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

Optimal. Leaf size=143 \[ \frac {x^3 \left (-2 c e (b d-a e)+b^2 e^2+c^2 d^2\right )}{3 e^3}-\frac {x (c d-b e) \left (c d^2-e (b d-2 a e)\right )}{e^4}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a e^2-b d e+c d^2\right )^2}{\sqrt {d} e^{9/2}}-\frac {c x^5 (c d-2 b e)}{5 e^2}+\frac {c^2 x^7}{7 e} \]

[Out]

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

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Rubi [A]  time = 0.14, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1153, 205} \[ \frac {x^3 \left (-2 c e (b d-a e)+b^2 e^2+c^2 d^2\right )}{3 e^3}-\frac {x (c d-b e) \left (c d^2-e (b d-2 a e)\right )}{e^4}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a e^2-b d e+c d^2\right )^2}{\sqrt {d} e^{9/2}}-\frac {c x^5 (c d-2 b e)}{5 e^2}+\frac {c^2 x^7}{7 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((c*d - b*e)*(c*d^2 - e*(b*d - 2*a*e))*x)/e^4) + ((c^2*d^2 + b^2*e^2 - 2*c*e*(b*d - a*e))*x^3)/(3*e^3) - (c*
(c*d - 2*b*e)*x^5)/(5*e^2) + (c^2*x^7)/(7*e) + ((c*d^2 - b*d*e + a*e^2)^2*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d
]*e^(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 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2+c x^4\right )^2}{d+e x^2} \, dx &=\int \left (-\frac {(c d-b e) \left (c d^2-e (b d-2 a e)\right )}{e^4}+\frac {\left (c^2 d^2+b^2 e^2-2 c e (b d-a e)\right ) x^2}{e^3}-\frac {c (c d-2 b e) x^4}{e^2}+\frac {c^2 x^6}{e}+\frac {c^2 d^4-2 b c d^3 e+b^2 d^2 e^2+2 a c d^2 e^2-2 a b d e^3+a^2 e^4}{e^4 \left (d+e x^2\right )}\right ) \, dx\\ &=-\frac {(c d-b e) \left (c d^2-e (b d-2 a e)\right ) x}{e^4}+\frac {\left (c^2 d^2+b^2 e^2-2 c e (b d-a e)\right ) x^3}{3 e^3}-\frac {c (c d-2 b e) x^5}{5 e^2}+\frac {c^2 x^7}{7 e}+\frac {\left (c d^2-b d e+a e^2\right )^2 \int \frac {1}{d+e x^2} \, dx}{e^4}\\ &=-\frac {(c d-b e) \left (c d^2-e (b d-2 a e)\right ) x}{e^4}+\frac {\left (c^2 d^2+b^2 e^2-2 c e (b d-a e)\right ) x^3}{3 e^3}-\frac {c (c d-2 b e) x^5}{5 e^2}+\frac {c^2 x^7}{7 e}+\frac {\left (c d^2-b d e+a e^2\right )^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 144, normalized size = 1.01 \[ \frac {x^3 \left (2 a c e^2+b^2 e^2-2 b c d e+c^2 d^2\right )}{3 e^3}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a e^2-b d e+c d^2\right )^2}{\sqrt {d} e^{9/2}}+\frac {x (b e-c d) \left (2 a e^2-b d e+c d^2\right )}{e^4}+\frac {c x^5 (2 b e-c d)}{5 e^2}+\frac {c^2 x^7}{7 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.69, size = 406, normalized size = 2.84 \[ \left [\frac {30 \, c^{2} d e^{4} x^{7} - 42 \, {\left (c^{2} d^{2} e^{3} - 2 \, b c d e^{4}\right )} x^{5} + 70 \, {\left (c^{2} d^{3} e^{2} - 2 \, b c d^{2} e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{4}\right )} x^{3} - 105 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) - 210 \, {\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} - 2 \, a b d e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} x}{210 \, d e^{5}}, \frac {15 \, c^{2} d e^{4} x^{7} - 21 \, {\left (c^{2} d^{2} e^{3} - 2 \, b c d e^{4}\right )} x^{5} + 35 \, {\left (c^{2} d^{3} e^{2} - 2 \, b c d^{2} e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{4}\right )} x^{3} + 105 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) - 105 \, {\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} - 2 \, a b d e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} x}{105 \, d e^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/210*(30*c^2*d*e^4*x^7 - 42*(c^2*d^2*e^3 - 2*b*c*d*e^4)*x^5 + 70*(c^2*d^3*e^2 - 2*b*c*d^2*e^3 + (b^2 + 2*a*c
)*d*e^4)*x^3 - 105*(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)*sqrt(-d*e)*log((e*x
^2 - 2*sqrt(-d*e)*x - d)/(e*x^2 + d)) - 210*(c^2*d^4*e - 2*b*c*d^3*e^2 - 2*a*b*d*e^4 + (b^2 + 2*a*c)*d^2*e^3)*
x)/(d*e^5), 1/105*(15*c^2*d*e^4*x^7 - 21*(c^2*d^2*e^3 - 2*b*c*d*e^4)*x^5 + 35*(c^2*d^3*e^2 - 2*b*c*d^2*e^3 + (
b^2 + 2*a*c)*d*e^4)*x^3 + 105*(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)*sqrt(d*e
)*arctan(sqrt(d*e)*x/d) - 105*(c^2*d^4*e - 2*b*c*d^3*e^2 - 2*a*b*d*e^4 + (b^2 + 2*a*c)*d^2*e^3)*x)/(d*e^5)]

