3.2.11 \(\int \frac {x^4 (c+d x^2+e x^4+f x^6)}{a+b x^2} \, dx\)

Optimal. Leaf size=172 \[ \frac {x^5 \left (a^2 f-a b e+b^2 d\right )}{5 b^3}-\frac {a x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^5}+\frac {x^3 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^4}+\frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^{11/2}}+\frac {x^7 (b e-a f)}{7 b^2}+\frac {f x^9}{9 b} \]

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Rubi [A]  time = 0.12, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1802, 205} \begin {gather*} \frac {x^3 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^4}-\frac {a x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^5}+\frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^{11/2}}+\frac {x^5 \left (a^2 f-a b e+b^2 d\right )}{5 b^3}+\frac {x^7 (b e-a f)}{7 b^2}+\frac {f x^9}{9 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^4*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2),x]

[Out]

-((a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/b^5) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^3)/(3*b^4) + ((b^2*d
- a*b*e + a^2*f)*x^5)/(5*b^3) + ((b*e - a*f)*x^7)/(7*b^2) + (f*x^9)/(9*b) + (a^(3/2)*(b^3*c - a*b^2*d + a^2*b
*e - a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(11/2)

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 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {align*} \int \frac {x^4 \left (c+d x^2+e x^4+f x^6\right )}{a+b x^2} \, dx &=\int \left (-\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{b^5}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^4}{b^3}+\frac {(b e-a f) x^6}{b^2}+\frac {f x^8}{b}+\frac {a^2 b^3 c-a^3 b^2 d+a^4 b e-a^5 f}{b^5 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^5}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3}{3 b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^5}{5 b^3}+\frac {(b e-a f) x^7}{7 b^2}+\frac {f x^9}{9 b}+\frac {\left (a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac {1}{a+b x^2} \, dx}{b^5}\\ &=-\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^5}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3}{3 b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^5}{5 b^3}+\frac {(b e-a f) x^7}{7 b^2}+\frac {f x^9}{9 b}+\frac {a^{3/2} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{11/2}}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 162, normalized size = 0.94 \begin {gather*} \frac {x \left (315 a^4 f-105 a^3 b \left (3 e+f x^2\right )+21 a^2 b^2 \left (15 d+5 e x^2+3 f x^4\right )-3 a b^3 \left (105 c+35 d x^2+21 e x^4+15 f x^6\right )+b^4 x^2 \left (105 c+63 d x^2+45 e x^4+35 f x^6\right )\right )}{315 b^5}-\frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{b^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2),x]

[Out]

(x*(315*a^4*f - 105*a^3*b*(3*e + f*x^2) + 21*a^2*b^2*(15*d + 5*e*x^2 + 3*f*x^4) - 3*a*b^3*(105*c + 35*d*x^2 +
21*e*x^4 + 15*f*x^6) + b^4*x^2*(105*c + 63*d*x^2 + 45*e*x^4 + 35*f*x^6)))/(315*b^5) - (a^(3/2)*(-(b^3*c) + a*b
^2*d - a^2*b*e + a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(11/2)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4 \left (c+d x^2+e x^4+f x^6\right )}{a+b x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(x^4*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2),x]

[Out]

IntegrateAlgebraic[(x^4*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2), x]

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fricas [A]  time = 0.67, size = 368, normalized size = 2.14 \begin {gather*} \left [\frac {70 \, b^{4} f x^{9} + 90 \, {\left (b^{4} e - a b^{3} f\right )} x^{7} + 126 \, {\left (b^{4} d - a b^{3} e + a^{2} b^{2} f\right )} x^{5} + 210 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{3} - 315 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - 630 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x}{630 \, b^{5}}, \frac {35 \, b^{4} f x^{9} + 45 \, {\left (b^{4} e - a b^{3} f\right )} x^{7} + 63 \, {\left (b^{4} d - a b^{3} e + a^{2} b^{2} f\right )} x^{5} + 105 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{3} + 315 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 315 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x}{315 \, b^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a),x, algorithm="fricas")

