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

Optimal. Leaf size=291 \[ \frac {d x \left (3 a f \left (-3 c^2 f^2-4 c d e f+15 d^2 e^2\right )-b e \left (3 c^2 f^2-100 c d e f+105 d^2 e^2\right )\right )}{24 e^2 f^4}+\frac {(d e-c f) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (b e \left (-c^2 f^2-10 c d e f+35 d^2 e^2\right )-3 a f \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )}{8 e^{5/2} f^{9/2}}+\frac {d x \left (c+d x^2\right ) (b e (35 d e-3 c f)-3 a f (3 c f+5 d e))}{24 e^2 f^3}-\frac {x \left (c+d x^2\right )^2 (b e (7 d e-c f)-3 a f (c f+d e))}{8 e^2 f^2 \left (e+f x^2\right )}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{4 e f \left (e+f x^2\right )^2} \]

[Out]

1/24*d*(3*a*f*(-3*c^2*f^2-4*c*d*e*f+15*d^2*e^2)-b*e*(3*c^2*f^2-100*c*d*e*f+105*d^2*e^2))*x/e^2/f^4+1/24*d*(b*e
*(-3*c*f+35*d*e)-3*a*f*(3*c*f+5*d*e))*x*(d*x^2+c)/e^2/f^3-1/4*(-a*f+b*e)*x*(d*x^2+c)^3/e/f/(f*x^2+e)^2-1/8*(b*
e*(-c*f+7*d*e)-3*a*f*(c*f+d*e))*x*(d*x^2+c)^2/e^2/f^2/(f*x^2+e)+1/8*(-c*f+d*e)*(b*e*(-c^2*f^2-10*c*d*e*f+35*d^
2*e^2)-3*a*f*(c^2*f^2+2*c*d*e*f+5*d^2*e^2))*arctan(x*f^(1/2)/e^(1/2))/e^(5/2)/f^(9/2)

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Rubi [A]  time = 0.42, antiderivative size = 291, normalized size of antiderivative = 1.00, 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} \[ \frac {d x \left (3 a f \left (-3 c^2 f^2-4 c d e f+15 d^2 e^2\right )-b e \left (3 c^2 f^2-100 c d e f+105 d^2 e^2\right )\right )}{24 e^2 f^4}+\frac {(d e-c f) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (b e \left (-c^2 f^2-10 c d e f+35 d^2 e^2\right )-3 a f \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )}{8 e^{5/2} f^{9/2}}-\frac {x \left (c+d x^2\right )^2 (b e (7 d e-c f)-3 a f (c f+d e))}{8 e^2 f^2 \left (e+f x^2\right )}+\frac {d x \left (c+d x^2\right ) (b e (35 d e-3 c f)-3 a f (3 c f+5 d e))}{24 e^2 f^3}-\frac {x \left (c+d x^2\right )^3 (b e-a f)}{4 e f \left (e+f x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(3*a*f*(15*d^2*e^2 - 4*c*d*e*f - 3*c^2*f^2) - b*e*(105*d^2*e^2 - 100*c*d*e*f + 3*c^2*f^2))*x)/(24*e^2*f^4)
+ (d*(b*e*(35*d*e - 3*c*f) - 3*a*f*(5*d*e + 3*c*f))*x*(c + d*x^2))/(24*e^2*f^3) - ((b*e - a*f)*x*(c + d*x^2)^3
)/(4*e*f*(e + f*x^2)^2) - ((b*e*(7*d*e - c*f) - 3*a*f*(d*e + c*f))*x*(c + d*x^2)^2)/(8*e^2*f^2*(e + f*x^2)) +
((d*e - c*f)*(b*e*(35*d^2*e^2 - 10*c*d*e*f - c^2*f^2) - 3*a*f*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2))*ArcTan[(Sqrt[
f]*x)/Sqrt[e]])/(8*e^(5/2)*f^(9/2))

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]

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 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]

