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

Optimal. Leaf size=242 \[ -\frac{d x \left (5 a f \left (3 c^2 f^2-22 c d e f+15 d^2 e^2\right )-b e \left (81 c^2 f^2-190 c d e f+105 d^2 e^2\right )\right )}{30 e f^4}-\frac{(d e-c f)^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) (b e (7 d e-c f)-a f (c f+5 d e))}{2 e^{3/2} f^{9/2}}+\frac{d x \left (c+d x^2\right )^2 (7 b e-5 a f)}{10 e f^2}-\frac{d x \left (c+d x^2\right ) (b e (35 d e-33 c f)-5 a f (5 d e-3 c f))}{30 e f^3}-\frac{x \left (c+d x^2\right )^3 (b e-a f)}{2 e f \left (e+f x^2\right )} \]

[Out]

-(d*(5*a*f*(15*d^2*e^2 - 22*c*d*e*f + 3*c^2*f^2) - b*e*(105*d^2*e^2 - 190*c*d*e*f + 81*c^2*f^2))*x)/(30*e*f^4)
- (d*(b*e*(35*d*e - 33*c*f) - 5*a*f*(5*d*e - 3*c*f))*x*(c + d*x^2))/(30*e*f^3) + (d*(7*b*e - 5*a*f)*x*(c + d*
x^2)^2)/(10*e*f^2) - ((b*e - a*f)*x*(c + d*x^2)^3)/(2*e*f*(e + f*x^2)) - ((d*e - c*f)^2*(b*e*(7*d*e - c*f) - a
*f*(5*d*e + c*f))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(2*e^(3/2)*f^(9/2))

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Rubi [A]  time = 0.398615, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {526, 528, 388, 205} \[ -\frac{d x \left (5 a f \left (3 c^2 f^2-22 c d e f+15 d^2 e^2\right )-b e \left (81 c^2 f^2-190 c d e f+105 d^2 e^2\right )\right )}{30 e f^4}-\frac{(d e-c f)^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) (b e (7 d e-c f)-a f (c f+5 d e))}{2 e^{3/2} f^{9/2}}+\frac{d x \left (c+d x^2\right )^2 (7 b e-5 a f)}{10 e f^2}-\frac{d x \left (c+d x^2\right ) (b e (35 d e-33 c f)-5 a f (5 d e-3 c f))}{30 e f^3}-\frac{x \left (c+d x^2\right )^3 (b e-a f)}{2 e f \left (e+f x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*(5*a*f*(15*d^2*e^2 - 22*c*d*e*f + 3*c^2*f^2) - b*e*(105*d^2*e^2 - 190*c*d*e*f + 81*c^2*f^2))*x)/(30*e*f^4)
- (d*(b*e*(35*d*e - 33*c*f) - 5*a*f*(5*d*e - 3*c*f))*x*(c + d*x^2))/(30*e*f^3) + (d*(7*b*e - 5*a*f)*x*(c + d*
x^2)^2)/(10*e*f^2) - ((b*e - a*f)*x*(c + d*x^2)^3)/(2*e*f*(e + f*x^2)) - ((d*e - c*f)^2*(b*e*(7*d*e - c*f) - a
*f*(5*d*e + c*f))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(2*e^(3/2)*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 528

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[
(f*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*(n*(p + q + 1) + 1)), x] + Dist[1/(b*(n*(p + q + 1) + 1)), Int[(a +
b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*
d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1
, 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 )^2} \, dx &=-\frac{(b e-a f) x \left (c+d x^2\right )^3}{2 e f \left (e+f x^2\right )}-\frac{\int \frac{\left (c+d x^2\right )^2 \left (-c (b e+a f)-d (7 b e-5 a f) x^2\right )}{e+f x^2} \, dx}{2 e f}\\ &=\frac{d (7 b e-5 a f) x \left (c+d x^2\right )^2}{10 e f^2}-\frac{(b e-a f) x \left (c+d x^2\right )^3}{2 e f \left (e+f x^2\right )}-\frac{\int \frac{\left (c+d x^2\right ) \left (c (b e (7 d e-5 c f)-5 a f (d e+c f))+d (b e (35 d e-33 c f)-5 a f (5 d e-3 c f)) x^2\right )}{e+f x^2} \, dx}{10 e f^2}\\ &=-\frac{d (b e (35 d e-33 c f)-5 a f (5 d e-3 c f)) x \left (c+d x^2\right )}{30 e f^3}+\frac{d (7 b e-5 a f) x \left (c+d x^2\right )^2}{10 e f^2}-\frac{(b e-a f) x \left (c+d x^2\right )^3}{2 e f \left (e+f x^2\right )}-\frac{\int \frac{c \left (5 a f \left (5 d^2 e^2-6 c d e f-3 c^2 f^2\right )-b e \left (35 d^2 e^2-54 c d e f+15 c^2 f^2\right )\right )+d \left (5 a f \left (15 d^2 e^2-22 c d e f+3 c^2 f^2\right )-b e \left (105 d^2 e^2-190 c d e f+81 c^2 f^2\right )\right ) x^2}{e+f x^2} \, dx}{30 e f^3}\\ &=-\frac{d \left (5 a f \left (15 d^2 e^2-22 c d e f+3 c^2 f^2\right )-b e \left (105 d^2 e^2-190 c d e f+81 c^2 f^2\right )\right ) x}{30 e f^4}-\frac{d (b e (35 d e-33 c f)-5 a f (5 d e-3 c f)) x \left (c+d x^2\right )}{30 e f^3}+\frac{d (7 b e-5 a f) x \left (c+d x^2\right )^2}{10 e f^2}-\frac{(b e-a f) x \left (c+d x^2\right )^3}{2 e f \left (e+f x^2\right )}-\frac{\left ((d e-c f)^2 (b e (7 d e-c f)-a f (5 d e+c f))\right ) \int \frac{1}{e+f x^2} \, dx}{2 e f^4}\\ &=-\frac{d \left (5 a f \left (15 d^2 e^2-22 c d e f+3 c^2 f^2\right )-b e \left (105 d^2 e^2-190 c d e f+81 c^2 f^2\right )\right ) x}{30 e f^4}-\frac{d (b e (35 d e-33 c f)-5 a f (5 d e-3 c f)) x \left (c+d x^2\right )}{30 e f^3}+\frac{d (7 b e-5 a f) x \left (c+d x^2\right )^2}{10 e f^2}-\frac{(b e-a f) x \left (c+d x^2\right )^3}{2 e f \left (e+f x^2\right )}-\frac{(d e-c f)^2 (b e (7 d e-c f)-a f (5 d e+c f)) \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )}{2 e^{3/2} f^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.128194, size = 176, normalized size = 0.73 \[ \frac{d^2 x^3 (a d f+3 b c f-2 b d e)}{3 f^3}-\frac{(d e-c f)^2 \tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ) (b e (7 d e-c f)-a f (c f+5 d e))}{2 e^{3/2} f^{9/2}}+\frac{x (b e-a f) (d e-c f)^3}{2 e f^4 \left (e+f x^2\right )}+\frac{d x \left (a d f (3 c f-2 d e)+3 b (d e-c f)^2\right )}{f^4}+\frac{b d^3 x^5}{5 f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(3*b*(d*e - c*f)^2 + a*d*f*(-2*d*e + 3*c*f))*x)/f^4 + (d^2*(-2*b*d*e + 3*b*c*f + a*d*f)*x^3)/(3*f^3) + (b*d
^3*x^5)/(5*f^2) + ((b*e - a*f)*(d*e - c*f)^3*x)/(2*e*f^4*(e + f*x^2)) - ((d*e - c*f)^2*(b*e*(7*d*e - c*f) - a*
f*(5*d*e + c*f))*ArcTan[(Sqrt[f]*x)/Sqrt[e]])/(2*e^(3/2)*f^(9/2))

