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

Optimal. Leaf size=126 \[ \frac {2 \sqrt {c-d} (a d+b c) \tan ^{-1}\left (\frac {x \sqrt {x^4-x} \sqrt {c-d}}{\sqrt {d} (x-1) \left (x^2+x+1\right )}\right )}{3 c^2 \sqrt {d}}+\frac {\tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4-x}}\right ) (-a c+2 a d+2 b c)}{3 c^2}+\frac {a \sqrt {x^4-x} x}{3 c} \]

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Rubi [A]  time = 0.37, antiderivative size = 164, normalized size of antiderivative = 1.30, number of steps used = 12, number of rules used = 11, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {2056, 581, 584, 329, 275, 217, 206, 466, 465, 377, 205} \begin {gather*} \frac {2 \sqrt {x^4-x} \sqrt {c-d} (a d+b c) \tan ^{-1}\left (\frac {x^{3/2} \sqrt {c-d}}{\sqrt {d} \sqrt {x^3-1}}\right )}{3 c^2 \sqrt {d} \sqrt {x^3-1} \sqrt {x}}-\frac {\sqrt {x^4-x} \tanh ^{-1}\left (\frac {x^{3/2}}{\sqrt {x^3-1}}\right ) (a c-2 a d-2 b c)}{3 c^2 \sqrt {x^3-1} \sqrt {x}}+\frac {a \sqrt {x^4-x} x}{3 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 206

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

Rule 217

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

Rule 275

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

Rule 329

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 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 581

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(f*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*g*(m + n*(p + q + 1) + 1)), x] + Dis
t[1/(b*(m + n*(p + q + 1) + 1)), Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*((b*e - a*f)*(m + 1) + b
*e*n*(p + q + 1)) + (d*(b*e - a*f)*(m + 1) + f*n*q*(b*c - a*d) + b*e*d*n*(p + q + 1))*x^n, x], x], x] /; FreeQ
[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0] && GtQ[q, 0] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^n, c + d*x^n])

Rule 584

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy
mbol] :> Int[ExpandIntegrand[((g*x)^m*(a + b*x^n)^p*(e + f*x^n))/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, g, m, p}, x] && IGtQ[n, 0]

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]

