3.27.63 \(\int \frac {x^6 \sqrt {x+x^4}}{b+a x^6} \, dx\) [2663]

Optimal. Leaf size=238 \[ \frac {x \sqrt {x+x^4}}{3 a}+\frac {\sqrt [4]{-1} \sqrt {\left (\sqrt {a}-i \sqrt {b}\right ) \sqrt {b}} \text {ArcTan}\left (\frac {(1+i) \sqrt {\sqrt {a} \sqrt {b}-i b} \sqrt {x+x^4}}{\sqrt {2} \left (\sqrt {a}-i \sqrt {b}\right ) x^2}\right )}{3 a^{3/2}}+\frac {(-1)^{3/4} \sqrt {\left (\sqrt {a}+i \sqrt {b}\right ) \sqrt {b}} \text {ArcTan}\left (\frac {(1+i) \sqrt {\sqrt {a} \sqrt {b}+i b} x \sqrt {x+x^4}}{\sqrt {2} \sqrt {b} (1+x) \left (1-x+x^2\right )}\right )}{3 a^{3/2}}+\frac {\tanh ^{-1}\left (\frac {x^2}{\sqrt {x+x^4}}\right )}{3 a} \]

[Out]

1/3*x*(x^4+x)^(1/2)/a+1/3*(-1)^(1/4)*((a^(1/2)-I*b^(1/2))*b^(1/2))^(1/2)*arctan((1/2+1/2*I)*(a^(1/2)*b^(1/2)-I
*b)^(1/2)*(x^4+x)^(1/2)*2^(1/2)/(a^(1/2)-I*b^(1/2))/x^2)/a^(3/2)+1/3*(-1)^(3/4)*((a^(1/2)+I*b^(1/2))*b^(1/2))^
(1/2)*arctan((1/2+1/2*I)*(a^(1/2)*b^(1/2)+I*b)^(1/2)*x*(x^4+x)^(1/2)*2^(1/2)/b^(1/2)/(1+x)/(x^2-x+1))/a^(3/2)+
1/3*arctanh(x^2/(x^4+x)^(1/2))/a

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Rubi [A]
time = 0.46, antiderivative size = 253, normalized size of antiderivative = 1.06, number of steps used = 15, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {2081, 1507, 1505, 1306, 201, 221, 1189, 399, 385, 214} \begin {gather*} \frac {\sqrt [4]{b} \sqrt {x^4+x} \sqrt {\sqrt {-a}+\sqrt {b}} \tanh ^{-1}\left (\frac {x^{3/2} \sqrt {\sqrt {-a}+\sqrt {b}}}{\sqrt [4]{b} \sqrt {x^3+1}}\right )}{3 (-a)^{3/2} \sqrt {x^3+1} \sqrt {x}}-\frac {\sqrt [4]{b} \sqrt {x^4+x} \sqrt {\sqrt {-a} \sqrt {b}+a} \tanh ^{-1}\left (\frac {x^{3/2} \sqrt {\sqrt {-a} \sqrt {b}+a}}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {x^3+1}}\right )}{3 (-a)^{7/4} \sqrt {x^3+1} \sqrt {x}}+\frac {\sqrt {x^4+x} x}{3 a}+\frac {\sqrt {x^4+x} \sinh ^{-1}\left (x^{3/2}\right )}{3 a \sqrt {x^3+1} \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[x + x^4])/(3*a) + (Sqrt[x + x^4]*ArcSinh[x^(3/2)])/(3*a*Sqrt[x]*Sqrt[1 + x^3]) + (Sqrt[Sqrt[-a] + Sqrt
[b]]*b^(1/4)*Sqrt[x + x^4]*ArcTanh[(Sqrt[Sqrt[-a] + Sqrt[b]]*x^(3/2))/(b^(1/4)*Sqrt[1 + x^3])])/(3*(-a)^(3/2)*
Sqrt[x]*Sqrt[1 + x^3]) - (Sqrt[a + Sqrt[-a]*Sqrt[b]]*b^(1/4)*Sqrt[x + x^4]*ArcTanh[(Sqrt[a + Sqrt[-a]*Sqrt[b]]
*x^(3/2))/((-a)^(1/4)*b^(1/4)*Sqrt[1 + x^3])])/(3*(-a)^(7/4)*Sqrt[x]*Sqrt[1 + x^3])

