3.1.71 \(\int \frac {x^3}{(a+c x^2)^{3/2} (d+e x+f x^2)} \, dx\) [71]

Optimal. Leaf size=499 \[ -\frac {1}{c f \sqrt {a+c x^2}}-\frac {e x}{a f^2 \sqrt {a+c x^2}}+\frac {a f \left (c d^2+a \left (e^2-d f\right )\right )+c e \left (c d^2+a \left (e^2-2 d f\right )\right ) x}{a f^2 \left (a c e^2+(c d-a f)^2\right ) \sqrt {a+c x^2}}-\frac {\left (2 a d e f-\left (e-\sqrt {e^2-4 d f}\right ) \left (c d^2+a \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e-\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \left (a c e^2+(c d-a f)^2\right ) \sqrt {2 a f^2+c \left (e^2-2 d f-e \sqrt {e^2-4 d f}\right )}}+\frac {\left (2 a d e f-\left (e+\sqrt {e^2-4 d f}\right ) \left (c d^2+a \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c \left (e+\sqrt {e^2-4 d f}\right ) x}{\sqrt {2} \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )} \sqrt {a+c x^2}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \left (a c e^2+(c d-a f)^2\right ) \sqrt {2 a f^2+c \left (e^2-2 d f+e \sqrt {e^2-4 d f}\right )}} \]

[Out]

-1/c/f/(c*x^2+a)^(1/2)-e*x/a/f^2/(c*x^2+a)^(1/2)+(a*f*(c*d^2+a*(-d*f+e^2))+c*e*(c*d^2+a*(-2*d*f+e^2))*x)/a/f^2
/(a*c*e^2+(-a*f+c*d)^2)/(c*x^2+a)^(1/2)-1/2*arctanh(1/2*(2*a*f-c*x*(e-(-4*d*f+e^2)^(1/2)))*2^(1/2)/(c*x^2+a)^(
1/2)/(2*a*f^2+c*(e^2-2*d*f-e*(-4*d*f+e^2)^(1/2)))^(1/2))*(2*a*d*e*f-(c*d^2+a*(-d*f+e^2))*(e-(-4*d*f+e^2)^(1/2)
))/(a*c*e^2+(-a*f+c*d)^2)*2^(1/2)/(-4*d*f+e^2)^(1/2)/(2*a*f^2+c*(e^2-2*d*f-e*(-4*d*f+e^2)^(1/2)))^(1/2)+1/2*ar
ctanh(1/2*(2*a*f-c*x*(e+(-4*d*f+e^2)^(1/2)))*2^(1/2)/(c*x^2+a)^(1/2)/(2*a*f^2+c*(e^2-2*d*f+e*(-4*d*f+e^2)^(1/2
)))^(1/2))*(2*a*d*e*f-(c*d^2+a*(-d*f+e^2))*(e+(-4*d*f+e^2)^(1/2)))/(a*c*e^2+(-a*f+c*d)^2)*2^(1/2)/(-4*d*f+e^2)
^(1/2)/(2*a*f^2+c*(e^2-2*d*f+e*(-4*d*f+e^2)^(1/2)))^(1/2)

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Rubi [A]
time = 1.34, antiderivative size = 499, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6860, 197, 267, 1031, 1048, 739, 212} \begin {gather*} \frac {c e x \left (a \left (e^2-2 d f\right )+c d^2\right )+a f \left (a \left (e^2-d f\right )+c d^2\right )}{a f^2 \sqrt {a+c x^2} \left ((c d-a f)^2+a c e^2\right )}-\frac {\left (2 a d e f-\left (e-\sqrt {e^2-4 d f}\right ) \left (a \left (e^2-d f\right )+c d^2\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c x \left (e-\sqrt {e^2-4 d f}\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \left ((c d-a f)^2+a c e^2\right ) \sqrt {2 a f^2+c \left (-e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}+\frac {\left (2 a d e f-\left (\sqrt {e^2-4 d f}+e\right ) \left (a \left (e^2-d f\right )+c d^2\right )\right ) \tanh ^{-1}\left (\frac {2 a f-c x \left (\sqrt {e^2-4 d f}+e\right )}{\sqrt {2} \sqrt {a+c x^2} \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}\right )}{\sqrt {2} \sqrt {e^2-4 d f} \left ((c d-a f)^2+a c e^2\right ) \sqrt {2 a f^2+c \left (e \sqrt {e^2-4 d f}-2 d f+e^2\right )}}-\frac {e x}{a f^2 \sqrt {a+c x^2}}-\frac {1}{c f \sqrt {a+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/((a + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]

