3.759 \(\int \frac {x^{7/2}}{(a+c x^4)^3} \, dx\)

Optimal. Leaf size=332 \[ -\frac {7 \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{15/8} c^{9/8}}+\frac {7 \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{15/8} c^{9/8}}-\frac {7 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{15/8} c^{9/8}}+\frac {7 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt {2} (-a)^{15/8} c^{9/8}}+\frac {7 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{15/8} c^{9/8}}+\frac {7 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{15/8} c^{9/8}}+\frac {\sqrt {x}}{64 a c \left (a+c x^4\right )}-\frac {\sqrt {x}}{8 c \left (a+c x^4\right )^2} \]

[Out]

7/256*arctan(c^(1/8)*x^(1/2)/(-a)^(1/8))/(-a)^(15/8)/c^(9/8)+7/256*arctanh(c^(1/8)*x^(1/2)/(-a)^(1/8))/(-a)^(1
5/8)/c^(9/8)+7/512*arctan(-1+c^(1/8)*2^(1/2)*x^(1/2)/(-a)^(1/8))/(-a)^(15/8)/c^(9/8)*2^(1/2)+7/512*arctan(1+c^
(1/8)*2^(1/2)*x^(1/2)/(-a)^(1/8))/(-a)^(15/8)/c^(9/8)*2^(1/2)-7/1024*ln((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8
)*2^(1/2)*x^(1/2))/(-a)^(15/8)/c^(9/8)*2^(1/2)+7/1024*ln((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*2^(1/2)*x^(1/
2))/(-a)^(15/8)/c^(9/8)*2^(1/2)-1/8*x^(1/2)/c/(c*x^4+a)^2+1/64*x^(1/2)/a/c/(c*x^4+a)

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Rubi [A]  time = 0.29, antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.867, Rules used = {288, 290, 329, 214, 212, 208, 205, 211, 1165, 628, 1162, 617, 204} \[ -\frac {7 \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{15/8} c^{9/8}}+\frac {7 \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{15/8} c^{9/8}}-\frac {7 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{15/8} c^{9/8}}+\frac {7 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{256 \sqrt {2} (-a)^{15/8} c^{9/8}}+\frac {7 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{15/8} c^{9/8}}+\frac {7 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{15/8} c^{9/8}}+\frac {\sqrt {x}}{64 a c \left (a+c x^4\right )}-\frac {\sqrt {x}}{8 c \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[x^(7/2)/(a + c*x^4)^3,x]

[Out]

-Sqrt[x]/(8*c*(a + c*x^4)^2) + Sqrt[x]/(64*a*c*(a + c*x^4)) - (7*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/
8)])/(256*Sqrt[2]*(-a)^(15/8)*c^(9/8)) + (7*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*Sqrt[2]*(-a
)^(15/8)*c^(9/8)) + (7*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(256*(-a)^(15/8)*c^(9/8)) + (7*ArcTanh[(c^(1/8)*S
qrt[x])/(-a)^(1/8)])/(256*(-a)^(15/8)*c^(9/8)) - (7*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1
/4)*x])/(512*Sqrt[2]*(-a)^(15/8)*c^(9/8)) + (7*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x
])/(512*Sqrt[2]*(-a)^(15/8)*c^(9/8))

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 205

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

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 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 214

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[r/(2*a), Int[1/(r - s*x^(n/2)), x], x] + Dist[r/(2*a), Int[1/(r + s*x^(n/2)), x], x]] /; FreeQ[{a,
b}, x] && IGtQ[n/4, 1] &&  !GtQ[a/b, 0]

