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

Optimal. Leaf size=348 \[ -\frac{x \left (c+d x^2\right ) \left (b e \left (-3 c^2 f^2-8 c d e f+35 d^2 e^2\right )-a f \left (15 c^2 f^2+4 c d e f+5 d^2 e^2\right )\right )}{48 e^3 f^3 \left (e+f x^2\right )}+\frac{d x \left (b e \left (-3 c^2 f^2-10 c d e f+105 d^2 e^2\right )-a f \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )}{48 e^3 f^4}-\frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (b e \left (-3 c^2 d e f^2-c^3 f^3-15 c d^2 e^2 f+35 d^3 e^3\right )-a f \left (3 c^2 d e f^2+5 c^3 f^3+3 c d^2 e^2 f+5 d^3 e^3\right )\right )}{16 e^{7/2} f^{9/2}}-\frac{x \left (c+d x^2\right )^2 (b e (7 d e-c f)-a f (5 c f+d e))}{24 e^2 f^2 \left (e+f x^2\right )^2}-\frac{x \left (c+d x^2\right )^3 (b e-a f)}{6 e f \left (e+f x^2\right )^3} \]

[Out]

(d*(b*e*(105*d^2*e^2 - 10*c*d*e*f - 3*c^2*f^2) - a*f*(15*d^2*e^2 + 14*c*d*e*f + 15*c^2*f^2))*x)/(48*e^3*f^4) -
((b*e - a*f)*x*(c + d*x^2)^3)/(6*e*f*(e + f*x^2)^3) - ((b*e*(7*d*e - c*f) - a*f*(d*e + 5*c*f))*x*(c + d*x^2)^
2)/(24*e^2*f^2*(e + f*x^2)^2) - ((b*e*(35*d^2*e^2 - 8*c*d*e*f - 3*c^2*f^2) - a*f*(5*d^2*e^2 + 4*c*d*e*f + 15*c
^2*f^2))*x*(c + d*x^2))/(48*e^3*f^3*(e + f*x^2)) - ((b*e*(35*d^3*e^3 - 15*c*d^2*e^2*f - 3*c^2*d*e*f^2 - c^3*f^
3) - a*f*(5*d^3*e^3 + 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 5*c^3*f^3))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(16*e^(7/2)*f^(
9/2))

________________________________________________________________________________________

Rubi [A]  time = 0.449597, antiderivative size = 348, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {526, 388, 205} \[ -\frac{x \left (c+d x^2\right ) \left (b e \left (-3 c^2 f^2-8 c d e f+35 d^2 e^2\right )-a f \left (15 c^2 f^2+4 c d e f+5 d^2 e^2\right )\right )}{48 e^3 f^3 \left (e+f x^2\right )}+\frac{d x \left (b e \left (-3 c^2 f^2-10 c d e f+105 d^2 e^2\right )-a f \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )}{48 e^3 f^4}-\frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (b e \left (-3 c^2 d e f^2-c^3 f^3-15 c d^2 e^2 f+35 d^3 e^3\right )-a f \left (3 c^2 d e f^2+5 c^3 f^3+3 c d^2 e^2 f+5 d^3 e^3\right )\right )}{16 e^{7/2} f^{9/2}}-\frac{x \left (c+d x^2\right )^2 (b e (7 d e-c f)-a f (5 c f+d e))}{24 e^2 f^2 \left (e+f x^2\right )^2}-\frac{x \left (c+d x^2\right )^3 (b e-a f)}{6 e f \left (e+f x^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(b*e*(105*d^2*e^2 - 10*c*d*e*f - 3*c^2*f^2) - a*f*(15*d^2*e^2 + 14*c*d*e*f + 15*c^2*f^2))*x)/(48*e^3*f^4) -
((b*e - a*f)*x*(c + d*x^2)^3)/(6*e*f*(e + f*x^2)^3) - ((b*e*(7*d*e - c*f) - a*f*(d*e + 5*c*f))*x*(c + d*x^2)^
2)/(24*e^2*f^2*(e + f*x^2)^2) - ((b*e*(35*d^2*e^2 - 8*c*d*e*f - 3*c^2*f^2) - a*f*(5*d^2*e^2 + 4*c*d*e*f + 15*c
^2*f^2))*x*(c + d*x^2))/(48*e^3*f^3*(e + f*x^2)) - ((b*e*(35*d^3*e^3 - 15*c*d^2*e^2*f - 3*c^2*d*e*f^2 - c^3*f^
3) - a*f*(5*d^3*e^3 + 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 5*c^3*f^3))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(16*e^(7/2)*f^(
9/2))