[Out]

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

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 212

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

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 1031

Int[((g_.) + (h_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp
[(a + c*x^2)^(p + 1)*((d + e*x + f*x^2)^(q + 1)/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1)))*(g*c*(2*a*c*e) +
((-a)*h)*(2*c^2*d - c*(2*a*f)) + c*(g*(2*c^2*d - c*(2*a*f)) - h*(-2*a*c*e))*x), x] + Dist[1/((-4*a*c)*(a*c*e^
2 + (c*d - a*f)^2)*(p + 1)), Int[(a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(-2*g*c)*((c*d - a*f)^2 - ((-a)*
e)*(c*e))*(p + 1) + (2*(g*c*(c*d - a*f) - a*((-h)*c*e)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((g*c)*(2*a*c*e) + ((
-a)*h)*(2*c^2*d - c*((Plus[2])*a*f)))*(p + q + 2) - (2*f*((g*c)*(2*a*c*e) + ((-a)*h)*(2*c^2*d + (-c)*((Plus[2]
)*a*f)))*(p + q + 2) - (2*(g*c*(c*d - a*f) - a*((-h)*c*e)))*((-c)*e*(2*p + q + 4)))*x - c*f*(2*(g*c*(c*d - a*f
) - a*((-h)*c*e)))*(2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, g, h, q}, x] && NeQ[e^2 - 4*d*f, 0
] && LtQ[p, -1] && NeQ[a*c*e^2 + (c*d - a*f)^2, 0] &&  !( !IntegerQ[p] && ILtQ[q, -1])

Rule 1048

Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q
= Rt[b^2 - 4*a*c, 2]}, Dist[(2*c*g - h*(b - q))/q, Int[1/((b - q + 2*c*x)*Sqrt[d + f*x^2]), x], x] - Dist[(2*c
*g - h*(b + q))/q, Int[1/((b + q + 2*c*x)*Sqrt[d + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, f, g, h}, x] && NeQ[
b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.61, size = 405, normalized size = 0.81 \begin {gather*} \frac {-a (-c d+a f+c e x)+c \sqrt {a+c x^2} \text {RootSum}\left [a^2 f+2 a \sqrt {c} e \text {$\#$1}+4 c d \text {$\#$1}^2-2 a f \text {$\#$1}^2-2 \sqrt {c} e \text {$\#$1}^3+f \text {$\#$1}^4\&,\frac {-a c d^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )-a^2 e^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )+a^2 d f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right )-2 a \sqrt {c} d e \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}+c d^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+a e^2 \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-a d f \log \left (-\sqrt {c} x+\sqrt {a+c x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{a \sqrt {c} e+4 c d \text {$\#$1}-2 a f \text {$\#$1}-3 \sqrt {c} e \text {$\#$1}^2+2 f \text {$\#$1}^3}\&\right ]}{c \left (c^2 d^2+a^2 f^2+a c \left (e^2-2 d f\right )\right ) \sqrt {a+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/((a + c*x^2)^(3/2)*(d + e*x + f*x^2)),x]

[Out]

(-(a*(-(c*d) + a*f + c*e*x)) + c*Sqrt[a + c*x^2]*RootSum[a^2*f + 2*a*Sqrt[c]*e*#1 + 4*c*d*#1^2 - 2*a*f*#1^2 -
2*Sqrt[c]*e*#1^3 + f*#1^4 & , (-(a*c*d^2*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]) - a^2*e^2*Log[-(Sqrt[c]*x)
+ Sqrt[a + c*x^2] - #1] + a^2*d*f*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1] - 2*a*Sqrt[c]*d*e*Log[-(Sqrt[c]*x)
+ Sqrt[a + c*x^2] - #1]*#1 + c*d^2*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1^2 + a*e^2*Log[-(Sqrt[c]*x) + Sq
rt[a + c*x^2] - #1]*#1^2 - a*d*f*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2] - #1]*#1^2)/(a*Sqrt[c]*e + 4*c*d*#1 - 2*a*
f*#1 - 3*Sqrt[c]*e*#1^2 + 2*f*#1^3) & ])/(c*(c^2*d^2 + a^2*f^2 + a*c*(e^2 - 2*d*f))*Sqrt[a + c*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1575\) vs. \(2(456)=912\).
time = 0.11, size = 1576, normalized size = 3.16