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {x^{7/2}}{\left (a+c x^4\right )^3} \, dx &=-\frac {\sqrt {x}}{8 c \left (a+c x^4\right )^2}+\frac {\int \frac {1}{\sqrt {x} \left (a+c x^4\right )^2} \, dx}{16 c}\\ &=-\frac {\sqrt {x}}{8 c \left (a+c x^4\right )^2}+\frac {\sqrt {x}}{64 a c \left (a+c x^4\right )}+\frac {7 \int \frac {1}{\sqrt {x} \left (a+c x^4\right )} \, dx}{128 a c}\\ &=-\frac {\sqrt {x}}{8 c \left (a+c x^4\right )^2}+\frac {\sqrt {x}}{64 a c \left (a+c x^4\right )}+\frac {7 \operatorname {Subst}\left (\int \frac {1}{a+c x^8} \, dx,x,\sqrt {x}\right )}{64 a c}\\ &=-\frac {\sqrt {x}}{8 c \left (a+c x^4\right )^2}+\frac {\sqrt {x}}{64 a c \left (a+c x^4\right )}+\frac {7 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a}-\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{128 (-a)^{3/2} c}+\frac {7 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{128 (-a)^{3/2} c}\\ &=-\frac {\sqrt {x}}{8 c \left (a+c x^4\right )^2}+\frac {\sqrt {x}}{64 a c \left (a+c x^4\right )}+\frac {7 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{256 (-a)^{7/4} c}+\frac {7 \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{256 (-a)^{7/4} c}+\frac {7 \operatorname {Subst}\left (\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{256 (-a)^{7/4} c}+\frac {7 \operatorname {Subst}\left (\int \frac {\sqrt [4]{-a}+\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{256 (-a)^{7/4} c}\\ &=-\frac {\sqrt {x}}{8 c \left (a+c x^4\right )^2}+\frac {\sqrt {x}}{64 a c \left (a+c x^4\right )}+\frac {7 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{15/8} c^{9/8}}+\frac {7 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{15/8} c^{9/8}}+\frac {7 \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{512 (-a)^{7/4} c^{5/4}}+\frac {7 \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}+x^2} \, dx,x,\sqrt {x}\right )}{512 (-a)^{7/4} c^{5/4}}-\frac {7 \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [8]{-a}}{\sqrt [8]{c}}+2 x}{-\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{512 \sqrt {2} (-a)^{15/8} c^{9/8}}-\frac {7 \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [8]{-a}}{\sqrt [8]{c}}-2 x}{-\frac {\sqrt [4]{-a}}{\sqrt [4]{c}}+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{c}}-x^2} \, dx,x,\sqrt {x}\right )}{512 \sqrt {2} (-a)^{15/8} c^{9/8}}\\ &=-\frac {\sqrt {x}}{8 c \left (a+c x^4\right )^2}+\frac {\sqrt {x}}{64 a c \left (a+c x^4\right )}+\frac {7 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{15/8} c^{9/8}}+\frac {7 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{15/8} c^{9/8}}-\frac {7 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{15/8} c^{9/8}}+\frac {7 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{15/8} c^{9/8}}+\frac {7 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{15/8} c^{9/8}}-\frac {7 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{15/8} c^{9/8}}\\ &=-\frac {\sqrt {x}}{8 c \left (a+c x^4\right )^2}+\frac {\sqrt {x}}{64 a c \left (a+c x^4\right )}-\frac {7 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{15/8} c^{9/8}}+\frac {7 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 \sqrt {2} (-a)^{15/8} c^{9/8}}+\frac {7 \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{15/8} c^{9/8}}+\frac {7 \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{256 (-a)^{15/8} c^{9/8}}-\frac {7 \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{15/8} c^{9/8}}+\frac {7 \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{512 \sqrt {2} (-a)^{15/8} c^{9/8}}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 64, normalized size = 0.19 \[ \frac {\sqrt {x} \left (7 \left (a+c x^4\right )^2 \, _2F_1\left (\frac {1}{8},1;\frac {9}{8};-\frac {c x^4}{a}\right )+a \left (c x^4-7 a\right )\right )}{64 a^2 c \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)/(a + c*x^4)^3,x]

[Out]

(Sqrt[x]*(a*(-7*a + c*x^4) + 7*(a + c*x^4)^2*Hypergeometric2F1[1/8, 1, 9/8, -((c*x^4)/a)]))/(64*a^2*c*(a + c*x
^4)^2)

