3.9.25 \(\int \frac {x^3}{(a+b x^4)^2 \sqrt {c+d x^4}} \, dx\) [825]

Optimal. Leaf size=87 \[ -\frac {\sqrt {c+d x^4}}{4 (b c-a d) \left (a+b x^4\right )}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{4 \sqrt {b} (b c-a d)^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 87, normalized size of antiderivative = 1.00, 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 = {455, 44, 65, 214} \begin {gather*} \frac {d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{4 \sqrt {b} (b c-a d)^{3/2}}-\frac {\sqrt {c+d x^4}}{4 \left (a+b x^4\right ) (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*Sqrt[c + d*x^4]/((b*c - a*d)*(a + b*x^4)) + (d*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^4])/Sqrt[b*c - a*d]])/(4*Sqr
t[b]*(b*c - a*d)^(3/2))

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

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 214

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

Rule 455

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^3}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx,x,x^4\right )\\ &=-\frac {\sqrt {c+d x^4}}{4 (b c-a d) \left (a+b x^4\right )}-\frac {d \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^4\right )}{8 (b c-a d)}\\ &=-\frac {\sqrt {c+d x^4}}{4 (b c-a d) \left (a+b x^4\right )}-\frac {\text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^4}\right )}{4 (b c-a d)}\\ &=-\frac {\sqrt {c+d x^4}}{4 (b c-a d) \left (a+b x^4\right )}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {b c-a d}}\right )}{4 \sqrt {b} (b c-a d)^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 86, normalized size = 0.99 \begin {gather*} \frac {1}{4} \left (-\frac {\sqrt {c+d x^4}}{(b c-a d) \left (a+b x^4\right )}+\frac {d \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^4}}{\sqrt {-b c+a d}}\right )}{\sqrt {b} (-b c+a d)^{3/2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(540\) vs. \(2(71)=142\).
time = 0.32, size = 541, normalized size = 6.22

method	result	size
default	\(-\frac {\sqrt {-a b}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{8 a b \left (a d -b c \right ) \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}-\frac {d \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{8 b \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}+\frac {\sqrt {-a b}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{8 a b \left (a d -b c \right ) \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}-\frac {d \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{8 b \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\)	\(541\)
elliptic	\(-\frac {\sqrt {-a b}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{8 a b \left (a d -b c \right ) \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}-\frac {d \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{8 b \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}+\frac {\sqrt {-a b}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{8 a b \left (a d -b c \right ) \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}-\frac {d \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{8 b \left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\)	\(541\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/a*(-a*b)^(1/2)/b/(a*d-b*c)/(x^2-1/b*(-a*b)^(1/2))*((x^2-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x^2-1/b
*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)-1/8/b*d/(a*d-b*c)/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b
*(x^2-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x^2-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x^2-1/b*(-a*b)^
(1/2))-(a*d-b*c)/b)^(1/2))/(x^2-1/b*(-a*b)^(1/2)))+1/8/a*(-a*b)^(1/2)/b/(a*d-b*c)/(x^2+1/b*(-a*b)^(1/2))*((x^2
+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x^2+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)-1/8/b*d/(a*d-b*c)/(-(a*d-b
*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x^2+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x^2+1/b*(-a
*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x^2+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x^2+1/b*(-a*b)^(1/2)))

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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^3/(b*x^4+a)^2/(d*x^4+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 detail

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (71) = 142\).
time = 2.36, size = 302, normalized size = 3.47 \begin {gather*} \left [-\frac {{\left (b d x^{4} + a d\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x^{4} + 2 \, b c - a d - 2 \, \sqrt {d x^{4} + c} \sqrt {b^{2} c - a b d}}{b x^{4} + a}\right ) + 2 \, \sqrt {d x^{4} + c} {\left (b^{2} c - a b d\right )}}{8 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4}\right )}}, -\frac {{\left (b d x^{4} + a d\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {d x^{4} + c} \sqrt {-b^{2} c + a b d}}{b d x^{4} + b c}\right ) + \sqrt {d x^{4} + c} {\left (b^{2} c - a b d\right )}}{4 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\left (a + b x^{4}\right )^{2} \sqrt {c + d x^{4}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3/((a + b*x**4)**2*sqrt(c + d*x**4)), x)

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Giac [A]
time = 1.62, size = 93, normalized size = 1.07 \begin {gather*} -\frac {d \arctan \left (\frac {\sqrt {d x^{4} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{4 \, \sqrt {-b^{2} c + a b d} {\left (b c - a d\right )}} - \frac {\sqrt {d x^{4} + c} d}{4 \, {\left ({\left (d x^{4} + c\right )} b - b c + a d\right )} {\left (b c - a d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*d*arctan(sqrt(d*x^4 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*(b*c - a*d)) - 1/4*sqrt(d*x^4 + c)
*d/(((d*x^4 + c)*b - b*c + a*d)*(b*c - a*d))

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Mupad [B]
time = 4.85, size = 84, normalized size = 0.97 \begin {gather*} \frac {d\,\sqrt {d\,x^4+c}}{2\,\left (a\,d-b\,c\right )\,\left (2\,b\,\left (d\,x^4+c\right )+2\,a\,d-2\,b\,c\right )}+\frac {d\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d\,x^4+c}}{\sqrt {a\,d-b\,c}}\right )}{4\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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