3.152 \(\int \frac{x^2 (c+d x^2+e x^4+f x^6)}{\sqrt{a+b x^2}} \, dx\)

Optimal. Leaf size=194 \[ \frac{x \sqrt{a+b x^2} \left (40 a^2 b e-35 a^3 f-48 a b^2 d+64 b^3 c\right )}{128 b^4}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \left (40 a^2 b e-35 a^3 f-48 a b^2 d+64 b^3 c\right )}{128 b^{9/2}}+\frac{x^3 \sqrt{a+b x^2} \left (35 a^2 f-40 a b e+48 b^2 d\right )}{192 b^3}+\frac{x^5 \sqrt{a+b x^2} (8 b e-7 a f)}{48 b^2}+\frac{f x^7 \sqrt{a+b x^2}}{8 b} \]

[Out]

((64*b^3*c - 48*a*b^2*d + 40*a^2*b*e - 35*a^3*f)*x*Sqrt[a + b*x^2])/(128*b^4) + ((48*b^2*d - 40*a*b*e + 35*a^2
*f)*x^3*Sqrt[a + b*x^2])/(192*b^3) + ((8*b*e - 7*a*f)*x^5*Sqrt[a + b*x^2])/(48*b^2) + (f*x^7*Sqrt[a + b*x^2])/
(8*b) - (a*(64*b^3*c - 48*a*b^2*d + 40*a^2*b*e - 35*a^3*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(128*b^(9/2))

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Rubi [A]  time = 0.207644, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {1809, 1267, 459, 321, 217, 206} \[ \frac{x \sqrt{a+b x^2} \left (40 a^2 b e-35 a^3 f-48 a b^2 d+64 b^3 c\right )}{128 b^4}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \left (40 a^2 b e-35 a^3 f-48 a b^2 d+64 b^3 c\right )}{128 b^{9/2}}+\frac{x^3 \sqrt{a+b x^2} \left (35 a^2 f-40 a b e+48 b^2 d\right )}{192 b^3}+\frac{x^5 \sqrt{a+b x^2} (8 b e-7 a f)}{48 b^2}+\frac{f x^7 \sqrt{a+b x^2}}{8 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(c + d*x^2 + e*x^4 + f*x^6))/Sqrt[a + b*x^2],x]

[Out]

((64*b^3*c - 48*a*b^2*d + 40*a^2*b*e - 35*a^3*f)*x*Sqrt[a + b*x^2])/(128*b^4) + ((48*b^2*d - 40*a*b*e + 35*a^2
*f)*x^3*Sqrt[a + b*x^2])/(192*b^3) + ((8*b*e - 7*a*f)*x^5*Sqrt[a + b*x^2])/(48*b^2) + (f*x^7*Sqrt[a + b*x^2])/
(8*b) - (a*(64*b^3*c - 48*a*b^2*d + 40*a^2*b*e - 35*a^3*f)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(128*b^(9/2))

Rule 1809

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[(f*(c*x)^(m + q - 1)*(a + b*x^2)^(p + 1))/(b*c^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 1267

