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

Optimal. Leaf size=125 \[ \frac {2 \sqrt {-c-d} (a d+b c) \tan ^{-1}\left (\frac {x \sqrt {x^4+x} \sqrt {-c-d}}{\sqrt {d} (x+1) \left (x^2-x+1\right )}\right )}{3 c^2 \sqrt {d}}+\frac {\tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right ) (a c+2 a d+2 b c)}{3 c^2}+\frac {a \sqrt {x^4+x} x}{3 c} \]

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Rubi [A]  time = 0.35, antiderivative size = 144, normalized size of antiderivative = 1.15, number of steps used = 11, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {2056, 581, 584, 329, 275, 215, 466, 465, 377, 208} \begin {gather*} \frac {\sqrt {x^4+x} \sinh ^{-1}\left (x^{3/2}\right ) (a (c+2 d)+2 b c)}{3 c^2 \sqrt {x^3+1} \sqrt {x}}-\frac {2 \sqrt {x^4+x} \sqrt {c+d} (a d+b c) \tanh ^{-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 {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*b*c + a*(c + 2*d))*Sqrt[x + x^4]*ArcSinh[x^(3/2)])/(3*c^2*Sqrt[x]*Sqrt[1 + x^3
]) - (2*Sqrt[c + d]*(b*c + a*d)*Sqrt[x + x^4]*ArcTanh[(Sqrt[c + d]*x^(3/2))/(Sqrt[d]*Sqrt[1 + x^3])])/(3*c^2*S
qrt[d]*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 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} (2 b c+a (c+2 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 (2 b c+a (c+2 d)) \sqrt {x}}{2 c \sqrt {1+x^3}}+\frac {\left (\frac {3}{2} c (2 b c+a d)+\frac {3}{2} d (2 b c+a (c+2 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 ((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 {\left ((2 b c+a (c+2 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 {a x \sqrt {x+x^4}}{3 c}+\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 {\left ((2 b c+a (c+2 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 {a x \sqrt {x+x^4}}{3 c}+\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 {\left ((2 b c+a (c+2 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 {a x \sqrt {x+x^4}}{3 c}+\frac {(2 b c+a (c+2 d)) \sqrt {x+x^4} \sinh ^{-1}\left (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}{-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 b c+a (c+2 d)) \sqrt {x+x^4} \sinh ^{-1}\left (x^{3/2}\right )}{3 c^2 \sqrt {x} \sqrt {1+x^3}}-\frac {2 \sqrt {c+d} (b c+a d) \sqrt {x+x^4} \tanh ^{-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}}\\ \end {align*}

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Mathematica [C]  time = 0.35, size = 170, normalized size = 1.36 \begin {gather*} -\frac {x \sqrt {x^4+x} \left (x^3 \sqrt {-\frac {x^3 (c+d)}{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 (a d+2 b c) \sin ^{-1}\left (\frac {\sqrt {-\frac {x^3 (c+d)}{d}}}{\sqrt {1-\frac {c x^3}{d}}}\right )-3 a d \sqrt {x^3+1} \sqrt {-\frac {x^3 (c+d)}{d}}\right )}{9 c d \sqrt {x^3+1} \sqrt {-\frac {x^3 (c+d)}{d}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.75, size = 125, 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 = 4.12, size = 266, normalized size = 2.13 \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 = 0.57, size = 128, normalized size = 1.02 \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)*l
og(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 = 5.61

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 +i \sqrt {3}\, d +3 \underline {\hspace {1.25 ex}}\alpha ^{2} c +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 +i \sqrt {3}\, d +3 \underline {\hspace {1.25 ex}}\alpha ^{2} c +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}\)	\(706\)
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 (-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 +i \sqrt {3}\, d +3 \underline {\hspace {1.25 ex}}\alpha ^{2} c +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}\right )}{c}\)	\(970\)

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*(_alpha^2-_al
pha+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+I*3^(1/2)*d+3*_alpha^2*c+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 {\left (a\,x^3+b\right )\,\sqrt {x^4+x}}{d-c\,x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(-((b + a*x^3)*(x + x^4)^(1/2))/(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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