3.1.21 \(\int \frac {(a+b x^2) (c+d x^2)^3}{(e+f x^2)^3} \, dx\) [21]

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

[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 = 0.27, 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 = {540, 542, 396, 211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) (d e-c f) \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 (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 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} \end {gather*}

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 211

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

Rule 396

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 540

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 542

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 = 0.11, size = 219, normalized size = 0.75 \begin {gather*} \frac {d^2 (-3 b d e+3 b c f+a d f) x}{f^4}+\frac {b d^3 x^3}{3 f^3}+\frac {(b e-a f) (d e-c f)^3 x}{4 e f^4 \left (e+f x^2\right )^2}-\frac {(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 \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 {gather*}

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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Maple [A]
time = 0.17, size = 344, normalized size = 1.18

method	result	size
default	\(\frac {d^{2} \left (\frac {1}{3} b d \,x^{3} f +a d f x +3 b c f x -3 b d e x \right )}{f^{4}}+\frac {\frac {\frac {f \left (3 a \,c^{3} f^{4}+3 a \,c^{2} d e \,f^{3}-15 a c \,d^{2} e^{2} f^{2}+9 a \,d^{3} e^{3} f +b \,c^{3} e \,f^{3}-15 b \,c^{2} d \,e^{2} f^{2}+27 b c \,d^{2} e^{3} f -13 b \,d^{3} e^{4}\right ) x^{3}}{8 e^{2}}+\frac {\left (5 a \,c^{3} f^{4}-3 a \,c^{2} d e \,f^{3}-9 a c \,d^{2} e^{2} f^{2}+7 a \,d^{3} e^{3} f -b \,c^{3} e \,f^{3}-9 b \,c^{2} d \,e^{2} f^{2}+21 b c \,d^{2} e^{3} f -11 b \,d^{3} e^{4}\right ) x}{8 e}}{\left (f \,x^{2}+e \right )^{2}}+\frac {\left (3 a \,c^{3} f^{4}+3 a \,c^{2} d e \,f^{3}+9 a c \,d^{2} e^{2} f^{2}-15 a \,d^{3} e^{3} f +b \,c^{3} e \,f^{3}+9 b \,c^{2} d \,e^{2} f^{2}-45 b c \,d^{2} e^{3} f +35 b \,d^{3} e^{4}\right ) \arctan \left (\frac {f x}{\sqrt {f e}}\right )}{8 e^{2} \sqrt {f e}}}{f^{4}}\)	\(344\)
risch	\(\frac {d^{3} b \,x^{3}}{3 f^{3}}+\frac {d^{3} a x}{f^{3}}+\frac {3 d^{2} b c x}{f^{3}}-\frac {3 d^{3} b e x}{f^{4}}+\frac {\frac {f \left (3 a \,c^{3} f^{4}+3 a \,c^{2} d e \,f^{3}-15 a c \,d^{2} e^{2} f^{2}+9 a \,d^{3} e^{3} f +b \,c^{3} e \,f^{3}-15 b \,c^{2} d \,e^{2} f^{2}+27 b c \,d^{2} e^{3} f -13 b \,d^{3} e^{4}\right ) x^{3}}{8 e^{2}}+\frac {\left (5 a \,c^{3} f^{4}-3 a \,c^{2} d e \,f^{3}-9 