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

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(b*e*(m + n*(p + 1) + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 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^2 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt{a+b x^2}} \, dx &=\frac{f x^7 \sqrt{a+b x^2}}{8 b}+\frac{\int \frac{x^2 \left (8 b c+8 b d x^2+(8 b e-7 a f) x^4\right )}{\sqrt{a+b x^2}} \, dx}{8 b}\\ &=\frac{(8 b e-7 a f) x^5 \sqrt{a+b x^2}}{48 b^2}+\frac{f x^7 \sqrt{a+b x^2}}{8 b}+\frac{\int \frac{x^2 \left (48 b^2 c+\left (48 b^2 d-40 a b e+35 a^2 f\right ) x^2\right )}{\sqrt{a+b x^2}} \, dx}{48 b^2}\\ &=\frac{\left (48 b^2 d-40 a b e+35 a^2 f\right ) x^3 \sqrt{a+b x^2}}{192 b^3}+\frac{(8 b e-7 a f) x^5 \sqrt{a+b x^2}}{48 b^2}+\frac{f x^7 \sqrt{a+b x^2}}{8 b}-\frac{1}{64} \left (-64 c+\frac{a \left (48 b^2 d-40 a b e+35 a^2 f\right )}{b^3}\right ) \int \frac{x^2}{\sqrt{a+b x^2}} \, dx\\ &=\frac{\left (64 c-\frac{a \left (48 b^2 d-40 a b e+35 a^2 f\right )}{b^3}\right ) x \sqrt{a+b x^2}}{128 b}+\frac{\left (48 b^2 d-40 a b e+35 a^2 f\right ) x^3 \sqrt{a+b x^2}}{192 b^3}+\frac{(8 b e-7 a f) x^5 \sqrt{a+b x^2}}{48 b^2}+\frac{f x^7 \sqrt{a+b x^2}}{8 b}-\frac{\left (a \left (64 b^3 c-48 a b^2 d+40 a^2 b e-35 a^3 f\right )\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{128 b^4}\\ &=\frac{\left (64 c-\frac{a \left (48 b^2 d-40 a b e+35 a^2 f\right )}{b^3}\right ) x \sqrt{a+b x^2}}{128 b}+\frac{\left (48 b^2 d-40 a b e+35 a^2 f\right ) x^3 \sqrt{a+b x^2}}{192 b^3}+\frac{(8 b e-7 a f) x^5 \sqrt{a+b x^2}}{48 b^2}+\frac{f x^7 \sqrt{a+b x^2}}{8 b}-\frac{\left (a \left (64 b^3 c-48 a b^2 d+40 a^2 b e-35 a^3 f\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{128 b^4}\\ &=\frac{\left (64 c-\frac{a \left (48 b^2 d-40 a b e+35 a^2 f\right )}{b^3}\right ) x \sqrt{a+b x^2}}{128 b}+\frac{\left (48 b^2 d-40 a b e+35 a^2 f\right ) x^3 \sqrt{a+b x^2}}{192 b^3}+\frac{(8 b e-7 a f) x^5 \sqrt{a+b x^2}}{48 b^2}+\frac{f x^7 \sqrt{a+b x^2}}{8 b}-\frac{a \left (64 b^3 c-48 a b^2 d+40 a^2 b e-35 a^3 f\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{128 b^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.158357, size = 149, normalized size = 0.77 \[ \frac{\sqrt{b} x \sqrt{a+b x^2} \left (10 a^2 b \left (12 e+7 f x^2\right )-105 a^3 f-8 a b^2 \left (18 d+10 e x^2+7 f x^4\right )+16 b^3 \left (12 c+6 d x^2+4 e x^4+3 f x^6\right )\right )+3 a \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \left (-40 a^2 b e+35 a^3 f+48 a b^2 d-64 b^3 c\right )}{384 b^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(c + d*x^2 + e*x^4 + f*x^6))/Sqrt[a + b*x^2],x]

[Out]

(Sqrt[b]*x*Sqrt[a + b*x^2]*(-105*a^3*f + 10*a^2*b*(12*e + 7*f*x^2) - 8*a*b^2*(18*d + 10*e*x^2 + 7*f*x^4) + 16*
b^3*(12*c + 6*d*x^2 + 4*e*x^4 + 3*f*x^6)) + 3*a*(-64*b^3*c + 48*a*b^2*d - 40*a^2*b*e + 35*a^3*f)*ArcTanh[(Sqrt
[b]*x)/Sqrt[a + b*x^2]])/(384*b^(9/2))

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Maple [A]  time = 0.008, size = 284, normalized size = 1.5 \begin{align*}{\frac{f{x}^{7}}{8\,b}\sqrt{b{x}^{2}+a}}-{\frac{7\,af{x}^{5}}{48\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{35\,{a}^{2}f{x}^{3}}{192\,{b}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{35\,{a}^{3}fx}{128\,{b}^{4}}\sqrt{b{x}^{2}+a}}+{\frac{35\,f{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{9}{2}}}}+{\frac{e{x}^{5}}{6\,b}\sqrt{b{x}^{2}+a}}-{\frac{5\,ae{x}^{3}}{24\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{5\,{a}^{2}ex}{16\,{b}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{5\,e{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}}+{\frac{d{x}^{3}}{4\,b}\sqrt{b{x}^{2}+a}}-{\frac{3\,adx}{8\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{3\,{a}^{2}d}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}+{\frac{cx}{2\,b}\sqrt{b{x}^{2}+a}}-{\frac{ac}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^(1/2),x)