[Out]

[1/630*(70*b^4*f*x^9 + 90*(b^4*e - a*b^3*f)*x^7 + 126*(b^4*d - a*b^3*e + a^2*b^2*f)*x^5 + 210*(b^4*c - a*b^3*d
+ a^2*b^2*e - a^3*b*f)*x^3 - 315*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-
a/b) - a)/(b*x^2 + a)) - 630*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*x)/b^5, 1/315*(35*b^4*f*x^9 + 45*(b^4*e -
a*b^3*f)*x^7 + 63*(b^4*d - a*b^3*e + a^2*b^2*f)*x^5 + 105*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*x^3 + 315*(
a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) - 315*(a*b^3*c - a^2*b^2*d + a^3*b*e
- a^4*f)*x)/b^5]

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giac [A]  time = 0.37, size = 200, normalized size = 1.16 \begin {gather*} \frac {{\left (a^{2} b^{3} c - a^{3} b^{2} d - a^{5} f + a^{4} b e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{5}} + \frac {35 \, b^{8} f x^{9} - 45 \, a b^{7} f x^{7} + 45 \, b^{8} x^{7} e + 63 \, b^{8} d x^{5} + 63 \, a^{2} b^{6} f x^{5} - 63 \, a b^{7} x^{5} e + 105 \, b^{8} c x^{3} - 105 \, a b^{7} d x^{3} - 105 \, a^{3} b^{5} f x^{3} + 105 \, a^{2} b^{6} x^{3} e - 315 \, a b^{7} c x + 315 \, a^{2} b^{6} d x + 315 \, a^{4} b^{4} f x - 315 \, a^{3} b^{5} x e}{315 \, b^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a),x, algorithm="giac")

[Out]

(a^2*b^3*c - a^3*b^2*d - a^5*f + a^4*b*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^5) + 1/315*(35*b^8*f*x^9 - 45*a*b
^7*f*x^7 + 45*b^8*x^7*e + 63*b^8*d*x^5 + 63*a^2*b^6*f*x^5 - 63*a*b^7*x^5*e + 105*b^8*c*x^3 - 105*a*b^7*d*x^3 -
105*a^3*b^5*f*x^3 + 105*a^2*b^6*x^3*e - 315*a*b^7*c*x + 315*a^2*b^6*d*x + 315*a^4*b^4*f*x - 315*a^3*b^5*x*e)/
b^9

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maple [A]  time = 0.01, size = 230, normalized size = 1.34 \begin {gather*} \frac {f \,x^{9}}{9 b}-\frac {a f \,x^{7}}{7 b^{2}}+\frac {e \,x^{7}}{7 b}+\frac {a^{2} f \,x^{5}}{5 b^{3}}-\frac {a e \,x^{5}}{5 b^{2}}+\frac {d \,x^{5}}{5 b}-\frac {a^{3} f \,x^{3}}{3 b^{4}}+\frac {a^{2} e \,x^{3}}{3 b^{3}}-\frac {a d \,x^{3}}{3 b^{2}}+\frac {c \,x^{3}}{3 b}-\frac {a^{5} f \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{5}}+\frac {a^{4} e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{4}}-\frac {a^{3} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{3}}+\frac {a^{2} c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{2}}+\frac {a^{4} f x}{b^{5}}-\frac {a^{3} e x}{b^{4}}+\frac {a^{2} d x}{b^{3}}-\frac {a c x}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a),x)

[Out]

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

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maxima [A]  time = 2.93, size = 172, normalized size = 1.00 \begin {gather*} \frac {{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{5}} + \frac {35 \, b^{4} f x^{9} + 45 \, {\left (b^{4} e - a b^{3} f\right )} x^{7} + 63 \, {\left (b^{4} d - a b^{3} e + a^{2} b^{2} f\right )} x^{5} + 105 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{3} - 315 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x}{315 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a),x, algorithm="maxima")

