3.8.38 \(\int \frac {x^{9/2}}{a+c x^4} \, dx\) [738]

Optimal. Leaf size=299 \[ \frac {2 x^{3/2}}{3 c}+\frac {(-a)^{3/8} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} c^{11/8}}-\frac {(-a)^{3/8} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} c^{11/8}}+\frac {(-a)^{3/8} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac {(-a)^{3/8} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac {(-a)^{3/8} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} c^{11/8}}+\frac {(-a)^{3/8} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} c^{11/8}} \]

[Out]

2/3*x^(3/2)/c+1/2*(-a)^(3/8)*arctan(c^(1/8)*x^(1/2)/(-a)^(1/8))/c^(11/8)-1/2*(-a)^(3/8)*arctanh(c^(1/8)*x^(1/2
)/(-a)^(1/8))/c^(11/8)-1/4*(-a)^(3/8)*arctan(-1+c^(1/8)*2^(1/2)*x^(1/2)/(-a)^(1/8))/c^(11/8)*2^(1/2)-1/4*(-a)^
(3/8)*arctan(1+c^(1/8)*2^(1/2)*x^(1/2)/(-a)^(1/8))/c^(11/8)*2^(1/2)-1/8*(-a)^(3/8)*ln((-a)^(1/4)+c^(1/4)*x-(-a
)^(1/8)*c^(1/8)*2^(1/2)*x^(1/2))/c^(11/8)*2^(1/2)+1/8*(-a)^(3/8)*ln((-a)^(1/4)+c^(1/4)*x+(-a)^(1/8)*c^(1/8)*2^
(1/2)*x^(1/2))/c^(11/8)*2^(1/2)

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Rubi [A]
time = 0.26, antiderivative size = 299, 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, 306, 303, 1176, 631, 210, 1179, 642, 304, 211, 214} \begin {gather*} \frac {(-a)^{3/8} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} c^{11/8}}-\frac {(-a)^{3/8} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{2 \sqrt {2} c^{11/8}}+\frac {(-a)^{3/8} \text {ArcTan}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac {(-a)^{3/8} \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt {2} c^{11/8}}+\frac {(-a)^{3/8} \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{4 \sqrt {2} c^{11/8}}-\frac {(-a)^{3/8} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}+\frac {2 x^{3/2}}{3 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(9/2)/(a + c*x^4),x]

[Out]

(2*x^(3/2))/(3*c) + ((-a)^(3/8)*ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*Sqrt[2]*c^(11/8)) - ((-a)
^(3/8)*ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*Sqrt[2]*c^(11/8)) + ((-a)^(3/8)*ArcTan[(c^(1/8)*Sq
rt[x])/(-a)^(1/8)])/(2*c^(11/8)) - ((-a)^(3/8)*ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)])/(2*c^(11/8)) - ((-a)^(3/
8)*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(4*Sqrt[2]*c^(11/8)) + ((-a)^(3/8)*Log[(-
a)^(1/4) + Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x])/(4*Sqrt[2]*c^(11/8))

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 303

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

Rule 304

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

Rule 306

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b
, 2]]}, Dist[r/(2*a), Int[x^m/(r + s*x^(n/2)), x], x] + Dist[r/(2*a), Int[x^m/(r - s*x^(n/2)), x], x]] /; Free
Q[{a, b}, x] && IGtQ[n/4, 0] && IGtQ[m, 0] && LtQ[m, n/2] &&  !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^{9/2}}{a+c x^4} \, dx &=\frac {2 x^{3/2}}{3 c}-\frac {a \int \frac {\sqrt {x}}{a+c x^4} \, dx}{c}\\ &=\frac {2 x^{3/2}}{3 c}-\frac {(2 a) \text {Subst}\left (\int \frac {x^2}{a+c x^8} \, dx,x,\sqrt {x}\right )}{c}\\ &=\frac {2 x^{3/2}}{3 c}-\frac {\sqrt {-a} \text {Subst}\left (\int \frac {x^2}{\sqrt {-a}-\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{c}-\frac {\sqrt {-a} \text {Subst}\left (\int \frac {x^2}{\sqrt {-a}+\sqrt {c} x^4} \, dx,x,\sqrt {x}\right )}{c}\\ &=\frac {2 x^{3/2}}{3 c}-\frac {\sqrt {-a} \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{2 c^{5/4}}+\frac {\sqrt {-a} \text {Subst}\left (\int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{c} x^2} \, dx,x,\sqrt {x}\right )}{2 c^{5/4}}+\frac {\sqrt {-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^{5/4}}-\frac {\sqrt {-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^{5/4}}\\ &=\frac {2 x^{3/2}}{3 c}+\frac {(-a)^{3/8} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac {(-a)^{3/8} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac {\sqrt {-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^{3/2}}-\frac {\sqrt {-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^{3/2}}-\frac {(-a)^{3/8} \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^{11/8}}-\frac {(-a)^{3/8} \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^{11/8}}\\ &=\frac {2 x^{3/2}}{3 c}+\frac {(-a)^{3/8} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac {(-a)^{3/8} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac {(-a)^{3/8} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} c^{11/8}}+\frac {(-a)^{3/8} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} c^{11/8}}-\frac {(-a)^{3/8} \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^{11/8}}+\frac {(-a)^{3/8} \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^{11/8}}\\ &=\frac {2 x^{3/2}}{3 c}+\frac {(-a)^{3/8} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} c^{11/8}}-\frac {(-a)^{3/8} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} c^{11/8}}+\frac {(-a)^{3/8} \tan ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac {(-a)^{3/8} \tanh ^{-1}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 c^{11/8}}-\frac {(-a)^{3/8} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} c^{11/8}}+\frac {(-a)^{3/8} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{c} x\right )}{4 \sqrt {2} c^{11/8}}\\ \end {align*}