Rule 201

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 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 221

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 385

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 399

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

Rule 1189

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 1306

Int[(((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(q_))/((a_) + (c_.)*(x_)^4), x_Symbol] :> Dist[f^4/c, Int[(f*x)
^(m - 4)*(d + e*x^2)^q, x], x] - Dist[a*(f^4/c), Int[(f*x)^(m - 4)*((d + e*x^2)^q/(a + c*x^4)), x], x] /; Free
Q[{a, c, d, e, f, q}, x] &&  !IntegerQ[q] && GtQ[m, 3]

Rule 1505

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = GCD[m +
1, n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(d + e*x^(n/k))^q*(a + c*x^(2*(n/k)))^p, x], x, x^k], x] /; k !=
1] /; FreeQ[{a, c, d, e, p, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IntegerQ[m]

Rule 1507

Int[((f_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = D
enominator[m]}, Dist[k/f, Subst[Int[x^(k*(m + 1) - 1)*(d + e*(x^(k*n)/f))^q*(a + c*(x^(2*k*n)/f))^p, x], x, (f
*x)^(1/k)], x]] /; FreeQ[{a, c, d, e, f, p, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && FractionQ[m] && IntegerQ[p
]

Rule 2081

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]

Rubi steps

\begin {align*} \int \frac {x^6 \sqrt {x+x^4}}{b+a x^6} \, dx &=\frac {\sqrt {x+x^4} \int \frac {x^{13/2} \sqrt {1+x^3}}{b+a x^6} \, dx}{\sqrt {x} \sqrt {1+x^3}}\\ &=\frac {\left (2 \sqrt {x+x^4}\right ) \text {Subst}\left (\int \frac {x^{14} \sqrt {1+x^6}}{b+a x^{12}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3}}\\ &=\frac {\left (2 \sqrt {x+x^4}\right ) \text {Subst}\left (\int \frac {x^4 \sqrt {1+x^2}}{b+a x^4} \, dx,x,x^{3/2}\right )}{3 \sqrt {x} \sqrt {1+x^3}}\\ &=\frac {\left (2 \sqrt {x+x^4}\right ) \text {Subst}\left (\int \sqrt {1+x^2} \, dx,x,x^{3/2}\right )}{3 a \sqrt {x} \sqrt {1+x^3}}-\frac {\left (2 b \sqrt {x+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt {1+x^2}}{b+a x^4} \, dx,x,x^{3/2}\right )}{3 a \sqrt {x} \sqrt {1+x^3}}\\ &=\frac {x \sqrt {x+x^4}}{3 a}+\frac {\sqrt {x+x^4} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^{3/2}\right )}{3 a \sqrt {x} \sqrt {1+x^3}}-\frac {\left (\sqrt {-a} \sqrt {b} \sqrt {x+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt {1+x^2}}{\sqrt {-a} \sqrt {b}-a x^2} \, dx,x,x^{3/2}\right )}{3 a \sqrt {x} \sqrt {1+x^3}}-\frac {\left (\sqrt {-a} \sqrt {b} \sqrt {x+x^4}\right ) \text {Subst}\left (\int \frac {\sqrt {1+x^2}}{\sqrt {-a} \sqrt {b}+a x^2} \, dx,x,x^{3/2}\right )}{3 a \sqrt {x} \sqrt {1+x^3}}\\ &=\frac {x \sqrt {x+x^4}}{3 a}+\frac {\sqrt {x+x^4} \sinh ^{-1}\left (x^{3/2}\right )}{3 a \sqrt {x} \sqrt {1+x^3}}+\frac {\left (\sqrt {-a} \left (-a+\sqrt {-a} \sqrt {b}\right ) \sqrt {b} \sqrt {x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2} \left (\sqrt {-a} \sqrt {b}+a x^2\right )} \, dx,x,x^{3/2}\right )}{3 a^2 \sqrt {x} \sqrt {1+x^3}}-\frac {\left (\sqrt {-a} \left (a+\sqrt {-a} \sqrt {b}\right ) \sqrt {b} \sqrt {x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2} \left (\sqrt {-a} \sqrt {b}-a x^2\right )} \, dx,x,x^{3/2}\right )}{3 a^2 \sqrt {x} \sqrt {1+x^3}}\\ &=\frac {x \sqrt {x+x^4}}{3 a}+\frac {\sqrt {x+x^4} \sinh ^{-1}\left (x^{3/2}\right )}{3 a \sqrt {x} \sqrt {1+x^3}}+\frac {\left (\sqrt {-a} \left (-a+\sqrt {-a} \sqrt {b}\right ) \sqrt {b} \sqrt {x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a} \sqrt {b}-\left (-a+\sqrt {-a} \sqrt {b}\right ) x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {1+x^3}}\right )}{3 a^2 \sqrt {x} \sqrt {1+x^3}}-\frac {\left (\sqrt {-a} \left (a+\sqrt {-a} \sqrt {b}\right ) \sqrt {b} \sqrt {x+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a} \sqrt {b}-\left (a+\sqrt {-a} \sqrt {b}\right ) x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {1+x^3}}\right )}{3 a^2 \sqrt {x} \sqrt {1+x^3}}\\ &=\frac {x \sqrt {x+x^4}}{3 a}+\frac {\sqrt {x+x^4} \sinh ^{-1}\left (x^{3/2}\right )}{3 a \sqrt {x} \sqrt {1+x^3}}+\frac {\sqrt {\sqrt {-a}+\sqrt {b}} \sqrt [4]{b} \sqrt {x+x^4} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {-a}+\sqrt {b}} x^{3/2}}{\sqrt [4]{b} \sqrt {1+x^3}}\right )}{3 (-a)^{3/2} \sqrt {x} \sqrt {1+x^3}}-\frac {\sqrt {a+\sqrt {-a} \sqrt {b}} \sqrt [4]{b} \sqrt {x+x^4} \tanh ^{-1}\left (\frac {\sqrt {a+\sqrt {-a} \sqrt {b}} x^{3/2}}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {1+x^3}}\right )}{3 (-a)^{7/4} \sqrt {x} \sqrt {1+x^3}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.25, size = 177, normalized size = 0.74 \begin {gather*} \frac {\sqrt {x+x^4} \left (x^{3/2} \sqrt {1+x^3}+\tanh ^{-1}\left (\frac {x^{3/2}}{\sqrt {1+x^3}}\right )-b \text {RootSum}\left [16 a+16 b-32 a \text {$\#$1}-32 b \text {$\#$1}+24 a \text {$\#$1}^2+16 b \text {$\#$1}^2-8 a \text {$\#$1}^3+a \text {$\#$1}^4\&,\frac {\log \left (2+2 x^3+2 x^{3/2} \sqrt {1+x^3}-\text {$\#$1}\right ) \text {$\#$1}^2}{-8 a-8 b+12 a \text {$\#$1}+8 b \text {$\#$1}-6 a \text {$\#$1}^2+a \text {$\#$1}^3}\&\right ]\right )}{3 a \sqrt {x} \sqrt {1+x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x + x^4]*(x^(3/2)*Sqrt[1 + x^3] + ArcTanh[x^(3/2)/Sqrt[1 + x^3]] - b*RootSum[16*a + 16*b - 32*a*#1 - 32*
b*#1 + 24*a*#1^2 + 16*b*#1^2 - 8*a*#1^3 + a*#1^4 & , (Log[2 + 2*x^3 + 2*x^(3/2)*Sqrt[1 + x^3] - #1]*#1^2)/(-8*
a - 8*b + 12*a*#1 + 8*b*#1 - 6*a*#1^2 + a*#1^3) & ]))/(3*a*Sqrt[x]*Sqrt[1 + x^3])

