3.27.11 \(\int \frac {\sqrt {x+x^4} (b+a x^6)}{-d+c x^6} \, dx\)

Optimal. Leaf size=227 \[ \frac {\sqrt {-\left (\sqrt {d} \left (\sqrt {c}+\sqrt {d}\right )\right )} (a d+b c) \tan ^{-1}\left (\frac {x \sqrt {x^4+x} \sqrt {-\sqrt {c} \sqrt {d}-d}}{\sqrt {d} (x+1) \left (x^2-x+1\right )}\right )}{3 c^{3/2} d}-\frac {\sqrt {\sqrt {d} \left (\sqrt {c}-\sqrt {d}\right )} (a d+b c) \tan ^{-1}\left (\frac {x \sqrt {x^4+x} \sqrt {\sqrt {c} \sqrt {d}-d}}{\sqrt {d} (x+1) \left (x^2-x+1\right )}\right )}{3 c^{3/2} d}+\frac {a \sqrt {x^4+x} x}{3 c}+\frac {a \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right )}{3 c} \]

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Rubi [A]  time = 0.74, antiderivative size = 250, normalized size of antiderivative = 1.10, number of steps used = 15, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {2056, 6715, 1693, 195, 215, 1175, 402, 377, 208, 205} \begin {gather*} -\frac {\sqrt {x^4+x} \sqrt {\sqrt {c}-\sqrt {d}} (a d+b c) \tan ^{-1}\left (\frac {x^{3/2} \sqrt {\sqrt {c}-\sqrt {d}}}{\sqrt [4]{d} \sqrt {x^3+1}}\right )}{3 c^{3/2} d^{3/4} \sqrt {x^3+1} \sqrt {x}}-\frac {\sqrt {x^4+x} \sqrt {\sqrt {c}+\sqrt {d}} (a d+b c) \tanh ^{-1}\left (\frac {x^{3/2} \sqrt {\sqrt {c}+\sqrt {d}}}{\sqrt [4]{d} \sqrt {x^3+1}}\right )}{3 c^{3/2} d^{3/4} \sqrt {x^3+1} \sqrt {x}}+\frac {a \sqrt {x^4+x} x}{3 c}+\frac {a \sqrt {x^4+x} \sinh ^{-1}\left (x^{3/2}\right )}{3 c \sqrt {x^3+1} \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[x + x^4]*(b + a*x^6))/(-d + c*x^6),x]

[Out]

(a*x*Sqrt[x + x^4])/(3*c) + (a*Sqrt[x + x^4]*ArcSinh[x^(3/2)])/(3*c*Sqrt[x]*Sqrt[1 + x^3]) - (Sqrt[Sqrt[c] - S
qrt[d]]*(b*c + a*d)*Sqrt[x + x^4]*ArcTan[(Sqrt[Sqrt[c] - Sqrt[d]]*x^(3/2))/(d^(1/4)*Sqrt[1 + x^3])])/(3*c^(3/2
)*d^(3/4)*Sqrt[x]*Sqrt[1 + x^3]) - (Sqrt[Sqrt[c] + Sqrt[d]]*(b*c + a*d)*Sqrt[x + x^4]*ArcTanh[(Sqrt[Sqrt[c] +
Sqrt[d]]*x^(3/2))/(d^(1/4)*Sqrt[1 + x^3])])/(3*c^(3/2)*d^(3/4)*Sqrt[x]*Sqrt[1 + x^3])

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 215

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

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Dist[b/d, Int[(a + b*x^2)^(p - 1), x], x]
- Dist[(b*c - a*d)/d, Int[(a + b*x^2)^(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d,
0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4])

Rule 1175

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{r = Rt[-(a*c), 2]}, -Dist[c/(2*r), In
t[(d + e*x^2)^q/(r - c*x^2), x], x] - Dist[c/(2*r), Int[(d + e*x^2)^q/(r + c*x^2), x], x]] /; FreeQ[{a, c, d,
e, q}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[q]

Rule 1693

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(d + e*
x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a, c, d, e, q}, x] && PolyQ[Px, x^2] && NeQ[c*d^2 + a*e^2, 0] && Intege
rQ[p]

