3.323 \(\int \frac{x^3 (a+b x^2+c x^4)^{3/2}}{d+e x^2} \, dx\)

Optimal. Leaf size=360 \[ -\frac{\sqrt{a+b x^2+c x^4} \left (-2 c e x^2 \left (-4 c e (2 b d-3 a e)-3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (2 b d-3 a e)+3 b^3 e^3+64 c^3 d^3\right )}{128 c^2 e^4}+\frac{\left (8 b^2 c e^3 (b d-3 a e)-192 c^3 d^2 e (b d-a e)+48 c^2 e^2 (b d-a e)^2+3 b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{256 c^{5/2} e^5}-\frac{d \left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^5}-\frac{\left (a+b x^2+c x^4\right )^{3/2} \left (-3 b e+8 c d-6 c e x^2\right )}{48 c e^2} \]

[Out]

-((64*c^3*d^3 + 3*b^3*e^3 - 16*c^2*d*e*(5*b*d - 4*a*e) + 4*b*c*e^2*(2*b*d - 3*a*e) - 2*c*e*(16*c^2*d^2 - 3*b^2
*e^2 - 4*c*e*(2*b*d - 3*a*e))*x^2)*Sqrt[a + b*x^2 + c*x^4])/(128*c^2*e^4) - ((8*c*d - 3*b*e - 6*c*e*x^2)*(a +
b*x^2 + c*x^4)^(3/2))/(48*c*e^2) + ((128*c^4*d^4 + 3*b^4*e^4 + 8*b^2*c*e^3*(b*d - 3*a*e) - 192*c^3*d^2*e*(b*d
- a*e) + 48*c^2*e^2*(b*d - a*e)^2)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(256*c^(5/2)*e^
5) - (d*(c*d^2 - b*d*e + a*e^2)^(3/2)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x^2)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]
*Sqrt[a + b*x^2 + c*x^4])])/(2*e^5)

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Rubi [A]  time = 0.69676, antiderivative size = 360, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {1251, 814, 843, 621, 206, 724} \[ -\frac{\sqrt{a+b x^2+c x^4} \left (-2 c e x^2 \left (-4 c e (2 b d-3 a e)-3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (2 b d-3 a e)+3 b^3 e^3+64 c^3 d^3\right )}{128 c^2 e^4}+\frac{\left (8 b^2 c e^3 (b d-3 a e)-192 c^3 d^2 e (b d-a e)+48 c^2 e^2 (b d-a e)^2+3 b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{256 c^{5/2} e^5}-\frac{d \left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^5}-\frac{\left (a+b x^2+c x^4\right )^{3/2} \left (-3 b e+8 c d-6 c e x^2\right )}{48 c e^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2),x]

[Out]

-((64*c^3*d^3 + 3*b^3*e^3 - 16*c^2*d*e*(5*b*d - 4*a*e) + 4*b*c*e^2*(2*b*d - 3*a*e) - 2*c*e*(16*c^2*d^2 - 3*b^2
*e^2 - 4*c*e*(2*b*d - 3*a*e))*x^2)*Sqrt[a + b*x^2 + c*x^4])/(128*c^2*e^4) - ((8*c*d - 3*b*e - 6*c*e*x^2)*(a +
b*x^2 + c*x^4)^(3/2))/(48*c*e^2) + ((128*c^4*d^4 + 3*b^4*e^4 + 8*b^2*c*e^3*(b*d - 3*a*e) - 192*c^3*d^2*e*(b*d
- a*e) + 48*c^2*e^2*(b*d - a*e)^2)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(256*c^(5/2)*e^
5) - (d*(c*d^2 - b*d*e + a*e^2)^(3/2)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x^2)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]
*Sqrt[a + b*x^2 + c*x^4])])/(2*e^5)

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 814

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*(a + b*x + c*x^
2)^p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