a c \,d^{2} e^{2} f^{2}+7 a \,d^{3} e^{3} f -b \,c^{3} e \,f^{3}-9 b \,c^{2} d \,e^{2} f^{2}+21 b c \,d^{2} e^{3} f -11 b \,d^{3} e^{4}\right ) x}{8 e}}{f^{4} \left (f \,x^{2}+e \right )^{2}}-\frac {3 \ln \left (f x +\sqrt {-f e}\right ) a \,c^{3}}{16 \sqrt {-f e}\, e^{2}}-\frac {3 \ln \left (f x +\sqrt {-f e}\right ) a \,c^{2} d}{16 f \sqrt {-f e}\, e}-\frac {9 \ln \left (f x +\sqrt {-f e}\right ) a c \,d^{2}}{16 f^{2} \sqrt {-f e}}+\frac {15 e \ln \left (f x +\sqrt {-f e}\right ) a \,d^{3}}{16 f^{3} \sqrt {-f e}}-\frac {\ln \left (f x +\sqrt {-f e}\right ) b \,c^{3}}{16 f \sqrt {-f e}\, e}-\frac {9 \ln \left (f x +\sqrt {-f e}\right ) b \,c^{2} d}{16 f^{2} \sqrt {-f e}}+\frac {45 e \ln \left (f x +\sqrt {-f e}\right ) b c \,d^{2}}{16 f^{3} \sqrt {-f e}}-\frac {35 e^{2} \ln \left (f x +\sqrt {-f e}\right ) b \,d^{3}}{16 f^{4} \sqrt {-f e}}+\frac {3 \ln \left (-f x +\sqrt {-f e}\right ) a \,c^{3}}{16 \sqrt {-f e}\, e^{2}}+\frac {3 \ln \left (-f x +\sqrt {-f e}\right ) a \,c^{2} d}{16 f \sqrt {-f e}\, e}+\frac {9 \ln \left (-f x +\sqrt {-f e}\right ) a c \,d^{2}}{16 f^{2} \sqrt {-f e}}-\frac {15 e \ln \left (-f x +\sqrt {-f e}\right ) a \,d^{3}}{16 f^{3} \sqrt {-f e}}+\frac {\ln \left (-f x +\sqrt {-f e}\right ) b \,c^{3}}{16 f \sqrt {-f e}\, e}+\frac {9 \ln \left (-f x +\sqrt {-f e}\right ) b \,c^{2} d}{16 f^{2} \sqrt {-f e}}-\frac {45 e \ln \left (-f x +\sqrt {-f e}\right ) b c \,d^{2}}{16 f^{3} \sqrt {-f e}}+\frac {35 e^{2} \ln \left (-f x +\sqrt {-f e}\right ) b \,d^{3}}{16 f^{4} \sqrt {-f e}}\)	\(701\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.57, size = 345, normalized size = 1.19 \begin {gather*} \frac {{\left (3 \, a c^{3} f^{5} - 13 \, b d^{3} f e^{4} + {\left (b c^{3} e + 3 \, a c^{2} d e\right )} f^{4} - 15 \, {\left (b c^{2} d e^{2} + a c d^{2} e^{2}\right )} f^{3} + 9 \, {\left (3 \, b c d^{2} e^{3} + a d^{3} e^{3}\right )} f^{2}\right )} x^{3} + {\left (5 \, a c^{3} f^{4} e - 11 \, b d^{3} e^{5} - {\left (b c^{3} e^{2} + 3 \, a c^{2} d e^{2}\right )} f^{3} - 9 \, {\left (b c^{2} d e^{3} + a c d^{2} e^{3}\right )} f^{2} + 7 \, {\left (3 \, b c d^{2} e^{4} + a d^{3} e^{4}\right )} f\right )} x}{8 \, {\left (f^{6} x^{4} e^{2} + 2 \, f^{5} x^{2} e^{3} + f^{4} e^{4}\right )}} + \frac {{\left (3 \, a c^{3} f^{4} + 35 \, b d^{3} e^{4} + {\left (b c^{3} e + 3 \, a c^{2} d e\right )} f^{3} + 9 \, {\left (b c^{2} d e^{2} + a c d^{2} e^{2}\right )} f^{2} - 15 \, {\left (3 \, b c d^{2} e^{3} + a d^{3} e^{3}\right )} f\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 {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}} \end {gather*}

