3.8.7 \(\int \frac {(-b+a x^3) \sqrt {-x+x^4}}{x^3} \, dx\)

Optimal. Leaf size=55 \[ \frac {1}{3} (-a-2 b) \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4-x}}\right )+\frac {\sqrt {x^4-x} \left (a x^3+2 b\right )}{3 x^2} \]

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Rubi [A]  time = 0.09, antiderivative size = 66, normalized size of antiderivative = 1.20, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2038, 2004, 2029, 206} \begin {gather*} \frac {1}{3} x \sqrt {x^4-x} (a+2 b)-\frac {1}{3} (a+2 b) \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4-x}}\right )-\frac {2 b \left (x^4-x\right )^{3/2}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2004

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 2029

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 2038

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Sim
p[(c*e^(j - 1)*(e*x)^(m - j + 1)*(a*x^j + b*x^(j + n))^(p + 1))/(a*(m + j*p + 1)), x] + Dist[(a*d*(m + j*p + 1
) - b*c*(m + n + p*(j + n) + 1))/(a*e^n*(m + j*p + 1)), Int[(e*x)^(m + n)*(a*x^j + b*x^(j + n))^p, x], x] /; F
reeQ[{a, b, c, d, e, j, p}, x] && EqQ[jn, j + n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && (LtQ[m
+ j*p, -1] || (IntegersQ[m - 1/2, p - 1/2] && LtQ[p, 0] && LtQ[m, -(n*p) - 1])) && (GtQ[e, 0] || IntegersQ[j,
n]) && NeQ[m + j*p + 1, 0] && NeQ[m - n + j*p + 1, 0]

Rubi steps

\begin {align*} \int \frac {\left (-b+a x^3\right ) \sqrt {-x+x^4}}{x^3} \, dx &=-\frac {2 b \left (-x+x^4\right )^{3/2}}{3 x^3}+(a+2 b) \int \sqrt {-x+x^4} \, dx\\ &=\frac {1}{3} (a+2 b) x \sqrt {-x+x^4}-\frac {2 b \left (-x+x^4\right )^{3/2}}{3 x^3}+\frac {1}{2} (-a-2 b) \int \frac {x}{\sqrt {-x+x^4}} \, dx\\ &=\frac {1}{3} (a+2 b) x \sqrt {-x+x^4}-\frac {2 b \left (-x+x^4\right )^{3/2}}{3 x^3}+\frac {1}{3} (-a-2 b) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^2}{\sqrt {-x+x^4}}\right )\\ &=\frac {1}{3} (a+2 b) x \sqrt {-x+x^4}-\frac {2 b \left (-x+x^4\right )^{3/2}}{3 x^3}-\frac {1}{3} (a+2 b) \tanh ^{-1}\left (\frac {x^2}{\sqrt {-x+x^4}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 68, normalized size = 1.24 \begin {gather*} \frac {\sqrt {x \left (x^3-1\right )} \left (x^{3/2} (a+2 b) \sin ^{-1}\left (x^{3/2}\right )+\sqrt {1-x^3} \left (a x^3+2 b\right )\right )}{3 x^2 \sqrt {1-x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x*(-1 + x^3)]*(Sqrt[1 - x^3]*(2*b + a*x^3) + (a + 2*b)*x^(3/2)*ArcSin[x^(3/2)]))/(3*x^2*Sqrt[1 - x^3])

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IntegrateAlgebraic [A]  time = 0.46, size = 55, normalized size = 1.00 \begin {gather*} \frac {\left (2 b+a x^3\right ) \sqrt {-x+x^4}}{3 x^2}+\frac {1}{3} (-a-2 b) \tanh ^{-1}\left (\frac {x^2}{\sqrt {-x+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.50, size = 55, normalized size = 1.00 \begin {gather*} \frac {{\left (a + 2 \, b\right )} x^{2} \log \left (2 \, x^{3} - 2 \, \sqrt {x^{4} - x} x - 1\right ) + 2 \, {\left (a x^{3} + 2 \, b\right )} \sqrt {x^{4} - x}}{6 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*((a + 2*b)*x^2*log(2*x^3 - 2*sqrt(x^4 - x)*x - 1) + 2*(a*x^3 + 2*b)*sqrt(x^4 - x))/x^2