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 )^3} \, dx &=-\frac {(b e-a f) x \left (c+d x^2\right )^3}{4 e f \left (e+f x^2\right )^2}-\frac {\int \frac {\left (c+d x^2\right )^2 \left (-c (b e+3 a f)-d (7 b e-3 a f) x^2\right )}{\left (e+f x^2\right )^2} \, dx}{4 e f}\\ &=-\frac {(b e-a f) x \left (c+d x^2\right )^3}{4 e f \left (e+f x^2\right )^2}-\frac {(b e (7 d e-c f)-3 a f (d e+c f)) x \left (c+d x^2\right )^2}{8 e^2 f^2 \left (e+f x^2\right )}+\frac {\int \frac {\left (c+d x^2\right ) \left (-c (3 a f (d e-c f)-b e (7 d e+c f))+d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x^2\right )}{e+f x^2} \, dx}{8 e^2 f^2}\\ &=\frac {d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x \left (c+d x^2\right )}{24 e^2 f^3}-\frac {(b e-a f) x \left (c+d x^2\right )^3}{4 e f \left (e+f x^2\right )^2}-\frac {(b e (7 d e-c f)-3 a f (d e+c f)) x \left (c+d x^2\right )^2}{8 e^2 f^2 \left (e+f x^2\right )}+\frac {\int \frac {-c \left (b e \left (35 d^2 e^2-24 c d e f-3 c^2 f^2\right )-3 a f \left (5 d^2 e^2+3 c^2 f^2\right )\right )+d \left (3 a f \left (15 d^2 e^2-4 c d e f-3 c^2 f^2\right )-b e \left (105 d^2 e^2-100 c d e f+3 c^2 f^2\right )\right ) x^2}{e+f x^2} \, dx}{24 e^2 f^3}\\ &=\frac {d \left (3 a f \left (15 d^2 e^2-4 c d e f-3 c^2 f^2\right )-b e \left (105 d^2 e^2-100 c d e f+3 c^2 f^2\right )\right ) x}{24 e^2 f^4}+\frac {d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x \left (c+d x^2\right )}{24 e^2 f^3}-\frac {(b e-a f) x \left (c+d x^2\right )^3}{4 e f \left (e+f x^2\right )^2}-\frac {(b e (7 d e-c f)-3 a f (d e+c f)) x \left (c+d x^2\right )^2}{8 e^2 f^2 \left (e+f x^2\right )}+\frac {\left ((d e-c f) \left (b e \left (35 d^2 e^2-10 c d e f-c^2 f^2\right )-3 a f \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right )\right ) \int \frac {1}{e+f x^2} \, dx}{8 e^2 f^4}\\ &=\frac {d \left (3 a f \left (15 d^2 e^2-4 c d e f-3 c^2 f^2\right )-b e \left (105 d^2 e^2-100 c d e f+3 c^2 f^2\right )\right ) x}{24 e^2 f^4}+\frac {d (b e (35 d e-3 c f)-3 a f (5 d e+3 c f)) x \left (c+d x^2\right )}{24 e^2 f^3}-\frac {(b e-a f) x \left (c+d x^2\right )^3}{4 e f \left (e+f x^2\right )^2}-\frac {(b e (7 d e-c f)-3 a f (d e+c f)) x \left (c+d x^2\right )^2}{8 e^2 f^2 \left (e+f x^2\right )}+\frac {(d e-c f) \left (b e \left (35 d^2 e^2-10 c d e f-c^2 f^2\right )-3 a f \left (5 d^2 e^2+2 c d e f+c^2 f^2\right )\right ) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{8 e^{5/2} f^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 219, normalized size = 0.75 \[ \frac {(d e-c f) \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (b e \left (-c^2 f^2-10 c d e f+35 d^2 e^2\right )-3 a f \left (c^2 f^2+2 c d e f+5 d^2 e^2\right )\right )}{8 e^{5/2} f^{9/2}}+\frac {d^2 x (a d f+3 b c f-3 b d e)}{f^4}-\frac {x (d e-c f)^2 (b e (13 d e-c f)-3 a f (c f+3 d e))}{8 e^2 f^4 \left (e+f x^2\right )}+\frac {x (b e-a f) (d e-c f)^3}{4 e f^4 \left (e+f x^2\right )^2}+\frac {b d^3 x^3}{3 f^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(-3*b*d*e + 3*b*c*f + a*d*f)*x)/f^4 + (b*d^3*x^3)/(3*f^3) + ((b*e - a*f)*(d*e - c*f)^3*x)/(4*e*f^4*(e + f
*x^2)^2) - ((d*e - c*f)^2*(b*e*(13*d*e - c*f) - 3*a*f*(3*d*e + c*f))*x)/(8*e^2*f^4*(e + f*x^2)) + ((d*e - c*f)
*(b*e*(35*d^2*e^2 - 10*c*d*e*f - c^2*f^2) - 3*a*f*(5*d^2*e^2 + 2*c*d*e*f + c^2*f^2))*ArcTan[(Sqrt[f]*x)/Sqrt[e
]])/(8*e^(5/2)*f^(9/2))