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rubi steps

\begin {align*} \int \frac {\sqrt {x+x^4} \left (b+a x^6\right )}{-d+c x^6} \, dx &=\frac {\sqrt {x+x^4} \int \frac {\sqrt {x} \sqrt {1+x^3} \left (b+a x^6\right )}{-d+c x^6} \, dx}{\sqrt {x} \sqrt {1+x^3}}\\ &=\frac {\left (2 \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2} \left (b+a x^4\right )}{-d+c x^4} \, dx,x,x^{3/2}\right )}{3 \sqrt {x} \sqrt {1+x^3}}\\ &=\frac {\left (2 \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {a \sqrt {1+x^2}}{c}+\frac {(b c+a d) \sqrt {1+x^2}}{c \left (-d+c x^4\right )}\right ) \, dx,x,x^{3/2}\right )}{3 \sqrt {x} \sqrt {1+x^3}}\\ &=\frac {\left (2 a \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \sqrt {1+x^2} \, dx,x,x^{3/2}\right )}{3 c \sqrt {x} \sqrt {1+x^3}}+\frac {\left (2 (b c+a d) \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{-d+c x^4} \, dx,x,x^{3/2}\right )}{3 c \sqrt {x} \sqrt {1+x^3}}\\ &=\frac {a x \sqrt {x+x^4}}{3 c}+\frac {\left (a \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^{3/2}\right )}{3 c \sqrt {x} \sqrt {1+x^3}}-\frac {\left ((b c+a d) \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\sqrt {c} \sqrt {d}-c x^2} \, dx,x,x^{3/2}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {x} \sqrt {1+x^3}}-\frac {\left ((b c+a d) \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^2}}{\sqrt {c} \sqrt {d}+c x^2} \, dx,x,x^{3/2}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {x} \sqrt {1+x^3}}\\ &=\frac {a x \sqrt {x+x^4}}{3 c}+\frac {a \sqrt {x+x^4} \sinh ^{-1}\left (x^{3/2}\right )}{3 c \sqrt {x} \sqrt {1+x^3}}+\frac {\left (\left (-1+\frac {\sqrt {d}}{\sqrt {c}}\right ) (b c+a d) \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2} \left (\sqrt {c} \sqrt {d}+c x^2\right )} \, dx,x,x^{3/2}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {x} \sqrt {1+x^3}}-\frac {\left (\left (1+\frac {\sqrt {d}}{\sqrt {c}}\right ) (b c+a d) \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2} \left (\sqrt {c} \sqrt {d}-c x^2\right )} \, dx,x,x^{3/2}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {x} \sqrt {1+x^3}}\\ &=\frac {a x \sqrt {x+x^4}}{3 c}+\frac {a \sqrt {x+x^4} \sinh ^{-1}\left (x^{3/2}\right )}{3 c \sqrt {x} \sqrt {1+x^3}}+\frac {\left (\left (-1+\frac {\sqrt {d}}{\sqrt {c}}\right ) (b c+a d) \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c} \sqrt {d}-\left (-c+\sqrt {c} \sqrt {d}\right ) x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {1+x^3}}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {x} \sqrt {1+x^3}}-\frac {\left (\left (1+\frac {\sqrt {d}}{\sqrt {c}}\right ) (b c+a d) \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c} \sqrt {d}-\left (c+\sqrt {c} \sqrt {d}\right ) x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {1+x^3}}\right )}{3 \sqrt {c} \sqrt {d} \sqrt {x} \sqrt {1+x^3}}\\ &=\frac {a x \sqrt {x+x^4}}{3 c}+\frac {a \sqrt {x+x^4} \sinh ^{-1}\left (x^{3/2}\right )}{3 c \sqrt {x} \sqrt {1+x^3}}-\frac {\sqrt {\sqrt {c}-\sqrt {d}} (b c+a d) \sqrt {x+x^4} \tan ^{-1}\left (\frac {\sqrt {\sqrt {c}-\sqrt {d}} x^{3/2}}{\sqrt [4]{d} \sqrt {1+x^3}}\right )}{3 c^{3/2} d^{3/4} \sqrt {x} \sqrt {1+x^3}}-\frac {\sqrt {\sqrt {c}+\sqrt {d}} (b c+a d) \sqrt {x+x^4} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c}+\sqrt {d}} x^{3/2}}{\sqrt [4]{d} \sqrt {1+x^3}}\right )}{3 c^{3/2} d^{3/4} \sqrt {x} \sqrt {1+x^3}}\\ \end {align*}