Rule 526

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

Rule 388

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

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right ) \left (c+d x^2\right )^3}{\left (e+f x^2\right )^4} \, dx &=-\frac{(b e-a f) x \left (c+d x^2\right )^3}{6 e f \left (e+f x^2\right )^3}-\frac{\int \frac{\left (c+d x^2\right )^2 \left (-c (b e+5 a f)-d (7 b e-a f) x^2\right )}{\left (e+f x^2\right )^3} \, dx}{6 e f}\\ &=-\frac{(b e-a f) x \left (c+d x^2\right )^3}{6 e f \left (e+f x^2\right )^3}-\frac{(b e (7 d e-c f)-a f (d e+5 c f)) x \left (c+d x^2\right )^2}{24 e^2 f^2 \left (e+f x^2\right )^2}+\frac{\int \frac{\left (c+d x^2\right ) \left (c (d e (7 b e-a f)+3 c f (b e+5 a f))+d (b e (35 d e-c f)-5 a f (d e+c f)) x^2\right )}{\left (e+f x^2\right )^2} \, dx}{24 e^2 f^2}\\ &=-\frac{(b e-a f) x \left (c+d x^2\right )^3}{6 e f \left (e+f x^2\right )^3}-\frac{(b e (7 d e-c f)-a f (d e+5 c f)) x \left (c+d x^2\right )^2}{24 e^2 f^2 \left (e+f x^2\right )^2}-\frac{\left (b e \left (35 d^2 e^2-8 c d e f-3 c^2 f^2\right )-a f \left (5 d^2 e^2+4 c d e f+15 c^2 f^2\right )\right ) x \left (c+d x^2\right )}{48 e^3 f^3 \left (e+f x^2\right )}-\frac{\int \frac{c \left (a f \left (5 d^2 e^2+6 c d e f-15 c^2 f^2\right )-b e \left (35 d^2 e^2+6 c d e f+3 c^2 f^2\right )\right )-d \left (b e \left (105 d^2 e^2-10 c d e f-3 c^2 f^2\right )-a f \left (15 d^2 e^2+14 c d e f+15 c^2 f^2\right )\right ) x^2}{e+f x^2} \, dx}{48 e^3 f^3}\\ &=\frac{d \left (b e \left (105 d^2 e^2-10 c d e f-3 c^2 f^2\right )-a f \left (15 d^2 e^2+14 c d e f+15 c^2 f^2\right )\right ) x}{48 e^3 f^4}-\frac{(b e-a f) x \left (c+d x^2\right )^3}{6 e f \left (e+f x^2\right )^3}-\frac{(b e (7 d e-c f)-a f (d e+5 c f)) x \left (c+d x^2\right )^2}{24 e^2 f^2 \left (e+f x^2\right )^2}-\frac{\left (b e \left (35 d^2 e^2-8 c d e f-3 c^2 f^2\right )-a f \left (5 d^2 e^2+4 c d e f+15 c^2 f^2\right )\right ) x \left (c+d x^2\right )}{48 e^3 f^3 \left (e+f x^2\right )}-\frac{\left (b e \left (35 d^3 e^3-15 c d^2 e^2 f-3 c^2 d e f^2-c^3 f^3\right )-a f \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )\right ) \int \frac{1}{e+f x^2} \, dx}{16 e^3 f^4}\\ &=\frac{d \left (b e \left (105 d^2 e^2-10 c d e f-3 c^2 f^2\right )-a f \left (15 d^2 e^2+14 c d e f+15 c^2 f^2\right )\right ) x}{48 e^3 f^4}-\frac{(b e-a f) x \left (c+d x^2\right )^3}{6 e f \left (e+f x^2\right )^3}-\frac{(b e (7 d e-c f)-a f (d e+5 c f)) x \left (c+d x^2\right )^2}{24 e^2 f^2 \left (e+f x^2\right )^2}-\frac{\left (b e \left (35 d^2 e^2-8 c d e f-3 c^2 f^2\right )-a f \left (5 d^2 e^2+4 c d e f+15 c^2 f^2\right )\right ) x \left (c+d x^2\right )}{48 e^3 f^3 \left (e+f x^2\right )}-\frac{\left (b e \left (35 d^3 e^3-15 c d^2 e^2 f-3 c^2 d e f^2-c^3 f^3\right )-a f \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )\right ) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{16 e^{7/2} f^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.209906, size = 295, normalized size = 0.85 \[ \frac{x (d e-c f) \left (b e \left (-c^2 f^2-4 c d e f+29 d^2 e^2\right )-a f \left (5 c^2 f^2+8 c d e f+11 d^2 e^2\right )\right )}{16 e^3 f^4 \left (e+f x^2\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) \left (b e \left (-3 c^2 d e f^2-c^3 f^3-15 c d^2 e^2 f+35 d^3 e^3\right )-a f \left (3 c^2 d e f^2+5 c^3 f^3+3 c d^2 e^2 f+5 d^3 e^3\right )\right )}{16 e^{7/2} f^{9/2}}-\frac{x (d e-c f)^2 (b e (19 d e-c f)-a f (5 c f+13 d e))}{24 e^2 f^4 \left (e+f x^2\right )^2}+\frac{x (b e-a f) (d e-c f)^3}{6 e f^4 \left (e+f x^2\right )^3}+\frac{b d^3 x}{f^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*d^3*x)/f^4 + ((b*e - a*f)*(d*e - c*f)^3*x)/(6*e*f^4*(e + f*x^2)^3) - ((d*e - c*f)^2*(b*e*(19*d*e - c*f) - a
*f*(13*d*e + 5*c*f))*x)/(24*e^2*f^4*(e + f*x^2)^2) + ((d*e - c*f)*(b*e*(29*d^2*e^2 - 4*c*d*e*f - c^2*f^2) - a*
f*(11*d^2*e^2 + 8*c*d*e*f + 5*c^2*f^2))*x)/(16*e^3*f^4*(e + f*x^2)) - ((b*e*(35*d^3*e^3 - 15*c*d^2*e^2*f - 3*c
^2*d*e*f^2 - c^3*f^3) - a*f*(5*d^3*e^3 + 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 5*c^3*f^3))*ArcTan[(Sqrt[f]*x)/Sqrt[e
]])/(16*e^(7/2)*f^(9/2))