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fricas [A]  time = 0.84, size = 1102, normalized size = 3.79 \[ \text {result too large to display} \]

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)^3,x, algorithm="fricas")

[Out]

[1/48*(16*b*d^3*e^3*f^4*x^7 - 16*(7*b*d^3*e^4*f^3 - 3*(3*b*c*d^2 + a*d^3)*e^3*f^4)*x^5 - 2*(175*b*d^3*e^5*f^2
- 9*a*c^3*e*f^6 - 75*(3*b*c*d^2 + a*d^3)*e^4*f^3 + 45*(b*c^2*d + a*c*d^2)*e^3*f^4 - 3*(b*c^3 + 3*a*c^2*d)*e^2*
f^5)*x^3 - 3*(35*b*d^3*e^6 + 3*a*c^3*e^2*f^4 - 15*(3*b*c*d^2 + a*d^3)*e^5*f + 9*(b*c^2*d + a*c*d^2)*e^4*f^2 +
(b*c^3 + 3*a*c^2*d)*e^3*f^3 + (35*b*d^3*e^4*f^2 + 3*a*c^3*f^6 - 15*(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 + (b*c^3 + 3*a*c^2*d)*e*f^5)*x^4 + 2*(35*b*d^3*e^5*f + 3*a*c^3*e*f^5 - 15*(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^2)*sqrt(-e*f)*log((f*x^2 - 2*sqrt(-e
*f)*x - e)/(f*x^2 + e)) - 6*(35*b*d^3*e^6*f - 5*a*c^3*e^2*f^5 - 15*(3*b*c*d^2 + a*d^3)*e^5*f^2 + 9*(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)/(e^3*f^7*x^4 + 2*e^4*f^6*x^2 + e^5*f^5), 1/24*(8*b*d^3*e^3*
f^4*x^7 - 8*(7*b*d^3*e^4*f^3 - 3*(3*b*c*d^2 + a*d^3)*e^3*f^4)*x^5 - (175*b*d^3*e^5*f^2 - 9*a*c^3*e*f^6 - 75*(3
*b*c*d^2 + a*d^3)*e^4*f^3 + 45*(b*c^2*d + a*c*d^2)*e^3*f^4 - 3*(b*c^3 + 3*a*c^2*d)*e^2*f^5)*x^3 + 3*(35*b*d^3*
e^6 + 3*a*c^3*e^2*f^4 - 15*(3*b*c*d^2 + a*d^3)*e^5*f + 9*(b*c^2*d + a*c*d^2)*e^4*f^2 + (b*c^3 + 3*a*c^2*d)*e^3
*f^3 + (35*b*d^3*e^4*f^2 + 3*a*c^3*f^6 - 15*(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 + (b*c
^3 + 3*a*c^2*d)*e*f^5)*x^4 + 2*(35*b*d^3*e^5*f + 3*a*c^3*e*f^5 - 15*(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^2)*sqrt(e*f)*arctan(sqrt(e*f)*x/e) - 3*(35*b*d^3*e^6*f - 5*
a*c^3*e^2*f^5 - 15*(3*b*c*d^2 + a*d^3)*e^5*f^2 + 9*(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)/(e^3*f^7*x^4 + 2*e^4*f^6*x^2 + e^5*f^5)]

