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

Optimal. Leaf size=159 \[ \frac{c x^2 (b d-2 a e)+a (2 c d-b e)}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}-\frac{d e \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.191106, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {1251, 822, 12, 724, 206} \[ \frac{c x^2 (b d-2 a e)+a (2 c d-b e)}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}-\frac{d e \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*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] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

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])

Rubi steps

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

Mathematica [A]  time = 0.207123, size = 162, normalized size = 1.02 \[ \frac{a \left (b e-2 c d+2 c e x^2\right )-b c d x^2}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4} \left (e (b d-a e)-c d^2\right )}+\frac{d e \tanh ^{-1}\left (\frac{2 a e-b d+b e x^2-2 c d x^2}{2 \sqrt{a+b x^2+c x^4} \sqrt{e (a e-b d)+c d^2}}\right )}{2 \left (e (a e-b d)+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.01, size = 506, normalized size = 3.2 \begin{align*}{\frac{2\,c{x}^{2}+b}{e \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+2\,{\frac{cd}{e \left ( e\sqrt{-4\,ac+{b}^{2}}-be+2\,cd \right ) \left ( -4\,ac+{b}^{2} \right ) }\sqrt{c \left ({x}^{2}-1/2\,{\frac{-b+\sqrt{-4\,ac+{b}^{2}}}{c}} \right ) ^{2}+\sqrt{-4\,ac+{b}^{2}} \left ({x}^{2}-1/2\,{\frac{-b+\sqrt{-4\,ac+{b}^{2}}}{c}} \right ) } \left ({x}^{2}+1/2\,{\frac{b}{c}}-1/2\,{\frac{\sqrt{-4\,ac+{b}^{2}}}{c}} \right ) ^{-1}}-2\,{\frac{cd}{ \left ( e\sqrt{-4\,ac+{b}^{2}}-be+2\,cd \right ) \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) }\ln \left ({ \left ( 2\,{\frac{a{e}^{2}-deb+c{d}^{2}}{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ({x}^{2}+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}-deb+c{d}^{2}}{{e}^{2}}}}\sqrt{c \left ({x}^{2}+{\frac{d}{e}} \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({x}^{2}+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}-deb+c{d}^{2}}{{e}^{2}}}} \right ) \left ({x}^{2}+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}-deb+c{d}^{2}}{{e}^{2}}}}}}}-2\,{\frac{cd}{e \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) \left ( -4\,ac+{b}^{2} \right ) }\sqrt{c \left ({x}^{2}+1/2\,{\frac{b+\sqrt{-4\,ac+{b}^{2}}}{c}} \right ) ^{2}-\sqrt{-4\,ac+{b}^{2}} \left ({x}^{2}+1/2\,{\frac{b+\sqrt{-4\,ac+{b}^{2}}}{c}} \right ) } \left ({x}^{2}+1/2\,{\frac{\sqrt{-4\,ac+{b}^{2}}}{c}}+1/2\,{\frac{b}{c}} \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e*(2*c*x^2+b)/(4*a*c-b^2)/(c*x^4+b*x^2+a)^(1/2)+2*d/e*c/(e*(-4*a*c+b^2)^(1/2)-b*e+2*c*d)/(-4*a*c+b^2)/(x^2+1
/2*b/c-1/2*(-4*a*c+b^2)^(1/2)/c)*(c*(x^2-1/2*(-b+(-4*a*c+b^2)^(1/2))/c)^2+(-4*a*c+b^2)^(1/2)*(x^2-1/2*(-b+(-4*
a*c+b^2)^(1/2))/c))^(1/2)-2*d*c/(e*(-4*a*c+b^2)^(1/2)-b*e+2*c*d)/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)/((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))-2*d/e*c/(e*(-4*a*c+b^2)^(1/2)+
b*e-2*c*d)/(-4*a*c+b^2)/(x^2+1/2*(-4*a*c+b^2)^(1/2)/c+1/2*b/c)*(c*(x^2+1/2*(b+(-4*a*c+b^2)^(1/2))/c)^2-(-4*a*c
+b^2)^(1/2)*(x^2+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 5.57832, size = 2778, normalized size = 17.47 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(((b^2*c - 4*a*c^2)*d*e*x^4 + (b^3 - 4*a*b*c)*d*e*x^2 + (a*b^2 - 4*a^2*c)*d*e)*sqrt(c*d^2 - b*d*e + a*e^2
