3.1.22 \(\int \frac {(a+b x^2) (c+d x^2)^3}{(e+f x^2)^4} \, dx\) [22]

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

[Out]

1/48*d*(b*e*(-3*c^2*f^2-10*c*d*e*f+105*d^2*e^2)-a*f*(15*c^2*f^2+14*c*d*e*f+15*d^2*e^2))*x/e^3/f^4-1/6*(-a*f+b*
e)*x*(d*x^2+c)^3/e/f/(f*x^2+e)^3-1/24*(b*e*(-c*f+7*d*e)-a*f*(5*c*f+d*e))*x*(d*x^2+c)^2/e^2/f^2/(f*x^2+e)^2-1/4
8*(b*e*(-3*c^2*f^2-8*c*d*e*f+35*d^2*e^2)-a*f*(15*c^2*f^2+4*c*d*e*f+5*d^2*e^2))*x*(d*x^2+c)/e^3/f^3/(f*x^2+e)-1
/16*(b*e*(-c^3*f^3-3*c^2*d*e*f^2-15*c*d^2*e^2*f+35*d^3*e^3)-a*f*(5*c^3*f^3+3*c^2*d*e*f^2+3*c*d^2*e^2*f+5*d^3*e
^3))*arctan(x*f^(1/2)/e^(1/2))/e^(7/2)/f^(9/2)

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Rubi [A]
time = 0.29, antiderivative size = 348, normalized size of antiderivative = 1.00, 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 = {540, 396, 211} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (b e \left (-c^3 f^3-3 c^2 d e f^2-15 c d^2 e^2 f+35 d^3 e^3\right )-a f \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )}{16 e^{7/2} f^{9/2}}+\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 {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 {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} \end {gather*}

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 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]

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*}

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Mathematica [A]
time = 0.14, size = 295, normalized size = 0.85 \begin {gather*} \frac {b d^3 x}{f^4}+\frac {(b e-a f) (d e-c f)^3 x}{6 e f^4 \left (e+f x^2\right )^3}-\frac {(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 \left (e+f x^2\right )^2}+\frac {(d e-c f) \left (b e \left (29 d^2 e^2-4 c d e f-c^2 f^2\right )-a f \left (11 d^2 e^2+8 c d e f+5 c^2 f^2\right )\right ) x}{16 e^3 f^4 \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 {gather*}

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))

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Maple [A]
time = 0.18, size = 417, normalized size = 1.20