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giac [A]  time = 0.39, size = 371, normalized size = 1.27 \[ \frac {{\left (3 \, a c^{3} f^{4} + b c^{3} f^{3} e + 3 \, a c^{2} d f^{3} e + 9 \, b c^{2} d f^{2} e^{2} + 9 \, a c d^{2} f^{2} e^{2} - 45 \, b c d^{2} f e^{3} - 15 \, a d^{3} f e^{3} + 35 \, b d^{3} e^{4}\right )} \arctan \left (\sqrt {f} x e^{\left (-\frac {1}{2}\right )}\right ) e^{\left (-\frac {5}{2}\right )}}{8 \, f^{\frac {9}{2}}} + \frac {{\left (3 \, a c^{3} f^{5} x^{3} + b c^{3} f^{4} x^{3} e + 3 \, a c^{2} d f^{4} x^{3} e - 15 \, b c^{2} d f^{3} x^{3} e^{2} - 15 \, a c d^{2} f^{3} x^{3} e^{2} + 27 \, b c d^{2} f^{2} x^{3} e^{3} + 9 \, a d^{3} f^{2} x^{3} e^{3} + 5 \, a c^{3} f^{4} x e - 13 \, b d^{3} f x^{3} e^{4} - b c^{3} f^{3} x e^{2} - 3 \, a c^{2} d f^{3} x e^{2} - 9 \, b c^{2} d f^{2} x e^{3} - 9 \, a c d^{2} f^{2} x e^{3} + 21 \, b c d^{2} f x e^{4} + 7 \, a d^{3} f x e^{4} - 11 \, b d^{3} x e^{5}\right )} e^{\left (-2\right )}}{8 \, {\left (f x^{2} + e\right )}^{2} f^{4}} + \frac {b d^{3} f^{6} x^{3} + 9 \, b c d^{2} f^{6} x + 3 \, a d^{3} f^{6} x - 9 \, b d^{3} f^{5} x e}{3 \, f^{9}} \]

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)^3,x, algorithm="giac")

[Out]

1/8*(3*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 - 45*b*c*d^2*f*e^3 -
15*a*d^3*f*e^3 + 35*b*d^3*e^4)*arctan(sqrt(f)*x*e^(-1/2))*e^(-5/2)/f^(9/2) + 1/8*(3*a*c^3*f^5*x^3 + b*c^3*f^4*
x^3*e + 3*a*c^2*d*f^4*x^3*e - 15*b*c^2*d*f^3*x^3*e^2 - 15*a*c*d^2*f^3*x^3*e^2 + 27*b*c*d^2*f^2*x^3*e^3 + 9*a*d
^3*f^2*x^3*e^3 + 5*a*c^3*f^4*x*e - 13*b*d^3*f*x^3*e^4 - b*c^3*f^3*x*e^2 - 3*a*c^2*d*f^3*x*e^2 - 9*b*c^2*d*f^2*
x*e^3 - 9*a*c*d^2*f^2*x*e^3 + 21*b*c*d^2*f*x*e^4 + 7*a*d^3*f*x*e^4 - 11*b*d^3*x*e^5)*e^(-2)/((f*x^2 + e)^2*f^4
) + 1/3*(b*d^3*f^6*x^3 + 9*b*c*d^2*f^6*x + 3*a*d^3*f^6*x - 9*b*d^3*f^5*x*e)/f^9

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maple [B]  time = 0.02, size = 589, normalized size = 2.02 \[ \frac {3 a \,c^{3} f \,x^{3}}{8 \left (f \,x^{2}+e \right )^{2} e^{2}}+\frac {3 a \,c^{2} d \,x^{3}}{8 \left (f \,x^{2}+e \right )^{2} e}-\frac {15 a c \,d^{2} x^{3}}{8 \left (f \,x^{2}+e \right )^{2} f}+\frac {9 a \,d^{3} e \,x^{3}}{8 \left (f \,x^{2}+e \right )^{2} f^{2}}+\frac {b \,c^{3} x^{3}}{8 \left (f \,x^{2}+e \right )^{2} e}-\frac {15 b \,c^{2} d \,x^{3}}{8 \left (f \,x^{2}+e \right )^{2} f}+\frac {27 b c \,d^{2} e \,x^{3}}{8 \left (f \,x^{2}+e \right )^{2} f^{2}}-\frac {13 b \,d^{3} e^{2} x^{3}}{8 \left (f \,x^{2}+e \right )^{2} f^{3}}+\frac {5 a \,c^{3} x}{8 \left (f \,x^{2}+e \right )^{2} e}-\frac {3 a \,c^{2} d x}{8 \left (f \,x^{2}+e \right )^{2} f}-\frac {9 a c \,d^{2} e x}{8 \left (f \,x^{2}+e \right )^{2} f^{2}}+\frac {7 a \,d^{3} e^{2} x}{8 \left (f \,x^{2}+e \right )^{2} f^{3}}-\frac {b \,c^{3} x}{8 \left (f \,x^{2}+e \right )^{2} f}-\frac {9 b \,c^{2} d e x}{8 \left (f \,x^{2}+e \right )^{2} f^{2}}+\frac {21 b c \,d^{2} e^{2} x}{8 \left (f \,x^{2}+e \right )^{2} f^{3}}-\frac {11 b \,d^{3} e^{3} x}{8 \left (f \,x^{2}+e \right )^{2} f^{4}}+\frac {b \,d^{3} x^{3}}{3 f^{3}}+\frac {3 a \,c^{3} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, e^{2}}+\frac {3 a \,c^{2} d \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, e f}+\frac {9 a c \,d^{2} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, f^{2}}-\frac {15 a \,d^{3} e \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, f^{3}}+\frac {b \,c^{3} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, e f}+\frac {9 b \,c^{2} d \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, f^{2}}-\frac {45 b c \,d^{2} e \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, f^{3}}+\frac {35 b \,d^{3} e^{2} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \sqrt {e f}\, f^{4}}+\frac {a \,d^{3} x}{f^{3}}+\frac {3 b c \,d^{2} x}{f^{3}}-\frac {3 b \,d^{3} e x}{f^{4}} \]