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 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{x^3 \left (a+b x^2+c x^4\right )^{3/2}}{d+e x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx,x,x^2\right )\\ &=-\frac{\left (8 c d-3 b e-6 c e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{48 c e^2}-\frac{\operatorname{Subst}\left (\int \frac{\left (\frac{1}{2} d \left (4 a c e-2 b \left (4 c d-\frac{3 b e}{2}\right )\right )-\frac{1}{2} \left (16 c^2 d^2-3 b^2 e^2-4 c e (2 b d-3 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{d+e x} \, dx,x,x^2\right )}{16 c e^2}\\ &=-\frac{\left (64 c^3 d^3+3 b^3 e^3-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (2 b d-3 a e)-2 c e \left (16 c^2 d^2-3 b^2 e^2-4 c e (2 b d-3 a e)\right ) x^2\right ) \sqrt{a+b x^2+c x^4}}{128 c^2 e^4}-\frac{\left (8 c d-3 b e-6 c e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{48 c e^2}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{1}{4} d \left (4 c e (b d-2 a e) \left (8 b c d-3 b^2 e-4 a c e\right )-\left (4 b c d-b^2 e-4 a c e\right ) \left (16 c^2 d^2-3 b^2 e^2-4 c e (2 b d-3 a e)\right )\right )+\frac{1}{4} \left (128 c^4 d^4+3 b^4 e^4+8 b^2 c e^3 (b d-3 a e)-192 c^3 d^2 e (b d-a e)+48 c^2 e^2 (b d-a e)^2\right ) x}{(d+e x) \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{64 c^2 e^4}\\ &=-\frac{\left (64 c^3 d^3+3 b^3 e^3-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (2 b d-3 a e)-2 c e \left (16 c^2 d^2-3 b^2 e^2-4 c e (2 b d-3 a e)\right ) x^2\right ) \sqrt{a+b x^2+c x^4}}{128 c^2 e^4}-\frac{\left (8 c d-3 b e-6 c e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{48 c e^2}-\frac{\left (d \left (c d^2-b d e+a e^2\right )^2\right ) \operatorname{Subst}\left (\int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{2 e^5}+\frac{\left (128 c^4 d^4+3 b^4 e^4+8 b^2 c e^3 (b d-3 a e)-192 c^3 d^2 e (b d-a e)+48 c^2 e^2 (b d-a e)^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{256 c^2 e^5}\\ &=-\frac{\left (64 c^3 d^3+3 b^3 e^3-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (2 b d-3 a e)-2 c e \left (16 c^2 d^2-3 b^2 e^2-4 c e (2 b d-3 a e)\right ) x^2\right ) \sqrt{a+b x^2+c x^4}}{128 c^2 e^4}-\frac{\left (8 c d-3 b e-6 c e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{48 c e^2}+\frac{\left (d \left (c d^2-b d e+a e^2\right )^2\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac{-b d+2 a e-(2 c d-b e) x^2}{\sqrt{a+b x^2+c x^4}}\right )}{e^5}+\frac{\left (128 c^4 d^4+3 b^4 e^4+8 b^2 c e^3 (b d-3 a e)-192 c^3 d^2 e (b d-a e)+48 c^2 e^2 (b d-a e)^2\right ) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x^2}{\sqrt{a+b x^2+c x^4}}\right )}{128 c^2 e^5}\\ &=-\frac{\left (64 c^3 d^3+3 b^3 e^3-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (2 b d-3 a e)-2 c e \left (16 c^2 d^2-3 b^2 e^2-4 c e (2 b d-3 a e)\right ) x^2\right ) \sqrt{a+b x^2+c x^4}}{128 c^2 e^4}-\frac{\left (8 c d-3 b e-6 c e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{48 c e^2}+\frac{\left (128 c^4 d^4+3 b^4 e^4+8 b^2 c e^3 (b d-3 a e)-192 c^3 d^2 e (b d-a e)+48 c^2 e^2 (b d-a e)^2\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{256 c^{5/2} e^5}-\frac{d \left (c d^2-b d e+a e^2\right )^{3/2} \tanh ^{-1}\left (\frac{b d-2 a e+(2 c d-b e) x^2}{2 \sqrt{c d^2-b d e+a e^2} \sqrt{a+b x^2+c x^4}}\right )}{2 e^5}\\ \end{align*}

Mathematica [A]  time = 0.603905, size = 344, normalized size = 0.96 \[ \frac{3 \left (8 b^2 c e^3 (b d-3 a e)-192 c^3 d^2 e (b d-a e)+48 c^2 e^2 (b d-a e)^2+3 b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )+2 \sqrt{c} \left (e \sqrt{a+b x^2+c x^4} \left (8 c^2 e \left (a e \left (15 e x^2-32 d\right )+b \left (30 d^2-14 d e x^2+9 e^2 x^4\right )\right )+6 b c e^2 \left (10 a e-4 b d+b e x^2\right )-9 b^3 e^3-16 c^3 \left (-6 d^2 e x^2+12 d^3+4 d e^2 x^4-3 e^3 x^6\right )\right )+192 c^2 d \left (e (a e-b d)+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{2 a e-b d+b e x^2-2 c d x^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{e (a e-b d)+c d^2}}\right )\right )}{768 c^{5/2} e^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2),x]