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Mathematica [C]  time = 0.79, size = 441, normalized size = 1.94 \begin {gather*} \frac {\sqrt {x^4+x} \left (-\left (x^3+1\right ) (a d+b c) \left (\sqrt [4]{d} \sinh ^{-1}\left (x^{3/2}\right )+\sqrt {\sqrt {c}-\sqrt {d}} \tan ^{-1}\left (\frac {-\sqrt [4]{d} x^{3/2}+i \sqrt [4]{c}}{\sqrt {x^3+1} \sqrt {\sqrt {c}-\sqrt {d}}}\right )\right )-\left (x^3+1\right ) (a d+b c) \left (\sqrt [4]{d} \sinh ^{-1}\left (x^{3/2}\right )-\sqrt {\sqrt {c}-\sqrt {d}} \tan ^{-1}\left (\frac {\sqrt [4]{d} x^{3/2}+i \sqrt [4]{c}}{\sqrt {x^3+1} \sqrt {\sqrt {c}-\sqrt {d}}}\right )\right )+\left (x^3+1\right ) (a d+b c) \left (\sqrt {\sqrt {c}+\sqrt {d}} \tanh ^{-1}\left (\frac {\sqrt [4]{c}-\sqrt [4]{d} x^{3/2}}{\sqrt {x^3+1} \sqrt {\sqrt {c}+\sqrt {d}}}\right )+\sqrt [4]{d} \sinh ^{-1}\left (x^{3/2}\right )\right )+\left (x^3+1\right ) (a d+b c) \left (\sqrt [4]{d} \sinh ^{-1}\left (x^{3/2}\right )-\sqrt {\sqrt {c}+\sqrt {d}} \tanh ^{-1}\left (\frac {\sqrt [4]{c}+\sqrt [4]{d} x^{3/2}}{\sqrt {x^3+1} \sqrt {\sqrt {c}+\sqrt {d}}}\right )\right )+2 a \sqrt {c} d^{3/4} \left (x^{3/2} \left (x^3+1\right )^{3/2}+\left (x^3+1\right ) \sinh ^{-1}\left (x^{3/2}\right )\right )\right )}{6 c^{3/2} d^{3/4} \sqrt {x} \left (x^3+1\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[x + x^4]*(b + a*x^6))/(-d + c*x^6),x]

[Out]

(Sqrt[x + x^4]*(2*a*Sqrt[c]*d^(3/4)*(x^(3/2)*(1 + x^3)^(3/2) + (1 + x^3)*ArcSinh[x^(3/2)]) - (b*c + a*d)*(1 +
x^3)*(d^(1/4)*ArcSinh[x^(3/2)] + Sqrt[Sqrt[c] - Sqrt[d]]*ArcTan[(I*c^(1/4) - d^(1/4)*x^(3/2))/(Sqrt[Sqrt[c] -
Sqrt[d]]*Sqrt[1 + x^3])]) - (b*c + a*d)*(1 + x^3)*(d^(1/4)*ArcSinh[x^(3/2)] - Sqrt[Sqrt[c] - Sqrt[d]]*ArcTan[(
I*c^(1/4) + d^(1/4)*x^(3/2))/(Sqrt[Sqrt[c] - Sqrt[d]]*Sqrt[1 + x^3])]) + (b*c + a*d)*(1 + x^3)*(d^(1/4)*ArcSin
h[x^(3/2)] + Sqrt[Sqrt[c] + Sqrt[d]]*ArcTanh[(c^(1/4) - d^(1/4)*x^(3/2))/(Sqrt[Sqrt[c] + Sqrt[d]]*Sqrt[1 + x^3
])]) + (b*c + a*d)*(1 + x^3)*(d^(1/4)*ArcSinh[x^(3/2)] - Sqrt[Sqrt[c] + Sqrt[d]]*ArcTanh[(c^(1/4) + d^(1/4)*x^
(3/2))/(Sqrt[Sqrt[c] + Sqrt[d]]*Sqrt[1 + x^3])])))/(6*c^(3/2)*d^(3/4)*Sqrt[x]*(1 + x^3)^(3/2))

