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

Optimal. Leaf size=49 \[ \frac {1}{3} (a-2 b) \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right )+\frac {\sqrt {x^4+x} \left (a x^3+2 b\right )}{3 x^2} \]

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 60, normalized size of antiderivative = 1.22, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, 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*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 62, normalized size = 1.27 \begin {gather*} \frac {\sqrt {x^4+x} \left (x^{3/2} (a-2 b) \sinh ^{-1}\left (x^{3/2}\right )+\sqrt {x^3+1} \left (a x^3+2 b\right )\right )}{3 x^2 \sqrt {x^3+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.45, size = 49, 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

________________________________________________________________________________________

fricas [A]  time = 0.58, size = 51, normalized size = 1.04 \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

________________________________________________________________________________________

giac [A]  time = 0.35, size = 57, normalized size = 1.16 \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)

________________________________________________________________________________________

maple [A]  time = 0.29, size = 48, normalized size = 0.98

method	result	size
trager	\(\frac {\left (a \,x^{3}+2 b \right ) \sqrt {x^{4}+x}}{3 x^{2}}+\frac {\left (a -2 b \right ) \ln \left (-2 x^{3}-2 x \sqrt {x^{4}+x}-1\right )}{6}\)	\(48\)
meijerg	\(-\frac {a \left (-2 \sqrt {\pi }\, x^{\frac {3}{2}} \sqrt {x^{3}+1}-2 \sqrt {\pi }\, \arcsinh \left (x^{\frac {3}{2}}\right )\right )}{6 \sqrt {\pi }}+\frac {b \left (\frac {4 \sqrt {\pi }\, \sqrt {x^{3}+1}}{x^{\frac {3}{2}}}-4 \sqrt {\pi }\, \arcsinh \left (x^{\frac {3}{2}}\right )\right )}{6 \sqrt {\pi }}\)	\(64\)
elliptic	\(\frac {2 b \sqrt {x^{4}+x}}{3 x^{2}}+\frac {a x \sqrt {x^{4}+x}}{3}-\frac {2 \left (\frac {a}{2}-b \right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(322\)
risch	\(\frac {\left (x^{3}+1\right ) \left (a \,x^{3}+2 b \right )}{3 x \sqrt {x \left (x^{3}+1\right )}}-\frac {2 \left (\frac {a}{2}-b \right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(326\)
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 \sqrt {-\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}{x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \EllipticF \left (\sqrt {-\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \sqrt {-\frac {i \sqrt {3}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticPi \left (\sqrt {-\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, -\frac {1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {-\frac {i \sqrt {3}}{\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 ) \left (\frac {1}{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 )\)	\(606\)

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)

________________________________________________________________________________________

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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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)

________________________________________________________________________________________