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)^3,x)

[Out]

-45/8/f^3*e/(e*f)^(1/2)*arctan(1/(e*f)^(1/2)*f*x)*b*c*d^2+27/8/f^2/(f*x^2+e)^2*x^3*b*c*d^2*e-9/8/f^2/(f*x^2+e)
^2*a*c*d^2*e*x-9/8/f^2/(f*x^2+e)^2*b*c^2*d*e*x+21/8/f^3/(f*x^2+e)^2*b*c*d^2*e^2*x+3/8/f/e/(e*f)^(1/2)*arctan(1
/(e*f)^(1/2)*f*x)*a*c^2*d+3/8/(f*x^2+e)^2/e*x^3*a*c^2*d-15/8/f/(f*x^2+e)^2*x^3*b*c^2*d-13/8/f^3/(f*x^2+e)^2*x^
3*b*d^3*e^2-3/8/f/(f*x^2+e)^2*a*c^2*d*x+7/8/f^3/(f*x^2+e)^2*a*d^3*e^2*x-11/8/f^4/(f*x^2+e)^2*b*d^3*e^3*x+9/8/f
^2/(e*f)^(1/2)*arctan(1/(e*f)^(1/2)*f*x)*a*c*d^2-15/8/f^3*e/(e*f)^(1/2)*arctan(1/(e*f)^(1/2)*f*x)*a*d^3+1/8/f/
e/(e*f)^(1/2)*arctan(1/(e*f)^(1/2)*f*x)*b*c^3+9/8/f^2/(e*f)^(1/2)*arctan(1/(e*f)^(1/2)*f*x)*b*c^2*d+35/8/f^4*e
^2/(e*f)^(1/2)*arctan(1/(e*f)^(1/2)*f*x)*b*d^3+3/8*f/(f*x^2+e)^2/e^2*x^3*a*c^3-15/8/f/(f*x^2+e)^2*x^3*a*c*d^2+
9/8/f^2/(f*x^2+e)^2*x^3*a*d^3*e+d^3/f^3*a*x+1/3*d^3/f^3*x^3*b-3*d^3/f^4*b*e*x-1/8/f/(f*x^2+e)^2*b*c^3*x+3/8/e^
2/(e*f)^(1/2)*arctan(1/(e*f)^(1/2)*f*x)*a*c^3+3*d^2/f^3*b*c*x+1/8/(f*x^2+e)^2/e*x^3*b*c^3+5/8/(f*x^2+e)^2/e*x*
a*c^3