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IntegrateAlgebraic [A]  time = 5.75, size = 227, normalized size = 1.00 \begin {gather*} \frac {a x \sqrt {x+x^4}}{3 c}+\frac {\sqrt {-\left (\left (\sqrt {c}+\sqrt {d}\right ) \sqrt {d}\right )} (b c+a d) \tan ^{-1}\left (\frac {\sqrt {-\sqrt {c} \sqrt {d}-d} x \sqrt {x+x^4}}{\sqrt {d} (1+x) \left (1-x+x^2\right )}\right )}{3 c^{3/2} d}-\frac {\sqrt {\left (\sqrt {c}-\sqrt {d}\right ) \sqrt {d}} (b c+a d) \tan ^{-1}\left (\frac {\sqrt {\sqrt {c} \sqrt {d}-d} x \sqrt {x+x^4}}{\sqrt {d} (1+x) \left (1-x+x^2\right )}\right )}{3 c^{3/2} d}+\frac {a \tanh ^{-1}\left (\frac {x^2}{\sqrt {x+x^4}}\right )}{3 c} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[x + x^4]*(b + a*x^6))/(-d + c*x^6),x]

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+x)^(1/2)*(a*x^6+b)/(c*x^6-d),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+x)^(1/2)*(a*x^6+b)/(c*x^6-d),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Warning, need to choose a branch for the
root of a polynomial with parameters. This might be wrong.The choice was done assuming [c,d]=[19,-71]Warning,
need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done
assuming [c,d]=[-57,-3]Warning, need to choose a branch for the root of a polynomial with parameters. This mi
ght be wrong.The choice was done assuming [c,d]=[-66,-2]Warning, need to choose a branch for the root of a pol
ynomial with parameters. This might be wrong.The choice was done assuming [c,d]=[22,71]Warning, need to choose
a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [c,d]
=[-36,13]-a/6/c*ln(abs(sqrt((1/x)^3+1)-1))+a/6/c*ln(sqrt((1/x)^3+1)+1)+((2*c*d^3-4*c*d*sqrt(-d^2-d*sqrt(c*d))*
sqrt(c*d)-5*d^2*sqrt(-d^2-d*sqrt(c*d))*sqrt(c*d)-2*d^2*c*d)*a*c^2*abs(d)+(-2*c^2*d^4+4*c^2*d^2*sqrt(-d^2-d*sqr
t(c*d))*sqrt(c*d)+5*c*d^3*sqrt(-d^2-d*sqrt(c*d))*sqrt(c*d)+2*c*d^3*c*d)*a*abs(d)+(2*c^2*d^2-4*c^2*sqrt(-d^2-d*
sqrt(c*d))*sqrt(c*d)-5*c*d*sqrt(-d^2-d*sqrt(c*d))*sqrt(c*d)-2*c*d*c*d)*b*c^2*abs(d)+(-2*c^3*d^3+4*c^3*d*sqrt(-
d^2-d*sqrt(c*d))*sqrt(c*d)+5*c^2*d^2*sqrt(-d^2-d*sqrt(c*d))*sqrt(c*d)+2*c^2*d^2*c*d)*b*abs(d))/(12*c^4*d^3+3*c
^3*d^4-15*c^2*d^5)/abs(c)*atan(sqrt((1/x)^3+1)/sqrt(-(6*c*d+sqrt(6*c*d*6*c*d-12*c*d*(-3*c^2+3*c*d)))/2/3/c/d))
-((2*c*d^3-4*c*d*sqrt(-d^2+d*sqrt(c*d))*sqrt(c*d)-5*d^2*sqrt(-d^2+d*sqrt(c*d))*sqrt(c*d)-2*d^2*c*d)*a*c^2*abs(
d)+(-2*c^2*d^4+4*c^2*d^2*sqrt(-d^2+d*sqrt(c*d))*sqrt(c*d)+5*c*d^3*sqrt(-d^2+d*sqrt(c*d))*sqrt(c*d)+2*c*d^3*c*d
)*a*abs(d)+(2*c^2*d^2-4*c^2*sqrt(-d^2+d*sqrt(c*d))*sqrt(c*d)-5*c*d*sqrt(-d^2+d*sqrt(c*d))*sqrt(c*d)-2*c*d*c*d)
*b*c^2*abs(d)+(-2*c^3*d^3+4*c^3*d*sqrt(-d^2+d*sqrt(c*d))*sqrt(c*d)+5*c^2*d^2*sqrt(-d^2+d*sqrt(c*d))*sqrt(c*d)+
2*c^2*d^2*c*d)*b*abs(d))/(12*c^4*d^3+3*c^3*d^4-15*c^2*d^5)/abs(c)*atan(sqrt((1/x)^3+1)/sqrt(-(6*c*d-sqrt(6*c*d
*6*c*d-12*c*d*(-3*c^2+3*c*d)))/2/3/c/d))+8*a*c*1/24/c^2*x*sqrt(x^4+x)