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fricas [B]  time = 0.76, size = 679, normalized size = 2.05 \[ \frac {28 \, \sqrt {2} {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac {1}{a^{15} c^{9}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {a^{4} c^{2} \left (-\frac {1}{a^{15} c^{9}}\right )^{\frac {1}{4}} + \sqrt {2} a^{2} c \sqrt {x} \left (-\frac {1}{a^{15} c^{9}}\right )^{\frac {1}{8}} + x} a^{13} c^{8} \left (-\frac {1}{a^{15} c^{9}}\right )^{\frac {7}{8}} - \sqrt {2} a^{13} c^{8} \sqrt {x} \left (-\frac {1}{a^{15} c^{9}}\right )^{\frac {7}{8}} + 1\right ) + 28 \, \sqrt {2} {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac {1}{a^{15} c^{9}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {2} \sqrt {a^{4} c^{2} \left (-\frac {1}{a^{15} c^{9}}\right )^{\frac {1}{4}} - \sqrt {2} a^{2} c \sqrt {x} \left (-\frac {1}{a^{15} c^{9}}\right )^{\frac {1}{8}} + x} a^{13} c^{8} \left (-\frac {1}{a^{15} c^{9}}\right )^{\frac {7}{8}} - \sqrt {2} a^{13} c^{8} \sqrt {x} \left (-\frac {1}{a^{15} c^{9}}\right )^{\frac {7}{8}} - 1\right ) + 7 \, \sqrt {2} {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac {1}{a^{15} c^{9}}\right )^{\frac {1}{8}} \log \left (a^{4} c^{2} \left (-\frac {1}{a^{15} c^{9}}\right )^{\frac {1}{4}} + \sqrt {2} a^{2} c \sqrt {x} \left (-\frac {1}{a^{15} c^{9}}\right )^{\frac {1}{8}} + x\right ) - 7 \, \sqrt {2} {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac {1}{a^{15} c^{9}}\right )^{\frac {1}{8}} \log \left (a^{4} c^{2} \left (-\frac {1}{a^{15} c^{9}}\right )^{\frac {1}{4}} - \sqrt {2} a^{2} c \sqrt {x} \left (-\frac {1}{a^{15} c^{9}}\right )^{\frac {1}{8}} + x\right ) + 56 \, {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac {1}{a^{15} c^{9}}\right )^{\frac {1}{8}} \arctan \left (\sqrt {a^{4} c^{2} \left (-\frac {1}{a^{15} c^{9}}\right )^{\frac {1}{4}} + x} a^{13} c^{8} \left (-\frac {1}{a^{15} c^{9}}\right )^{\frac {7}{8}} - a^{13} c^{8} \sqrt {x} \left (-\frac {1}{a^{15} c^{9}}\right )^{\frac {7}{8}}\right ) + 14 \, {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac {1}{a^{15} c^{9}}\right )^{\frac {1}{8}} \log \left (a^{2} c \left (-\frac {1}{a^{15} c^{9}}\right )^{\frac {1}{8}} + \sqrt {x}\right ) - 14 \, {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac {1}{a^{15} c^{9}}\right )^{\frac {1}{8}} \log \left (-a^{2} c \left (-\frac {1}{a^{15} c^{9}}\right )^{\frac {1}{8}} + \sqrt {x}\right ) + 16 \, {\left (c x^{4} - 7 \, a\right )} \sqrt {x}}{1024 \, {\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)/(c*x^4+a)^3,x, algorithm="fricas")

[Out]