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Mathematica [A]
time = 0.79, size = 268, normalized size = 0.90 \begin {gather*} \frac {8 c^{3/8} x^{3/2}-3 \sqrt {2-\sqrt {2}} a^{3/8} \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 )+3 \sqrt {2+\sqrt {2}} a^{3/8} \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 )-3 \sqrt {2-\sqrt {2}} a^{3/8} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )+3 \sqrt {2+\sqrt {2}} a^{3/8} \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 )}{12 c^{11/8}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(9/2)/(a + c*x^4),x]

[Out]

(8*c^(3/8)*x^(3/2) - 3*Sqrt[2 - Sqrt[2]]*a^(3/8)*ArcTan[(Sqrt[1 - 1/Sqrt[2]]*(a^(1/4) - c^(1/4)*x))/(a^(1/8)*c
^(1/8)*Sqrt[x])] + 3*Sqrt[2 + Sqrt[2]]*a^(3/8)*ArcTan[(Sqrt[1 + 1/Sqrt[2]]*(a^(1/4) - c^(1/4)*x))/(a^(1/8)*c^(
1/8)*Sqrt[x])] - 3*Sqrt[2 - Sqrt[2]]*a^(3/8)*ArcTanh[(Sqrt[2 + Sqrt[2]]*a^(1/8)*c^(1/8)*Sqrt[x])/(a^(1/4) + c^
(1/4)*x)] + 3*Sqrt[2 + Sqrt[2]]*a^(3/8)*ArcTanh[(a^(1/8)*c^(1/8)*Sqrt[-((-2 + Sqrt[2])*x)])/(a^(1/4) + c^(1/4)
*x)])/(12*c^(11/8))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.16, size = 39, normalized size = 0.13

method	result	size
derivativedivides	\(\frac {2 x^{\frac {3}{2}}}{3 c}-\frac {a \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{5}}\right )}{4 c^{2}}\)	\(39\)
default	\(\frac {2 x^{\frac {3}{2}}}{3 c}-\frac {a \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{5}}\right )}{4 c^{2}}\)	\(39\)
risch	\(\frac {2 x^{\frac {3}{2}}}{3 c}-\frac {a \left (\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{5}}\right )}{4 c^{2}}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(9/2)/(c*x^4+a),x,method=_RETURNVERBOSE)

[Out]

2/3*x^(3/2)/c-1/4/c^2*a*sum(1/_R^5*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^(9/2)/(c*x^4+a),x, algorithm="maxima")

[Out]