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

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Fricas [A]
time = 1.47, size = 1104, normalized size = 3.79 \begin {gather*} \left [\frac {18 \, a c^{3} f^{6} x^{3} e - 210 \, b d^{3} f x e^{6} - 3 \, {\left (3 \, a c^{3} f^{6} x^{4} + 35 \, b d^{3} e^{6} + 5 \, {\left (14 \, b d^{3} f x^{2} - 3 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f\right )} e^{5} + {\left (35 \, b d^{3} f^{2} x^{4} - 30 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{2} x^{2} + 9 \, {\left (b c^{2} d + a c d^{2}\right )} f^{2}\right )} e^{4} - {\left (15 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{3} x^{4} - 18 \, {\left (b c^{2} d + a c d^{2}\right )} f^{3} x^{2} - {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{3}\right )} e^{3} + {\left (9 \, {\left (b c^{2} d + a c d^{2}\right )} f^{4} x^{4} + 3 \, a c^{3} f^{4} + 2 \, {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{4} x^{2}\right )} e^{2} + {\left (6 \, a c^{3} f^{5} x^{2} + {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{5} x^{4}\right )} e\right )} \sqrt {-f e} \log \left (\frac {f x^{2} - 2 \, \sqrt {-f e} x - e}{f x^{2} + e}\right ) - 10 \, {\left (35 \, b d^{3} f^{2} x^{3} - 9 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{2} x\right )} e^{5} - 2 \, {\left (56 \, b d^{3} f^{3} x^{5} - 75 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{3} x^{3} + 27 \, {\left (b c^{2} d + a c d^{2}\right )} f^{3} x\right )} e^{4} + 2 \, {\left (8 \, b d^{3} f^{4} x^{7} + 24 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{4} x^{5} - 45 \, {\left (b c^{2} d + a c d^{2}\right )} f^{4} x^{3} - 3 \, {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{4} x\right )} e^{3} + 6 \, {\left (5 \, a c^{3} f^{5} x + {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{5} x^{3}\right )} e^{2}}{48 \, {\left (f^{7} x^{4} e^{3} + 2 \, f^{6} x^{2} e^{4} + f^{5} e^{5}\right )}}, \frac {9 \, a c^{3} f^{6} x^{3} e - 105 \, b d^{3} f x e^{6} + 3 \, {\left (3 \, a c^{3} f^{6} x^{4} + 35 \, b d^{3} e^{6} + 5 \, {\left (14 \, b d^{3} f x^{2} - 3 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f\right )} e^{5} + {\left (35 \, b d^{3} f^{2} x^{4} - 30 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{2} x^{2} + 9 \, {\left (b c^{2} d + a c d^{2}\right )} f^{2}\right )} e^{4} - {\left (15 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{3} x^{4} - 18 \, {\left (b c^{2} d + a c d^{2}\right )} f^{3} x^{2} - {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{3}\right )} e^{3} + {\left (9 \, {\left (b c^{2} d + a c d^{2}\right )} f^{4} x^{4} + 3 \, a c^{3} f^{4} + 2 \, {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{4} x^{2}\right )} e^{2} + {\left (6 \, a c^{3} f^{5} x^{2} + {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{5} x^{4}\right )} e\right )} \sqrt {f} \arctan \left (\sqrt {f} x e^{\left (-\frac {1}{2}\right )}\right ) e^{\frac {1}{2}} - 5 \, {\left (35 \, b d^{3} f^{2} x^{3} - 9 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{2} x\right )} e^{5} - {\left (56 \, b d^{3} f^{3} x^{5} - 75 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{3} x^{3} + 27 \, {\left (b c^{2} d + a c d^{2}\right )} f^{3} x\right )} e^{4} + {\left (8 \, b d^{3} f^{4} x^{7} + 24 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{4} x^{5} - 45 \, {\left (b c^{2} d + a c d^{2}\right )} f^{4} x^{3} - 3 \, {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{4} x\right )} e^{3} + 3 \, {\left (5 \, a c^{3} f^{5} x + {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{5} x^{3}\right )} e^{2}}{24 \, {\left (f^{7} x^{4} e^{3} + 2 \, f^{6} x^{2} e^{4} + f^{5} e^{5}\right )}}\right ] \end {gather*}

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*(18*a*c^3*f^6*x^3*e - 210*b*d^3*f*x*e^6 - 3*(3*a*c^3*f^6*x^4 + 35*b*d^3*e^6 + 5*(14*b*d^3*f*x^2 - 3*(3*b
*c*d^2 + a*d^3)*f)*e^5 + (35*b*d^3*f^2*x^4 - 30*(3*b*c*d^2 + a*d^3)*f^2*x^2 + 9*(b*c^2*d + a*c*d^2)*f^2)*e^4 -
(15*(3*b*c*d^2 + a*d^3)*f^3*x^4 - 18*(b*c^2*d + a*c*d^2)*f^3*x^2 - (b*c^3 + 3*a*c^2*d)*f^3)*e^3 + (9*(b*c^2*d
+ a*c*d^2)*f^4*x^4 + 3*a*c^3*f^4 + 2*(b*c^3 + 3*a*c^2*d)*f^4*x^2)*e^2 + (6*a*c^3*f^5*x^2 + (b*c^3 + 3*a*c^2*d
)*f^5*x^4)*e)*sqrt(-f*e)*log((f*x^2 - 2*sqrt(-f*e)*x - e)/(f*x^2 + e)) - 10*(35*b*d^3*f^2*x^3 - 9*(3*b*c*d^2 +
a*d^3)*f^2*x)*e^5 - 2*(56*b*d^3*f^3*x^5 - 75*(3*b*c*d^2 + a*d^3)*f^3*x^3 + 27*(b*c^2*d + a*c*d^2)*f^3*x)*e^4
+ 2*(8*b*d^3*f^4*x^7 + 24*(3*b*c*d^2 + a*d^3)*f^4*x^5 - 45*(b*c^2*d + a*c*d^2)*f^4*x^3 - 3*(b*c^3 + 3*a*c^2*d)
*f^4*x)*e^3 + 6*(5*a*c^3*f^5*x + (b*c^3 + 3*a*c^2*d)*f^5*x^3)*e^2)/(f^7*x^4*e^3 + 2*f^6*x^2*e^4 + f^5*e^5), 1/
24*(9*a*c^3*f^6*x^3*e - 105*b*d^3*f*x*e^6 + 3*(3*a*c^3*f^6*x^4 + 35*b*d^3*e^6 + 5*(14*b*d^3*f*x^2 - 3*(3*b*c*d
^2 + a*d^3)*f)*e^5 + (35*b*d^3*f^2*x^4 - 30*(3*b*c*d^2 + a*d^3)*f^2*x^2 + 9*(b*c^2*d + a*c*d^2)*f^2)*e^4 - (15
*(3*b*c*d^2 + a*d^3)*f^3*x^4 - 18*(b*c^2*d + a*c*d^2)*f^3*x^2 - (b*c^3 + 3*a*c^2*d)*f^3)*e^3 + (9*(b*c^2*d + a
*c*d^2)*f^4*x^4 + 3*a*c^3*f^4 + 2*(b*c^3 + 3*a*c^2*d)*f^4*x^2)*e^2 + (6*a*c^3*f^5*x^2 + (b*c^3 + 3*a*c^2*d)*f^
5*x^4)*e)*sqrt(f)*arctan(sqrt(f)*x*e^(-1/2))*e^(1/2) - 5*(35*b*d^3*f^2*x^3 - 9*(3*b*c*d^2 + a*d^3)*f^2*x)*e^5
- (56*b*d^3*f^3*x^5 - 75*(3*b*c*d^2 + a*d^3)*f^3*x^3 + 27*(b*c^2*d + a*c*d^2)*f^3*x)*e^4 + (8*b*d^3*f^4*x^7 +
24*(3*b*c*d^2 + a*d^3)*f^4*x^5 - 45*(b*c^2*d + a*c*d^2)*f^4*x^3 - 3*(b*c^3 + 3*a*c^2*d)*f^4*x)*e^3 + 3*(5*a*c^
3*f^5*x + (b*c^3 + 3*a*c^2*d)*f^5*x^3)*e^2)/(f^7*x^4*e^3 + 2*f^6*x^2*e^4 + f^5*e^5)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 865 vs. \(2 (291) = 582\).
time = 59.62, size = 865, normalized size = 2.97 \begin {gather*} \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} \cdot \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}} \end {gather*}

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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Giac [A]
time = 1.68, size = 371, normalized size = 1.27 \begin {gather*} \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}} \end {gather*}

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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Mupad [B]
time = 0.22, size = 495, normalized size = 1.70 \begin {gather*} \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}} \end {gather*}

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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