________________________________________________________________________________________

Maple [B]  time = 0.013, size = 735, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*d^3/f^4*x+15/16/f^3/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*c*d^2+1/16/f/e^2/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2
))*b*c^3-35/16/f^4*e/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*d^3+5/16*f^2/(f*x^2+e)^3/e^3*x^5*a*c^3+1/16*f/(f*x^
2+e)^3/e^2*x^5*b*c^3-33/16/f/(f*x^2+e)^3*x^5*b*c*d^2+29/16/f^2/(f*x^2+e)^3*x^5*b*d^3*e+5/6*f/(f*x^2+e)^3/e^2*x
^3*a*c^3-1/2/f/(f*x^2+e)^3*x^3*a*c*d^2-5/6/f^2/(f*x^2+e)^3*x^3*a*d^3*e-1/2/f/(f*x^2+e)^3*x^3*b*c^2*d-1/16/f/(f
*x^2+e)^3*b*c^3*x-11/16/f/(f*x^2+e)^3*x^5*a*d^3+5/16/f^3/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*a*d^3+5/16/e^3/(e
*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*a*c^3+1/6/(f*x^2+e)^3/e*x^3*b*c^3+11/16/(f*x^2+e)^3/e*x*a*c^3+17/6/f^3/(f*x^
2+e)^3*x^3*b*d^3*e^2-3/16/f/(f*x^2+e)^3*a*c^2*d*x-5/16/f^3/(f*x^2+e)^3*a*d^3*e^2*x+3/16/(f*x^2+e)^3/e*x^5*a*c*
d^2+3/16/(f*x^2+e)^3/e*x^5*b*c^2*d+1/2/(f*x^2+e)^3/e*x^3*a*c^2*d+19/16/f^4/(f*x^2+e)^3*b*d^3*e^3*x-3/16/f^2/(f
*x^2+e)^3*b*c^2*d*e*x-15/16/f^3/(f*x^2+e)^3*b*c*d^2*e^2*x+3/16*f/(f*x^2+e)^3/e^2*x^5*a*c^2*d-5/2/f^2/(f*x^2+e)
^3*x^3*b*c*d^2*e-3/16/f^2/(f*x^2+e)^3*a*c*d^2*e*x+3/16/f/e^2/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*a*c^2*d+3/16/
f^2/e/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*a*c*d^2+3/16/f^2/e/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*c^2*d