[Out]

(3*(128*c^4*d^4 + 3*b^4*e^4 + 8*b^2*c*e^3*(b*d - 3*a*e) - 192*c^3*d^2*e*(b*d - a*e) + 48*c^2*e^2*(b*d - a*e)^2
)*ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])] + 2*Sqrt[c]*(e*Sqrt[a + b*x^2 + c*x^4]*(-9*b^3*e^
3 + 6*b*c*e^2*(-4*b*d + 10*a*e + b*e*x^2) - 16*c^3*(12*d^3 - 6*d^2*e*x^2 + 4*d*e^2*x^4 - 3*e^3*x^6) + 8*c^2*e*
(a*e*(-32*d + 15*e*x^2) + b*(30*d^2 - 14*d*e*x^2 + 9*e^2*x^4))) + 192*c^2*d*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*A
rcTanh[(-(b*d) + 2*a*e - 2*c*d*x^2 + b*e*x^2)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + b*x^2 + c*x^4])]))/(7
68*c^(5/2)*e^5)

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Maple [B]  time = 0.01, size = 1696, normalized size = 4.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*d^2/e^3*a*c^(1/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))+3/16*d^2/e^3*b^2*ln((1/2*b+c*x^2)/c^(1/2
)+(c*x^4+b*x^2+a)^(1/2))/c^(1/2)-3/4*d^3/e^4*b*c^(1/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))+1/2*d/e
^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x^2+d/e)+2*((a*e^2-b*d*e+c*d^2
)/e^2)^(1/2)*(c*(x^2+d/e)^2+(b*e-2*c*d)/e*(x^2+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x^2+d/e))*a^2+1/2*d^3/e^4
/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x^2+d/e)+2*((a*e^2-b*d*e+c*d^2)/
e^2)^(1/2)*(c*(x^2+d/e)^2+(b*e-2*c*d)/e*(x^2+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x^2+d/e))*b^2+1/2*d^5/e^6/(
(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x^2+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^
2)^(1/2)*(c*(x^2+d/e)^2+(b*e-2*c*d)/e*(x^2+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x^2+d/e))*c^2-1/6*d/e^2*c*x^4
*(c*x^4+b*x^2+a)^(1/2)+5/32/e*a*b/c*(c*x^4+b*x^2+a)^(1/2)-3/32/e*a*b^2/c^(3/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4
+b*x^2+a)^(1/2))+1/64/e*b^2*x^2/c*(c*x^4+b*x^2+a)^(1/2)+1/32*d/e^2*b^3/c^(3/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4
+b*x^2+a)^(1/2))+1/4*d^2/e^3*x^2*c*(c*x^4+b*x^2+a)^(1/2)-7/24*d/e^2*b*x^2*(c*x^4+b*x^2+a)^(1/2)-1/16*d/e^2/c*b
^2*(c*x^4+b*x^2+a)^(1/2)+5/8*d^2/e^3*b*(c*x^4+b*x^2+a)^(1/2)-1/2*d^3/e^4*c*(c*x^4+b*x^2+a)^(1/2)+1/2*d^4/e^5*c
^(3/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))-2/3*d/e^2*a*(c*x^4+b*x^2+a)^(1/2)+5/16/e*a*x^2*(c*x^4+b
*x^2+a)^(1/2)-3/128/e*b^3/c^2*(c*x^4+b*x^2+a)^(1/2)+3/256/e*b^4/c^(5/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+
a)^(1/2))+3/16/e*a^2*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))/c^(1/2)+1/8/e*c*x^6*(c*x^4+b*x^2+a)^(1/2)
+3/16/e*b*x^4*(c*x^4+b*x^2+a)^(1/2)-3/8*d/e^2*a*b*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))/c^(1/2)-d^2/
e^3/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x^2+d/e)+2*((a*e^2-b*d*e+c*d^
2)/e^2)^(1/2)*(c*(x^2+d/e)^2+(b*e-2*c*d)/e*(x^2+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x^2+d/e))*a*b+d^3/e^4/((
a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x^2+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2
)^(1/2)*(c*(x^2+d/e)^2+(b*e-2*c*d)/e*(x^2+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x^2+d/e))*a*c-d^4/e^5/((a*e^2-
b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x^2+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2
)*(c*(x^2+d/e)^2+(b*e-2*c*d)/e*(x^2+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x^2+d/e))*b*c

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{d + e x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*(a + b*x**2 + c*x**4)**(3/2)/(d + e*x**2), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError