3.8.39 \(\int \frac {x^{7/2}}{a+c x^4} \, dx\) [739]

Optimal. Leaf size=297 \[ \frac {2 \sqrt {x}}{c}+\frac {\sqrt [8]{-a} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} c^{9/8}}-\frac {\sqrt [8]{-a} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} c^{9/8}}-\frac {\sqrt [8]{-a} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{9/8}}-\frac {\sqrt [8]{-a} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{9/8}}+\frac {\sqrt [8]{-a} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} c^{9/8}}-\frac {\sqrt [8]{-a} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} c^{9/8}} \]

[Out]

-1/2*(-a)^(1/8)*arctan(c^(1/8)*x^(1/2)/(-a)^(1/8))/c^(9/8)-1/2*(-a)^(1/8)*arctanh(c^(1/8)*x^(1/2)/(-a)^(1/8))/
c^(9/8)-1/4*(-a)^(1/8)*arctan(-1+c^(1/8)*2^(1/2)*x^(1/2)/(-a)^(1/8))/c^(9/8)*2^(1/2)-1/4*(-a)^(1/8)*arctan(1+c
^(1/8)*2^(1/2)*x^(1/2)/(-a)^(1/8))/c^(9/8)*2^(1/2)+1/8*(-a)^(1/8)*ln((-a)^(1/4)+c^(1/4)*x-(-a)^(1/8)*c^(1/8)*2
^(1/2)*x^(1/2))/c^(9/8)*2^(1/2)-1/8*(-a)^(1/8)*ln((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*2^(1/2)*x^(1/2))/c^(
9/8)*2^(1/2)+2*x^(1/2)/c

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Rubi [A]
time = 0.21, antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {327, 335, 220, 218, 214, 211, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {\sqrt [8]{-a} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} c^{9/8}}-\frac {\sqrt [8]{-a} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt {2} c^{9/8}}-\frac {\sqrt [8]{-a} \text {ArcTan}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{9/8}}+\frac {\sqrt [8]{-a} \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt {2} c^{9/8}}-\frac {\sqrt [8]{-a} \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt {2} c^{9/8}}-\frac {\sqrt [8]{-a} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{9/8}}+\frac {2 \sqrt {x}}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(7/2)/(a + c*x^4),x]

[Out]

(2*Sqrt[x])/c + ((-a)^(1/8)*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*Sqrt[2]*c^(9/8)) - ((-a)^(1/8
)*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*Sqrt[2]*c^(9/8)) - ((-a)^(1/8)*ArcTan[(c^(1/8)*Sqrt[x])
/(-a)^(1/8)])/(2*c^(9/8)) - ((-a)^(1/8)*ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*c^(9/8)) + ((-a)^(1/8)*Log[(
-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(4*Sqrt[2]*c^(9/8)) - ((-a)^(1/8)*Log[(-a)^(1/4)
+ Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(4*Sqrt[2]*c^(9/8))

Rule 210

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

Rule 211

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

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 217

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 218

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] &&  !Gt
Q[a/b, 0]

Rule 220

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 327

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*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 335

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 631

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 642

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 1176

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 1179

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}}{a+c x^4} \, dx &=\frac {2 \sqrt {x}}{c}-\frac {a \int \frac {1}{\sqrt {x} \left (a+c x^4\right )} \, dx}{c}\\ &=\frac {2 \sqrt {x}}{c}-\frac {(2 a) \text {Subst}\left (\int \frac {1}{a+c x^8} \, dx,x,\sqrt {x}\right )}{c}\\ &=\frac {2 \sqrt {x}}{c}-\frac {\sqrt {-a} \text {Subst}\left (\int \frac {1}{\sqrt {-a}-\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{c}-\frac {\sqrt {-a} \text {Subst}\left (\int \frac {1}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{c}\\ &=\frac {2 \sqrt {x}}{c}-\frac {\sqrt [4]{-a} \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{2 c}-\frac {\sqrt [4]{-a} \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{2 c}-\frac {\sqrt [4]{-a} \text {Subst}\left (\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{2 c}-\frac {\sqrt [4]{-a} \text {Subst}\left (\int \frac {\sqrt [4]{-a}+\sqrt [4]{c} x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{2 c}\\ &=\frac {2 \sqrt {x}}{c}-\frac {\sqrt [8]{-a} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{9/8}}-\frac {\sqrt [8]{-a} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{9/8}}-\frac {\sqrt [4]{-a} \text {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 )}{4 c^{5/4}}-\frac {\sqrt [4]{-a} \text {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 )}{4 c^{5/4}}+\frac {\sqrt [8]{-a} \text {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 )}{4 \sqrt {2} c^{9/8}}+\frac {\sqrt [8]{-a} \text {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 )}{4 \sqrt {2} c^{9/8}}\\ &=\frac {2 \sqrt {x}}{c}-\frac {\sqrt [8]{-a} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{9/8}}-\frac {\sqrt [8]{-a} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{9/8}}+\frac {\sqrt [8]{-a} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} c^{9/8}}-\frac {\sqrt [8]{-a} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} c^{9/8}}-\frac {\sqrt [8]{-a} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} c^{9/8}}+\frac {\sqrt [8]{-a} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} c^{9/8}}\\ &=\frac {2 \sqrt {x}}{c}+\frac {\sqrt [8]{-a} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} c^{9/8}}-\frac {\sqrt [8]{-a} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} c^{9/8}}-\frac {\sqrt [8]{-a} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{9/8}}-\frac {\sqrt [8]{-a} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{9/8}}+\frac {\sqrt [8]{-a} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} c^{9/8}}-\frac {\sqrt [8]{-a} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} c^{9/8}}\\ \end {align*}

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Mathematica [A]
time = 0.69, size = 266, normalized size = 0.90 \begin {gather*} \frac {8 \sqrt [8]{c} \sqrt {x}+\sqrt {2+\sqrt {2}} \sqrt [8]{a} \tan ^{-1}\left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )+\sqrt {2-\sqrt {2}} \sqrt [8]{a} \tan ^{-1}\left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )-\sqrt {2+\sqrt {2}} \sqrt [8]{a} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )-\sqrt {2-\sqrt {2}} \sqrt [8]{a} \tanh ^{-1}\left (\frac {\sqrt [8]{a} \sqrt [8]{c} \sqrt {-\left (\left (-2+\sqrt {2}\right ) x\right )}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{4 c^{9/8}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(7/2)/(a + c*x^4),x]

[Out]

(8*c^(1/8)*Sqrt[x] + Sqrt[2 + Sqrt[2]]*a^(1/8)*ArcTan[(Sqrt[1 - 1/Sqrt[2]]*(a^(1/4) - c^(1/4)*x))/(a^(1/8)*c^(
1/8)*Sqrt[x])] + Sqrt[2 - Sqrt[2]]*a^(1/8)*ArcTan[(Sqrt[1 + 1/Sqrt[2]]*(a^(1/4) - c^(1/4)*x))/(a^(1/8)*c^(1/8)
*Sqrt[x])] - Sqrt[2 + Sqrt[2]]*a^(1/8)*ArcTanh[(Sqrt[2 + Sqrt[2]]*a^(1/8)*c^(1/8)*Sqrt[x])/(a^(1/4) + c^(1/4)*
x)] - Sqrt[2 - Sqrt[2]]*a^(1/8)*ArcTanh[(a^(1/8)*c^(1/8)*Sqrt[-((-2 + Sqrt[2])*x)])/(a^(1/4) + c^(1/4)*x)])/(4
*c^(9/8))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.15, size = 39, normalized size = 0.13

method	result	size
derivativedivides	\(\frac {2 \sqrt {x}}{c}-\frac {a \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{7}}\right )}{4 c^{2}}\)	\(39\)
default	\(\frac {2 \sqrt {x}}{c}-\frac {a \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{7}}\right )}{4 c^{2}}\)	\(39\)
risch	\(\frac {2 \sqrt {x}}{c}-\frac {a \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{7}}\right )}{4 c^{2}}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)/(c*x^4+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^(7/2)/(c*x^4 + a), x)

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Fricas [A]
time = 0.39, size = 392, normalized size = 1.32 \begin {gather*} -\frac {4 \, \sqrt {2} c \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} \arctan \left (\frac {\sqrt {2} \sqrt {c^{2} \left (-\frac {a}{c^{9}}\right )^{\frac {1}{4}} + \sqrt {2} c \sqrt {x} \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} + x} c^{8} \left (-\frac {a}{c^{9}}\right )^{\frac {7}{8}} - \sqrt {2} c^{8} \sqrt {x} \left (-\frac {a}{c^{9}}\right )^{\frac {7}{8}} + a}{a}\right ) + 4 \, \sqrt {2} c \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} \arctan \left (\frac {\sqrt {2} \sqrt {c^{2} \left (-\frac {a}{c^{9}}\right )^{\frac {1}{4}} - \sqrt {2} c \sqrt {x} \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} + x} c^{8} \left (-\frac {a}{c^{9}}\right )^{\frac {7}{8}} - \sqrt {2} c^{8} \sqrt {x} \left (-\frac {a}{c^{9}}\right )^{\frac {7}{8}} - a}{a}\right ) + \sqrt {2} c \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} \log \left (c^{2} \left (-\frac {a}{c^{9}}\right )^{\frac {1}{4}} + \sqrt {2} c \sqrt {x} \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} + x\right ) - \sqrt {2} c \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} \log \left (c^{2} \left (-\frac {a}{c^{9}}\right )^{\frac {1}{4}} - \sqrt {2} c \sqrt {x} \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} + x\right ) + 8 \, c \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} \arctan \left (\frac {\sqrt {c^{2} \left (-\frac {a}{c^{9}}\right )^{\frac {1}{4}} + x} c^{8} \left (-\frac {a}{c^{9}}\right )^{\frac {7}{8}} - c^{8} \sqrt {x} \left (-\frac {a}{c^{9}}\right )^{\frac {7}{8}}}{a}\right ) + 2 \, c \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} \log \left (c \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} + \sqrt {x}\right ) - 2 \, c \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} \log \left (-c \left (-\frac {a}{c^{9}}\right )^{\frac {1}{8}} + \sqrt {x}\right ) - 16 \, \sqrt {x}}{8 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(4*sqrt(2)*c*(-a/c^9)^(1/8)*arctan((sqrt(2)*sqrt(c^2*(-a/c^9)^(1/4) + sqrt(2)*c*sqrt(x)*(-a/c^9)^(1/8) +
x)*c^8*(-a/c^9)^(7/8) - sqrt(2)*c^8*sqrt(x)*(-a/c^9)^(7/8) + a)/a) + 4*sqrt(2)*c*(-a/c^9)^(1/8)*arctan((sqrt(2
)*sqrt(c^2*(-a/c^9)^(1/4) - sqrt(2)*c*sqrt(x)*(-a/c^9)^(1/8) + x)*c^8*(-a/c^9)^(7/8) - sqrt(2)*c^8*sqrt(x)*(-a
/c^9)^(7/8) - a)/a) + sqrt(2)*c*(-a/c^9)^(1/8)*log(c^2*(-a/c^9)^(1/4) + sqrt(2)*c*sqrt(x)*(-a/c^9)^(1/8) + x)
- sqrt(2)*c*(-a/c^9)^(1/8)*log(c^2*(-a/c^9)^(1/4) - sqrt(2)*c*sqrt(x)*(-a/c^9)^(1/8) + x) + 8*c*(-a/c^9)^(1/8)
*arctan((sqrt(c^2*(-a/c^9)^(1/4) + x)*c^8*(-a/c^9)^(7/8) - c^8*sqrt(x)*(-a/c^9)^(7/8))/a) + 2*c*(-a/c^9)^(1/8)
*log(c*(-a/c^9)^(1/8) + sqrt(x)) - 2*c*(-a/c^9)^(1/8)*log(-c*(-a/c^9)^(1/8) + sqrt(x)) - 16*sqrt(x))/c