method	result	size
default	\(\text {Expression too large to display}\)	\(1576\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(c*x^2+a)^(3/2)/(f*x^2+e*x+d),x,method=_RETURNVERBOSE)

[Out]

-1/c/f/(c*x^2+a)^(1/2)-e*x/a/f^2/(c*x^2+a)^(1/2)+1/2*(e^3-3*d*e*f+e^2*(-4*d*f+e^2)^(1/2)-d*f*(-4*d*f+e^2)^(1/2
))/f^3/(-4*d*f+e^2)^(1/2)*(2/((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)*f^2/((x+1/2*(e+(-4*d*f+e^2)^(1/2))
/f)^2*c-c*(e+(-4*d*f+e^2)^(1/2))/f*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*
f+c*e^2)/f^2)^(1/2)+2*c*(e+(-4*d*f+e^2)^(1/2))*f/((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)*(2*c*(x+1/2*(e
+(-4*d*f+e^2)^(1/2))/f)-c*(e+(-4*d*f+e^2)^(1/2))/f)/(2*c*((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2-c^
2*(e+(-4*d*f+e^2)^(1/2))^2/f^2)/((x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)^2*c-c*(e+(-4*d*f+e^2)^(1/2))/f*(x+1/2*(e+(-4
*d*f+e^2)^(1/2))/f)+1/2*((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2)-2/((-4*d*f+e^2)^(1/2)*c*e+2*
a*f^2-2*c*d*f+c*e^2)*f^2*2^(1/2)/(((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2)*ln((((-4*d*f+e^2)^
(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2-c*(e+(-4*d*f+e^2)^(1/2))/f*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+1/2*2^(1/2)*(
((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2)*(4*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)^2*c-4*c*(e+(-4*d
*f+e^2)^(1/2))/f*(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)+2*((-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2))
/(x+1/2*(e+(-4*d*f+e^2)^(1/2))/f)))+1/2*(-d*f*(-4*d*f+e^2)^(1/2)+e^2*(-4*d*f+e^2)^(1/2)+3*d*e*f-e^3)/f^3/(-4*d
*f+e^2)^(1/2)*(2/(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)*f^2/((x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))^2*c-c*
(e-(-4*d*f+e^2)^(1/2))/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+1/2*(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)
/f^2)^(1/2)+2*c*(e-(-4*d*f+e^2)^(1/2))*f/(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)*(2*c*(x-1/2/f*(-e+(-4
*d*f+e^2)^(1/2)))-c*(e-(-4*d*f+e^2)^(1/2))/f)/(2*c*(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2-c^2*(e-
(-4*d*f+e^2)^(1/2))^2/f^2)/((x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))^2*c-c*(e-(-4*d*f+e^2)^(1/2))/f*(x-1/2/f*(-e+(-4*
d*f+e^2)^(1/2)))+1/2*(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2)-2/(-(-4*d*f+e^2)^(1/2)*c*e+2*a
*f^2-2*c*d*f+c*e^2)*f^2*2^(1/2)/((-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2)*ln(((-(-4*d*f+e^2)
^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2-c*(e-(-4*d*f+e^2)^(1/2))/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+1/2*2^(1/2)
*((-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^(1/2)*(4*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))^2*c-4*c*(e-(
-4*d*f+e^2)^(1/2))/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))+2*(-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-2*c*d*f+c*e^2)/f^2)^
(1/2))/(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2)))))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(c*x^2+a)^(3/2)/(f*x^2+e*x+d),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*d*f-%e^2>0)', see `assume?`
for more det