method	result	size
default	\(\frac {b \,d^{3} x}{f^{4}}+\frac {\frac {\frac {f^{2} \left (5 a \,c^{3} f^{4}+3 a \,c^{2} d e \,f^{3}+3 a c \,d^{2} e^{2} f^{2}-11 a \,d^{3} e^{3} f +b \,c^{3} e \,f^{3}+3 b \,c^{2} d \,e^{2} f^{2}-33 b c \,d^{2} e^{3} f +29 b \,d^{3} e^{4}\right ) x^{5}}{16 e^{3}}+\frac {f \left (5 a \,c^{3} f^{4}+3 a \,c^{2} d e \,f^{3}-3 a c \,d^{2} e^{2} f^{2}-5 a \,d^{3} e^{3} f +b \,c^{3} e \,f^{3}-3 b \,c^{2} d \,e^{2} f^{2}-15 b c \,d^{2} e^{3} f +17 b \,d^{3} e^{4}\right ) x^{3}}{6 e^{2}}+\frac {\left (11 a \,c^{3} f^{4}-3 a \,c^{2} d e \,f^{3}-3 a c \,d^{2} e^{2} f^{2}-5 a \,d^{3} e^{3} f -b \,c^{3} e \,f^{3}-3 b \,c^{2} d \,e^{2} f^{2}-15 b c \,d^{2} e^{3} f +19 b \,d^{3} e^{4}\right ) x}{16 e}}{\left (f \,x^{2}+e \right )^{3}}+\frac {\left (5 a \,c^{3} f^{4}+3 a \,c^{2} d e \,f^{3}+3 a c \,d^{2} e^{2} f^{2}+5 a \,d^{3} e^{3} f +b \,c^{3} e \,f^{3}+3 b \,c^{2} d \,e^{2} f^{2}+15 b c \,d^{2} e^{3} f -35 b \,d^{3} e^{4}\right ) \arctan \left (\frac {f x}{\sqrt {f e}}\right )}{16 e^{3} \sqrt {f e}}}{f^{4}}\)	\(417\)
risch	\(\frac {b \,d^{3} x}{f^{4}}+\frac {\frac {f^{2} \left (5 a \,c^{3} f^{4}+3 a \,c^{2} d e \,f^{3}+3 a c \,d^{2} e^{2} f^{2}-11 a \,d^{3} e^{3} f +b \,c^{3} e \,f^{3}+3 b \,c^{2} d \,e^{2} f^{2}-33 b c \,d^{2} e^{3} f +29 b \,d^{3} e^{4}\right ) x^{5}}{16 e^{3}}+\frac {f \left (5 a \,c^{3} f^{4}+3 a \,c^{2} d e \,f^{3}-3 a c \,d^{2} e^{2} f^{2}-5 a \,d^{3} e^{3} f +b \,c^{3} e \,f^{3}-3 b \,c^{2} d \,e^{2} f^{2}-15 b c \,d^{2} e^{3} f +17 b \,d^{3} e^{4}\right ) x^{3}}{6 e^{2}}+\frac {\left (11 a \,c^{3} f^{4}-3 a \,c^{2} d e \,f^{3}-3 a c \,d^{2} e^{2} f^{2}-5 a \,d^{3} e^{3} f -b \,c^{3} e \,f^{3}-3 b \,c^{2} d \,e^{2} f^{2}-15 b c \,d^{2} e^{3} f +19 b \,d^{3} e^{4}\right ) x}{16 e}}{f^{4} \left (f \,x^{2}+e \right )^{3}}-\frac {5 \ln \left (f x +\sqrt {-f e}\right ) a \,c^{3}}{32 \sqrt {-f e}\, e^{3}}-\frac {3 \ln \left (f x +\sqrt {-f e}\right ) a \,c^{2} d}{32 f \sqrt {-f e}\, e^{2}}-\frac {3 \ln \left (f x +\sqrt {-f e}\right ) a c \,d^{2}}{32 f^{2} \sqrt {-f e}\, e}-\frac {5 \ln \left (f x +\sqrt {-f e}\right ) a \,d^{3}}{32 f^{3} \sqrt {-f e}}-\frac {\ln \left (f x +\sqrt {-f e}\right ) b \,c^{3}}{32 f \sqrt {-f e}\, e^{2}}-\frac {3 \ln \left (f x +\sqrt {-f e}\right ) b \,c^{2} d}{32 f^{2} \sqrt {-f e}\, e}-\frac {15 \ln \left (f x +\sqrt {-f e}\right ) b c \,d^{2}}{32 f^{3} \sqrt {-f e}}+\frac {35 e \ln \left (f x +\sqrt {-f e}\right ) b \,d^{3}}{32 f^{4} \sqrt {-f e}}+\frac {5 \ln \left (-f x +\sqrt {-f e}\right ) a \,c^{3}}{32 \sqrt {-f e}\, e^{3}}+\frac {3 \ln \left (-f x +\sqrt {-f e}\right ) a \,c^{2} d}{32 f \sqrt {-f e}\, e^{2}}+\frac {3 \ln \left (-f x +\sqrt {-f e}\right ) a c \,d^{2}}{32 f^{2} \sqrt {-f e}\, e}+\frac {5 \ln \left (-f x +\sqrt {-f e}\right ) a \,d^{3}}{32 f^{3} \sqrt {-f e}}+\frac {\ln \left (-f x +\sqrt {-f e}\right ) b \,c^{3}}{32 f \sqrt {-f e}\, e^{2}}+\frac {3 \ln \left (-f x +\sqrt {-f e}\right ) b \,c^{2} d}{32 f^{2} \sqrt {-f e}\, e}+\frac {15 \ln \left (-f x +\sqrt {-f e}\right ) b c \,d^{2}}{32 f^{3} \sqrt {-f e}}-\frac {35 e \ln \left (-f x +\sqrt {-f e}\right ) b \,d^{3}}{32 f^{4} \sqrt {-f e}}\)	\(768\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.49, size = 416, normalized size = 1.20 \begin {gather*} \frac {b d^{3} x}{f^{4}} + \frac {3 \, {\left (5 \, a c^{3} f^{6} + 29 \, b d^{3} f^{2} e^{4} + {\left (b c^{3} e + 3 \, a c^{2} d e\right )} f^{5} + 3 \, {\left (b c^{2} d e^{2} + a c d^{2} e^{2}\right )} f^{4} - 11 \, {\left (3 \, b c d^{2} e^{3} + a d^{3} e^{3}\right )} f^{3}\right )} x^{5} + 8 \, {\left (5 \, a c^{3} f^{5} e + 17 \, b d^{3} f e^{5} + {\left (b c^{3} e^{2} + 3 \, a c^{2} d e^{2}\right )} f^{4} - 3 \, {\left (b c^{2} d e^{3} + a c d^{2} e^{3}\right )} f^{3} - 5 \, {\left (3 \, b c d^{2} e^{4} + a d^{3} e^{4}\right )} f^{2}\right )} x^{3} + 3 \, {\left (11 \, a c^{3} f^{4} e^{2} + 19 \, b d^{3} e^{6} - {\left (b c^{3} e^{3} + 3 \, a c^{2} d e^{3}\right )} f^{3} - 3 \, {\left (b c^{2} d e^{4} + a c d^{2} e^{4}\right )} f^{2} - 5 \, {\left (3 \, b c d^{2} e^{5} + a d^{3} e^{5}\right )} f\right )} x}{48 \, {\left (f^{7} x^{6} e^{3} + 3 \, f^{6} x^{4} e^{4} + 3 \, f^{5} x^{2} e^{5} + f^{4} e^{6}\right )}} + \frac {{\left (5 \, 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} + 3 \, {\left (b c^{2} d e^{2} + a c d^{2} e^{2}\right )} f^{2} + 5 \, {\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 {7}{2}\right )}}{16 \, f^{\frac {9}{2}}} \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)^4,x, algorithm="maxima")