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giac [A]  time = 0.23, size = 65, normalized size = 1.18 \begin {gather*} \frac {1}{3} \, \sqrt {x^{4} - x} a x - \frac {1}{6} \, {\left (a + 2 \, b\right )} \log \left (\sqrt {-\frac {1}{x^{3}} + 1} + 1\right ) + \frac {1}{6} \, {\left (a + 2 \, b\right )} \log \left ({\left | \sqrt {-\frac {1}{x^{3}} + 1} - 1 \right |}\right ) + \frac {2}{3} \, b \sqrt {-\frac {1}{x^{3}} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(x^4 - x)*a*x - 1/6*(a + 2*b)*log(sqrt(-1/x^3 + 1) + 1) + 1/6*(a + 2*b)*log(abs(sqrt(-1/x^3 + 1) - 1))
+ 2/3*b*sqrt(-1/x^3 + 1)

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maple [A]  time = 0.34, size = 52, normalized size = 0.95

method	result	size
trager	\(\frac {\left (a \,x^{3}+2 b \right ) \sqrt {x^{4}-x}}{3 x^{2}}-\frac {\left (a +2 b \right ) \ln \left (-2 x^{3}-2 x \sqrt {x^{4}-x}+1\right )}{6}\)	\(52\)
meijerg	\(\frac {i a \sqrt {\mathrm {signum}\left (x^{3}-1\right )}\, \left (-2 i \sqrt {\pi }\, x^{\frac {3}{2}} \sqrt {-x^{3}+1}-2 i \sqrt {\pi }\, \arcsin \left (x^{\frac {3}{2}}\right )\right )}{6 \sqrt {\pi }\, \sqrt {-\mathrm {signum}\left (x^{3}-1\right )}}+\frac {i b \sqrt {\mathrm {signum}\left (x^{3}-1\right )}\, \left (-\frac {4 i \sqrt {\pi }\, \sqrt {-x^{3}+1}}{x^{\frac {3}{2}}}-4 i \sqrt {\pi }\, \arcsin \left (x^{\frac {3}{2}}\right )\right )}{6 \sqrt {\pi }\, \sqrt {-\mathrm {signum}\left (x^{3}-1\right )}}\)	\(110\)
risch	\(\frac {\left (x^{3}-1\right ) \left (a \,x^{3}+2 b \right )}{3 x \sqrt {x \left (x^{3}-1\right )}}+\frac {2 \left (-\frac {a}{2}-b \right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(324\)
elliptic	\(\frac {2 b \sqrt {x^{4}-x}}{3 x^{2}}+\frac {a x \sqrt {x^{4}-x}}{3}+\frac {2 \left (-\frac {a}{2}-b \right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(324\)
default	\(a \left (\frac {x \sqrt {x^{4}-x}}{3}-\frac {\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 (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (\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 (-1+x \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 )-\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 (-1+x \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 (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-b \left (-\frac {2 \sqrt {x^{4}-x}}{3 x^{2}}+\frac {2 \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 (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (\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 (-1+x \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 )-\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 (-1+x \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 (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\)	\(609\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(a*x^3+2*b)*(x^4-x)^(1/2)/x^2-1/6*(a+2*b)*ln(-2*x^3-2*x*(x^4-x)^(1/2)+1)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{3} - b\right )} \sqrt {x^{4} - x}}{x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} -\int \frac {\sqrt {x^4-x}\,\left (b-a\,x^3\right )}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x^4 - x)^(1/2)*(b - a*x^3))/x^3,x)

[Out]

-int(((x^4 - x)^(1/2)*(b - a*x^3))/x^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (a x^{3} - b\right )}{x^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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