3.342 \(\int \frac{x^7}{(d+e x^2) (a+b x^2+c x^4)^{3/2}} \, dx\)

Optimal. Leaf size=236 \[ \frac{x^2 \left (2 a^2 c e-a b^2 e-3 a b c d+b^3 d\right )+a \left (-a b e-2 a c d+b^2 d\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{2 c^{3/2} e}-\frac{d^3 \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e \left (a e^2-b d e+c d^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.473671, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {1251, 1646, 843, 621, 206, 724} \[ \frac{x^2 \left (2 a^2 c e-a b^2 e-3 a b c d+b^3 d\right )+a \left (-a b e-2 a c d+b^2 d\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4} \left (a e^2-b d e+c d^2\right )}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{2 c^{3/2} e}-\frac{d^3 \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e \left (a e^2-b d e+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^7/((d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)),x]

[Out]

(a*(b^2*d - 2*a*c*d - a*b*e) + (b^3*d - 3*a*b*c*d - a*b^2*e + 2*a^2*c*e)*x^2)/(c*(b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)*Sqrt[a + b*x^2 + c*x^4]) + ArcTanh[(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])]/(2*c^(3/2)*e) -
(d^3*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
*(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 1646

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[(d + e*x)^m*Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2,
x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2
*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^m*(a + b*x + c*x^2)^(p + 1)*ExpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*f - b*
g))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]

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^7}{\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^3}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac{a \left (b^2 d-2 a c d-a b e\right )+\left (b^3 d-3 a b c d-a b^2 e+2 a^2 c e\right ) x^2}{c \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{\frac{\left (b^2-4 a c\right ) d (b d-a e)}{2 c \left (c d^2-b d e+a e^2\right )}-\frac{\left (b^2-4 a c\right ) x}{2 c}}{(d+e x) \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{b^2-4 a c}\\ &=\frac{a \left (b^2 d-2 a c d-a b e\right )+\left (b^3 d-3 a b c d-a b^2 e+2 a^2 c e\right ) x^2}{c \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{1}{\sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{2 c e}-\frac{d^3 \operatorname{Subst}\left (\int \frac{1}{(d+e x) \sqrt{a+b x+c x^2}} \, dx,x,x^2\right )}{2 e \left (c d^2-b d e+a e^2\right )}\\ &=\frac{a \left (b^2 d-2 a c d-a b e\right )+\left (b^3 d-3 a b c d-a b^2 e+2 a^2 c e\right ) x^2}{c \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{1}{4 c-x^2} \, dx,x,\frac{b+2 c x^2}{\sqrt{a+b x^2+c x^4}}\right )}{c e}+\frac{d^3 \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 \left (c d^2-b d e+a e^2\right )}\\ &=\frac{a \left (b^2 d-2 a c d-a b e\right )+\left (b^3 d-3 a b c d-a b^2 e+2 a^2 c e\right ) x^2}{c \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{\tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{2 c^{3/2} e}-\frac{d^3 \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 \left (c d^2-b d e+a e^2\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.841386, size = 271, normalized size = 1.15 \[ \frac{1}{2} \left (\frac{2 \left (a^2 \left (b e+2 c \left (d-e x^2\right )\right )+a b \left (-b d+b e x^2+3 c d x^2\right )+b^3 (-d) x^2\right )}{c \left (4 a c-b^2\right ) \sqrt{a+b x^2+c x^4} \left (e (a e-b d)+c d^2\right )}+\frac{\log \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4}+b+2 c x^2\right )}{c^{3/2} e}+\frac{d^3 \log \left (2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x^2-2 c d x^2\right )}{e \left (e (a e-b d)+c d^2\right )^{3/2}}-\frac{d^3 \log \left (d+e x^2\right )}{e \left (e (a e-b d)+c d^2\right )^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/((d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2)),x]

[Out]

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

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Maple [B]  time = 0.066, size = 720, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(e*x^2+d)/(c*x^4+b*x^2+a)^(3/2),x)

[Out]

-1/2/e*x^2/c/(c*x^4+b*x^2+a)^(1/2)+1/4/e*b/c^2/(c*x^4+b*x^2+a)^(1/2)+1/2/e*b^2/c/(4*a*c-b^2)/(c*x^4+b*x^2+a)^(
1/2)*x^2+1/4/e*b^3/c^2/(4*a*c-b^2)/(c*x^4+b*x^2+a)^(1/2)+1/2/e/c^(3/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a
)^(1/2))+d/e^2/(c*x^4+b*x^2+a)^(1/2)*(b*x^2+2*a)/(4*a*c-b^2)+d^2/e^3*(2*c*x^2+b)/(4*a*c-b^2)/(c*x^4+b*x^2+a)^(
1/2)+2*d^3/e^3*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^3/e^2*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^3/e^3*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^{7}}{{\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^7/(e*x^2+d)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^7/((c*x^4 + b*x^2 + a)^(3/2)*(e*x^2 + d)), x)

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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^7/(e*x^2+d)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{7}}{\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**7/(e*x**2+d)/(c*x**4+b*x**2+a)**(3/2),x)

[Out]

Integral(x**7/((d + e*x**2)*(a + b*x**2 + c*x**4)**(3/2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{7}}{{\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^7/(e*x^2+d)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="giac")

[Out]

integrate(x^7/((c*x^4 + b*x^2 + a)^(3/2)*(e*x^2 + d)), x)