1/1024*(28*sqrt(2)*(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c)*(-1/(a^15*c^9))^(1/8)*arctan(sqrt(2)*sqrt(a^4*c^2*(-1/(
a^15*c^9))^(1/4) + sqrt(2)*a^2*c*sqrt(x)*(-1/(a^15*c^9))^(1/8) + x)*a^13*c^8*(-1/(a^15*c^9))^(7/8) - sqrt(2)*a
^13*c^8*sqrt(x)*(-1/(a^15*c^9))^(7/8) + 1) + 28*sqrt(2)*(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c)*(-1/(a^15*c^9))^(1
/8)*arctan(sqrt(2)*sqrt(a^4*c^2*(-1/(a^15*c^9))^(1/4) - sqrt(2)*a^2*c*sqrt(x)*(-1/(a^15*c^9))^(1/8) + x)*a^13*
c^8*(-1/(a^15*c^9))^(7/8) - sqrt(2)*a^13*c^8*sqrt(x)*(-1/(a^15*c^9))^(7/8) - 1) + 7*sqrt(2)*(a*c^3*x^8 + 2*a^2
*c^2*x^4 + a^3*c)*(-1/(a^15*c^9))^(1/8)*log(a^4*c^2*(-1/(a^15*c^9))^(1/4) + sqrt(2)*a^2*c*sqrt(x)*(-1/(a^15*c^
9))^(1/8) + x) - 7*sqrt(2)*(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c)*(-1/(a^15*c^9))^(1/8)*log(a^4*c^2*(-1/(a^15*c^9
))^(1/4) - sqrt(2)*a^2*c*sqrt(x)*(-1/(a^15*c^9))^(1/8) + x) + 56*(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c)*(-1/(a^15
*c^9))^(1/8)*arctan(sqrt(a^4*c^2*(-1/(a^15*c^9))^(1/4) + x)*a^13*c^8*(-1/(a^15*c^9))^(7/8) - a^13*c^8*sqrt(x)*
(-1/(a^15*c^9))^(7/8)) + 14*(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c)*(-1/(a^15*c^9))^(1/8)*log(a^2*c*(-1/(a^15*c^9)
)^(1/8) + sqrt(x)) - 14*(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c)*(-1/(a^15*c^9))^(1/8)*log(-a^2*c*(-1/(a^15*c^9))^(
1/8) + sqrt(x)) + 16*(c*x^4 - 7*a)*sqrt(x))/(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c)

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giac [B]  time = 0.73, size = 498, normalized size = 1.50 \[ \frac {7 \, \left (\frac {a}{c}\right )^{\frac {1}{8}} \arctan \left (\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{256 \, a^{2} c \sqrt {-2 \, \sqrt {2} + 4}} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {1}{8}} \arctan \left (-\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{256 \, a^{2} c \sqrt {-2 \, \sqrt {2} + 4}} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {1}{8}} \arctan \left (\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{256 \, a^{2} c \sqrt {2 \, \sqrt {2} + 4}} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {1}{8}} \arctan \left (-\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{256 \, a^{2} c \sqrt {2 \, \sqrt {2} + 4}} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {1}{8}} \log \left (\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{512 \, a^{2} c \sqrt {-2 \, \sqrt {2} + 4}} - \frac {7 \, \left (\frac {a}{c}\right )^{\frac {1}{8}} \log \left (-\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{512 \, a^{2} c \sqrt {-2 \, \sqrt {2} + 4}} + \frac {7 \, \left (\frac {a}{c}\right )^{\frac {1}{8}} \log \left (\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{512 \, a^{2} c \sqrt {2 \, \sqrt {2} + 4}} - \frac {7 \, \left (\frac {a}{c}\right )^{\frac {1}{8}} \log \left (-\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{512 \, a^{2} c \sqrt {2 \, \sqrt {2} + 4}} + \frac {c x^{\frac {9}{2}} - 7 \, a \sqrt {x}}{64 \, {\left (c x^{4} + a\right )}^{2} a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)/(c*x^4+a)^3,x, algorithm="giac")

[Out]