)*log(-((8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^4 - 8*a*b*d*e + 8*a^2*e^2 + (b^2 + 4*a*c)*d^2 + 2*(4*b*c
*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x^2 - 4*sqrt(c*x^4 + b*x^2 + a)*sqrt(c*d^2 - b*d*e + a*e^2)*((2*c*d -
b*e)*x^2 + b*d - 2*a*e))/(e^2*x^4 + 2*d*e*x^2 + d^2)) + 4*(2*a*c^2*d^3 - 3*a*b*c*d^2*e - a^2*b*e^3 + (a*b^2 +
2*a^2*c)*d*e^2 + (b*c^2*d^3 + 3*a*b*c*d*e^2 - 2*a^2*c*e^3 - (b^2*c + 2*a*c^2)*d^2*e)*x^2)*sqrt(c*x^4 + b*x^2 +
a))/((a*b^2*c^2 - 4*a^2*c^3)*d^4 - 2*(a*b^3*c - 4*a^2*b*c^2)*d^3*e + (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^2*e^
2 - 2*(a^2*b^3 - 4*a^3*b*c)*d*e^3 + (a^3*b^2 - 4*a^4*c)*e^4 + ((b^2*c^3 - 4*a*c^4)*d^4 - 2*(b^3*c^2 - 4*a*b*c^
3)*d^3*e + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d^2*e^2 - 2*(a*b^3*c - 4*a^2*b*c^2)*d*e^3 + (a^2*b^2*c - 4*a^3*c^
2)*e^4)*x^4 + ((b^3*c^2 - 4*a*b*c^3)*d^4 - 2*(b^4*c - 4*a*b^2*c^2)*d^3*e + (b^5 - 2*a*b^3*c - 8*a^2*b*c^2)*d^2
*e^2 - 2*(a*b^4 - 4*a^2*b^2*c)*d*e^3 + (a^2*b^3 - 4*a^3*b*c)*e^4)*x^2), -1/2*(((b^2*c - 4*a*c^2)*d*e*x^4 + (b^
3 - 4*a*b*c)*d*e*x^2 + (a*b^2 - 4*a^2*c)*d*e)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(c*x^4 + b*x^2 + a)
*sqrt(-c*d^2 + b*d*e - a*e^2)*((2*c*d - b*e)*x^2 + b*d - 2*a*e)/((c^2*d^2 - b*c*d*e + a*c*e^2)*x^4 + a*c*d^2 -
a*b*d*e + a^2*e^2 + (b*c*d^2 - b^2*d*e + a*b*e^2)*x^2)) - 2*(2*a*c^2*d^3 - 3*a*b*c*d^2*e - a^2*b*e^3 + (a*b^2
+ 2*a^2*c)*d*e^2 + (b*c^2*d^3 + 3*a*b*c*d*e^2 - 2*a^2*c*e^3 - (b^2*c + 2*a*c^2)*d^2*e)*x^2)*sqrt(c*x^4 + b*x^
2 + a))/((a*b^2*c^2 - 4*a^2*c^3)*d^4 - 2*(a*b^3*c - 4*a^2*b*c^2)*d^3*e + (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^2
*e^2 - 2*(a^2*b^3 - 4*a^3*b*c)*d*e^3 + (a^3*b^2 - 4*a^4*c)*e^4 + ((b^2*c^3 - 4*a*c^4)*d^4 - 2*(b^3*c^2 - 4*a*b
*c^3)*d^3*e + (b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d^2*e^2 - 2*(a*b^3*c - 4*a^2*b*c^2)*d*e^3 + (a^2*b^2*c - 4*a^3
*c^2)*e^4)*x^4 + ((b^3*c^2 - 4*a*b*c^3)*d^4 - 2*(b^4*c - 4*a*b^2*c^2)*d^3*e + (b^5 - 2*a*b^3*c - 8*a^2*b*c^2)*
d^2*e^2 - 2*(a*b^4 - 4*a^2*b^2*c)*d*e^3 + (a^2*b^3 - 4*a^3*b*c)*e^4)*x^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.23176, size = 595, normalized size = 3.74 \begin{align*} -\frac{d \arctan \left (-\frac{{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} + b d e - a e^{2}}}\right ) e}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{-c d^{2} + b d e - a e^{2}}} + \frac{\frac{{\left (b c^{2} d^{3} - b^{2} c d^{2} e - 2 \, a c^{2} d^{2} e + 3 \, a b c d e^{2} - 2 \, a^{2} c e^{3}\right )} x^{2}}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}} + \frac{2 \, a c^{2} d^{3} - 3 \, a b c d^{2} e + a b^{2} d e^{2} + 2 \, a^{2} c d e^{2} - a^{2} b e^{3}}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}}}{\sqrt{c x^{4} + b x^{2} + a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d*arctan(-((sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))*e/((c*d^2 - b
*d*e + a*e^2)*sqrt(-c*d^2 + b*d*e - a*e^2)) + ((b*c^2*d^3 - b^2*c*d^2*e - 2*a*c^2*d^2*e + 3*a*b*c*d*e^2 - 2*a^
2*c*e^3)*x^2/(b^2*c^2*d^4 - 4*a*c^3*d^4 - 2*b^3*c*d^3*e + 8*a*b*c^2*d^3*e + b^4*d^2*e^2 - 2*a*b^2*c*d^2*e^2 -
8*a^2*c^2*d^2*e^2 - 2*a*b^3*d*e^3 + 8*a^2*b*c*d*e^3 + a^2*b^2*e^4 - 4*a^3*c*e^4) + (2*a*c^2*d^3 - 3*a*b*c*d^2*
e + a*b^2*d*e^2 + 2*a^2*c*d*e^2 - a^2*b*e^3)/(b^2*c^2*d^4 - 4*a*c^3*d^4 - 2*b^3*c*d^3*e + 8*a*b*c^2*d^3*e + b^
4*d^2*e^2 - 2*a*b^2*c*d^2*e^2 - 8*a^2*c^2*d^2*e^2 - 2*a*b^3*d*e^3 + 8*a^2*b*c*d*e^3 + a^2*b^2*e^4 - 4*a^3*c*e^
4))/sqrt(c*x^4 + b*x^2 + a)