Integrand size = 26, antiderivative size = 112 \[ \int \frac {x^3 \sqrt {-x+x^4}}{-b+a x^3} \, dx=\frac {x \sqrt {-x+x^4}}{3 a}+\frac {2 \sqrt {a-b} \sqrt {b} \arctan \left (\frac {\sqrt {a-b} x \sqrt {-x+x^4}}{\sqrt {b} (-1+x) \left (1+x+x^2\right )}\right )}{3 a^2}+\frac {(-a+2 b) \text {arctanh}\left (\frac {x^2}{\sqrt {-x+x^4}}\right )}{3 a^2} \]
1/3*x*(x^4-x)^(1/2)/a+2/3*(a-b)^(1/2)*b^(1/2)*arctan((a-b)^(1/2)*x*(x^4-x) ^(1/2)/b^(1/2)/(-1+x)/(x^2+x+1))/a^2+1/3*(-a+2*b)*arctanh(x^2/(x^4-x)^(1/2 ))/a^2
Time = 0.62 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.15 \[ \int \frac {x^3 \sqrt {-x+x^4}}{-b+a x^3} \, dx=\frac {\sqrt {x} \sqrt {-1+x^3} \left (a x^{3/2} \sqrt {-1+x^3}+2 \sqrt {a-b} \sqrt {b} \arctan \left (\frac {b-a x^{3/2} \left (x^{3/2}+\sqrt {-1+x^3}\right )}{\sqrt {a-b} \sqrt {b}}\right )-(a-2 b) \log \left (x^{3/2}+\sqrt {-1+x^3}\right )\right )}{3 a^2 \sqrt {x \left (-1+x^3\right )}} \]
(Sqrt[x]*Sqrt[-1 + x^3]*(a*x^(3/2)*Sqrt[-1 + x^3] + 2*Sqrt[a - b]*Sqrt[b]* ArcTan[(b - a*x^(3/2)*(x^(3/2) + Sqrt[-1 + x^3]))/(Sqrt[a - b]*Sqrt[b])] - (a - 2*b)*Log[x^(3/2) + Sqrt[-1 + x^3]]))/(3*a^2*Sqrt[x*(-1 + x^3)])
Time = 0.35 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.14, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {1948, 25, 966, 965, 380, 25, 398, 224, 219, 291, 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.
\(\displaystyle \int \frac {x^3 \sqrt {x^4-x}}{a x^3-b} \, dx\) |
\(\Big \downarrow \) 1948 |
\(\displaystyle \frac {\sqrt {x^4-x} \int -\frac {x^{7/2} \sqrt {x^3-1}}{b-a x^3}dx}{\sqrt {x} \sqrt {x^3-1}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {x^4-x} \int \frac {x^{7/2} \sqrt {x^3-1}}{b-a x^3}dx}{\sqrt {x} \sqrt {x^3-1}}\) |
\(\Big \downarrow \) 966 |
\(\displaystyle -\frac {2 \sqrt {x^4-x} \int \frac {x^4 \sqrt {x^3-1}}{b-a x^3}d\sqrt {x}}{\sqrt {x} \sqrt {x^3-1}}\) |
\(\Big \downarrow \) 965 |
\(\displaystyle -\frac {2 \sqrt {x^4-x} \int \frac {\sqrt {x-1} x}{b-a x}dx^{3/2}}{3 \sqrt {x} \sqrt {x^3-1}}\) |
\(\Big \downarrow \) 380 |
\(\displaystyle -\frac {2 \sqrt {x^4-x} \left (\frac {\int -\frac {b+(a-2 b) x}{\sqrt {x-1} (b-a x)}dx^{3/2}}{2 a}-\frac {\sqrt {x-1} x^{3/2}}{2 a}\right )}{3 \sqrt {x} \sqrt {x^3-1}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {2 \sqrt {x^4-x} \left (-\frac {\int \frac {b+(a-2 b) x}{\sqrt {x-1} (b-a x)}dx^{3/2}}{2 a}-\frac {\sqrt {x-1} x^{3/2}}{2 a}\right )}{3 \sqrt {x} \sqrt {x^3-1}}\) |
\(\Big \downarrow \) 398 |
\(\displaystyle -\frac {2 \sqrt {x^4-x} \left (-\frac {\frac {2 b (a-b) \int \frac {1}{\sqrt {x-1} (b-a x)}dx^{3/2}}{a}-\frac {(a-2 b) \int \frac {1}{\sqrt {x-1}}dx^{3/2}}{a}}{2 a}-\frac {\sqrt {x-1} x^{3/2}}{2 a}\right )}{3 \sqrt {x} \sqrt {x^3-1}}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -\frac {2 \sqrt {x^4-x} \left (-\frac {\frac {2 b (a-b) \int \frac {1}{\sqrt {x-1} (b-a x)}dx^{3/2}}{a}-\frac {(a-2 b) \int \frac {1}{1-x}d\frac {x^{3/2}}{\sqrt {x-1}}}{a}}{2 a}-\frac {\sqrt {x-1} x^{3/2}}{2 a}\right )}{3 \sqrt {x} \sqrt {x^3-1}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {2 \sqrt {x^4-x} \left (-\frac {\frac {2 b (a-b) \int \frac {1}{\sqrt {x-1} (b-a x)}dx^{3/2}}{a}-\frac {(a-2 b) \text {arctanh}\left (\frac {x^{3/2}}{\sqrt {x-1}}\right )}{a}}{2 a}-\frac {\sqrt {x-1} x^{3/2}}{2 a}\right )}{3 \sqrt {x} \sqrt {x^3-1}}\) |