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.58, size = 676, normalized size = 2.84 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/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*b/a*4^(1/2)*sum((-_alpha^3-1)/_alpha^4*(1+x)^2*(_alpha^5-_alpha^4+_alpha^3-_alpha^2+_alpha-1)/(a+b)
*(-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)*(-I*3^(1/2)+2*x-1)/(1+I*3^(1/2)))^(1/2)/(3+I*3^(1/2))/(x*(1+x)*(I*3^(1/2)+2*x-1)*(-I*3^(1/2)+2*x-1))^
(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*a/b*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*_alpha^5*3^(1/2)*a+3*_alpha^5*a-I*3^(1/2)*b-3*b)/b,((-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*a+b))

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1784 vs. \(2 (162) = 324\).
time = 53.52, size = 1784, normalized size = 7.50 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(a*sqrt((a^3*sqrt(-b/a^5) - b)/a^3)*log((2*((9*a^4*b + 73*a^3*b^2 + 279*a^2*b^3 + 567*a*b^4)*x^4 - (a^4*
b - 12*a^3*b^2 - 120*a^2*b^3 - 216*a*b^4 + 243*b^5)*x + ((a^7 - 12*a^6*b - 120*a^5*b^2 - 216*a^4*b^3 + 243*a^3
*b^4)*x^4 + (9*a^6*b + 73*a^5*b^2 + 279*a^4*b^3 + 567*a^3*b^4)*x)*sqrt(-b/a^5))*sqrt(x^4 + x) + ((a^6 - 2*a^5*
b - 69*a^4*b^2 - 108*a^3*b^3 + 486*a^2*b^4)*x^6 - a^5*b + 22*a^4*b^2 + 171*a^3*b^3 + 324*a^2*b^4 + 2*(9*a^5*b
+ 73*a^4*b^2 + 279*a^3*b^3 + 567*a^2*b^4)*x^3 + (10*a^7*b + 51*a^6*b^2 + 108*a^5*b^3 + 243*a^4*b^4 - (8*a^8 +
95*a^7*b + 450*a^6*b^2 + 891*a^5*b^3)*x^6 + 2*(a^8 - 12*a^7*b - 120*a^6*b^2 - 216*a^5*b^3 + 243*a^4*b^4)*x^3)*
sqrt(-b/a^5))*sqrt((a^3*sqrt(-b/a^5) - b)/a^3))/(a*x^6 + b)) - a*sqrt((a^3*sqrt(-b/a^5) - b)/a^3)*log((2*((9*a
^4*b + 73*a^3*b^2 + 279*a^2*b^3 + 567*a*b^4)*x^4 - (a^4*b - 12*a^3*b^2 - 120*a^2*b^3 - 216*a*b^4 + 243*b^5)*x
+ ((a^7 - 12*a^6*b - 120*a^5*b^2 - 216*a^4*b^3 + 243*a^3*b^4)*x^4 + (9*a^6*b + 73*a^5*b^2 + 279*a^4*b^3 + 567*
a^3*b^4)*x)*sqrt(-b/a^5))*sqrt(x^4 + x) - ((a^6 - 2*a^5*b - 69*a^4*b^2 - 108*a^3*b^3 + 486*a^2*b^4)*x^6 - a^5*
b + 22*a^4*b^2 + 171*a^3*b^3 + 324*a^2*b^4 + 2*(9*a^5*b + 73*a^4*b^2 + 279*a^3*b^3 + 567*a^2*b^4)*x^3 + (10*a^