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maple [C]  time = 0.58, size = 686, normalized size = 3.02

method	result	size
default	\(\frac {a \left (\frac {x \sqrt {x^{4}+x}}{3}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )}{c}+\frac {\left (a d +b c \right ) \sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (c \,\textit {\_Z}^{6}-d \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3}-1\right ) \left (1+x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{5}-\underline {\hspace {1.25 ex}}\alpha ^{4}+\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha -1\right ) \left (-1-i \sqrt {3}\right ) \sqrt {\frac {x \left (3+i \sqrt {3}\right )}{\left (1+x \right ) \left (1+i \sqrt {3}\right )}}\, \sqrt {-\frac {i \sqrt {3}+2 x -1}{\left (1+x \right ) \left (1-i \sqrt {3}\right )}}\, \sqrt {-\frac {-1+2 x -i \sqrt {3}}{\left (1+x \right ) \left (1+i \sqrt {3}\right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\frac {\underline {\hspace {1.25 ex}}\alpha ^{5} c \EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} c +3 \underline {\hspace {1.25 ex}}\alpha ^{5} c +i \sqrt {3}\, d +3 d}{6 d}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )}{d}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{4} \left (c -d \right ) \left (3+i \sqrt {3}\right ) \sqrt {x \left (1+x \right ) \left (i \sqrt {3}+2 x -1\right ) \left (-1+2 x -i \sqrt {3}\right )}}\right )}{3 c}\)	\(686\)
elliptic	\(\frac {a x \sqrt {x^{4}+x}}{3 c}-\frac {a \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{c \left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}-\frac {\sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (c \,\textit {\_Z}^{6}-d \right )}{\sum }\frac {\left (a d +b c \right ) \left (\underline {\hspace {1.25 ex}}\alpha ^{3}+1\right ) \left (1+x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{5}-\underline {\hspace {1.25 ex}}\alpha ^{4}+\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha -1\right ) \left (-1-i \sqrt {3}\right ) \sqrt {\frac {x \left (3+i \sqrt {3}\right )}{\left (1+x \right ) \left (1+i \sqrt {3}\right )}}\, \sqrt {-\frac {i \sqrt {3}+2 x -1}{\left (1+x \right ) \left (1-i \sqrt {3}\right )}}\, \sqrt {-\frac {-1+2 x -i \sqrt {3}}{\left (1+x \right ) \left (1+i \sqrt {3}\right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\frac {\underline {\hspace {1.25 ex}}\alpha ^{5} c \EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} c +3 \underline {\hspace {1.25 ex}}\alpha ^{5} c +i \sqrt {3}\, d +3 d}{6 d}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )}{d}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{4} \left (c -d \right ) \left (3+i \sqrt {3}\right ) \sqrt {x \left (1+x \right ) \left (i \sqrt {3}+2 x -1\right ) \left (-1+2 x -i \sqrt {3}\right )}}\right )}{3 c}\)	\(686\)
risch	\(\frac {a \,x^{2} \left (x^{3}+1\right )}{3 c \sqrt {x \left (x^{3}+1\right )}}+\frac {-\frac {2 a \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}+\frac {2 \sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (c \,\textit {\_Z}^{6}-d \right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3} a d -\underline {\hspace {1.25 ex}}\alpha ^{3} b c -a d -b c \right ) \left (1+x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{5}-\underline {\hspace {1.25 ex}}\alpha ^{4}+\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha -1\right ) \left (-1-i \sqrt {3}\right ) \sqrt {\frac {x \left (3+i \sqrt {3}\right )}{\left (1+x \right ) \left (1+i \sqrt {3}\right )}}\, \sqrt {-\frac {i \sqrt {3}+2 x -1}{\left (1+x \right ) \left (1-i \sqrt {3}\right )}}\, \sqrt {-\frac {-1+2 x -i \sqrt {3}}{\left (1+x \right ) \left (1+i \sqrt {3}\right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\frac {\underline {\hspace {1.25 ex}}\alpha ^{5} c \EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} c +3 \underline {\hspace {1.25 ex}}\alpha ^{5} c +i \sqrt {3}\, d +3 d}{6 d}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )}{d}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{4} \left (c -d \right ) \left (3+i \sqrt {3}\right ) \sqrt {x \left (1+x \right ) \left (i \sqrt {3}+2 x -1\right ) \left (-1+2 x -i \sqrt {3}\right )}}\right )}{3}}{2 c}\)	\(706\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+x)^(1/2)*(a*x^6+b)/(c*x^6-d),x,method=_RETURNVERBOSE)

