3.4.22 \(\int \frac {x^2}{\sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx\)

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

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Rubi [A]  time = 0.06, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {444, 63, 217, 206} \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^3}}{\sqrt {b} \sqrt {c+d x^3}}\right )}{3 \sqrt {b} \sqrt {d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^3])/(Sqrt[b]*Sqrt[c + d*x^3])])/(3*Sqrt[b]*Sqrt[d])

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 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 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(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] && EqQ[m
- n + 1, 0]

Rubi steps

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

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Mathematica [A]  time = 0.10, size = 85, normalized size = 1.77 \begin {gather*} \frac {2 \sqrt {c+d x^3} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^3}}{\sqrt {b c-a d}}\right )}{3 \sqrt {d} \sqrt {b c-a d} \sqrt {\frac {b \left (c+d x^3\right )}{b c-a d}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.85, size = 48, normalized size = 1.00 \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {d} \sqrt {a+b x^3}}\right )}{3 \sqrt {b} \sqrt {d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/(Sqrt[d]*Sqrt[a + b*x^3])])/(3*Sqrt[b]*Sqrt[d])

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fricas [B]  time = 0.64, size = 194, normalized size = 4.04 \begin {gather*} \left [\frac {\sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{6} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{3} + 4 \, {\left (2 \, b d x^{3} + b c + a d\right )} \sqrt {b x^{3} + a} \sqrt {d x^{3} + c} \sqrt {b d}\right )}{6 \, b d}, -\frac {\sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x^{3} + b c + a d\right )} \sqrt {b x^{3} + a} \sqrt {d x^{3} + c} \sqrt {-b d}}{2 \, {\left (b^{2} d^{2} x^{6} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x^{3}\right )}}\right )}{3 \, b d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*sqrt(b*d)*log(8*b^2*d^2*x^6 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 8*(b^2*c*d + a*b*d^2)*x^3 + 4*(2*b*d*x^3 +
b*c + a*d)*sqrt(b*x^3 + a)*sqrt(d*x^3 + c)*sqrt(b*d))/(b*d), -1/3*sqrt(-b*d)*arctan(1/2*(2*b*d*x^3 + b*c + a*d
)*sqrt(b*x^3 + a)*sqrt(d*x^3 + c)*sqrt(-b*d)/(b^2*d^2*x^6 + a*b*c*d + (b^2*c*d + a*b*d^2)*x^3))/(b*d)]

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giac [A]  time = 0.18, size = 54, normalized size = 1.12 \begin {gather*} -\frac {2 \, b \log \left ({\left | -\sqrt {b x^{3} + a} \sqrt {b d} + \sqrt {b^{2} c + {\left (b x^{3} + a\right )} b d - a b d} \right |}\right )}{3 \, \sqrt {b d} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*b*log(abs(-sqrt(b*x^3 + a)*sqrt(b*d) + sqrt(b^2*c + (b*x^3 + a)*b*d - a*b*d)))/(sqrt(b*d)*abs(b))

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maple [F]  time = 0.65, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\sqrt {b \,x^{3}+a}\, \sqrt {d \,x^{3}+c}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x)

[Out]

int(x^2/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more details)Is a*d-b*c zero or nonzero?

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mupad [B]  time = 5.04, size = 49, normalized size = 1.02 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {d\,x^3+c}-\sqrt {c}\right )}{\sqrt {-b\,d}\,\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}\right )}{3\,\sqrt {-b\,d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/((a + b*x^3)^(1/2)*(c + d*x^3)^(1/2)),x)

[Out]

-(4*atan((b*((c + d*x^3)^(1/2) - c^(1/2)))/((-b*d)^(1/2)*((a + b*x^3)^(1/2) - a^(1/2)))))/(3*(-b*d)^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\sqrt {a + b x^{3}} \sqrt {c + d x^{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(b*x**3+a)**(1/2)/(d*x**3+c)**(1/2),x)

[Out]

Integral(x**2/(sqrt(a + b*x**3)*sqrt(c + d*x**3)), x)

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