[Out]

1/8*f*x^7*(b*x^2+a)^(1/2)/b-7/48*f/b^2*a*x^5*(b*x^2+a)^(1/2)+35/192*f/b^3*a^2*x^3*(b*x^2+a)^(1/2)-35/128*f/b^4
*a^3*x*(b*x^2+a)^(1/2)+35/128*f/b^(9/2)*a^4*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+1/6*e*x^5/b*(b*x^2+a)^(1/2)-5/24*e/b
^2*a*x^3*(b*x^2+a)^(1/2)+5/16*e/b^3*a^2*x*(b*x^2+a)^(1/2)-5/16*e/b^(7/2)*a^3*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+1/4
*d*x^3/b*(b*x^2+a)^(1/2)-3/8*d/b^2*a*x*(b*x^2+a)^(1/2)+3/8*d/b^(5/2)*a^2*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+1/2*c*x
/b*(b*x^2+a)^(1/2)-1/2*c*a/b^(3/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))

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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^2*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.71605, size = 779, normalized size = 4.02 \begin{align*} \left [-\frac{3 \,{\left (64 \, a b^{3} c - 48 \, a^{2} b^{2} d + 40 \, a^{3} b e - 35 \, a^{4} f\right )} \sqrt{b} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left (48 \, b^{4} f x^{7} + 8 \,{\left (8 \, b^{4} e - 7 \, a b^{3} f\right )} x^{5} + 2 \,{\left (48 \, b^{4} d - 40 \, a b^{3} e + 35 \, a^{2} b^{2} f\right )} x^{3} + 3 \,{\left (64 \, b^{4} c - 48 \, a b^{3} d + 40 \, a^{2} b^{2} e - 35 \, a^{3} b f\right )} x\right )} \sqrt{b x^{2} + a}}{768 \, b^{5}}, \frac{3 \,{\left (64 \, a b^{3} c - 48 \, a^{2} b^{2} d + 40 \, a^{3} b e - 35 \, a^{4} f\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (48 \, b^{4} f x^{7} + 8 \,{\left (8 \, b^{4} e - 7 \, a b^{3} f\right )} x^{5} + 2 \,{\left (48 \, b^{4} d - 40 \, a b^{3} e + 35 \, a^{2} b^{2} f\right )} x^{3} + 3 \,{\left (64 \, b^{4} c - 48 \, a b^{3} d + 40 \, a^{2} b^{2} e - 35 \, a^{3} b f\right )} x\right )} \sqrt{b x^{2} + a}}{384 \, b^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/768*(3*(64*a*b^3*c - 48*a^2*b^2*d + 40*a^3*b*e - 35*a^4*f)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b
)*x - a) - 2*(48*b^4*f*x^7 + 8*(8*b^4*e - 7*a*b^3*f)*x^5 + 2*(48*b^4*d - 40*a*b^3*e + 35*a^2*b^2*f)*x^3 + 3*(6
4*b^4*c - 48*a*b^3*d + 40*a^2*b^2*e - 35*a^3*b*f)*x)*sqrt(b*x^2 + a))/b^5, 1/384*(3*(64*a*b^3*c - 48*a^2*b^2*d
+ 40*a^3*b*e - 35*a^4*f)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (48*b^4*f*x^7 + 8*(8*b^4*e - 7*a*b^3*f
)*x^5 + 2*(48*b^4*d - 40*a*b^3*e + 35*a^2*b^2*f)*x^3 + 3*(64*b^4*c - 48*a*b^3*d + 40*a^2*b^2*e - 35*a^3*b*f)*x
)*sqrt(b*x^2 + a))/b^5]