Rubi steps

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

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Mathematica [C]  time = 0.61, size = 186, normalized size = 1.48 \begin {gather*} \frac {\sqrt {1-x^3} x^5 \sqrt {\frac {x^3 (d-c)}{d}} (a (c-2 d)-2 b c) F_1\left (\frac {3}{2};\frac {1}{2},1;\frac {5}{2};x^3,\frac {c x^3}{d}\right )+3 x^2 \left (\sqrt {1-x^3} (a d+2 b c) \sin ^{-1}\left (\frac {\sqrt {\frac {x^3 (d-c)}{d}}}{\sqrt {1-\frac {c x^3}{d}}}\right )+a d \left (x^3-1\right ) \sqrt {\frac {x^3 (d-c)}{d}}\right )}{9 c d \sqrt {x \left (x^3-1\right )} \sqrt {\frac {x^3 (d-c)}{d}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.66, size = 126, normalized size = 1.00 \begin {gather*} \frac {a x \sqrt {-x+x^4}}{3 c}+\frac {2 \sqrt {c-d} (b c+a d) \tan ^{-1}\left (\frac {\sqrt {c-d} x \sqrt {-x+x^4}}{\sqrt {d} (-1+x) \left (1+x+x^2\right )}\right )}{3 c^2 \sqrt {d}}+\frac {(-a c+2 b c+2 a d) \tanh ^{-1}\left (\frac {x^2}{\sqrt {-x+x^4}}\right )}{3 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 4.08, size = 288, normalized size = 2.29 \begin {gather*} \left [\frac {2 \, \sqrt {x^{4} - x} a c x + {\left (b c + a d\right )} \sqrt {-\frac {c - d}{d}} \log \left (-\frac {{\left (c^{2} - 8 \, c d + 8 \, d^{2}\right )} x^{6} + 2 \, {\left (3 \, c d - 4 \, d^{2}\right )} x^{3} + d^{2} + 4 \, {\left ({\left (c d - 2 \, d^{2}\right )} x^{4} + d^{2} x\right )} \sqrt {x^{4} - x} \sqrt {-\frac {c - d}{d}}}{c^{2} x^{6} - 2 \, c d x^{3} + d^{2}}\right ) - {\left ({\left (a - 2 \, b\right )} c - 2 \, a d\right )} \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} - x} x + 1\right )}{6 \, c^{2}}, \frac {2 \, \sqrt {x^{4} - x} a c x + 2 \, {\left (b c + a d\right )} \sqrt {\frac {c - d}{d}} \arctan \left (-\frac {2 \, \sqrt {x^{4} - x} d x \sqrt {\frac {c - d}{d}}}{{\left (c - 2 \, d\right )} x^{3} + d}\right ) - {\left ({\left (a - 2 \, b\right )} c - 2 \, a d\right )} \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} - x} x + 1\right )}{6 \, c^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.72, size = 136, normalized size = 1.08 \begin {gather*} \frac {\sqrt {x^{4} - x} a x}{3 \, c} - \frac {{\left (a c - 2 \, b c - 2 \, a d\right )} \log \left (\sqrt {-\frac {1}{x^{3}} + 1} + 1\right )}{6 \, c^{2}} + \frac {{\left (a c - 2 \, b c - 2 \, a d\right )} \log \left ({\left | \sqrt {-\frac {1}{x^{3}} + 1} - 1 \right |}\right )}{6 \, c^{2}} - \frac {2 \, {\left (b c^{2} + a c d - b c d - a d^{2}\right )} \arctan \left (\frac {d \sqrt {-\frac {1}{x^{3}} + 1}}{\sqrt {c d - d^{2}}}\right )}{3 \, \sqrt {c d - d^{2}} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.40, size = 701, normalized size = 5.56

method	result	size
elliptic	\(\frac {a x \sqrt {x^{4}-x}}{3 c}+\frac {2 \left (-\frac {a c -a d -b c}{c^{2}}+\frac {a}{2 c}\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 \sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (c \,\textit {\_Z}^{3}-d \right )}{\sum }\frac {\left (a c d -a \,d^{2}+b \,c^{2}-b c d \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} 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 \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}\, c -3 \underline {\hspace {1.25 ex}}\alpha ^{2} 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 \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^{2}}\)	\(701\)
risch	\(\frac {a \,x^{2} \left (x^{3}-1\right )}{3 c \sqrt {x \left (x^{3}-1\right )}}-\frac {\frac {2 \left (a c -2 a d -2 b c \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 )}{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 {4 \left (a c d -a \,d^{2}+b \,c^{2}-b c d \right ) \sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (c \,\textit {\_Z}^{3}-d \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} 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 \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}\, c -3 \underline {\hspace {1.25 ex}}\alpha ^{2} 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 \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}}{2 c}\)	\(705\)
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 ) \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 )}{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 {2 \sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (c \,\textit {\_Z}^{3}-d \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} 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 \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}\, c -3 \underline {\hspace {1.25 ex}}\alpha ^{2} 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 \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}\right )}{c}\)	\(953\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*x*(x^4-x)^(1/2)/c+2*(-(a*c-a*d-b*c)/c^2+1/2/c*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)*(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)))-2/3/c^2*4^(1/2)*sum((a*c*d-a*d^2+b*c^2-b*c*d)/_alpha*(-1+x)^2*(_alph
a^2+_alpha+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^2*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*_alpha^2*3^(1/2)*c-3*_alpha^2*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^3*c-
d))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(-((x^4 - x)^(1/2)*(b + a*x^3))/(d - c*x^3), 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^{3} + b\right )}{c x^{3} - d}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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