[Out]

1/c*a*(1/3*x*(x^4+x)^(1/2)-(-1/2-1/2*I*3^(1/2))*((3/2+1/2*I*3^(1/2))*x/(1/2+1/2*I*3^(1/2))/(1+x))^(1/2)*(1+x)^
2*(-(x-1/2+1/2*I*3^(1/2))/(1/2-1/2*I*3^(1/2))/(1+x))^(1/2)*(-(x-1/2-1/2*I*3^(1/2))/(1/2+1/2*I*3^(1/2))/(1+x))^
(1/2)/(3/2+1/2*I*3^(1/2))/(x*(1+x)*(x-1/2+1/2*I*3^(1/2))*(x-1/2-1/2*I*3^(1/2)))^(1/2)*(-EllipticF(((3/2+1/2*I*
3^(1/2))*x/(1/2+1/2*I*3^(1/2))/(1+x))^(1/2),((-3/2+1/2*I*3^(1/2))*(-1/2-1/2*I*3^(1/2))/(-1/2+1/2*I*3^(1/2))/(-
3/2-1/2*I*3^(1/2)))^(1/2))+EllipticPi(((3/2+1/2*I*3^(1/2))*x/(1/2+1/2*I*3^(1/2))/(1+x))^(1/2),(1/2+1/2*I*3^(1/
2))/(3/2+1/2*I*3^(1/2)),((-3/2+1/2*I*3^(1/2))*(-1/2-1/2*I*3^(1/2))/(-1/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^
(1/2))))+1/3*(a*d+b*c)/c*4^(1/2)*sum((-_alpha^3-1)/_alpha^4*(1+x)^2*(_alpha^5-_alpha^4+_alpha^3-_alpha^2+_alph
a-1)/(c-d)*(-1-I*3^(1/2))*(x/(1+x)*(3+I*3^(1/2))/(1+I*3^(1/2)))^(1/2)*(-1/(1+x)*(I*3^(1/2)+2*x-1)/(1-I*3^(1/2)
))^(1/2)*(-1/(1+x)*(-1+2*x-I*3^(1/2))/(1+I*3^(1/2)))^(1/2)/(3+I*3^(1/2))/(x*(1+x)*(I*3^(1/2)+2*x-1)*(-1+2*x-I*
3^(1/2)))^(1/2)*(EllipticF(((3/2+1/2*I*3^(1/2))*x/(1/2+1/2*I*3^(1/2))/(1+x))^(1/2),((-3/2+1/2*I*3^(1/2))*(-1/2
-1/2*I*3^(1/2))/(-1/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))+_alpha^5*c/d*EllipticPi(((3/2+1/2*I*3^(1/2))
*x/(1/2+1/2*I*3^(1/2))/(1+x))^(1/2),1/6*(I*3^(1/2)*_alpha^5*c+3*_alpha^5*c+I*3^(1/2)*d+3*d)/d,((-3/2+1/2*I*3^(
1/2))*(-1/2-1/2*I*3^(1/2))/(-1/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))),_alpha=RootOf(_Z^6*c-d))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{6} + b\right )} \sqrt {x^{4} + x}}{c x^{6} - d}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+x)^(1/2)*(a*x^6+b)/(c*x^6-d),x, algorithm="maxima")

[Out]

integrate((a*x^6 + b)*sqrt(x^4 + x)/(c*x^6 - d), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {\left (a\,x^6+b\right )\,\sqrt {x^4+x}}{d-c\,x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((b + a*x^6)*(x + x^4)^(1/2))/(d - c*x^6),x)

[Out]

int(-((b + a*x^6)*(x + x^4)^(1/2))/(d - c*x^6), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (a x^{6} + b\right )}{c x^{6} - d}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+x)**(1/2)*(a*x**6+b)/(c*x**6-d),x)

[Out]

Integral(sqrt(x*(x + 1)*(x**2 - x + 1))*(a*x**6 + b)/(c*x**6 - d), x)

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