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Sympy [B]  time = 21.593, size = 444, normalized size = 2.29 \begin{align*} - \frac{35 a^{\frac{7}{2}} f x}{128 b^{4} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 a^{\frac{5}{2}} e x}{16 b^{3} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{35 a^{\frac{5}{2}} f x^{3}}{384 b^{3} \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{3 a^{\frac{3}{2}} d x}{8 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 a^{\frac{3}{2}} e x^{3}}{48 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{7 a^{\frac{3}{2}} f x^{5}}{192 b^{2} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{\sqrt{a} c x \sqrt{1 + \frac{b x^{2}}{a}}}{2 b} - \frac{\sqrt{a} d x^{3}}{8 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{\sqrt{a} e x^{5}}{24 b \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{\sqrt{a} f x^{7}}{48 b \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{35 a^{4} f \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{128 b^{\frac{9}{2}}} - \frac{5 a^{3} e \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 b^{\frac{7}{2}}} + \frac{3 a^{2} d \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 b^{\frac{5}{2}}} - \frac{a c \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 b^{\frac{3}{2}}} + \frac{d x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{e x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{f x^{9}}{8 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(f*x**6+e*x**4+d*x**2+c)/(b*x**2+a)**(1/2),x)

[Out]

-35*a**(7/2)*f*x/(128*b**4*sqrt(1 + b*x**2/a)) + 5*a**(5/2)*e*x/(16*b**3*sqrt(1 + b*x**2/a)) - 35*a**(5/2)*f*x
**3/(384*b**3*sqrt(1 + b*x**2/a)) - 3*a**(3/2)*d*x/(8*b**2*sqrt(1 + b*x**2/a)) + 5*a**(3/2)*e*x**3/(48*b**2*sq
rt(1 + b*x**2/a)) + 7*a**(3/2)*f*x**5/(192*b**2*sqrt(1 + b*x**2/a)) + sqrt(a)*c*x*sqrt(1 + b*x**2/a)/(2*b) - s
qrt(a)*d*x**3/(8*b*sqrt(1 + b*x**2/a)) - sqrt(a)*e*x**5/(24*b*sqrt(1 + b*x**2/a)) - sqrt(a)*f*x**7/(48*b*sqrt(
1 + b*x**2/a)) + 35*a**4*f*asinh(sqrt(b)*x/sqrt(a))/(128*b**(9/2)) - 5*a**3*e*asinh(sqrt(b)*x/sqrt(a))/(16*b**
(7/2)) + 3*a**2*d*asinh(sqrt(b)*x/sqrt(a))/(8*b**(5/2)) - a*c*asinh(sqrt(b)*x/sqrt(a))/(2*b**(3/2)) + d*x**5/(
4*sqrt(a)*sqrt(1 + b*x**2/a)) + e*x**7/(6*sqrt(a)*sqrt(1 + b*x**2/a)) + f*x**9/(8*sqrt(a)*sqrt(1 + b*x**2/a))

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Giac [A]  time = 1.22415, size = 236, normalized size = 1.22 \begin{align*} \frac{1}{384} \,{\left (2 \,{\left (4 \,{\left (\frac{6 \, f x^{2}}{b} - \frac{7 \, a b^{5} f - 8 \, b^{6} e}{b^{7}}\right )} x^{2} + \frac{48 \, b^{6} d + 35 \, a^{2} b^{4} f - 40 \, a b^{5} e}{b^{7}}\right )} x^{2} + \frac{3 \,{\left (64 \, b^{6} c - 48 \, a b^{5} d - 35 \, a^{3} b^{3} f + 40 \, a^{2} b^{4} e\right )}}{b^{7}}\right )} \sqrt{b x^{2} + a} x + \frac{{\left (64 \, a b^{3} c - 48 \, a^{2} b^{2} d - 35 \, a^{4} f + 40 \, a^{3} b e\right )} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{128 \, b^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^(1/2),x, algorithm="giac")

[Out]

1/384*(2*(4*(6*f*x^2/b - (7*a*b^5*f - 8*b^6*e)/b^7)*x^2 + (48*b^6*d + 35*a^2*b^4*f - 40*a*b^5*e)/b^7)*x^2 + 3*
(64*b^6*c - 48*a*b^5*d - 35*a^3*b^3*f + 40*a^2*b^4*e)/b^7)*sqrt(b*x^2 + a)*x + 1/128*(64*a*b^3*c - 48*a^2*b^2*
d - 35*a^4*f + 40*a^3*b*e)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(9/2)