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 26071 vs. \(2 (465) = 930\).
time = 79.47, size = 26071, normalized size = 52.25 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(c*x^2+a)^(3/2)/(f*x^2+e*x+d),x, algorithm="fricas")

[Out]

-1/4*(sqrt(2)*(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2 + (c^4*d^2 - 2*a*c^3*d*f + a^2*c^2*f^2)*x^2 + (a*c^3*x^2
+ a^2*c^2)*e^2)*sqrt(-(2*c^3*d^6 - 6*a*c^2*d^5*f + 6*a^2*c*d^4*f^2 - 2*a^3*d^3*f^3 + a^3*e^6 + 3*(a^2*c*d^2 -
2*a^3*d*f)*e^4 + 3*(a*c^2*d^4 - 4*a^2*c*d^3*f + 3*a^3*d^2*f^2)*e^2 + (4*c^6*d^7*f - 24*a*c^5*d^6*f^2 + 60*a^2*
c^4*d^5*f^3 - 80*a^3*c^3*d^4*f^4 + 60*a^4*c^2*d^3*f^5 - 24*a^5*c*d^2*f^6 + 4*a^6*d*f^7 - a^3*c^3*e^8 - (3*a^2*
c^4*d^2 - 10*a^3*c^3*d*f + 3*a^4*c^2*f^2)*e^6 - 3*(a*c^5*d^4 - 8*a^2*c^4*d^3*f + 14*a^3*c^3*d^2*f^2 - 8*a^4*c^
2*d*f^3 + a^5*c*f^4)*e^4 - (c^6*d^6 - 18*a*c^5*d^5*f + 63*a^2*c^4*d^4*f^2 - 92*a^3*c^3*d^3*f^3 + 63*a^4*c^2*d^
2*f^4 - 18*a^5*c*d*f^5 + a^6*f^6)*e^2)*sqrt(-(a^6*e^10 + 2*(3*a^5*c*d^2 - 4*a^6*d*f)*e^8 + (15*a^4*c^2*d^4 - 3
6*a^5*c*d^3*f + 22*a^6*d^2*f^2)*e^6 + 6*(3*a^3*c^3*d^6 - 10*a^4*c^2*d^5*f + 11*a^5*c*d^4*f^2 - 4*a^6*d^3*f^3)*
e^4 + 9*(a^2*c^4*d^8 - 4*a^3*c^3*d^7*f + 6*a^4*c^2*d^6*f^2 - 4*a^5*c*d^5*f^3 + a^6*d^4*f^4)*e^2)/(4*c^12*d^13*
f - 48*a*c^11*d^12*f^2 + 264*a^2*c^10*d^11*f^3 - 880*a^3*c^9*d^10*f^4 + 1980*a^4*c^8*d^9*f^5 - 3168*a^5*c^7*d^
8*f^6 + 3696*a^6*c^6*d^7*f^7 - 3168*a^7*c^5*d^6*f^8 + 1980*a^8*c^4*d^5*f^9 - 880*a^9*c^3*d^4*f^10 + 264*a^10*c
^2*d^3*f^11 - 48*a^11*c*d^2*f^12 + 4*a^12*d*f^13 - a^6*c^6*e^14 - 2*(3*a^5*c^7*d^2 - 8*a^6*c^6*d*f + 3*a^7*c^5
*f^2)*e^12 - 3*(5*a^4*c^8*d^4 - 28*a^5*c^7*d^3*f + 46*a^6*c^6*d^2*f^2 - 28*a^7*c^5*d*f^3 + 5*a^8*c^4*f^4)*e^10
- 20*(a^3*c^9*d^6 - 9*a^4*c^8*d^5*f + 27*a^5*c^7*d^4*f^2 - 38*a^6*c^6*d^3*f^3 + 27*a^7*c^5*d^2*f^4 - 9*a^8*c^
4*d*f^5 + a^9*c^3*f^6)*e^8 - 5*(3*a^2*c^10*d^8 - 40*a^3*c^9*d^7*f + 180*a^4*c^8*d^6*f^2 - 408*a^5*c^7*d^5*f^3