[Out]

b*d^3*x/f^4 + 1/48*(3*(5*a*c^3*f^6 + 29*b*d^3*f^2*e^4 + (b*c^3*e + 3*a*c^2*d*e)*f^5 + 3*(b*c^2*d*e^2 + a*c*d^2
*e^2)*f^4 - 11*(3*b*c*d^2*e^3 + a*d^3*e^3)*f^3)*x^5 + 8*(5*a*c^3*f^5*e + 17*b*d^3*f*e^5 + (b*c^3*e^2 + 3*a*c^2
*d*e^2)*f^4 - 3*(b*c^2*d*e^3 + a*c*d^2*e^3)*f^3 - 5*(3*b*c*d^2*e^4 + a*d^3*e^4)*f^2)*x^3 + 3*(11*a*c^3*f^4*e^2
+ 19*b*d^3*e^6 - (b*c^3*e^3 + 3*a*c^2*d*e^3)*f^3 - 3*(b*c^2*d*e^4 + a*c*d^2*e^4)*f^2 - 5*(3*b*c*d^2*e^5 + a*d
^3*e^5)*f)*x)/(f^7*x^6*e^3 + 3*f^6*x^4*e^4 + 3*f^5*x^2*e^5 + f^4*e^6) + 1/16*(5*a*c^3*f^4 - 35*b*d^3*e^4 + (b*
c^3*e + 3*a*c^2*d*e)*f^3 + 3*(b*c^2*d*e^2 + a*c*d^2*e^2)*f^2 + 5*(3*b*c*d^2*e^3 + a*d^3*e^3)*f)*arctan(sqrt(f)
*x*e^(-1/2))*e^(-7/2)/f^(9/2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 697 vs. \(2 (326) = 652\).
time = 1.45, size = 1424, normalized size = 4.09 \begin {gather*} \left [\frac {30 \, a c^{3} f^{7} x^{5} e + 210 \, b d^{3} f x e^{7} + 3 \, {\left (5 \, a c^{3} f^{7} x^{6} - 35 \, b d^{3} e^{7} - 5 \, {\left (21 \, b d^{3} f x^{2} - {\left (3 \, b c d^{2} + a d^{3}\right )} f\right )} e^{6} - 3 \, {\left (35 \, b d^{3} f^{2} x^{4} - 5 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{2} x^{2} - {\left (b c^{2} d + a c d^{2}\right )} f^{2}\right )} e^{5} - {\left (35 \, b d^{3} f^{3} x^{6} - 15 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{3} x^{4} - 9 \, {\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^{4} + {\left (5 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{4} x^{6} + 9 \, {\left (b c^{2} d + a c d^{2}\right )} f^{4} x^{4} + 5 \, a c^{3} f^{4} + 3 \, {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{4} x^{2}\right )} e^{3} + 3 \, {\left ({\left (b c^{2} d + a c d^{2}\right )} f^{5} x^{6} + 5 \, a c^{3} f^{5} x^{2} + {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{5} x^{4}\right )} e^{2} + {\left (15 \, a c^{3} f^{6} x^{4} + {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{6} x^{6}\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 (56 \, b d^{3} f^{2} x^{3} - 3 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{2} x\right )} e^{6} + 2 \, {\left (231 \, b d^{3} f^{3} x^{5} - 40 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{3} x^{3} - 9 \, {\left (b c^{2} d + a c d^{2}\right )} f^{3} x\right )} e^{5} + 6 \, {\left (16 \, b d^{3} f^{4} x^{7} - 11 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{4} x^{5} - 8 \, {\left (b c^{2} d + a c d^{2}\right )} f^{4} x^{3} - {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{4} x\right )} e^{4} + 2 \, {\left (9 \, {\left (b c^{2} d + a c d^{2}\right )} f^{5} x^{5} + 33 \, a c^{3} f^{5} x + 8 \, {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{5} x^{3}\right )} e^{3} + 2 \, {\left (40 \, a c^{3} f^{6} x^{3} + 3 \, {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{6} x^{5}\right )} e^{2}}{96 \, {\left (f^{8} x^{6} e^{4} + 3 \, f^{7} x^{4} e^{5} + 3 \, f^{6} x^{2} e^{6} + f^{5} e^{7}\right )}}, \frac {15 \, a c^{3} f^{7} x^{5} e + 105 \, b d^{3} f x e^{7} + 3 \, {\left (5 \, a c^{3} f^{7} x^{6} - 35 \, b d^{3} e^{7} - 5 \, {\left (21 \, b d^{3} f x^{2} - {\left (3 \, b c d^{2} + a d^{3}\right )} f\right )} e^{6} - 3 \, {\left (35 \, b d^{3} f^{2} x^{4} - 5 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{2} x^{2} - {\left (b c^{2} d + a c d^{2}\right )} f^{2}\right )} e^{5} - {\left (35 \, b d^{3} f^{3} x^{6} - 15 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{3} x^{4} - 9 \, {\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^{4} + {\left (5 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{4} x^{6} + 9 \, {\left (b c^{2} d + a c d^{2}\right )} f^{4} x^{4} + 5 \, a c^{3} f^{4} + 3 \, {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{4} x^{2}\right )} e^{3} + 3 \, {\left ({\left (b c^{2} d + a c d^{2}\right )} f^{5} x^{6} + 5 \, a c^{3} f^{5} x^{2} + {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{5} x^{4}\right )} e^{2} + {\left (15 \, a c^{3} f^{6} x^{4} + {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{6} x^{6}\right )} e\right )} \sqrt {f} \arctan \left (\sqrt {f} x e^{\left (-\frac {1}{2}\right )}\right ) e^{\frac {1}{2}} + 5 \, {\left (56 \, b d^{3} f^{2} x^{3} - 3 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{2} x\right )} e^{6} + {\left (231 \, b d^{3} f^{3} x^{5} - 40 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{3} x^{3} - 9 \, {\left (b c^{2} d + a c d^{2}\right )} f^{3} x\right )} e^{5} + 3 \, {\left (16 \, b d^{3} f^{4} x^{7} - 11 \, {\left (3 \, b c d^{2} + a d^{3}\right )} f^{4} x^{5} - 8 \, {\left (b c^{2} d + a c d^{2}\right )} f^{4} x^{3} - {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{4} x\right )} e^{4} + {\left (9 \, {\left (b c^{2} d + a c d^{2}\right )} f^{5} x^{5} + 33 \, a c^{3} f^{5} x + 8 \, {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{5} x^{3}\right )} e^{3} + {\left (40 \, a c^{3} f^{6} x^{3} + 3 \, {\left (b c^{3} + 3 \, a c^{2} d\right )} f^{6} x^{5}\right )} e^{2}}{48 \, {\left (f^{8} x^{6} e^{4} + 3 \, f^{7} x^{4} e^{5} + 3 \, f^{6} x^{2} e^{6} + f^{5} e^{7}\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)^4,x, algorithm="fricas")

[Out]

