\(\int (b+a x^3) \sqrt {-x+x^4} \, dx\) [749]

Optimal result

Integrand size = 19, antiderivative size = 58 \[ \int \left (b+a x^3\right ) \sqrt {-x+x^4} \, dx=\frac {1}{12} \sqrt {-x+x^4} \left (-a x+4 b x+2 a x^4\right )+\frac {1}{12} (-a-4 b) \text {arctanh}\left (\frac {x^2}{\sqrt {-x+x^4}}\right ) \]

[Out]

1/12*(x^4-x)^(1/2)*(2*a*x^4-a*x+4*b*x)+1/12*(-a-4*b)*arctanh(x^2/(x^4-x)^(1/2))

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.66, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2078, 2029, 2054, 212, 2046, 2049} \[ \int \left (b+a x^3\right ) \sqrt {-x+x^4} \, dx=-\frac {1}{12} a \text {arctanh}\left (\frac {x^2}{\sqrt {x^4-x}}\right )+\frac {1}{6} a \sqrt {x^4-x} x^4-\frac {1}{12} a \sqrt {x^4-x} x-\frac {1}{3} b \text {arctanh}\left (\frac {x^2}{\sqrt {x^4-x}}\right )+\frac {1}{3} b \sqrt {x^4-x} x \]

[In]

Int[(b + a*x^3)*Sqrt[-x + x^4],x]

[Out]

-1/12*(a*x*Sqrt[-x + x^4]) + (b*x*Sqrt[-x + x^4])/3 + (a*x^4*Sqrt[-x + x^4])/6 - (a*ArcTanh[x^2/Sqrt[-x + x^4]
])/12 - (b*ArcTanh[x^2/Sqrt[-x + x^4]])/3

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2029

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[x*((a*x^j + b*x^n)^p/(n*p + 1)), x] + Dist[a
*(n - j)*(p/(n*p + 1)), Int[x^j*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] &&  !IntegerQ[p] && LtQ[0,
j, n] && GtQ[p, 0] && NeQ[n*p + 1, 0]

Rule 2046

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] + Dist[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] && (IntegersQ[j, n] || GtQ[c, 0]) && G
tQ[p, 0] && NeQ[m + n*p + 1, 0]

Rule 2049

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] - Dist[a*c^(n - j)*((m + j*p - n + j + 1)/(b*(m + n*p + 1))
), Int[(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]) && GtQ[m + j*p + 1 - n + j, 0] && NeQ[m + n*p + 1, 0]

Rule 2054

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[-2/(n - j), Subst[Int[1/(1 - a*x^2
), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]

Rule 2078

Int[(Pq_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Pq*(a*x^j + b*x^n)^p, x]
, x] /; FreeQ[{a, b, j, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !IntegerQ[p] && NeQ[n, j]

Rubi steps \begin{align*} \text {integral}& = \int \left (b \sqrt {-x+x^4}+a x^3 \sqrt {-x+x^4}\right ) \, dx \\ & = a \int x^3 \sqrt {-x+x^4} \, dx+b \int \sqrt {-x+x^4} \, dx \\ & = \frac {1}{3} b x \sqrt {-x+x^4}+\frac {1}{6} a x^4 \sqrt {-x+x^4}-\frac {1}{4} a \int \frac {x^4}{\sqrt {-x+x^4}} \, dx-\frac {1}{2} b \int \frac {x}{\sqrt {-x+x^4}} \, dx \\ & = -\frac {1}{12} a x \sqrt {-x+x^4}+\frac {1}{3} b x \sqrt {-x+x^4}+\frac {1}{6} a x^4 \sqrt {-x+x^4}-\frac {1}{8} a \int \frac {x}{\sqrt {-x+x^4}} \, dx-\frac {1}{3} b \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^2}{\sqrt {-x+x^4}}\right ) \\ & = -\frac {1}{12} a x \sqrt {-x+x^4}+\frac {1}{3} b x \sqrt {-x+x^4}+\frac {1}{6} a x^4 \sqrt {-x+x^4}-\frac {1}{3} b \text {arctanh}\left (\frac {x^2}{\sqrt {-x+x^4}}\right )-\frac {1}{12} a \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^2}{\sqrt {-x+x^4}}\right ) \\ & = -\frac {1}{12} a x \sqrt {-x+x^4}+\frac {1}{3} b x \sqrt {-x+x^4}+\frac {1}{6} a x^4 \sqrt {-x+x^4}-\frac {1}{12} a \text {arctanh}\left (\frac {x^2}{\sqrt {-x+x^4}}\right )-\frac {1}{3} b \text {arctanh}\left (\frac {x^2}{\sqrt {-x+x^4}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.29 \[ \int \left (b+a x^3\right ) \sqrt {-x+x^4} \, dx=\frac {x^2 \left (-1+x^3\right ) \left (4 b+a \left (-1+2 x^3\right )\right )-(a+4 b) \sqrt {x} \sqrt {-1+x^3} \log \left (x^{3/2}+\sqrt {-1+x^3}\right )}{12 \sqrt {x \left (-1+x^3\right )}} \]