+ 530*a^6*c^6*d^4*f^4 - 408*a^7*c^5*d^3*f^5 + 180*a^8*c^4*d^2*f^6 - 40*a^9*c^3*d*f^7 + 3*a^10*c^2*f^8)*e^6 - 6
*(a*c^11*d^10 - 20*a^2*c^10*d^9*f + 125*a^3*c^9*d^8*f^2 - 400*a^4*c^8*d^7*f^3 + 770*a^5*c^7*d^6*f^4 - 952*a^6*
c^6*d^5*f^5 + 770*a^7*c^5*d^4*f^6 - 400*a^8*c^4*d^3*f^7 + 125*a^9*c^3*d^2*f^8 - 20*a^10*c^2*d*f^9 + a^11*c*f^1
0)*e^4 - (c^12*d^12 - 36*a*c^11*d^11*f + 306*a^2*c^10*d^10*f^2 - 1300*a^3*c^9*d^9*f^3 + 3375*a^4*c^8*d^8*f^4 -
5832*a^5*c^7*d^7*f^5 + 6972*a^6*c^6*d^6*f^6 - 5832*a^7*c^5*d^5*f^7 + 3375*a^8*c^4*d^4*f^8 - 1300*a^9*c^3*d^3*
f^9 + 306*a^10*c^2*d^2*f^10 - 36*a^11*c*d*f^11 + a^12*f^12)*e^2)))/(4*c^6*d^7*f - 24*a*c^5*d^6*f^2 + 60*a^2*c^
4*d^5*f^3 - 80*a^3*c^3*d^4*f^4 + 60*a^4*c^2*d^3*f^5 - 24*a^5*c*d^2*f^6 + 4*a^6*d*f^7 - a^3*c^3*e^8 - (3*a^2*c^
4*d^2 - 10*a^3*c^3*d*f + 3*a^4*c^2*f^2)*e^6 - 3*(a*c^5*d^4 - 8*a^2*c^4*d^3*f + 14*a^3*c^3*d^2*f^2 - 8*a^4*c^2*
d*f^3 + a^5*c*f^4)*e^4 - (c^6*d^6 - 18*a*c^5*d^5*f + 63*a^2*c^4*d^4*f^2 - 92*a^3*c^3*d^3*f^3 + 63*a^4*c^2*d^2*
f^4 - 18*a^5*c*d*f^5 + a^6*f^6)*e^2))*log((4*a^3*c*d^4*x*e^5 - 2*a^4*d^3*e^6 + 4*(3*a^2*c^2*d^6 - 4*a^3*c*d^5*
f)*x*e^3 + 12*(a*c^3*d^8 - 2*a^2*c^2*d^7*f + a^3*c*d^6*f^2)*x*e + sqrt(2)*(a^5*e^10 + 5*(a^4*c*d^2 - 2*a^5*d*f
)*e^8 + (9*a^3*c^2*d^4 - 40*a^4*c*d^3*f + 35*a^5*d^2*f^2)*e^6 + 2*(3*a^2*c^3*d^6 - 27*a^3*c^2*d^5*f + 49*a^4*c
*d^4*f^2 - 25*a^5*d^3*f^3)*e^4 - 24*(a^2*c^3*d^7*f - 3*a^3*c^2*d^6*f^2 + 3*a^4*c*d^5*f^3 - a^5*d^4*f^4)*e^2 -
(8*c^8*d^11*f - 64*a*c^7*d^10*f^2 + 224*a^2*c^6*d^9*f^3 - 448*a^3*c^5*d^8*f^4 + 560*a^4*c^4*d^7*f^5 - 448*a^5*
c^3*d^6*f^6 + 224*a^6*c^2*d^5*f^7 - 64*a^7*c*d^4*f^8 + 8*a^8*d^3*f^9 - a^5*c^3*e^12 - (5*a^4*c^4*d^2 - 14*a^5*
c^3*d*f + 3*a^6*c^2*f^2)*e^10 - (11*a^3*c^5*d^4 - 60*a^4*c^4*d^3*f + 90*a^5*c^3*d^2*f^2 - 36*a^6*c^2*d*f^3 + 3
*a^7*c*f^4)*e^8 - (13*a^2*c^6*d^6 - 110*a^3*c^5*d^5*f + 295*a^4*c^4*d^4*f^2 - 340*a^5*c^3*d^3*f^3 + 171*a^6*c^
2*d^2*f^4 - 30*a^7*c*d*f^5 + a^8*f^6)*e^6 - 8*(a*c^7*d^8 - 13*a^2*c^6*d^7*f + 51*a^3*c^5*d^6*f^2 - 95*a^4*c^4*