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.71072, size = 2913, normalized size = 8.37 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/96*(96*b*d^3*e^4*f^4*x^7 + 6*(77*b*d^3*e^5*f^3 + 5*a*c^3*e*f^7 - 11*(3*b*c*d^2 + a*d^3)*e^4*f^4 + 3*(b*c^2*
d + a*c*d^2)*e^3*f^5 + (b*c^3 + 3*a*c^2*d)*e^2*f^6)*x^5 + 16*(35*b*d^3*e^6*f^2 + 5*a*c^3*e^2*f^6 - 5*(3*b*c*d^
2 + a*d^3)*e^5*f^3 - 3*(b*c^2*d + a*c*d^2)*e^4*f^4 + (b*c^3 + 3*a*c^2*d)*e^3*f^5)*x^3 + 3*(35*b*d^3*e^7 - 5*a*
c^3*e^3*f^4 - 5*(3*b*c*d^2 + a*d^3)*e^6*f - 3*(b*c^2*d + a*c*d^2)*e^5*f^2 - (b*c^3 + 3*a*c^2*d)*e^4*f^3 + (35*
b*d^3*e^4*f^3 - 5*a*c^3*f^7 - 5*(3*b*c*d^2 + a*d^3)*e^3*f^4 - 3*(b*c^2*d + a*c*d^2)*e^2*f^5 - (b*c^3 + 3*a*c^2
*d)*e*f^6)*x^6 + 3*(35*b*d^3*e^5*f^2 - 5*a*c^3*e*f^6 - 5*(3*b*c*d^2 + a*d^3)*e^4*f^3 - 3*(b*c^2*d + a*c*d^2)*e
^3*f^4 - (b*c^3 + 3*a*c^2*d)*e^2*f^5)*x^4 + 3*(35*b*d^3*e^6*f - 5*a*c^3*e^2*f^5 - 5*(3*b*c*d^2 + a*d^3)*e^5*f^
2 - 3*(b*c^2*d + a*c*d^2)*e^4*f^3 - (b*c^3 + 3*a*c^2*d)*e^3*f^4)*x^2)*sqrt(-e*f)*log((f*x^2 - 2*sqrt(-e*f)*x -
e)/(f*x^2 + e)) + 6*(35*b*d^3*e^7*f + 11*a*c^3*e^3*f^5 - 5*(3*b*c*d^2 + a*d^3)*e^6*f^2 - 3*(b*c^2*d + a*c*d^2
)*e^5*f^3 - (b*c^3 + 3*a*c^2*d)*e^4*f^4)*x)/(e^4*f^8*x^6 + 3*e^5*f^7*x^4 + 3*e^6*f^6*x^2 + e^7*f^5), 1/48*(48*
b*d^3*e^4*f^4*x^7 + 3*(77*b*d^3*e^5*f^3 + 5*a*c^3*e*f^7 - 11*(3*b*c*d^2 + a*d^3)*e^4*f^4 + 3*(b*c^2*d + a*c*d^
2)*e^3*f^5 + (b*c^3 + 3*a*c^2*d)*e^2*f^6)*x^5 + 8*(35*b*d^3*e^6*f^2 + 5*a*c^3*e^2*f^6 - 5*(3*b*c*d^2 + a*d^3)*
e^5*f^3 - 3*(b*c^2*d + a*c*d^2)*e^4*f^4 + (b*c^3 + 3*a*c^2*d)*e^3*f^5)*x^3 - 3*(35*b*d^3*e^7 - 5*a*c^3*e^3*f^4
- 5*(3*b*c*d^2 + a*d^3)*e^6*f - 3*(b*c^2*d + a*c*d^2)*e^5*f^2 - (b*c^3 + 3*a*c^2*d)*e^4*f^3 + (35*b*d^3*e^4*f
^3 - 5*a*c^3*f^7 - 5*(3*b*c*d^2 + a*d^3)*e^3*f^4 - 3*(b*c^2*d + a*c*d^2)*e^2*f^5 - (b*c^3 + 3*a*c^2*d)*e*f^6)*
x^6 + 3*(35*b*d^3*e^5*f^2 - 5*a*c^3*e*f^6 - 5*(3*b*c*d^2 + a*d^3)*e^4*f^3 - 3*(b*c^2*d + a*c*d^2)*e^3*f^4 - (b
*c^3 + 3*a*c^2*d)*e^2*f^5)*x^4 + 3*(35*b*d^3*e^6*f - 5*a*c^3*e^2*f^5 - 5*(3*b*c*d^2 + a*d^3)*e^5*f^2 - 3*(b*c^
2*d + a*c*d^2)*e^4*f^3 - (b*c^3 + 3*a*c^2*d)*e^3*f^4)*x^2)*sqrt(e*f)*arctan(sqrt(e*f)*x/e) + 3*(35*b*d^3*e^7*f
+ 11*a*c^3*e^3*f^5 - 5*(3*b*c*d^2 + a*d^3)*e^6*f^2 - 3*(b*c^2*d + a*c*d^2)*e^5*f^3 - (b*c^3 + 3*a*c^2*d)*e^4*
f^4)*x)/(e^4*f^8*x^6 + 3*e^5*f^7*x^4 + 3*e^6*f^6*x^2 + e^7*f^5)]