\(\Big \downarrow \) 291 |
\(\displaystyle -\frac {2 \sqrt {x^4-x} \left (-\frac {\frac {2 b (a-b) \int \frac {1}{b-(b-a) x}d\frac {x^{3/2}}{\sqrt {x-1}}}{a}-\frac {(a-2 b) \text {arctanh}\left (\frac {x^{3/2}}{\sqrt {x-1}}\right )}{a}}{2 a}-\frac {\sqrt {x-1} x^{3/2}}{2 a}\right )}{3 \sqrt {x} \sqrt {x^3-1}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle -\frac {2 \sqrt {x^4-x} \left (-\frac {\frac {2 \sqrt {b} \sqrt {a-b} \arctan \left (\frac {x^{3/2} \sqrt {a-b}}{\sqrt {b} \sqrt {x-1}}\right )}{a}-\frac {(a-2 b) \text {arctanh}\left (\frac {x^{3/2}}{\sqrt {x-1}}\right )}{a}}{2 a}-\frac {\sqrt {x-1} x^{3/2}}{2 a}\right )}{3 \sqrt {x} \sqrt {x^3-1}}\) |
(-2*Sqrt[-x + x^4]*(-1/2*(Sqrt[-1 + x]*x^(3/2))/a - ((2*Sqrt[a - b]*Sqrt[b ]*ArcTan[(Sqrt[a - b]*x^(3/2))/(Sqrt[b]*Sqrt[-1 + x])])/a - ((a - 2*b)*Arc Tanh[x^(3/2)/Sqrt[-1 + x]])/a)/(2*a)))/(3*Sqrt[x]*Sqrt[-1 + x^3])
3.17.55.3.1 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[((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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b* (m + 2*(p + q) + 1))), x] - Simp[e^2/(b*(m + 2*(p + q) + 1)) Int[(e*x)^(m - 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[a*c*(m - 1) + (a*d*(m - 1) - 2 *q*(b*c - a*d))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 0] && GtQ[m, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /; Free Q[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) )^(q_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*( m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*x)^( 1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[ n, 0] && FractionQ[m] && IntegerQ[p]
Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Simp[e^IntPart[m]*(e*x)^FracPart[m]*( (a*x^j + b*x^(j + n))^FracPart[p]/(x^(FracPart[m] + j*FracPart[p])*(a + b*x ^n)^FracPart[p])) Int[x^(m + j*p)*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, j, m, n, p, q}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && !(EqQ[n, 1] && EqQ[j, 1])
Time = 1.57 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.93
method | result | size |
risch | \(\frac {x^{2} \left (x^{3}-1\right )}{3 a \sqrt {x \left (x^{3}-1\right )}}-\frac {\frac {\left (a -2 b \right ) \ln \left (-2 x^{3}-2 x \sqrt {x^{4}-x}+1\right )}{3 a}+\frac {4 \left (a -b \right ) b \arctan \left (\frac {\sqrt {x^{4}-x}\, b}{x^{2} \sqrt {\left (a -b \right ) b}}\right )}{3 a \sqrt {\left (a -b \right ) b}}}{2 a}\) | \(104\) |
pseudoelliptic | \(-\frac {2 \left (\left (a -b \right ) b \arctan \left (\frac {\sqrt {x^{4}-x}\, b}{x^{2} \sqrt {\left (a -b \right ) b}}\right )-\frac {\sqrt {\left (a -b \right ) b}\, \left (\left (\frac {a}{2}-b \right ) \ln \left (\frac {-x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )+\left (-\frac {a}{2}+b \right ) \ln \left (\frac {x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )+\sqrt {x^{4}-x}\, a x \right )}{2}\right )}{3 \sqrt {\left (a -b \right ) b}\, a^{2}}\) | \(123\) |