-a*integrate(sqrt(x)/(c^2*x^4 + a*c), x) + 2/3*x^(3/2)/c

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 491 vs. \(2 (198) = 396\).
time = 0.40, size = 491, normalized size = 1.64 \begin {gather*} \frac {12 \, \sqrt {2} c \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {1}{8}} \arctan \left (-\frac {\sqrt {2} a c^{7} \sqrt {x} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {5}{8}} - \sqrt {2} \sqrt {c^{8} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {3}{4}} + \sqrt {2} a c^{4} \sqrt {x} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {3}{8}} + a^{2} x} c^{7} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {5}{8}} - a^{3}}{a^{3}}\right ) + 12 \, \sqrt {2} c \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {1}{8}} \arctan \left (-\frac {\sqrt {2} a c^{7} \sqrt {x} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {5}{8}} - \sqrt {2} \sqrt {c^{8} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {3}{4}} - \sqrt {2} a c^{4} \sqrt {x} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {3}{8}} + a^{2} x} c^{7} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {5}{8}} + a^{3}}{a^{3}}\right ) - 3 \, \sqrt {2} c \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {1}{8}} \log \left (c^{8} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {3}{4}} + \sqrt {2} a c^{4} \sqrt {x} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {3}{8}} + a^{2} x\right ) + 3 \, \sqrt {2} c \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {1}{8}} \log \left (c^{8} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {3}{4}} - \sqrt {2} a c^{4} \sqrt {x} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {3}{8}} + a^{2} x\right ) - 24 \, c \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {1}{8}} \arctan \left (-\frac {a c^{7} \sqrt {x} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {5}{8}} - \sqrt {c^{8} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {3}{4}} + a^{2} x} c^{7} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {5}{8}}}{a^{3}}\right ) + 6 \, c \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {1}{8}} \log \left (c^{4} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {3}{8}} + a \sqrt {x}\right ) - 6 \, c \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {1}{8}} \log \left (-c^{4} \left (-\frac {a^{3}}{c^{11}}\right )^{\frac {3}{8}} + a \sqrt {x}\right ) + 16 \, x^{\frac {3}{2}}}{24 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(9/2)/(c*x^4+a),x, algorithm="fricas")

[Out]

1/24*(12*sqrt(2)*c*(-a^3/c^11)^(1/8)*arctan(-(sqrt(2)*a*c^7*sqrt(x)*(-a^3/c^11)^(5/8) - sqrt(2)*sqrt(c^8*(-a^3
/c^11)^(3/4) + sqrt(2)*a*c^4*sqrt(x)*(-a^3/c^11)^(3/8) + a^2*x)*c^7*(-a^3/c^11)^(5/8) - a^3)/a^3) + 12*sqrt(2)
*c*(-a^3/c^11)^(1/8)*arctan(-(sqrt(2)*a*c^7*sqrt(x)*(-a^3/c^11)^(5/8) - sqrt(2)*sqrt(c^8*(-a^3/c^11)^(3/4) - s
qrt(2)*a*c^4*sqrt(x)*(-a^3/c^11)^(3/8) + a^2*x)*c^7*(-a^3/c^11)^(5/8) + a^3)/a^3) - 3*sqrt(2)*c*(-a^3/c^11)^(1
/8)*log(c^8*(-a^3/c^11)^(3/4) + sqrt(2)*a*c^4*sqrt(x)*(-a^3/c^11)^(3/8) + a^2*x) + 3*sqrt(2)*c*(-a^3/c^11)^(1/
8)*log(c^8*(-a^3/c^11)^(3/4) - sqrt(2)*a*c^4*sqrt(x)*(-a^3/c^11)^(3/8) + a^2*x) - 24*c*(-a^3/c^11)^(1/8)*arcta
n(-(a*c^7*sqrt(x)*(-a^3/c^11)^(5/8) - sqrt(c^8*(-a^3/c^11)^(3/4) + a^2*x)*c^7*(-a^3/c^11)^(5/8))/a^3) + 6*c*(-
a^3/c^11)^(1/8)*log(c^4*(-a^3/c^11)^(3/8) + a*sqrt(x)) - 6*c*(-a^3/c^11)^(1/8)*log(-c^4*(-a^3/c^11)^(3/8) + a*
sqrt(x)) + 16*x^(3/2))/c

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Sympy [A]
time = 48.65, size = 299, normalized size = 1.00 \begin {gather*} \begin {cases} \tilde {\infty } x^{\frac {3}{2}} & \text {for}\: a = 0 \wedge c = 0 \\\frac {2 x^{\frac {11}{2}}}{11 a} & \text {for}\: c = 0 \\\frac {2 x^{\frac {3}{2}}}{3 c} & \text {for}\: a = 0 \\\frac {2 x^{\frac {3}{2}}}{3 c} + \frac {\left (- \frac {a}{c}\right )^{\frac {3}{8}} \log {\left (\sqrt {x} - \sqrt [8]{- \frac {a}{c}} \right )}}{4 c} - \frac {\left (- \frac {a}{c}\right )^{\frac {3}{8}} \log {\left (\sqrt {x} + \sqrt [8]{- \frac {a}{c}} \right )}}{4 c} - \frac {\sqrt {2} \left (- \frac {a}{c}\right )^{\frac {3}{8}} \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} \left (- \frac {a}{c}\right )^{\frac {3}{8}} \log {\left (4 \sqrt {2} \sqrt {x} \sqrt [8]{- \frac {a}{c}} + 4 x + 4 \sqrt [4]{- \frac {a}{c}} \right )}}{8 c} + \frac {\left (- \frac {a}{c}\right )^{\frac {3}{8}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} \right )}}{2 c} - \frac {\sqrt {2} \left (- \frac {a}{c}\right )^{\frac {3}{8}} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} - 1 \right )}}{4 c} - \frac {\sqrt {2} \left (- \frac {a}{c}\right )^{\frac {3}{8}} \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**(9/2)/(c*x**4+a),x)

