3.2.28 \(\int \frac {(-2 b+a x^3) \sqrt {b+a x^3}}{x^4} \, dx\)

Optimal. Leaf size=18 \[ \frac {2 \left (a x^3+b\right )^{3/2}}{3 x^3} \]

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {446, 74} \begin {gather*} \frac {2 \left (a x^3+b\right )^{3/2}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b + a*x^3)^(3/2))/(3*x^3)

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} \frac {2 \left (a x^3+b\right )^{3/2}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b + a*x^3)^(3/2))/(3*x^3)

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IntegrateAlgebraic [A]  time = 0.06, size = 18, normalized size = 1.00 \begin {gather*} \frac {2 \left (b+a x^3\right )^{3/2}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b + a*x^3)^(3/2))/(3*x^3)

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fricas [A]  time = 0.45, size = 14, normalized size = 0.78 \begin {gather*} \frac {2 \, {\left (a x^{3} + b\right )}^{\frac {3}{2}}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(a*x^3 + b)^(3/2)/x^3

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giac [B]  time = 0.31, size = 34, normalized size = 1.89 \begin {gather*} \frac {2 \, {\left (\sqrt {a x^{3} + b} a^{2} + \frac {\sqrt {a x^{3} + b} a b}{x^{3}}\right )}}{3 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(sqrt(a*x^3 + b)*a^2 + sqrt(a*x^3 + b)*a*b/x^3)/a

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maple [A]  time = 0.36, size = 15, normalized size = 0.83

method	result	size
gosper	\(\frac {2 \left (a \,x^{3}+b \right )^{\frac {3}{2}}}{3 x^{3}}\)	\(15\)
trager	\(\frac {2 \left (a \,x^{3}+b \right )^{\frac {3}{2}}}{3 x^{3}}\)	\(15\)
risch	\(\frac {2 \left (a \,x^{3}+b \right )^{\frac {3}{2}}}{3 x^{3}}\)	\(15\)
elliptic	\(\frac {2 b \sqrt {a \,x^{3}+b}}{3 x^{3}}+\frac {2 a \sqrt {a \,x^{3}+b}}{3}\)	\(29\)
default	\(a \left (\frac {2 \sqrt {a \,x^{3}+b}}{3}-\frac {2 \sqrt {b}\, \arctanh \left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{3}\right )-2 b \left (-\frac {\sqrt {a \,x^{3}+b}}{3 x^{3}}-\frac {a \arctanh \left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{3 \sqrt {b}}\right )\)	\(73\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(a*x^3+b)^(3/2)/x^3

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maxima [B]  time = 0.45, size = 107, normalized size = 5.94 \begin {gather*} \frac {1}{3} \, {\left (\sqrt {b} \log \left (\frac {\sqrt {a x^{3} + b} - \sqrt {b}}{\sqrt {a x^{3} + b} + \sqrt {b}}\right ) + 2 \, \sqrt {a x^{3} + b}\right )} a - \frac {1}{3} \, {\left (\frac {a \log \left (\frac {\sqrt {a x^{3} + b} - \sqrt {b}}{\sqrt {a x^{3} + b} + \sqrt {b}}\right )}{\sqrt {b}} - \frac {2 \, \sqrt {a x^{3} + b}}{x^{3}}\right )} b \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.25, size = 14, normalized size = 0.78 \begin {gather*} \frac {2\,{\left (a\,x^3+b\right )}^{3/2}}{3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(b + a*x^3)^(3/2))/(3*x^3)

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sympy [B]  time = 26.24, size = 78, normalized size = 4.33 \begin {gather*} \frac {2 a^{\frac {3}{2}} x^{\frac {3}{2}}}{3 \sqrt {1 + \frac {b}{a x^{3}}}} + \frac {2 \sqrt {a} b \sqrt {1 + \frac {b}{a x^{3}}}}{3 x^{\frac {3}{2}}} + \frac {2 \sqrt {a} b}{3 x^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x^{3}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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