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maxima [A]  time = 1.94, size = 343, normalized size = 1.18 \[ -\frac {{\left (13 \, b d^{3} e^{4} f - 3 \, a c^{3} f^{5} - 9 \, {\left (3 \, b c d^{2} + a d^{3}\right )} e^{3} f^{2} + 15 \, {\left (b c^{2} d + a c d^{2}\right )} e^{2} f^{3} - {\left (b c^{3} + 3 \, a c^{2} d\right )} e f^{4}\right )} x^{3} + {\left (11 \, b d^{3} e^{5} - 5 \, a c^{3} e f^{4} - 7 \, {\left (3 \, b c d^{2} + a d^{3}\right )} e^{4} f + 9 \, {\left (b c^{2} d + a c d^{2}\right )} e^{3} f^{2} + {\left (b c^{3} + 3 \, a c^{2} d\right )} e^{2} f^{3}\right )} x}{8 \, {\left (e^{2} f^{6} x^{4} + 2 \, e^{3} f^{5} x^{2} + e^{4} f^{4}\right )}} + \frac {b d^{3} f x^{3} - 3 \, {\left (3 \, b d^{3} e - {\left (3 \, b c d^{2} + a d^{3}\right )} f\right )} x}{3 \, f^{4}} + \frac {{\left (35 \, b d^{3} e^{4} + 3 \, a c^{3} f^{4} - 15 \, {\left (3 \, b c d^{2} + a d^{3}\right )} e^{3} f + 9 \, {\left (b c^{2} d + a c d^{2}\right )} e^{2} f^{2} + {\left (b c^{3} + 3 \, a c^{2} d\right )} e f^{3}\right )} \arctan \left (\frac {f x}{\sqrt {e f}}\right )}{8 \, \sqrt {e f} e^{2} f^{4}} \]

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)^3,x, algorithm="maxima")

[Out]

-1/8*((13*b*d^3*e^4*f - 3*a*c^3*f^5 - 9*(3*b*c*d^2 + a*d^3)*e^3*f^2 + 15*(b*c^2*d + a*c*d^2)*e^2*f^3 - (b*c^3
+ 3*a*c^2*d)*e*f^4)*x^3 + (11*b*d^3*e^5 - 5*a*c^3*e*f^4 - 7*(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)*x)/(e^2*f^6*x^4 + 2*e^3*f^5*x^2 + e^4*f^4) + 1/3*(b*d^3*f*x^3 - 3*(3*b*
d^3*e - (3*b*c*d^2 + a*d^3)*f)*x)/f^4 + 1/8*(35*b*d^3*e^4 + 3*a*c^3*f^4 - 15*(3*b*c*d^2 + a*d^3)*e^3*f + 9*(b*
c^2*d + a*c*d^2)*e^2*f^2 + (b*c^3 + 3*a*c^2*d)*e*f^3)*arctan(f*x/sqrt(e*f))/(sqrt(e*f)*e^2*f^4)