________________________________________________________________________________________

Sympy [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((b*x**2+a)*(d*x**2+c)**3/(f*x**2+e)**4,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.20757, size = 603, normalized size = 1.73 \begin{align*} \frac{b d^{3} x}{f^{4}} + \frac{{\left (5 \, a c^{3} f^{4} + b c^{3} f^{3} e + 3 \, a c^{2} d f^{3} e + 3 \, b c^{2} d f^{2} e^{2} + 3 \, a c d^{2} f^{2} e^{2} + 15 \, b c d^{2} f e^{3} + 5 \, a d^{3} f e^{3} - 35 \, b d^{3} e^{4}\right )} \arctan \left (\sqrt{f} x e^{\left (-\frac{1}{2}\right )}\right ) e^{\left (-\frac{7}{2}\right )}}{16 \, f^{\frac{9}{2}}} + \frac{{\left (15 \, a c^{3} f^{6} x^{5} + 3 \, b c^{3} f^{5} x^{5} e + 9 \, a c^{2} d f^{5} x^{5} e + 9 \, b c^{2} d f^{4} x^{5} e^{2} + 9 \, a c d^{2} f^{4} x^{5} e^{2} - 99 \, b c d^{2} f^{3} x^{5} e^{3} - 33 \, a d^{3} f^{3} x^{5} e^{3} + 40 \, a c^{3} f^{5} x^{3} e + 87 \, b d^{3} f^{2} x^{5} e^{4} + 8 \, b c^{3} f^{4} x^{3} e^{2} + 24 \, a c^{2} d f^{4} x^{3} e^{2} - 24 \, b c^{2} d f^{3} x^{3} e^{3} - 24 \, a c d^{2} f^{3} x^{3} e^{3} - 120 \, b c d^{2} f^{2} x^{3} e^{4} - 40 \, a d^{3} f^{2} x^{3} e^{4} + 33 \, a c^{3} f^{4} x e^{2} + 136 \, b d^{3} f x^{3} e^{5} - 3 \, b c^{3} f^{3} x e^{3} - 9 \, a c^{2} d f^{3} x e^{3} - 9 \, b c^{2} d f^{2} x e^{4} - 9 \, a c d^{2} f^{2} x e^{4} - 45 \, b c d^{2} f x e^{5} - 15 \, a d^{3} f x e^{5} + 57 \, b d^{3} x e^{6}\right )} e^{\left (-3\right )}}{48 \,{\left (f x^{2} + e\right )}^{3} f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*d^3*x/f^4 + 1/16*(5*a*c^3*f^4 + b*c^3*f^3*e + 3*a*c^2*d*f^3*e + 3*b*c^2*d*f^2*e^2 + 3*a*c*d^2*f^2*e^2 + 15*b
*c*d^2*f*e^3 + 5*a*d^3*f*e^3 - 35*b*d^3*e^4)*arctan(sqrt(f)*x*e^(-1/2))*e^(-7/2)/f^(9/2) + 1/48*(15*a*c^3*f^6*
x^5 + 3*b*c^3*f^5*x^5*e + 9*a*c^2*d*f^5*x^5*e + 9*b*c^2*d*f^4*x^5*e^2 + 9*a*c*d^2*f^4*x^5*e^2 - 99*b*c*d^2*f^3
*x^5*e^3 - 33*a*d^3*f^3*x^5*e^3 + 40*a*c^3*f^5*x^3*e + 87*b*d^3*f^2*x^5*e^4 + 8*b*c^3*f^4*x^3*e^2 + 24*a*c^2*d
*f^4*x^3*e^2 - 24*b*c^2*d*f^3*x^3*e^3 - 24*a*c*d^2*f^3*x^3*e^3 - 120*b*c*d^2*f^2*x^3*e^4 - 40*a*d^3*f^2*x^3*e^
4 + 33*a*c^3*f^4*x*e^2 + 136*b*d^3*f*x^3*e^5 - 3*b*c^3*f^3*x*e^3 - 9*a*c^2*d*f^3*x*e^3 - 9*b*c^2*d*f^2*x*e^4 -
9*a*c*d^2*f^2*x*e^4 - 45*b*c*d^2*f*x*e^5 - 15*a*d^3*f*x*e^5 + 57*b*d^3*x*e^6)*e^(-3)/((f*x^2 + e)^3*f^4)