[1/96*(30*a*c^3*f^7*x^5*e + 210*b*d^3*f*x*e^7 + 3*(5*a*c^3*f^7*x^6 - 35*b*d^3*e^7 - 5*(21*b*d^3*f*x^2 - (3*b*c
*d^2 + a*d^3)*f)*e^6 - 3*(35*b*d^3*f^2*x^4 - 5*(3*b*c*d^2 + a*d^3)*f^2*x^2 - (b*c^2*d + a*c*d^2)*f^2)*e^5 - (3
5*b*d^3*f^3*x^6 - 15*(3*b*c*d^2 + a*d^3)*f^3*x^4 - 9*(b*c^2*d + a*c*d^2)*f^3*x^2 - (b*c^3 + 3*a*c^2*d)*f^3)*e^
4 + (5*(3*b*c*d^2 + a*d^3)*f^4*x^6 + 9*(b*c^2*d + a*c*d^2)*f^4*x^4 + 5*a*c^3*f^4 + 3*(b*c^3 + 3*a*c^2*d)*f^4*x
^2)*e^3 + 3*((b*c^2*d + a*c*d^2)*f^5*x^6 + 5*a*c^3*f^5*x^2 + (b*c^3 + 3*a*c^2*d)*f^5*x^4)*e^2 + (15*a*c^3*f^6*
x^4 + (b*c^3 + 3*a*c^2*d)*f^6*x^6)*e)*sqrt(-f*e)*log((f*x^2 + 2*sqrt(-f*e)*x - e)/(f*x^2 + e)) + 10*(56*b*d^3*
f^2*x^3 - 3*(3*b*c*d^2 + a*d^3)*f^2*x)*e^6 + 2*(231*b*d^3*f^3*x^5 - 40*(3*b*c*d^2 + a*d^3)*f^3*x^3 - 9*(b*c^2*
d + a*c*d^2)*f^3*x)*e^5 + 6*(16*b*d^3*f^4*x^7 - 11*(3*b*c*d^2 + a*d^3)*f^4*x^5 - 8*(b*c^2*d + a*c*d^2)*f^4*x^3
- (b*c^3 + 3*a*c^2*d)*f^4*x)*e^4 + 2*(9*(b*c^2*d + a*c*d^2)*f^5*x^5 + 33*a*c^3*f^5*x + 8*(b*c^3 + 3*a*c^2*d)*
f^5*x^3)*e^3 + 2*(40*a*c^3*f^6*x^3 + 3*(b*c^3 + 3*a*c^2*d)*f^6*x^5)*e^2)/(f^8*x^6*e^4 + 3*f^7*x^4*e^5 + 3*f^6*
x^2*e^6 + f^5*e^7), 1/48*(15*a*c^3*f^7*x^5*e + 105*b*d^3*f*x*e^7 + 3*(5*a*c^3*f^7*x^6 - 35*b*d^3*e^7 - 5*(21*b
*d^3*f*x^2 - (3*b*c*d^2 + a*d^3)*f)*e^6 - 3*(35*b*d^3*f^2*x^4 - 5*(3*b*c*d^2 + a*d^3)*f^2*x^2 - (b*c^2*d + a*c
*d^2)*f^2)*e^5 - (35*b*d^3*f^3*x^6 - 15*(3*b*c*d^2 + a*d^3)*f^3*x^4 - 9*(b*c^2*d + a*c*d^2)*f^3*x^2 - (b*c^3 +
3*a*c^2*d)*f^3)*e^4 + (5*(3*b*c*d^2 + a*d^3)*f^4*x^6 + 9*(b*c^2*d + a*c*d^2)*f^4*x^4 + 5*a*c^3*f^4 + 3*(b*c^3
+ 3*a*c^2*d)*f^4*x^2)*e^3 + 3*((b*c^2*d + a*c*d^2)*f^5*x^6 + 5*a*c^3*f^5*x^2 + (b*c^3 + 3*a*c^2*d)*f^5*x^4)*e
^2 + (15*a*c^3*f^6*x^4 + (b*c^3 + 3*a*c^2*d)*f^6*x^6)*e)*sqrt(f)*arctan(sqrt(f)*x*e^(-1/2))*e^(1/2) + 5*(56*b*
d^3*f^2*x^3 - 3*(3*b*c*d^2 + a*d^3)*f^2*x)*e^6 + (231*b*d^3*f^3*x^5 - 40*(3*b*c*d^2 + a*d^3)*f^3*x^3 - 9*(b*c^
2*d + a*c*d^2)*f^3*x)*e^5 + 3*(16*b*d^3*f^4*x^7 - 11*(3*b*c*d^2 + a*d^3)*f^4*x^5 - 8*(b*c^2*d + a*c*d^2)*f^4*x
^3 - (b*c^3 + 3*a*c^2*d)*f^4*x)*e^4 + (9*(b*c^2*d + a*c*d^2)*f^5*x^5 + 33*a*c^3*f^5*x + 8*(b*c^3 + 3*a*c^2*d)*
f^5*x^3)*e^3 + (40*a*c^3*f^6*x^3 + 3*(b*c^3 + 3*a*c^2*d)*f^6*x^5)*e^2)/(f^8*x^6*e^4 + 3*f^7*x^4*e^5 + 3*f^6*x^
2*e^6 + f^5*e^7)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**4,x)

