3.17.11 \(\int \frac {x^3 \sqrt {x+x^4}}{-b+a x^3} \, dx\)

Optimal. Leaf size=110 \[ \frac {2 \sqrt {b} \sqrt {-a-b} \tan ^{-1}\left (\frac {x \sqrt {x^4+x} \sqrt {-a-b}}{\sqrt {b} (x+1) \left (x^2-x+1\right )}\right )}{3 a^2}+\frac {(a+2 b) \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right )}{3 a^2}+\frac {\sqrt {x^4+x} x}{3 a} \]

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Rubi [A]  time = 0.26, antiderivative size = 129, normalized size of antiderivative = 1.17, number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2042, 466, 465, 478, 523, 215, 377, 208} \begin {gather*} \frac {\sqrt {x^4+x} (a+2 b) \sinh ^{-1}\left (x^{3/2}\right )}{3 a^2 \sqrt {x^3+1} \sqrt {x}}-\frac {2 \sqrt {b} \sqrt {x^4+x} \sqrt {a+b} \tanh ^{-1}\left (\frac {x^{3/2} \sqrt {a+b}}{\sqrt {b} \sqrt {x^3+1}}\right )}{3 a^2 \sqrt {x^3+1} \sqrt {x}}+\frac {\sqrt {x^4+x} x}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3*Sqrt[x + x^4])/(-b + a*x^3),x]

[Out]

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

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 465

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

Rule 466

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/e^n)^p*(c + (d*x^(k*n))/e^n)^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 478

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

Rule 523

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 2042

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.))^(q_.), x_Symbol]
:> Dist[(e^IntPart[m]*(e*x)^FracPart[m]*(a*x^j + b*x^(j + n))^FracPart[p])/(x^(FracPart[m] + j*FracPart[p])*(a
+ b*x^n)^FracPart[p]), Int[x^(m + j*p)*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, j, m, n,
p, q}, x] && EqQ[jn, j + n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] &&  !(EqQ[n, 1] && EqQ[j, 1])

Rubi steps

\begin {align*} \int \frac {x^3 \sqrt {x+x^4}}{-b+a x^3} \, dx &=\frac {\sqrt {x+x^4} \int \frac {x^{7/2} \sqrt {1+x^3}}{-b+a x^3} \, dx}{\sqrt {x} \sqrt {1+x^3}}\\ &=\frac {\left (2 \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^8 \sqrt {1+x^6}}{-b+a x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3}}\\ &=\frac {\left (2 \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1+x^2}}{-b+a x^2} \, dx,x,x^{3/2}\right )}{3 \sqrt {x} \sqrt {1+x^3}}\\ &=\frac {x \sqrt {x+x^4}}{3 a}-\frac {\sqrt {x+x^4} \operatorname {Subst}\left (\int \frac {-b+(-a-2 b) x^2}{\sqrt {1+x^2} \left (-b+a x^2\right )} \, dx,x,x^{3/2}\right )}{3 a \sqrt {x} \sqrt {1+x^3}}\\ &=\frac {x \sqrt {x+x^4}}{3 a}-\frac {\left ((-a-2 b) \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^{3/2}\right )}{3 a^2 \sqrt {x} \sqrt {1+x^3}}+\frac {\left (2 b (a+b) \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2} \left (-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 {(a+2 b) \sqrt {x+x^4} \sinh ^{-1}\left (x^{3/2}\right )}{3 a^2 \sqrt {x} \sqrt {1+x^3}}+\frac {\left (2 b (a+b) \sqrt {x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{-b-(-a-b) 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 {(a+2 b) \sqrt {x+x^4} \sinh ^{-1}\left (x^{3/2}\right )}{3 a^2 \sqrt {x} \sqrt {1+x^3}}-\frac {2 \sqrt {b} \sqrt {a+b} \sqrt {x+x^4} \tanh ^{-1}\left (\frac {\sqrt {a+b} x^{3/2}}{\sqrt {b} \sqrt {1+x^3}}\right )}{3 a^2 \sqrt {x} \sqrt {1+x^3}}\\ \end {align*}