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mupad [B]  time = 0.22, size = 495, normalized size = 1.70 \[ \frac {\frac {x^3\,\left (b\,c^3\,e\,f^4+3\,a\,c^3\,f^5-15\,b\,c^2\,d\,e^2\,f^3+3\,a\,c^2\,d\,e\,f^4+27\,b\,c\,d^2\,e^3\,f^2-15\,a\,c\,d^2\,e^2\,f^3-13\,b\,d^3\,e^4\,f+9\,a\,d^3\,e^3\,f^2\right )}{8\,e^2}-\frac {x\,\left (b\,c^3\,e\,f^3-5\,a\,c^3\,f^4+9\,b\,c^2\,d\,e^2\,f^2+3\,a\,c^2\,d\,e\,f^3-21\,b\,c\,d^2\,e^3\,f+9\,a\,c\,d^2\,e^2\,f^2+11\,b\,d^3\,e^4-7\,a\,d^3\,e^3\,f\right )}{8\,e}}{e^2\,f^4+2\,e\,f^5\,x^2+f^6\,x^4}+x\,\left (\frac {a\,d^3+3\,b\,c\,d^2}{f^3}-\frac {3\,b\,d^3\,e}{f^4}\right )+\frac {b\,d^3\,x^3}{3\,f^3}+\frac {\mathrm {atan}\left (\frac {\sqrt {f}\,x\,\left (c\,f-d\,e\right )\,\left (b\,c^2\,e\,f^2+3\,a\,c^2\,f^3+10\,b\,c\,d\,e^2\,f+6\,a\,c\,d\,e\,f^2-35\,b\,d^2\,e^3+15\,a\,d^2\,e^2\,f\right )}{\sqrt {e}\,\left (b\,c^3\,e\,f^3+3\,a\,c^3\,f^4+9\,b\,c^2\,d\,e^2\,f^2+3\,a\,c^2\,d\,e\,f^3-45\,b\,c\,d^2\,e^3\,f+9\,a\,c\,d^2\,e^2\,f^2+35\,b\,d^3\,e^4-15\,a\,d^3\,e^3\,f\right )}\right )\,\left (c\,f-d\,e\right )\,\left (b\,c^2\,e\,f^2+3\,a\,c^2\,f^3+10\,b\,c\,d\,e^2\,f+6\,a\,c\,d\,e\,f^2-35\,b\,d^2\,e^3+15\,a\,d^2\,e^2\,f\right )}{8\,e^{5/2}\,f^{9/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x^3*(3*a*c^3*f^5 + 9*a*d^3*e^3*f^2 + b*c^3*e*f^4 - 13*b*d^3*e^4*f + 3*a*c^2*d*e*f^4 - 15*a*c*d^2*e^2*f^3 + 2
7*b*c*d^2*e^3*f^2 - 15*b*c^2*d*e^2*f^3))/(8*e^2) - (x*(11*b*d^3*e^4 - 5*a*c^3*f^4 - 7*a*d^3*e^3*f + b*c^3*e*f^
3 + 3*a*c^2*d*e*f^3 - 21*b*c*d^2*e^3*f + 9*a*c*d^2*e^2*f^2 + 9*b*c^2*d*e^2*f^2))/(8*e))/(e^2*f^4 + f^6*x^4 + 2
*e*f^5*x^2) + x*((a*d^3 + 3*b*c*d^2)/f^3 - (3*b*d^3*e)/f^4) + (b*d^3*x^3)/(3*f^3) + (atan((f^(1/2)*x*(c*f - d*
e)*(3*a*c^2*f^3 - 35*b*d^2*e^3 + 15*a*d^2*e^2*f + b*c^2*e*f^2 + 6*a*c*d*e*f^2 + 10*b*c*d*e^2*f))/(e^(1/2)*(3*a
*c^3*f^4 + 35*b*d^3*e^4 - 15*a*d^3*e^3*f + b*c^3*e*f^3 + 3*a*c^2*d*e*f^3 - 45*b*c*d^2*e^3*f + 9*a*c*d^2*e^2*f^
2 + 9*b*c^2*d*e^2*f^2)))*(c*f - d*e)*(3*a*c^2*f^3 - 35*b*d^2*e^3 + 15*a*d^2*e^2*f + b*c^2*e*f^2 + 6*a*c*d*e*f^
2 + 10*b*c*d*e^2*f))/(8*e^(5/2)*f^(9/2))

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sympy [B]  time = 26.16, size = 865, normalized size = 2.97 \[ \frac {b d^{3} x^{3}}{3 f^{3}} + x \left (\frac {a d^{3}}{f^{3}} + \frac {3 b c d^{2}}{f^{3}} - \frac {3 b d^{3} e}{f^{4}}\right ) - \frac {\sqrt {- \frac {1}{e^{5} f^{9}}} \left (c f - d e\right ) \left (3 a c^{2} f^{3} + 6 a c d e f^{2} + 15 a d^{2} e^{2} f + b c^{2} e f^{2} + 10 b c d e^{2} f - 35 b d^{2} e^{3}\right ) \log {\left (- \frac {e^{3} f^{4} \sqrt {- \frac {1}{e^{5} f^{9}}} \left (c f - d e\right ) \left (3 a c^{2} f^{3} + 6 a c d e f^{2} + 15 a d^{2} e^{2} f + b c^{2} e f^{2} + 10 b c d e^{2} f - 35 b d^{2} e^{3}\right )}{3 a c^{3} f^{4} + 3 a c^{2} d e f^{3} + 9 a c d^{2} e^{2} f^{2} - 15 a d^{3} e^{3} f + b c^{3} e f^{3} + 9 b c^{2} d e^{2} f^{2} - 45 b c d^{2} e^{3} f + 35 b d^{3} e^{4}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{e^{5} f^{9}}} \left (c f - d e\right ) \left (3 a c^{2} f^{3} + 6 a c d e f^{2} + 15 a d^{2} e^{2} f + b c^{2} e f^{2} + 10 b c d e^{2} f - 35 b d^{2} e^{3}\right ) \log {\left (\frac {e^{3} f^{4} \sqrt {- \frac {1}{e^{5} f^{9}}} \left (c f - d e\right ) \left (3 a c^{2} f^{3} + 6 a c d e f^{2} + 15 a d^{2} e^{2} f + b c^{2} e f^{2} + 10 b c d e^{2} f - 35 b d^{2} e^{3}\right )}{3 a c^{3} f^{4} + 3 a c^{2} d e f^{3} + 9 a c d^{2} e^{2} f^{2} - 15 a d^{3} e^{3} f + b c^{3} e f^{3} + 9 b c^{2} d e^{2} f^{2} - 45 b c d^{2} e^{3} f + 35 b d^{3} e^{4}} + x \right )}}{16} + \frac {x^{3} \left (3 a c^{3} f^{5} + 3 a c^{2} d e f^{4} - 15 a c d^{2} e^{2} f^{3} + 9 a d^{3} e^{3} f^{2} + b c^{3} e f^{4} - 15 b c^{2} d e^{2} f^{3} + 27 b c d^{2} e^{3} f^{2} - 13 b d^{3} e^{4} f\right ) + x \left (5 a c^{3} e f^{4} - 3 a c^{2} d e^{2} f^{3} - 9 a c d^{2} e^{3} f^{2} + 7 a d^{3} e^{4} f - b c^{3} e^{2} f^{3} - 9 b c^{2} d e^{3} f^{2} + 21 b c d^{2} e^{4} f - 11 b d^{3} e^{5}\right )}{8 e^{4} f^{4} + 16 e^{3} f^{5} x^{2} + 8 e^{2} f^{6} x^{4}} \]