[In]

Integrate[(b + a*x^3)*Sqrt[-x + x^4],x]

[Out]

(x^2*(-1 + x^3)*(4*b + a*(-1 + 2*x^3)) - (a + 4*b)*Sqrt[x]*Sqrt[-1 + x^3]*Log[x^(3/2) + Sqrt[-1 + x^3]])/(12*S
qrt[x*(-1 + x^3)])

Maple [A] (verified)

Time = 3.40 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.93

method	result	size
trager	\(\frac {x \left (2 a \,x^{3}-a +4 b \right ) \sqrt {x^{4}-x}}{12}+\frac {\left (a +4 b \right ) \ln \left (2 x^{3}-2 x \sqrt {x^{4}-x}-1\right )}{24}\)	\(54\)
risch	\(\frac {x^{2} \left (2 a \,x^{3}-a +4 b \right ) \left (x^{3}-1\right )}{12 \sqrt {x \left (x^{3}-1\right )}}-\frac {\left (-\frac {b}{2}-\frac {a}{8}\right ) \ln \left (2 x^{3}-2 x \sqrt {x^{4}-x}-1\right )}{3}\)	\(63\)
pseudoelliptic	\(\frac {\left (4 \sqrt {x^{4}-x}\, x \left (a \,x^{3}-\frac {1}{2} a +2 b \right )+\left (a +4 b \right ) \left (\ln \left (\frac {-x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )-\ln \left (\frac {x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )\right )\right ) x^{2}}{24 \left (x^{2}+\sqrt {x^{4}-x}\right )^{2} \left (x^{2}-\sqrt {x^{4}-x}\right )^{2}}\)	\(110\)
meijerg	\(-\frac {i a \sqrt {\operatorname {signum}\left (x^{3}-1\right )}\, \left (-\frac {i \sqrt {\pi }\, x^{\frac {3}{2}} \left (-6 x^{3}+3\right ) \sqrt {-x^{3}+1}}{6}+\frac {i \sqrt {\pi }\, \arcsin \left (x^{\frac {3}{2}}\right )}{2}\right )}{6 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{3}-1\right )}}+\frac {i b \sqrt {\operatorname {signum}\left (x^{3}-1\right )}\, \left (-2 i \sqrt {\pi }\, x^{\frac {3}{2}} \sqrt {-x^{3}+1}-2 i \sqrt {\pi }\, \arcsin \left (x^{\frac {3}{2}}\right )\right )}{6 \sqrt {\pi }\, \sqrt {-\operatorname {signum}\left (x^{3}-1\right )}}\)	\(117\)
default	\(b \left (\frac {x \sqrt {x^{4}-x}}{3}-\frac {\ln \left (\frac {x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )}{6}+\frac {\ln \left (\frac {-x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )}{6}\right )+\frac {a \,x^{2} \left (\left (4 x^{4}-2 x \right ) \sqrt {x^{4}-x}+\ln \left (\frac {-x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )-\ln \left (\frac {x^{2}+\sqrt {x^{4}-x}}{x^{2}}\right )\right )}{24 \left (x^{2}+\sqrt {x^{4}-x}\right )^{2} \left (x^{2}-\sqrt {x^{4}-x}\right )^{2}}\)	\(157\)
elliptic	\(\frac {a \,x^{4} \sqrt {x^{4}-x}}{6}+\left (-\frac {a}{12}+\frac {b}{3}\right ) x \sqrt {x^{4}-x}+\frac {2 \left (-\frac {b}{2}-\frac {a}{8}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}\, \left (x -1\right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}\, \left (\operatorname {EllipticF}\left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\operatorname {EllipticPi}\left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -1\right )}}, \frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (x -1\right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(329\)