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giac [A]  time = 0.16, size = 185, normalized size = 1.29 \[ \frac {{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {9}{2}\right )}}{\sqrt {d}} + \frac {1}{105} \, {\left (15 \, c^{2} x^{7} e^{6} - 21 \, c^{2} d x^{5} e^{5} + 42 \, b c x^{5} e^{6} + 35 \, c^{2} d^{2} x^{3} e^{4} - 70 \, b c d x^{3} e^{5} - 105 \, c^{2} d^{3} x e^{3} + 35 \, b^{2} x^{3} e^{6} + 70 \, a c x^{3} e^{6} + 210 \, b c d^{2} x e^{4} - 105 \, b^{2} d x e^{5} - 210 \, a c d x e^{5} + 210 \, a b x e^{6}\right )} e^{\left (-7\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*arctan(x*e^(1/2)/sqrt(d))*e^(-9/
2)/sqrt(d) + 1/105*(15*c^2*x^7*e^6 - 21*c^2*d*x^5*e^5 + 42*b*c*x^5*e^6 + 35*c^2*d^2*x^3*e^4 - 70*b*c*d*x^3*e^5
- 105*c^2*d^3*x*e^3 + 35*b^2*x^3*e^6 + 70*a*c*x^3*e^6 + 210*b*c*d^2*x*e^4 - 105*b^2*d*x*e^5 - 210*a*c*d*x*e^5
+ 210*a*b*x*e^6)*e^(-7)

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maple [B]  time = 0.00, size = 267, normalized size = 1.87 \[ \frac {c^{2} x^{7}}{7 e}+\frac {2 b c \,x^{5}}{5 e}-\frac {c^{2} d \,x^{5}}{5 e^{2}}+\frac {2 a c \,x^{3}}{3 e}+\frac {b^{2} x^{3}}{3 e}-\frac {2 b c d \,x^{3}}{3 e^{2}}+\frac {c^{2} d^{2} x^{3}}{3 e^{3}}+\frac {a^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}-\frac {2 a b d \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}\, e}+\frac {2 a c \,d^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}\, e^{2}}+\frac {b^{2} d^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}\, e^{2}}-\frac {2 b c \,d^{3} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}\, e^{3}}+\frac {c^{2} d^{4} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}\, e^{4}}+\frac {2 a b x}{e}-\frac {2 a c d x}{e^{2}}-\frac {b^{2} d x}{e^{2}}+\frac {2 b c \,d^{2} x}{e^{3}}-\frac {c^{2} d^{3} x}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.38, size = 176, normalized size = 1.23 \[ \frac {{\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e} e^{4}} + \frac {15 \, c^{2} e^{3} x^{7} - 21 \, {\left (c^{2} d e^{2} - 2 \, b c e^{3}\right )} x^{5} + 35 \, {\left (c^{2} d^{2} e - 2 \, b c d e^{2} + {\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x^{3} - 105 \, {\left (c^{2} d^{3} - 2 \, b c d^{2} e - 2 \, a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x}{105 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)*arctan(e*x/sqrt(d*e))/(sqrt(d*e)*e^4)
+ 1/105*(15*c^2*e^3*x^7 - 21*(c^2*d*e^2 - 2*b*c*e^3)*x^5 + 35*(c^2*d^2*e - 2*b*c*d*e^2 + (b^2 + 2*a*c)*e^3)*x^
3 - 105*(c^2*d^3 - 2*b*c*d^2*e - 2*a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*x)/e^4