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Mathematica [C]  time = 0.19, size = 156, normalized size = 1.42 \begin {gather*} -\frac {x \sqrt {x^4+x} \left (x^3 (a+2 b) \sqrt {-\frac {x^3 (a+b)}{b}} F_1\left (\frac {3}{2};\frac {1}{2},1;\frac {5}{2};-x^3,\frac {a x^3}{b}\right )-3 b \left (\sqrt {x^3+1} \sqrt {-\frac {x^3 (a+b)}{b}}-\sin ^{-1}\left (\frac {\sqrt {-\frac {x^3 (a+b)}{b}}}{\sqrt {1-\frac {a x^3}{b}}}\right )\right )\right )}{9 a b \sqrt {x^3+1} \sqrt {-\frac {x^3 (a+b)}{b}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^3*Sqrt[x + x^4])/(-b + a*x^3),x]

[Out]

-1/9*(x*Sqrt[x + x^4]*((a + 2*b)*x^3*Sqrt[-(((a + b)*x^3)/b)]*AppellF1[3/2, 1/2, 1, 5/2, -x^3, (a*x^3)/b] - 3*
b*(Sqrt[-(((a + b)*x^3)/b)]*Sqrt[1 + x^3] - ArcSin[Sqrt[-(((a + b)*x^3)/b)]/Sqrt[1 - (a*x^3)/b]])))/(a*b*Sqrt[
-(((a + b)*x^3)/b)]*Sqrt[1 + x^3])

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IntegrateAlgebraic [A]  time = 0.59, size = 110, normalized size = 1.00 \begin {gather*} \frac {x \sqrt {x+x^4}}{3 a}+\frac {2 \sqrt {-a-b} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {-a-b} x \sqrt {x+x^4}}{\sqrt {b} (1+x) \left (1-x+x^2\right )}\right )}{3 a^2}+\frac {(a+2 b) \tanh ^{-1}\left (\frac {x^2}{\sqrt {x+x^4}}\right )}{3 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^3*Sqrt[x + x^4])/(-b + a*x^3),x]

[Out]

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

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fricas [A]  time = 1.86, size = 233, normalized size = 2.12 \begin {gather*} \left [\frac {2 \, \sqrt {x^{4} + x} a x + {\left (a + 2 \, b\right )} \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} + x} x - 1\right ) + \sqrt {a b + b^{2}} \log \left (-\frac {{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} x^{6} + 2 \, {\left (3 \, a b + 4 \, b^{2}\right )} x^{3} - 4 \, {\left ({\left (a + 2 \, b\right )} x^{4} + b x\right )} \sqrt {x^{4} + x} \sqrt {a b + b^{2}} + b^{2}}{a^{2} x^{6} - 2 \, a b x^{3} + b^{2}}\right )}{6 \, a^{2}}, \frac {2 \, \sqrt {x^{4} + x} a x + {\left (a + 2 \, b\right )} \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} + x} x - 1\right ) + 2 \, \sqrt {-a b - b^{2}} \arctan \left (\frac {2 \, \sqrt {x^{4} + x} \sqrt {-a b - b^{2}} x}{{\left (a + 2 \, b\right )} x^{3} + b}\right )}{6 \, a^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(x^4+x)^(1/2)/(a*x^3-b),x, algorithm="fricas")

[Out]

[1/6*(2*sqrt(x^4 + x)*a*x + (a + 2*b)*log(-2*x^3 - 2*sqrt(x^4 + x)*x - 1) + sqrt(a*b + b^2)*log(-((a^2 + 8*a*b
+ 8*b^2)*x^6 + 2*(3*a*b + 4*b^2)*x^3 - 4*((a + 2*b)*x^4 + b*x)*sqrt(x^4 + x)*sqrt(a*b + b^2) + b^2)/(a^2*x^6
- 2*a*b*x^3 + b^2)))/a^2, 1/6*(2*sqrt(x^4 + x)*a*x + (a + 2*b)*log(-2*x^3 - 2*sqrt(x^4 + x)*x - 1) + 2*sqrt(-a
*b - b^2)*arctan(2*sqrt(x^4 + x)*sqrt(-a*b - b^2)*x/((a + 2*b)*x^3 + b)))/a^2]