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Maple [B]  time = 0.013, size = 475, normalized size = 2. \begin{align*}{\frac{15\,bc{d}^{2}{e}^{2}}{2\,{f}^{3}}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}-{\frac{3\,{e}^{2}xbc{d}^{2}}{2\,{f}^{3} \left ( f{x}^{2}+e \right ) }}-{\frac{9\,b{c}^{2}de}{2\,{f}^{2}}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}-{\frac{{e}^{2}xa{d}^{3}}{2\,{f}^{3} \left ( f{x}^{2}+e \right ) }}+{\frac{{e}^{3}xb{d}^{3}}{2\,{f}^{4} \left ( f{x}^{2}+e \right ) }}+{\frac{3\,a{c}^{2}d}{2\,f}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}+{\frac{5\,a{d}^{3}{e}^{2}}{2\,{f}^{3}}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}-{\frac{9\,ac{d}^{2}e}{2\,{f}^{2}}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}+{\frac{3\,acex{d}^{2}}{2\,{f}^{2} \left ( f{x}^{2}+e \right ) }}+{\frac{3\,bxe{c}^{2}d}{2\,{f}^{2} \left ( f{x}^{2}+e \right ) }}-{\frac{bx{c}^{3}}{2\,f \left ( f{x}^{2}+e \right ) }}-{\frac{2\,{d}^{3}{x}^{3}be}{3\,{f}^{3}}}+{\frac{{d}^{2}{x}^{3}bc}{{f}^{2}}}-{\frac{3\,ax{c}^{2}d}{2\,f \left ( f{x}^{2}+e \right ) }}+{\frac{a{c}^{3}}{2\,e}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}+{\frac{b{c}^{3}}{2\,f}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}}+{\frac{a{c}^{3}x}{2\,e \left ( f{x}^{2}+e \right ) }}+{\frac{b{d}^{3}{x}^{5}}{5\,{f}^{2}}}+{\frac{{d}^{3}{x}^{3}a}{3\,{f}^{2}}}+3\,{\frac{b{d}^{3}{e}^{2}x}{{f}^{4}}}+3\,{\frac{b{c}^{2}dx}{{f}^{2}}}+3\,{\frac{ac{d}^{2}x}{{f}^{2}}}-2\,{\frac{a{d}^{3}ex}{{f}^{3}}}-6\,{\frac{bc{d}^{2}ex}{{f}^{3}}}-{\frac{7\,b{d}^{3}{e}^{3}}{2\,{f}^{4}}\arctan \left ({fx{\frac{1}{\sqrt{ef}}}} \right ){\frac{1}{\sqrt{ef}}}} \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)^2,x)

[Out]

15/2/f^3*e^2/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*c*d^2-3/2/f^3*e^2*x/(f*x^2+e)*b*c*d^2-9/2/f^2*e/(e*f)^(1/2)
*arctan(x*f/(e*f)^(1/2))*b*c^2*d-1/2/f^3*e^2*x/(f*x^2+e)*a*d^3+1/2/f^4*e^3*x/(f*x^2+e)*b*d^3+3/2/f/(e*f)^(1/2)
*arctan(x*f/(e*f)^(1/2))*a*c^2*d+5/2/f^3*e^2/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*a*d^3-9/2/f^2*e/(e*f)^(1/2)*a
rctan(x*f/(e*f)^(1/2))*a*c*d^2+3/2/f^2*e*x/(f*x^2+e)*a*c*d^2+3/2/f^2*e*x/(f*x^2+e)*b*c^2*d-1/2/f*x/(f*x^2+e)*b
*c^3-2/3*d^3/f^3*x^3*b*e+d^2/f^2*x^3*b*c-3/2/f*x/(f*x^2+e)*a*c^2*d+1/2/e/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*a
*c^3+1/2/f/(e*f)^(1/2)*arctan(x*f/(e*f)^(1/2))*b*c^3+1/2/e*x/(f*x^2+e)*a*c^3+1/5*d^3/f^2*b*x^5+1/3*d^3/f^2*x^3
*a+3*d^3/f^4*b*e^2*x+3*d/f^2*b*c^2*x+3*d^2/f^2*a*c*x-2*d^3/f^3*a*e*x-6*d^2/f^3*b*c*e*x-7/2/f^4*e^3/(e*f)^(1/2)
*arctan(x*f/(e*f)^(1/2))*b*d^3