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mupad [B]  time = 4.47, size = 229, normalized size = 1.60 \[ x^3\,\left (\frac {b^2+2\,a\,c}{3\,e}+\frac {d\,\left (\frac {c^2\,d}{e^2}-\frac {2\,b\,c}{e}\right )}{3\,e}\right )-x\,\left (\frac {d\,\left (\frac {b^2+2\,a\,c}{e}+\frac {d\,\left (\frac {c^2\,d}{e^2}-\frac {2\,b\,c}{e}\right )}{e}\right )}{e}-\frac {2\,a\,b}{e}\right )-x^5\,\left (\frac {c^2\,d}{5\,e^2}-\frac {2\,b\,c}{5\,e}\right )+\frac {c^2\,x^7}{7\,e}+\frac {\mathrm {atan}\left (\frac {\sqrt {e}\,x\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{\sqrt {d}\,\left (a^2\,e^4-2\,a\,b\,d\,e^3+2\,a\,c\,d^2\,e^2+b^2\,d^2\,e^2-2\,b\,c\,d^3\,e+c^2\,d^4\right )}\right )\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{\sqrt {d}\,e^{9/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^3*((2*a*c + b^2)/(3*e) + (d*((c^2*d)/e^2 - (2*b*c)/e))/(3*e)) - x*((d*((2*a*c + b^2)/e + (d*((c^2*d)/e^2 - (
2*b*c)/e))/e))/e - (2*a*b)/e) - x^5*((c^2*d)/(5*e^2) - (2*b*c)/(5*e)) + (c^2*x^7)/(7*e) + (atan((e^(1/2)*x*(a*
e^2 + c*d^2 - b*d*e)^2)/(d^(1/2)*(a^2*e^4 + c^2*d^4 + b^2*d^2*e^2 - 2*a*b*d*e^3 - 2*b*c*d^3*e + 2*a*c*d^2*e^2)
))*(a*e^2 + c*d^2 - b*d*e)^2)/(d^(1/2)*e^(9/2))

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sympy [B]  time = 1.53, size = 371, normalized size = 2.59 \[ \frac {c^{2} x^{7}}{7 e} + x^{5} \left (\frac {2 b c}{5 e} - \frac {c^{2} d}{5 e^{2}}\right ) + x^{3} \left (\frac {2 a c}{3 e} + \frac {b^{2}}{3 e} - \frac {2 b c d}{3 e^{2}} + \frac {c^{2} d^{2}}{3 e^{3}}\right ) + x \left (\frac {2 a b}{e} - \frac {2 a c d}{e^{2}} - \frac {b^{2} d}{e^{2}} + \frac {2 b c d^{2}}{e^{3}} - \frac {c^{2} d^{3}}{e^{4}}\right ) - \frac {\sqrt {- \frac {1}{d e^{9}}} \left (a e^{2} - b d e + c d^{2}\right )^{2} \log {\left (- \frac {d e^{4} \sqrt {- \frac {1}{d e^{9}}} \left (a e^{2} - b d e + c d^{2}\right )^{2}}{a^{2} e^{4} - 2 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} - 2 b c d^{3} e + c^{2} d^{4}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{d e^{9}}} \left (a e^{2} - b d e + c d^{2}\right )^{2} \log {\left (\frac {d e^{4} \sqrt {- \frac {1}{d e^{9}}} \left (a e^{2} - b d e + c d^{2}\right )^{2}}{a^{2} e^{4} - 2 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} - 2 b c d^{3} e + c^{2} d^{4}} + x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)**2/(e*x**2+d),x)

[Out]

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

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