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giac [A]  time = 0.33, size = 101, normalized size = 0.92 \begin {gather*} \frac {\sqrt {x^{4} + x} x}{3 \, a} + \frac {{\left (a + 2 \, b\right )} \log \left (\sqrt {\frac {1}{x^{3}} + 1} + 1\right )}{6 \, a^{2}} - \frac {{\left (a + 2 \, b\right )} \log \left ({\left | \sqrt {\frac {1}{x^{3}} + 1} - 1 \right |}\right )}{6 \, a^{2}} + \frac {2 \, {\left (a b + b^{2}\right )} \arctan \left (\frac {b \sqrt {\frac {1}{x^{3}} + 1}}{\sqrt {-a b - b^{2}}}\right )}{3 \, \sqrt {-a b - b^{2}} a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(x^4+x)^(1/2)/(a*x^3-b),x, algorithm="giac")

[Out]

1/3*sqrt(x^4 + x)*x/a + 1/6*(a + 2*b)*log(sqrt(1/x^3 + 1) + 1)/a^2 - 1/6*(a + 2*b)*log(abs(sqrt(1/x^3 + 1) - 1
))/a^2 + 2/3*(a*b + b^2)*arctan(b*sqrt(1/x^3 + 1)/sqrt(-a*b - b^2))/(sqrt(-a*b - b^2)*a^2)

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maple [C]  time = 0.39, size = 665, normalized size = 6.05

method	result	size
elliptic	\(\frac {x \sqrt {x^{4}+x}}{3 a}-\frac {2 \left (\frac {a +b}{a^{2}}-\frac {1}{2 a}\right ) \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 b \sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}-b \right )}{\sum }\frac {\left (1+x \right )^{2} \left (\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 ^{2} a \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 \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}\, a +3 \underline {\hspace {1.25 ex}}\alpha ^{2} a +i \sqrt {3}\, b +3 b}{6 b}, \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 )}{b}\right )}{\underline {\hspace {1.25 ex}}\alpha \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 a^{2}}\)	\(665\)
risch	\(\frac {x^{2} \left (x^{3}+1\right )}{3 a \sqrt {x \left (x^{3}+1\right )}}+\frac {-\frac {2 \left (a +2 b \right ) \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 )}{a \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 {4 b \left (a +b \right ) \sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}-b \right )}{\sum }\frac {\left (1+x \right )^{2} \left (\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 ^{2} a \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 \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}\, a +3 \underline {\hspace {1.25 ex}}\alpha ^{2} a +i \sqrt {3}\, b +3 b}{6 b}, \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 )}{b}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (a +b \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 a}}{2 a}\)	\(683\)
default	\(\frac {\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 )}}}{a}+\frac {b \left (-\frac {2 \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 )}{a \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 (a \,\textit {\_Z}^{3}-b \right )}{\sum }\frac {\left (-a -b \right ) \left (1+x \right )^{2} \left (\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 ^{2} a \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 \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}\, a +3 \underline {\hspace {1.25 ex}}\alpha ^{2} a +i \sqrt {3}\, b +3 b}{6 b}, \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 )}{b}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (a +b \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 a}\right )}{a}\)	\(963\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(x^4+x)^(1/2)/(a*x^3-b),x,method=_RETURNVERBOSE)

[Out]

1/3*x*(x^4+x)^(1/2)/a-2*((a+b)/a^2-1/2/a)*(-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)*(-Ellipti
cF(((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)))+2/3/a^2*b*4^(1/2)*sum(1/_alpha*(1+x)^2*(_alpha^2-_alpha+1)*(-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^2*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^2*3^(1/2)*a+3*_alpha^2*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^3*a-b))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{4} + x} x^{3}}{a x^{3} - b}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(x^4+x)^(1/2)/(a*x^3-b),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {x^3\,\sqrt {x^4+x}}{b-a\,x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^3*(x + x^4)^(1/2))/(b - a*x^3),x)

[Out]

-int((x^3*(x + x^4)^(1/2))/(b - a*x^3), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \sqrt {x \left (x + 1\right ) \left (x^{2} - x + 1\right )}}{a x^{3} - b}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(x**4+x)**(1/2)/(a*x**3-b),x)

[Out]

Integral(x**3*sqrt(x*(x + 1)*(x**2 - x + 1))/(a*x**3 - b), x)

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