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Sympy [A]
time = 28.14, size = 296, normalized size = 1.00 \begin {gather*} \begin {cases} \tilde {\infty } \sqrt {x} & \text {for}\: a = 0 \wedge c = 0 \\\frac {2 x^{\frac {9}{2}}}{9 a} & \text {for}\: c = 0 \\\frac {2 \sqrt {x}}{c} & \text {for}\: a = 0 \\\frac {2 \sqrt {x}}{c} + \frac {\sqrt [8]{- \frac {a}{c}} \log {\left (\sqrt {x} - \sqrt [8]{- \frac {a}{c}} \right )}}{4 c} - \frac {\sqrt [8]{- \frac {a}{c}} \log {\left (\sqrt {x} + \sqrt [8]{- \frac {a}{c}} \right )}}{4 c} + \frac {\sqrt {2} \sqrt [8]{- \frac {a}{c}} \log {\left (- 4 \sqrt {2} \sqrt {x} \sqrt [8]{- \frac {a}{c}} + 4 x + 4 \sqrt [4]{- \frac {a}{c}} \right )}}{8 c} - \frac {\sqrt {2} \sqrt [8]{- \frac {a}{c}} \log {\left (4 \sqrt {2} \sqrt {x} \sqrt [8]{- \frac {a}{c}} + 4 x + 4 \sqrt [4]{- \frac {a}{c}} \right )}}{8 c} - \frac {\sqrt [8]{- \frac {a}{c}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} \right )}}{2 c} - \frac {\sqrt {2} \sqrt [8]{- \frac {a}{c}} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} - 1 \right )}}{4 c} - \frac {\sqrt {2} \sqrt [8]{- \frac {a}{c}} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} + 1 \right )}}{4 c} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*sqrt(x), Eq(a, 0) & Eq(c, 0)), (2*x**(9/2)/(9*a), Eq(c, 0)), (2*sqrt(x)/c, Eq(a, 0)), (2*sqrt(x
)/c + (-a/c)**(1/8)*log(sqrt(x) - (-a/c)**(1/8))/(4*c) - (-a/c)**(1/8)*log(sqrt(x) + (-a/c)**(1/8))/(4*c) + sq
rt(2)*(-a/c)**(1/8)*log(-4*sqrt(2)*sqrt(x)*(-a/c)**(1/8) + 4*x + 4*(-a/c)**(1/4))/(8*c) - sqrt(2)*(-a/c)**(1/8
)*log(4*sqrt(2)*sqrt(x)*(-a/c)**(1/8) + 4*x + 4*(-a/c)**(1/4))/(8*c) - (-a/c)**(1/8)*atan(sqrt(x)/(-a/c)**(1/8
))/(2*c) - sqrt(2)*(-a/c)**(1/8)*atan(sqrt(2)*sqrt(x)/(-a/c)**(1/8) - 1)/(4*c) - sqrt(2)*(-a/c)**(1/8)*atan(sq
rt(2)*sqrt(x)/(-a/c)**(1/8) + 1)/(4*c), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 453 vs. \(2 (198) = 396\).
time = 1.04, size = 453, normalized size = 1.53 \begin {gather*} -\frac {\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 )}{2 \, c \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\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 )}{2 \, c \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\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 )}{2 \, c \sqrt {2 \, \sqrt {2} + 4}} - \frac {\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 )}{2 \, c \sqrt {2 \, \sqrt {2} + 4}} - \frac {\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 )}{4 \, c \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\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 )}{4 \, c \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\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 )}{4 \, c \sqrt {2 \, \sqrt {2} + 4}} + \frac {\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 )}{4 \, c \sqrt {2 \, \sqrt {2} + 4}} + \frac {2 \, \sqrt {x}}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a/c)^(1/8)*arctan((sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + 2*sqrt(x))/(sqrt(sqrt(2) + 2)*(a/c)^(1/8)))/(c*sqrt(
-2*sqrt(2) + 4)) - 1/2*(a/c)^(1/8)*arctan(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(sqrt(2) + 2)*(a/
c)^(1/8)))/(c*sqrt(-2*sqrt(2) + 4)) - 1/2*(a/c)^(1/8)*arctan((sqrt(sqrt(2) + 2)*(a/c)^(1/8) + 2*sqrt(x))/(sqrt
(-sqrt(2) + 2)*(a/c)^(1/8)))/(c*sqrt(2*sqrt(2) + 4)) - 1/2*(a/c)^(1/8)*arctan(-(sqrt(sqrt(2) + 2)*(a/c)^(1/8)
- 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(c*sqrt(2*sqrt(2) + 4)) - 1/4*(a/c)^(1/8)*log(sqrt(x)*sqrt(sqrt
(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(c*sqrt(-2*sqrt(2) + 4)) + 1/4*(a/c)^(1/8)*log(-sqrt(x)*sqrt(sqrt(2) +
2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(c*sqrt(-2*sqrt(2) + 4)) - 1/4*(a/c)^(1/8)*log(sqrt(x)*sqrt(-sqrt(2) + 2)*(
a/c)^(1/8) + x + (a/c)^(1/4))/(c*sqrt(2*sqrt(2) + 4)) + 1/4*(a/c)^(1/8)*log(-sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^
(1/8) + x + (a/c)^(1/4))/(c*sqrt(2*sqrt(2) + 4)) + 2*sqrt(x)/c

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Mupad [B]
time = 1.10, size = 126, normalized size = 0.42 \begin {gather*} \frac {2\,\sqrt {x}}{c}-\frac {{\left (-a\right )}^{1/8}\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{2\,c^{9/8}}+\frac {{\left (-a\right )}^{1/8}\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,1{}\mathrm {i}}{2\,c^{9/8}}+\frac {\sqrt {2}\,{\left (-a\right )}^{1/8}\,\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 {1}{4}-\frac {1}{4}{}\mathrm {i}\right )}{c^{9/8}}+\frac {\sqrt {2}\,{\left (-a\right )}^{1/8}\,\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 {1}{4}+\frac {1}{4}{}\mathrm {i}\right )}{c^{9/8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)/(a + c*x^4),x)

[Out]

(2*x^(1/2))/c - ((-a)^(1/8)*atan((c^(1/8)*x^(1/2))/(-a)^(1/8)))/(2*c^(9/8)) + ((-a)^(1/8)*atan((c^(1/8)*x^(1/2
)*1i)/(-a)^(1/8))*1i)/(2*c^(9/8)) - (2^(1/2)*(-a)^(1/8)*atan((2^(1/2)*c^(1/8)*x^(1/2)*(1/2 - 1i/2))/(-a)^(1/8)
)*(1/4 + 1i/4))/c^(9/8) - (2^(1/2)*(-a)^(1/8)*atan((2^(1/2)*c^(1/8)*x^(1/2)*(1/2 + 1i/2))/(-a)^(1/8))*(1/4 - 1
i/4))/c^(9/8)

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