d^5*f^3 + 95*a^5*c^3*d^4*f^4 - 51*a^6*c^2*d^3*f^5 + 13*a^7*c*d^2*f^6 - a^8*d*f^7)*e^4 - 2*(c^8*d^10 - 24*a*c^7
*d^9*f + 132*a^2*c^6*d^8*f^2 - 344*a^3*c^5*d^7*f^3 + 510*a^4*c^4*d^6*f^4 - 456*a^5*c^3*d^5*f^5 + 244*a^6*c^2*d
^4*f^6 - 72*a^7*c*d^3*f^7 + 9*a^8*d^2*f^8)*e^2)*sqrt(-(a^6*e^10 + 2*(3*a^5*c*d^2 - 4*a^6*d*f)*e^8 + (15*a^4*c^
2*d^4 - 36*a^5*c*d^3*f + 22*a^6*d^2*f^2)*e^6 + 6*(3*a^3*c^3*d^6 - 10*a^4*c^2*d^5*f + 11*a^5*c*d^4*f^2 - 4*a^6*
d^3*f^3)*e^4 + 9*(a^2*c^4*d^8 - 4*a^3*c^3*d^7*f + 6*a^4*c^2*d^6*f^2 - 4*a^5*c*d^5*f^3 + a^6*d^4*f^4)*e^2)/(4*c
^12*d^13*f - 48*a*c^11*d^12*f^2 + 264*a^2*c^10*d^11*f^3 - 880*a^3*c^9*d^10*f^4 + 1980*a^4*c^8*d^9*f^5 - 3168*a
^5*c^7*d^8*f^6 + 3696*a^6*c^6*d^7*f^7 - 3168*a^7*c^5*d^6*f^8 + 1980*a^8*c^4*d^5*f^9 - 880*a^9*c^3*d^4*f^10 + 2
64*a^10*c^2*d^3*f^11 - 48*a^11*c*d^2*f^12 + 4*a^12*d*f^13 - a^6*c^6*e^14 - 2*(3*a^5*c^7*d^2 - 8*a^6*c^6*d*f +
3*a^7*c^5*f^2)*e^12 - 3*(5*a^4*c^8*d^4 - 28*a^5*c^7*d^3*f + 46*a^6*c^6*d^2*f^2 - 28*a^7*c^5*d*f^3 + 5*a^8*c^4*
f^4)*e^10 - 20*(a^3*c^9*d^6 - 9*a^4*c^8*d^5*f + 27*a^5*c^7*d^4*f^2 - 38*a^6*c^6*d^3*f^3 + 27*a^7*c^5*d^2*f^4 -
9*a^8*c^4*d*f^5 + a^9*c^3*f^6)*e^8 - 5*(3*a^2*c^10*d^8 - 40*a^3*c^9*d^7*f + 180*a^4*c^8*d^6*f^2 - 408*a^5*c^7
*d^5*f^3 + 530*a^6*c^6*d^4*f^4 - 408*a^7*c^5*d^3*f^5 + 180*a^8*c^4*d^2*f^6 - 40*a^9*c^3*d*f^7 + 3*a^10*c^2*f^8
)*e^6 - 6*(a*c^11*d^10 - 20*a^2*c^10*d^9*f + 12...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x + f x^{2}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(c*x**2+a)**(3/2)/(f*x**2+e*x+d),x)

[Out]

Integral(x**3/((a + c*x**2)**(3/2)*(d + e*x + f*x**2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(c*x^2+a)^(3/2)/(f*x^2+e*x+d),x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3}{{\left (c\,x^2+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/((a + c*x^2)^(3/2)*(d + e*x + f*x^2)),x)

[Out]

int(x^3/((a + c*x^2)^(3/2)*(d + e*x + f*x^2)), x)

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