[Out]

Timed out

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Giac [A]
time = 1.26, size = 447, normalized size = 1.28 \begin {gather*} \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 {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)^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)

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Mupad [B]
time = 1.19, size = 444, normalized size = 1.28 \begin {gather*} \frac {\frac {x^3\,\left (b\,c^3\,e\,f^4+5\,a\,c^3\,f^5-3\,b\,c^2\,d\,e^2\,f^3+3\,a\,c^2\,d\,e\,f^4-15\,b\,c\,d^2\,e^3\,f^2-3\,a\,c\,d^2\,e^2\,f^3+17\,b\,d^3\,e^4\,f-5\,a\,d^3\,e^3\,f^2\right )}{6\,e^2}+\frac {x^5\,\left (b\,c^3\,e\,f^5+5\,a\,c^3\,f^6+3\,b\,c^2\,d\,e^2\,f^4+3\,a\,c^2\,d\,e\,f^5-33\,b\,c\,d^2\,e^3\,f^3+3\,a\,c\,d^2\,e^2\,f^4+29\,b\,d^3\,e^4\,f^2-11\,a\,d^3\,e^3\,f^3\right )}{16\,e^3}-\frac {x\,\left (b\,c^3\,e\,f^3-11\,a\,c^3\,f^4+3\,b\,c^2\,d\,e^2\,f^2+3\,a\,c^2\,d\,e\,f^3+15\,b\,c\,d^2\,e^3\,f+3\,a\,c\,d^2\,e^2\,f^2-19\,b\,d^3\,e^4+5\,a\,d^3\,e^3\,f\right )}{16\,e}}{e^3\,f^4+3\,e^2\,f^5\,x^2+3\,e\,f^6\,x^4+f^7\,x^6}+\frac {b\,d^3\,x}{f^4}+\frac {\mathrm {atan}\left (\frac {\sqrt {f}\,x}{\sqrt {e}}\right )\,\left (b\,c^3\,e\,f^3+5\,a\,c^3\,f^4+3\,b\,c^2\,d\,e^2\,f^2+3\,a\,c^2\,d\,e\,f^3+15\,b\,c\,d^2\,e^3\,f+3\,a\,c\,d^2\,e^2\,f^2-35\,b\,d^3\,e^4+5\,a\,d^3\,e^3\,f\right )}{16\,e^{7/2}\,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)^4,x)

[Out]

((x^3*(5*a*c^3*f^5 - 5*a*d^3*e^3*f^2 + b*c^3*e*f^4 + 17*b*d^3*e^4*f + 3*a*c^2*d*e*f^4 - 3*a*c*d^2*e^2*f^3 - 15
*b*c*d^2*e^3*f^2 - 3*b*c^2*d*e^2*f^3))/(6*e^2) + (x^5*(5*a*c^3*f^6 - 11*a*d^3*e^3*f^3 + 29*b*d^3*e^4*f^2 + b*c
^3*e*f^5 + 3*a*c^2*d*e*f^5 + 3*a*c*d^2*e^2*f^4 - 33*b*c*d^2*e^3*f^3 + 3*b*c^2*d*e^2*f^4))/(16*e^3) - (x*(5*a*d
^3*e^3*f - 19*b*d^3*e^4 - 11*a*c^3*f^4 + b*c^3*e*f^3 + 3*a*c^2*d*e*f^3 + 15*b*c*d^2*e^3*f + 3*a*c*d^2*e^2*f^2
+ 3*b*c^2*d*e^2*f^2))/(16*e))/(e^3*f^4 + f^7*x^6 + 3*e*f^6*x^4 + 3*e^2*f^5*x^2) + (b*d^3*x)/f^4 + (atan((f^(1/
2)*x)/e^(1/2))*(5*a*c^3*f^4 - 35*b*d^3*e^4 + 5*a*d^3*e^3*f + b*c^3*e*f^3 + 3*a*c^2*d*e*f^3 + 15*b*c*d^2*e^3*f
+ 3*a*c*d^2*e^2*f^2 + 3*b*c^2*d*e^2*f^2))/(16*e^(7/2)*f^(9/2))

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