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)**3,x)

[Out]

b*d**3*x**3/(3*f**3) + x*(a*d**3/f**3 + 3*b*c*d**2/f**3 - 3*b*d**3*e/f**4) - sqrt(-1/(e**5*f**9))*(c*f - d*e)*
(3*a*c**2*f**3 + 6*a*c*d*e*f**2 + 15*a*d**2*e**2*f + b*c**2*e*f**2 + 10*b*c*d*e**2*f - 35*b*d**2*e**3)*log(-e*
*3*f**4*sqrt(-1/(e**5*f**9))*(c*f - d*e)*(3*a*c**2*f**3 + 6*a*c*d*e*f**2 + 15*a*d**2*e**2*f + b*c**2*e*f**2 +
10*b*c*d*e**2*f - 35*b*d**2*e**3)/(3*a*c**3*f**4 + 3*a*c**2*d*e*f**3 + 9*a*c*d**2*e**2*f**2 - 15*a*d**3*e**3*f
+ b*c**3*e*f**3 + 9*b*c**2*d*e**2*f**2 - 45*b*c*d**2*e**3*f + 35*b*d**3*e**4) + x)/16 + sqrt(-1/(e**5*f**9))*
(c*f - d*e)*(3*a*c**2*f**3 + 6*a*c*d*e*f**2 + 15*a*d**2*e**2*f + b*c**2*e*f**2 + 10*b*c*d*e**2*f - 35*b*d**2*e
**3)*log(e**3*f**4*sqrt(-1/(e**5*f**9))*(c*f - d*e)*(3*a*c**2*f**3 + 6*a*c*d*e*f**2 + 15*a*d**2*e**2*f + b*c**
2*e*f**2 + 10*b*c*d*e**2*f - 35*b*d**2*e**3)/(3*a*c**3*f**4 + 3*a*c**2*d*e*f**3 + 9*a*c*d**2*e**2*f**2 - 15*a*
d**3*e**3*f + b*c**3*e*f**3 + 9*b*c**2*d*e**2*f**2 - 45*b*c*d**2*e**3*f + 35*b*d**3*e**4) + x)/16 + (x**3*(3*a
*c**3*f**5 + 3*a*c**2*d*e*f**4 - 15*a*c*d**2*e**2*f**3 + 9*a*d**3*e**3*f**2 + b*c**3*e*f**4 - 15*b*c**2*d*e**2
*f**3 + 27*b*c*d**2*e**3*f**2 - 13*b*d**3*e**4*f) + x*(5*a*c**3*e*f**4 - 3*a*c**2*d*e**2*f**3 - 9*a*c*d**2*e**
3*f**2 + 7*a*d**3*e**4*f - b*c**3*e**2*f**3 - 9*b*c**2*d*e**3*f**2 + 21*b*c*d**2*e**4*f - 11*b*d**3*e**5))/(8*
e**4*f**4 + 16*e**3*f**5*x**2 + 8*e**2*f**6*x**4)

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