7/256*(a/c)^(1/8)*arctan((sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + 2*sqrt(x))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/(a^2*c*
sqrt(-2*sqrt(2) + 4)) + 7/256*(a/c)^(1/8)*arctan(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(sqrt(2) +
2)*(a/c)^(1/8)))/(a^2*c*sqrt(-2*sqrt(2) + 4)) + 7/256*(a/c)^(1/8)*arctan((sqrt(sqrt(2) + 2)*(a/c)^(1/8) + 2*s
qrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(a^2*c*sqrt(2*sqrt(2) + 4)) + 7/256*(a/c)^(1/8)*arctan(-(sqrt(sqrt(2
) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(a^2*c*sqrt(2*sqrt(2) + 4)) + 7/512*(a/c)^(1
/8)*log(sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^2*c*sqrt(-2*sqrt(2) + 4)) - 7/512*(a/c)^(1
/8)*log(-sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^2*c*sqrt(-2*sqrt(2) + 4)) + 7/512*(a/c)^(
1/8)*log(sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^2*c*sqrt(2*sqrt(2) + 4)) - 7/512*(a/c)^(
1/8)*log(-sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a^2*c*sqrt(2*sqrt(2) + 4)) + 1/64*(c*x^(9
/2) - 7*a*sqrt(x))/((c*x^4 + a)^2*a*c)

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maple [C]  time = 0.02, size = 61, normalized size = 0.18 \[ \frac {7 \ln \left (-\RootOf \left (c \,\textit {\_Z}^{8}+a \right )+\sqrt {x}\right )}{512 a \,c^{2} \RootOf \left (c \,\textit {\_Z}^{8}+a \right )^{7}}+\frac {\frac {x^{\frac {9}{2}}}{64 a}-\frac {7 \sqrt {x}}{64 c}}{\left (c \,x^{4}+a \right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)/(c*x^4+a)^3,x)

[Out]

2*(-7/128/c*x^(1/2)+1/128/a*x^(9/2))/(c*x^4+a)^2+7/512/c^2/a*sum(1/_R^7*ln(-_R+x^(1/2)),_R=RootOf(_Z^8*c+a))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {7 \, c x^{\frac {17}{2}} + 15 \, a x^{\frac {9}{2}}}{64 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )}} - 7 \, \int \frac {x^{\frac {7}{2}}}{128 \, {\left (a^{2} c x^{4} + a^{3}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)/(c*x^4+a)^3,x, algorithm="maxima")

[Out]

1/64*(7*c*x^(17/2) + 15*a*x^(9/2))/(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4) - 7*integrate(1/128*x^(7/2)/(a^2*c*x^4 +
a^3), x)

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mupad [B]  time = 1.10, size = 156, normalized size = 0.47 \[ \frac {\frac {x^{9/2}}{64\,a}-\frac {7\,\sqrt {x}}{64\,c}}{a^2+2\,a\,c\,x^4+c^2\,x^8}+\frac {7\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{256\,{\left (-a\right )}^{15/8}\,c^{9/8}}-\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,7{}\mathrm {i}}{256\,{\left (-a\right )}^{15/8}\,c^{9/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {7}{512}+\frac {7}{512}{}\mathrm {i}\right )}{{\left (-a\right )}^{15/8}\,c^{9/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (\frac {7}{512}-\frac {7}{512}{}\mathrm {i}\right )}{{\left (-a\right )}^{15/8}\,c^{9/8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)/(a + c*x^4)^3,x)

[Out]

(x^(9/2)/(64*a) - (7*x^(1/2))/(64*c))/(a^2 + c^2*x^8 + 2*a*c*x^4) + (7*atan((c^(1/8)*x^(1/2))/(-a)^(1/8)))/(25
6*(-a)^(15/8)*c^(9/8)) - (atan((c^(1/8)*x^(1/2)*1i)/(-a)^(1/8))*7i)/(256*(-a)^(15/8)*c^(9/8)) + (2^(1/2)*atan(
(2^(1/2)*c^(1/8)*x^(1/2)*(1/2 - 1i/2))/(-a)^(1/8))*(7/512 + 7i/512))/((-a)^(15/8)*c^(9/8)) + (2^(1/2)*atan((2^
(1/2)*c^(1/8)*x^(1/2)*(1/2 + 1i/2))/(-a)^(1/8))*(7/512 - 7i/512))/((-a)^(15/8)*c^(9/8))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)/(c*x**4+a)**3,x)

[Out]

Timed out

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