3.2.90 \(\int \frac {A+B x^3}{x \sqrt {a+b x^3}} \, dx\)

Optimal. Leaf size=48 \[ \frac {2 B \sqrt {a+b x^3}}{3 b}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 \sqrt {a}} \]

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Rubi [A]  time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, 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 = {446, 80, 63, 208} \begin {gather*} \frac {2 B \sqrt {a+b x^3}}{3 b}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x^3)/(x*Sqrt[a + b*x^3]),x]

[Out]

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 80

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] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 208

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

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

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Mathematica [A]  time = 0.05, size = 48, normalized size = 1.00 \begin {gather*} \frac {2 B \sqrt {a+b x^3}}{3 b}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^3)/(x*Sqrt[a + b*x^3]),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.04, size = 48, normalized size = 1.00 \begin {gather*} \frac {2 B \sqrt {a+b x^3}}{3 b}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x^3)/(x*Sqrt[a + b*x^3]),x]

[Out]

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

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fricas [A]  time = 1.01, size = 105, normalized size = 2.19 \begin {gather*} \left [\frac {A \sqrt {a} b \log \left (\frac {b x^{3} - 2 \, \sqrt {b x^{3} + a} \sqrt {a} + 2 \, a}{x^{3}}\right ) + 2 \, \sqrt {b x^{3} + a} B a}{3 \, a b}, \frac {2 \, {\left (A \sqrt {-a} b \arctan \left (\frac {\sqrt {b x^{3} + a} \sqrt {-a}}{a}\right ) + \sqrt {b x^{3} + a} B a\right )}}{3 \, a b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/3*(A*sqrt(a)*b*log((b*x^3 - 2*sqrt(b*x^3 + a)*sqrt(a) + 2*a)/x^3) + 2*sqrt(b*x^3 + a)*B*a)/(a*b), 2/3*(A*sq
rt(-a)*b*arctan(sqrt(b*x^3 + a)*sqrt(-a)/a) + sqrt(b*x^3 + a)*B*a)/(a*b)]

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giac [A]  time = 0.19, size = 40, normalized size = 0.83 \begin {gather*} \frac {2 \, A \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{3 \, \sqrt {-a}} + \frac {2 \, \sqrt {b x^{3} + a} B}{3 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*A*arctan(sqrt(b*x^3 + a)/sqrt(-a))/sqrt(-a) + 2/3*sqrt(b*x^3 + a)*B/b

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maple [A]  time = 0.04, size = 37, normalized size = 0.77 \begin {gather*} -\frac {2 A \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 \sqrt {a}}+\frac {2 \sqrt {b \,x^{3}+a}\, B}{3 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^3+A)/x/(b*x^3+a)^(1/2),x)

[Out]

-2/3*A*arctanh((b*x^3+a)^(1/2)/a^(1/2))/a^(1/2)+2/3*B*(b*x^3+a)^(1/2)/b

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maxima [A]  time = 1.08, size = 54, normalized size = 1.12 \begin {gather*} \frac {A \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{3 \, \sqrt {a}} + \frac {2 \, \sqrt {b x^{3} + a} B}{3 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.72, size = 57, normalized size = 1.19 \begin {gather*} \frac {2\,B\,\sqrt {b\,x^3+a}}{3\,b}+\frac {A\,\ln \left (\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^3\,\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}{x^6}\right )}{3\,\sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^3)/(x*(a + b*x^3)^(1/2)),x)

[Out]

(2*B*(a + b*x^3)^(1/2))/(3*b) + (A*log((((a + b*x^3)^(1/2) - a^(1/2))^3*((a + b*x^3)^(1/2) + a^(1/2)))/x^6))/(
3*a^(1/2))

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sympy [A]  time = 11.27, size = 65, normalized size = 1.35 \begin {gather*} \frac {2 A \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{a}} \sqrt {a + b x^{3}}} \right )}}{3 a \sqrt {- \frac {1}{a}}} - \frac {B \left (\begin {cases} - \frac {x^{3}}{\sqrt {a}} & \text {for}\: b = 0 \\- \frac {2 \sqrt {a + b x^{3}}}{b} & \text {otherwise} \end {cases}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**3+A)/x/(b*x**3+a)**(1/2),x)

[Out]

2*A*atan(1/(sqrt(-1/a)*sqrt(a + b*x**3)))/(3*a*sqrt(-1/a)) - B*Piecewise((-x**3/sqrt(a), Eq(b, 0)), (-2*sqrt(a
+ b*x**3)/b, True))/3

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