[In]

int((a*x^3+b)*(x^4-x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/12*x*(2*a*x^3-a+4*b)*(x^4-x)^(1/2)+1/24*(a+4*b)*ln(2*x^3-2*x*(x^4-x)^(1/2)-1)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.93 \[ \int \left (b+a x^3\right ) \sqrt {-x+x^4} \, dx=\frac {1}{24} \, {\left (a + 4 \, b\right )} \log \left (2 \, x^{3} - 2 \, \sqrt {x^{4} - x} x - 1\right ) + \frac {1}{12} \, {\left (2 \, a x^{4} - {\left (a - 4 \, b\right )} x\right )} \sqrt {x^{4} - x} \]

[In]

integrate((a*x^3+b)*(x^4-x)^(1/2),x, algorithm="fricas")

[Out]

1/24*(a + 4*b)*log(2*x^3 - 2*sqrt(x^4 - x)*x - 1) + 1/12*(2*a*x^4 - (a - 4*b)*x)*sqrt(x^4 - x)

Sympy [F]

\[ \int \left (b+a x^3\right ) \sqrt {-x+x^4} \, dx=\int \sqrt {x \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (a x^{3} + b\right )\, dx \]

[In]

integrate((a*x**3+b)*(x**4-x)**(1/2),x)

[Out]

Integral(sqrt(x*(x - 1)*(x**2 + x + 1))*(a*x**3 + b), x)

Maxima [F]

\[ \int \left (b+a x^3\right ) \sqrt {-x+x^4} \, dx=\int { {\left (a x^{3} + b\right )} \sqrt {x^{4} - x} \,d x } \]

[In]

integrate((a*x^3+b)*(x^4-x)^(1/2),x, algorithm="maxima")

[Out]

integrate((a*x^3 + b)*sqrt(x^4 - x), x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.05 \[ \int \left (b+a x^3\right ) \sqrt {-x+x^4} \, dx=\frac {1}{12} \, {\left (2 \, a x^{3} - a + 4 \, b\right )} \sqrt {x^{4} - x} x - \frac {1}{24} \, {\left (a + 4 \, b\right )} {\left (\log \left (\sqrt {-\frac {1}{x^{3}} + 1} + 1\right ) - \log \left ({\left | \sqrt {-\frac {1}{x^{3}} + 1} - 1 \right |}\right )\right )} \]

[In]

integrate((a*x^3+b)*(x^4-x)^(1/2),x, algorithm="giac")

[Out]

1/12*(2*a*x^3 - a + 4*b)*sqrt(x^4 - x)*x - 1/24*(a + 4*b)*(log(sqrt(-1/x^3 + 1) + 1) - log(abs(sqrt(-1/x^3 + 1
) - 1)))

Mupad [F(-1)]

Timed out. \[ \int \left (b+a x^3\right ) \sqrt {-x+x^4} \, dx=\int \sqrt {x^4-x}\,\left (a\,x^3+b\right ) \,d x \]

[In]

int((x^4 - x)^(1/2)*(b + a*x^3),x)

[Out]

int((x^4 - x)^(1/2)*(b + a*x^3), x)