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.55181, size = 1701, normalized size = 7.03 \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)^2,x, algorithm="fricas")

[Out]

[1/60*(12*b*d^3*e^2*f^4*x^7 - 4*(7*b*d^3*e^3*f^3 - 5*(3*b*c*d^2 + a*d^3)*e^2*f^4)*x^5 + 20*(7*b*d^3*e^4*f^2 -
5*(3*b*c*d^2 + a*d^3)*e^3*f^3 + 9*(b*c^2*d + a*c*d^2)*e^2*f^4)*x^3 + 15*(7*b*d^3*e^5 - a*c^3*e*f^4 - 5*(3*b*c*
d^2 + a*d^3)*e^4*f + 9*(b*c^2*d + a*c*d^2)*e^3*f^2 - (b*c^3 + 3*a*c^2*d)*e^2*f^3 + (7*b*d^3*e^4*f - a*c^3*f^5
- 5*(3*b*c*d^2 + a*d^3)*e^3*f^2 + 9*(b*c^2*d + a*c*d^2)*e^2*f^3 - (b*c^3 + 3*a*c^2*d)*e*f^4)*x^2)*sqrt(-e*f)*l
og((f*x^2 - 2*sqrt(-e*f)*x - e)/(f*x^2 + e)) + 30*(7*b*d^3*e^5*f + a*c^3*e*f^5 - 5*(3*b*c*d^2 + a*d^3)*e^4*f^2
+ 9*(b*c^2*d + a*c*d^2)*e^3*f^3 - (b*c^3 + 3*a*c^2*d)*e^2*f^4)*x)/(e^2*f^6*x^2 + e^3*f^5), 1/30*(6*b*d^3*e^2*
f^4*x^7 - 2*(7*b*d^3*e^3*f^3 - 5*(3*b*c*d^2 + a*d^3)*e^2*f^4)*x^5 + 10*(7*b*d^3*e^4*f^2 - 5*(3*b*c*d^2 + a*d^3
)*e^3*f^3 + 9*(b*c^2*d + a*c*d^2)*e^2*f^4)*x^3 - 15*(7*b*d^3*e^5 - a*c^3*e*f^4 - 5*(3*b*c*d^2 + a*d^3)*e^4*f +
9*(b*c^2*d + a*c*d^2)*e^3*f^2 - (b*c^3 + 3*a*c^2*d)*e^2*f^3 + (7*b*d^3*e^4*f - a*c^3*f^5 - 5*(3*b*c*d^2 + a*d
^3)*e^3*f^2 + 9*(b*c^2*d + a*c*d^2)*e^2*f^3 - (b*c^3 + 3*a*c^2*d)*e*f^4)*x^2)*sqrt(e*f)*arctan(sqrt(e*f)*x/e)
+ 15*(7*b*d^3*e^5*f + a*c^3*e*f^5 - 5*(3*b*c*d^2 + a*d^3)*e^4*f^2 + 9*(b*c^2*d + a*c*d^2)*e^3*f^3 - (b*c^3 + 3
*a*c^2*d)*e^2*f^4)*x)/(e^2*f^6*x^2 + e^3*f^5)]

