrt {a} \sqrt {x}+1}d\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}}{2 \sqrt {a}}\right )}{a \left (a x^4+b x^3\right )^{3/4}}\right )}{8 a}\right )+\frac {1}{3} x^2 \sqrt [4]{a x^4+b x^3}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{12} b \left (\frac {x \sqrt [4]{a x^4+b x^3}}{2 a}-\frac {7 b \left (\frac {\sqrt [4]{a x^4+b x^3}}{a}-\frac {3 b x^{9/4} (a x+b)^{3/4} \left (\frac {\int \frac {1}{1-\sqrt {a} \sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}}{2 \sqrt {a}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{3/4}}\right )}{a \left (a x^4+b x^3\right )^{3/4}}\right )}{8 a}\right )+\frac {1}{3} x^2 \sqrt [4]{a x^4+b x^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{12} b \left (\frac {x \sqrt [4]{a x^4+b x^3}}{2 a}-\frac {7 b \left (\frac {\sqrt [4]{a x^4+b x^3}}{a}-\frac {3 b x^{9/4} (a x+b)^{3/4} \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{3/4}}\right )}{a \left (a x^4+b x^3\right )^{3/4}}\right )}{8 a}\right )+\frac {1}{3} x^2 \sqrt [4]{a x^4+b x^3}\) |
(x^2*(b*x^3 + a*x^4)^(1/4))/3 + (b*((x*(b*x^3 + a*x^4)^(1/4))/(2*a) - (7*b *((b*x^3 + a*x^4)^(1/4)/a - (3*b*x^(9/4)*(b + a*x)^(3/4)*(-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))))/(a*(b*x^3 + a*x^4)^(3/4))))/(8*a)))/12
3.17.58.3.1 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_)^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[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[(c*x)^(m + 1)*((a*x^j + b*x^n)^p/(c*(m + n*p + 1))), x] + Simp[a* (n - j)*(p/(c^j*(m + n*p + 1))) Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p - 1) , x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[p] && LtQ[0, j, n] && (Int egersQ[j, n] || GtQ[c, 0]) && GtQ[p, 0] && NeQ[m + n*p + 1, 0]
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a*x^j + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[a*c^(n - j)*((m + j*p - n + j + 1)/(b*(m + n*p + 1))) I nt[(c*x)^(m - (n - j))*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && Gt Q[m + j*p - n + j + 1, 0] && NeQ[m + n*p + 1, 0]
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[c^IntPart[m]*(c*x)^FracPart[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(F racPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p])) Int[x^(m + j* p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && !Inte gerQ[p] && NeQ[n, j] && PosQ[n - j]
Time = 0.42 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.17
| method | result | size |
| pseudoelliptic | \(\frac {128 \left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}} a^{\frac {11}{4}} x^{2}+16 a^{\frac {7}{4}} b x \left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}-28 b^{2} \left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}} a^{\frac {3}{4}}+21 \ln \left (\frac {a^{\frac {1}{4}} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}\right ) b^{3}+42 \arctan \left (\frac {\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) b^{3}}{384 a^{\frac {11}{4}}}\) | \(131\) |
1/384/a^(11/4)*(128*(x^3*(a*x+b))^(1/4)*a^(11/4)*x^2+16*a^(7/4)*b*x*(x^3*( a*x+b))^(1/4)-28*b^2*(x^3*(a*x+b))^(1/4)*a^(3/4)+21*ln((a^(1/4)*x+(x^3*(a* x+b))^(1/4))/(-a^(1/4)*x+(x^3*(a*x+b))^(1/4)))*b^3+42*arctan(1/a^(1/4)/x*( x^3*(a*x+b))^(1/4))*b^3)
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.26 \[ \int x \sqrt [4]{b x^3+a x^4} \, dx=\frac {21 \, a^{2} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} \log \left (\frac {7 \, {\left (a^{3} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{3}\right )}}{x}\right ) - 21 \, a^{2} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} \log \left (-\frac {7 \, {\left (a^{3} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{3}\right )}}{x}\right ) - 21 i \, a^{2} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} \log \left (-\frac {7 \, {\left (i \, a^{3} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{3}\right )}}{x}\right ) + 21 i \, a^{2} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} \log \left (-\frac {7 \, {\left (-i \, a^{3} \left (\frac {b^{12}}{a^{11}}\right )^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{3}\right )}}{x}\right ) + 4 \, {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (32 \, a^{2} x^{2} + 4 \, a b x - 7 \, b^{2}\right )}}{384 \, a^{2}} \]
1/384*(21*a^2*(b^12/a^11)^(1/4)*log(7*(a^3*(b^12/a^11)^(1/4)*x + (a*x^4 + b*x^3)^(1/4)*b^3)/x) - 21*a^2*(b^12/a^11)^(1/4)*log(-7*(a^3*(b^12/a^11)^(1 /4)*x - (a*x^4 + b*x^3)^(1/4)*b^3)/x) - 21*I*a^2*(b^12/a^11)^(1/4)*log(-7* (I*a^3*(b^12/a^11)^(1/4)*x - (a*x^4 + b*x^3)^(1/4)*b^3)/x) + 21*I*a^2*(b^1 2/a^11)^(1/4)*log(-7*(-I*a^3*(b^12/a^11)^(1/4)*x - (a*x^4 + b*x^3)^(1/4)*b ^3)/x) + 4*(a*x^4 + b*x^3)^(1/4)*(32*a^2*x^2 + 4*a*b*x - 7*b^2))/a^2
\[ \int x \sqrt [4]{b x^3+a x^4} \, dx=\int x \sqrt [4]{x^{3} \left (a x + b\right )}\, dx \]
\[ \int x \sqrt [4]{b x^3+a x^4} \, dx=\int { {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} x \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 261 vs. \(2 (92) = 184\).
Time = 0.32 (sec) , antiderivative size = 261, normalized size of antiderivative = 2.33 \[ \int x \sqrt [4]{b x^3+a x^4} \, dx=\frac {\frac {42 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{4} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a^{3}} + \frac {42 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{4} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a^{3}} + \frac {21 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{4} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{a^{3}} + \frac {21 \, \sqrt {2} b^{4} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{\left (-a\right )^{\frac {3}{4}} a^{2}} - \frac {8 \, {\left (7 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{4}} b^{4} - 18 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} a b^{4} - 21 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} a^{2} b^{4}\right )} x^{3}}{a^{2} b^{3}}}{768 \, b} \]
1/768*(42*sqrt(2)*(-a)^(1/4)*b^4*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a + b/x)^(1/4))/(-a)^(1/4))/a^3 + 42*sqrt(2)*(-a)^(1/4)*b^4*arctan(-1/2 *sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(a + b/x)^(1/4))/(-a)^(1/4))/a^3 + 21*sqr t(2)*(-a)^(1/4)*b^4*log(sqrt(2)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(-a) + sq rt(a + b/x))/a^3 + 21*sqrt(2)*b^4*log(-sqrt(2)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x))/((-a)^(3/4)*a^2) - 8*(7*(a + b/x)^(9/4)*b^4 - 18*(a + b/x)^(5/4)*a*b^4 - 21*(a + b/x)^(1/4)*a^2*b^4)*x^3/(a^2*b^3))/b
Timed out. \[ \int x \sqrt [4]{b x^3+a x^4} \, dx=\int x\,{\left (a\,x^4+b\,x^3\right )}^{1/4} \,d x \]