7*b + 51*a^6*b^2 + 108*a^5*b^3 + 243*a^4*b^4 - (8*a^8 + 95*a^7*b + 450*a^6*b^2 + 891*a^5*b^3)*x^6 + 2*(a^8 - 1
2*a^7*b - 120*a^6*b^2 - 216*a^5*b^3 + 243*a^4*b^4)*x^3)*sqrt(-b/a^5))*sqrt((a^3*sqrt(-b/a^5) - b)/a^3))/(a*x^6
+ b)) + a*sqrt(-(a^3*sqrt(-b/a^5) + b)/a^3)*log((2*((9*a^4*b + 73*a^3*b^2 + 279*a^2*b^3 + 567*a*b^4)*x^4 - (a
^4*b - 12*a^3*b^2 - 120*a^2*b^3 - 216*a*b^4 + 243*b^5)*x - ((a^7 - 12*a^6*b - 120*a^5*b^2 - 216*a^4*b^3 + 243*
a^3*b^4)*x^4 + (9*a^6*b + 73*a^5*b^2 + 279*a^4*b^3 + 567*a^3*b^4)*x)*sqrt(-b/a^5))*sqrt(x^4 + x) + ((a^6 - 2*a
^5*b - 69*a^4*b^2 - 108*a^3*b^3 + 486*a^2*b^4)*x^6 - a^5*b + 22*a^4*b^2 + 171*a^3*b^3 + 324*a^2*b^4 + 2*(9*a^5
*b + 73*a^4*b^2 + 279*a^3*b^3 + 567*a^2*b^4)*x^3 - (10*a^7*b + 51*a^6*b^2 + 108*a^5*b^3 + 243*a^4*b^4 - (8*a^8
+ 95*a^7*b + 450*a^6*b^2 + 891*a^5*b^3)*x^6 + 2*(a^8 - 12*a^7*b - 120*a^6*b^2 - 216*a^5*b^3 + 243*a^4*b^4)*x^
3)*sqrt(-b/a^5))*sqrt(-(a^3*sqrt(-b/a^5) + b)/a^3))/(a*x^6 + b)) - a*sqrt(-(a^3*sqrt(-b/a^5) + b)/a^3)*log((2*
((9*a^4*b + 73*a^3*b^2 + 279*a^2*b^3 + 567*a*b^4)*x^4 - (a^4*b - 12*a^3*b^2 - 120*a^2*b^3 - 216*a*b^4 + 243*b^
5)*x - ((a^7 - 12*a^6*b - 120*a^5*b^2 - 216*a^4*b^3 + 243*a^3*b^4)*x^4 + (9*a^6*b + 73*a^5*b^2 + 279*a^4*b^3 +
567*a^3*b^4)*x)*sqrt(-b/a^5))*sqrt(x^4 + x) - ((a^6 - 2*a^5*b - 69*a^4*b^2 - 108*a^3*b^3 + 486*a^2*b^4)*x^6 -
a^5*b + 22*a^4*b^2 + 171*a^3*b^3 + 324*a^2*b^4 + 2*(9*a^5*b + 73*a^4*b^2 + 279*a^3*b^3 + 567*a^2*b^4)*x^3 - (
10*a^7*b + 51*a^6*b^2 + 108*a^5*b^3 + 243*a^4*b^4 - (8*a^8 + 95*a^7*b + 450*a^6*b^2 + 891*a^5*b^3)*x^6 + 2*(a^
8 - 12*a^7*b - 120*a^6*b^2 - 216*a^5*b^3 + 243*a^4*b^4)*x^3)*sqrt(-b/a^5))*sqrt(-(a^3*sqrt(-b/a^5) + b)/a^3))/
(a*x^6 + b)) - 4*sqrt(x^4 + x)*x - 2*log(-2*x^3 - 2*sqrt(x^4 + x)*x - 1))/a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^6*(x^4+x)^(1/2)/(a*x^6+b),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 [sageVARa,sageVARb]=[-
96,98]Warning, need t

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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