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Sympy [B]  time = 7.2812, size = 654, normalized size = 2.7 \begin{align*} \frac{b d^{3} x^{5}}{5 f^{2}} + \frac{x \left (a c^{3} f^{4} - 3 a c^{2} d e f^{3} + 3 a c d^{2} e^{2} f^{2} - a d^{3} e^{3} f - b c^{3} e f^{3} + 3 b c^{2} d e^{2} f^{2} - 3 b c d^{2} e^{3} f + b d^{3} e^{4}\right )}{2 e^{2} f^{4} + 2 e f^{5} x^{2}} - \frac{\sqrt{- \frac{1}{e^{3} f^{9}}} \left (c f - d e\right )^{2} \left (a c f^{2} + 5 a d e f + b c e f - 7 b d e^{2}\right ) \log{\left (- \frac{e^{2} f^{4} \sqrt{- \frac{1}{e^{3} f^{9}}} \left (c f - d e\right )^{2} \left (a c f^{2} + 5 a d e f + b c e f - 7 b d e^{2}\right )}{a c^{3} f^{4} + 3 a c^{2} d e f^{3} - 9 a c d^{2} e^{2} f^{2} + 5 a d^{3} e^{3} f + b c^{3} e f^{3} - 9 b c^{2} d e^{2} f^{2} + 15 b c d^{2} e^{3} f - 7 b d^{3} e^{4}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{e^{3} f^{9}}} \left (c f - d e\right )^{2} \left (a c f^{2} + 5 a d e f + b c e f - 7 b d e^{2}\right ) \log{\left (\frac{e^{2} f^{4} \sqrt{- \frac{1}{e^{3} f^{9}}} \left (c f - d e\right )^{2} \left (a c f^{2} + 5 a d e f + b c e f - 7 b d e^{2}\right )}{a c^{3} f^{4} + 3 a c^{2} d e f^{3} - 9 a c d^{2} e^{2} f^{2} + 5 a d^{3} e^{3} f + b c^{3} e f^{3} - 9 b c^{2} d e^{2} f^{2} + 15 b c d^{2} e^{3} f - 7 b d^{3} e^{4}} + x \right )}}{4} + \frac{x^{3} \left (a d^{3} f + 3 b c d^{2} f - 2 b d^{3} e\right )}{3 f^{3}} + \frac{x \left (3 a c d^{2} f^{2} - 2 a d^{3} e f + 3 b c^{2} d f^{2} - 6 b c d^{2} e f + 3 b d^{3} e^{2}\right )}{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)**2,x)

[Out]

b*d**3*x**5/(5*f**2) + x*(a*c**3*f**4 - 3*a*c**2*d*e*f**3 + 3*a*c*d**2*e**2*f**2 - a*d**3*e**3*f - b*c**3*e*f*
*3 + 3*b*c**2*d*e**2*f**2 - 3*b*c*d**2*e**3*f + b*d**3*e**4)/(2*e**2*f**4 + 2*e*f**5*x**2) - sqrt(-1/(e**3*f**
9))*(c*f - d*e)**2*(a*c*f**2 + 5*a*d*e*f + b*c*e*f - 7*b*d*e**2)*log(-e**2*f**4*sqrt(-1/(e**3*f**9))*(c*f - d*
e)**2*(a*c*f**2 + 5*a*d*e*f + b*c*e*f - 7*b*d*e**2)/(a*c**3*f**4 + 3*a*c**2*d*e*f**3 - 9*a*c*d**2*e**2*f**2 +
5*a*d**3*e**3*f + b*c**3*e*f**3 - 9*b*c**2*d*e**2*f**2 + 15*b*c*d**2*e**3*f - 7*b*d**3*e**4) + x)/4 + sqrt(-1/
(e**3*f**9))*(c*f - d*e)**2*(a*c*f**2 + 5*a*d*e*f + b*c*e*f - 7*b*d*e**2)*log(e**2*f**4*sqrt(-1/(e**3*f**9))*(
c*f - d*e)**2*(a*c*f**2 + 5*a*d*e*f + b*c*e*f - 7*b*d*e**2)/(a*c**3*f**4 + 3*a*c**2*d*e*f**3 - 9*a*c*d**2*e**2
*f**2 + 5*a*d**3*e**3*f + b*c**3*e*f**3 - 9*b*c**2*d*e**2*f**2 + 15*b*c*d**2*e**3*f - 7*b*d**3*e**4) + x)/4 +
x**3*(a*d**3*f + 3*b*c*d**2*f - 2*b*d**3*e)/(3*f**3) + x*(3*a*c*d**2*f**2 - 2*a*d**3*e*f + 3*b*c**2*d*f**2 - 6
*b*c*d**2*e*f + 3*b*d**3*e**2)/f**4