default | \(\frac {\frac {x \sqrt {x^{4}-x}}{3}+\frac {\ln \left (\frac {-x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )}{6}-\frac {\ln \left (\frac {x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )}{6}}{a}-\frac {b \left (\left (2 a -2 b \right ) \arctan \left (\frac {\sqrt {x^{4}-x}\, b}{x^{2} \sqrt {\left (a -b \right ) b}}\right )+\sqrt {\left (a -b \right ) b}\, \left (\ln \left (\frac {-x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )-\ln \left (\frac {x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )\right )\right )}{3 a^{2} \sqrt {\left (a -b \right ) b}}\) | \(160\) |
elliptic | \(\text {Expression too large to display}\) | \(665\) |
1/3*x^2/a*(x^3-1)/(x*(x^3-1))^(1/2)-1/2/a*(1/3*(a-2*b)/a*ln(-2*x^3-2*x*(x^ 4-x)^(1/2)+1)+4/3*(a-b)*b/a/((a-b)*b)^(1/2)*arctan((x^4-x)^(1/2)/x^2*b/((a -b)*b)^(1/2)))
Time = 1.02 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.21 \[ \int \frac {x^3 \sqrt {-x+x^4}}{-b+a x^3} \, dx=\left [\frac {2 \, \sqrt {x^{4} - x} a x - {\left (a - 2 \, b\right )} \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} - x} x + 1\right ) + \sqrt {-a b + b^{2}} \log \left (-\frac {{\left (a^{2} - 8 \, a b + 8 \, b^{2}\right )} x^{6} + 2 \, {\left (3 \, a b - 4 \, b^{2}\right )} x^{3} + 4 \, {\left ({\left (a - 2 \, b\right )} x^{4} + b x\right )} \sqrt {x^{4} - x} \sqrt {-a b + b^{2}} + b^{2}}{a^{2} x^{6} - 2 \, a b x^{3} + b^{2}}\right )}{6 \, a^{2}}, \frac {2 \, \sqrt {x^{4} - x} a x - {\left (a - 2 \, b\right )} \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} - x} x + 1\right ) + 2 \, \sqrt {a b - b^{2}} \arctan \left (-\frac {2 \, \sqrt {x^{4} - x} \sqrt {a b - b^{2}} x}{{\left (a - 2 \, b\right )} x^{3} + b}\right )}{6 \, a^{2}}\right ] \]
[1/6*(2*sqrt(x^4 - x)*a*x - (a - 2*b)*log(-2*x^3 - 2*sqrt(x^4 - x)*x + 1) + sqrt(-a*b + b^2)*log(-((a^2 - 8*a*b + 8*b^2)*x^6 + 2*(3*a*b - 4*b^2)*x^3 + 4*((a - 2*b)*x^4 + b*x)*sqrt(x^4 - x)*sqrt(-a*b + b^2) + b^2)/(a^2*x^6 - 2*a*b*x^3 + b^2)))/a^2, 1/6*(2*sqrt(x^4 - x)*a*x - (a - 2*b)*log(-2*x^3 - 2*sqrt(x^4 - x)*x + 1) + 2*sqrt(a*b - b^2)*arctan(-2*sqrt(x^4 - x)*sqrt( a*b - b^2)*x/((a - 2*b)*x^3 + b)))/a^2]
\[ \int \frac {x^3 \sqrt {-x+x^4}}{-b+a x^3} \, dx=\int \frac {x^{3} \sqrt {x \left (x - 1\right ) \left (x^{2} + x + 1\right )}}{a x^{3} - b}\, dx \]
\[ \int \frac {x^3 \sqrt {-x+x^4}}{-b+a x^3} \, dx=\int { \frac {\sqrt {x^{4} - x} x^{3}}{a x^{3} - b} \,d x } \]
Time = 0.38 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.89 \[ \int \frac {x^3 \sqrt {-x+x^4}}{-b+a x^3} \, dx=\frac {\sqrt {x^{4} - x} x}{3 \, a} - \frac {{\left (a - 2 \, b\right )} \log \left (\sqrt {-\frac {1}{x^{3}} + 1} + 1\right )}{6 \, a^{2}} + \frac {{\left (a - 2 \, b\right )} \log \left ({\left | \sqrt {-\frac {1}{x^{3}} + 1} - 1 \right |}\right )}{6 \, a^{2}} - \frac {2 \, \sqrt {a b - b^{2}} \arctan \left (\frac {b \sqrt {-\frac {1}{x^{3}} + 1}}{\sqrt {a b - b^{2}}}\right )}{3 \, a^{2}} \]
1/3*sqrt(x^4 - x)*x/a - 1/6*(a - 2*b)*log(sqrt(-1/x^3 + 1) + 1)/a^2 + 1/6* (a - 2*b)*log(abs(sqrt(-1/x^3 + 1) - 1))/a^2 - 2/3*sqrt(a*b - b^2)*arctan( b*sqrt(-1/x^3 + 1)/sqrt(a*b - b^2))/a^2
Timed out. \[ \int \frac {x^3 \sqrt {-x+x^4}}{-b+a x^3} \, dx=-\int \frac {x^3\,\sqrt {x^4-x}}{b-a\,x^3} \,d x \]