[Out]

Piecewise((zoo*x**(3/2), Eq(a, 0) & Eq(c, 0)), (2*x**(11/2)/(11*a), Eq(c, 0)), (2*x**(3/2)/(3*c), Eq(a, 0)), (
2*x**(3/2)/(3*c) + (-a/c)**(3/8)*log(sqrt(x) - (-a/c)**(1/8))/(4*c) - (-a/c)**(3/8)*log(sqrt(x) + (-a/c)**(1/8
))/(4*c) - sqrt(2)*(-a/c)**(3/8)*log(-4*sqrt(2)*sqrt(x)*(-a/c)**(1/8) + 4*x + 4*(-a/c)**(1/4))/(8*c) + sqrt(2)
*(-a/c)**(3/8)*log(4*sqrt(2)*sqrt(x)*(-a/c)**(1/8) + 4*x + 4*(-a/c)**(1/4))/(8*c) + (-a/c)**(3/8)*atan(sqrt(x)
/(-a/c)**(1/8))/(2*c) - sqrt(2)*(-a/c)**(3/8)*atan(sqrt(2)*sqrt(x)/(-a/c)**(1/8) - 1)/(4*c) - sqrt(2)*(-a/c)**
(3/8)*atan(sqrt(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 = 1.43, size = 453, normalized size = 1.52 \begin {gather*} \frac {2 \, x^{\frac {3}{2}}}{3 \, c} + \frac {\left (\frac {a}{c}\right )^{\frac {3}{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 {3}{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 {3}{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 {3}{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 {3}{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 {3}{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 {3}{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 {3}{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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(9/2)/(c*x^4+a),x, algorithm="giac")

[Out]

2/3*x^(3/2)/c + 1/2*(a/c)^(3/8)*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)^(3/8)*arctan(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(sq
rt(2) + 2)*(a/c)^(1/8)))/(c*sqrt(2*sqrt(2) + 4)) - 1/2*(a/c)^(3/8)*arctan((sqrt(sqrt(2) + 2)*(a/c)^(1/8) + 2*s
qrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(c*sqrt(-2*sqrt(2) + 4)) - 1/2*(a/c)^(3/8)*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)^(3/8)*log(sq
rt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(c*sqrt(2*sqrt(2) + 4)) + 1/4*(a/c)^(3/8)*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)^(3/8)*log(sqrt(x)*sqrt(-s
qrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(c*sqrt(-2*sqrt(2) + 4)) - 1/4*(a/c)^(3/8)*log(-sqrt(x)*sqrt(-sqrt(
2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(c*sqrt(-2*sqrt(2) + 4))

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Mupad [B]
time = 1.11, size = 126, normalized size = 0.42 \begin {gather*} \frac {2\,x^{3/2}}{3\,c}+\frac {{\left (-a\right )}^{3/8}\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{2\,c^{11/8}}+\frac {{\left (-a\right )}^{3/8}\,\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,1{}\mathrm {i}}{2\,c^{11/8}}+\frac {\sqrt {2}\,{\left (-a\right )}^{3/8}\,\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^{11/8}}+\frac {\sqrt {2}\,{\left (-a\right )}^{3/8}\,\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^{11/8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(9/2)/(a + c*x^4),x)

[Out]

(2*x^(3/2))/(3*c) + ((-a)^(3/8)*atan((c^(1/8)*x^(1/2))/(-a)^(1/8)))/(2*c^(11/8)) + ((-a)^(3/8)*atan((c^(1/8)*x
^(1/2)*1i)/(-a)^(1/8))*1i)/(2*c^(11/8)) - (2^(1/2)*(-a)^(3/8)*atan((2^(1/2)*c^(1/8)*x^(1/2)*(1/2 - 1i/2))/(-a)
^(1/8))*(1/4 - 1i/4))/c^(11/8) - (2^(1/2)*(-a)^(3/8)*atan((2^(1/2)*c^(1/8)*x^(1/2)*(1/2 + 1i/2))/(-a)^(1/8))*(
1/4 + 1i/4))/c^(11/8)

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