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Giac [A]  time = 1.21962, size = 433, normalized size = 1.79 \begin{align*} \frac{{\left (a c^{3} f^{4} + b c^{3} f^{3} e + 3 \, a c^{2} d f^{3} e - 9 \, b c^{2} d f^{2} e^{2} - 9 \, a c d^{2} f^{2} e^{2} + 15 \, b c d^{2} f e^{3} + 5 \, a d^{3} f e^{3} - 7 \, b d^{3} e^{4}\right )} \arctan \left (\sqrt{f} x e^{\left (-\frac{1}{2}\right )}\right ) e^{\left (-\frac{3}{2}\right )}}{2 \, f^{\frac{9}{2}}} + \frac{{\left (a c^{3} f^{4} x - b c^{3} f^{3} x e - 3 \, a c^{2} d f^{3} x e + 3 \, b c^{2} d f^{2} x e^{2} + 3 \, a c d^{2} f^{2} x e^{2} - 3 \, b c d^{2} f x e^{3} - a d^{3} f x e^{3} + b d^{3} x e^{4}\right )} e^{\left (-1\right )}}{2 \,{\left (f x^{2} + e\right )} f^{4}} + \frac{3 \, b d^{3} f^{8} x^{5} + 15 \, b c d^{2} f^{8} x^{3} + 5 \, a d^{3} f^{8} x^{3} - 10 \, b d^{3} f^{7} x^{3} e + 45 \, b c^{2} d f^{8} x + 45 \, a c d^{2} f^{8} x - 90 \, b c d^{2} f^{7} x e - 30 \, a d^{3} f^{7} x e + 45 \, b d^{3} f^{6} x e^{2}}{15 \, f^{10}} \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)^2,x, algorithm="giac")

[Out]

1/2*(a*c^3*f^4 + b*c^3*f^3*e + 3*a*c^2*d*f^3*e - 9*b*c^2*d*f^2*e^2 - 9*a*c*d^2*f^2*e^2 + 15*b*c*d^2*f*e^3 + 5*
a*d^3*f*e^3 - 7*b*d^3*e^4)*arctan(sqrt(f)*x*e^(-1/2))*e^(-3/2)/f^(9/2) + 1/2*(a*c^3*f^4*x - b*c^3*f^3*x*e - 3*
a*c^2*d*f^3*x*e + 3*b*c^2*d*f^2*x*e^2 + 3*a*c*d^2*f^2*x*e^2 - 3*b*c*d^2*f*x*e^3 - a*d^3*f*x*e^3 + b*d^3*x*e^4)
*e^(-1)/((f*x^2 + e)*f^4) + 1/15*(3*b*d^3*f^8*x^5 + 15*b*c*d^2*f^8*x^3 + 5*a*d^3*f^8*x^3 - 10*b*d^3*f^7*x^3*e
+ 45*b*c^2*d*f^8*x + 45*a*c*d^2*f^8*x - 90*b*c*d^2*f^7*x*e - 30*a*d^3*f^7*x*e + 45*b*d^3*f^6*x*e^2)/f^10