[Out]

(a^2*b^3*c - a^3*b^2*d + a^4*b*e - a^5*f)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^5) + 1/315*(35*b^4*f*x^9 + 45*(b^
4*e - a*b^3*f)*x^7 + 63*(b^4*d - a*b^3*e + a^2*b^2*f)*x^5 + 105*(b^4*c - a*b^3*d + a^2*b^2*e - a^3*b*f)*x^3 -
315*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*x)/b^5

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mupad [B]  time = 0.94, size = 243, normalized size = 1.41 \begin {gather*} x^7\,\left (\frac {e}{7\,b}-\frac {a\,f}{7\,b^2}\right )+x^5\,\left (\frac {d}{5\,b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{5\,b}\right )+x^3\,\left (\frac {c}{3\,b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{3\,b}\right )+\frac {f\,x^9}{9\,b}-\frac {a\,x\,\left (\frac {c}{b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{b}\right )}{b}-\frac {a^{3/2}\,\mathrm {atan}\left (\frac {a^{3/2}\,\sqrt {b}\,x\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{f\,a^5-e\,a^4\,b+d\,a^3\,b^2-c\,a^2\,b^3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{b^{11/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2),x)

[Out]

x^7*(e/(7*b) - (a*f)/(7*b^2)) + x^5*(d/(5*b) - (a*(e/b - (a*f)/b^2))/(5*b)) + x^3*(c/(3*b) - (a*(d/b - (a*(e/b
- (a*f)/b^2))/b))/(3*b)) + (f*x^9)/(9*b) - (a*x*(c/b - (a*(d/b - (a*(e/b - (a*f)/b^2))/b))/b))/b - (a^(3/2)*a
tan((a^(3/2)*b^(1/2)*x*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/(a^5*f - a^2*b^3*c + a^3*b^2*d - a^4*b*e))*(b^3*c
- a^3*f - a*b^2*d + a^2*b*e))/b^(11/2)

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sympy [B]  time = 1.32, size = 337, normalized size = 1.96 \begin {gather*} x^{7} \left (- \frac {a f}{7 b^{2}} + \frac {e}{7 b}\right ) + x^{5} \left (\frac {a^{2} f}{5 b^{3}} - \frac {a e}{5 b^{2}} + \frac {d}{5 b}\right ) + x^{3} \left (- \frac {a^{3} f}{3 b^{4}} + \frac {a^{2} e}{3 b^{3}} - \frac {a d}{3 b^{2}} + \frac {c}{3 b}\right ) + x \left (\frac {a^{4} f}{b^{5}} - \frac {a^{3} e}{b^{4}} + \frac {a^{2} d}{b^{3}} - \frac {a c}{b^{2}}\right ) + \frac {\sqrt {- \frac {a^{3}}{b^{11}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (- \frac {b^{5} \sqrt {- \frac {a^{3}}{b^{11}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{4} f - a^{3} b e + a^{2} b^{2} d - a b^{3} c} + x \right )}}{2} - \frac {\sqrt {- \frac {a^{3}}{b^{11}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (\frac {b^{5} \sqrt {- \frac {a^{3}}{b^{11}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{4} f - a^{3} b e + a^{2} b^{2} d - a b^{3} c} + x \right )}}{2} + \frac {f x^{9}}{9 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(f*x**6+e*x**4+d*x**2+c)/(b*x**2+a),x)

[Out]

x**7*(-a*f/(7*b**2) + e/(7*b)) + x**5*(a**2*f/(5*b**3) - a*e/(5*b**2) + d/(5*b)) + x**3*(-a**3*f/(3*b**4) + a*
*2*e/(3*b**3) - a*d/(3*b**2) + c/(3*b)) + x*(a**4*f/b**5 - a**3*e/b**4 + a**2*d/b**3 - a*c/b**2) + sqrt(-a**3/
b**11)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(-b**5*sqrt(-a**3/b**11)*(a**3*f - a**2*b*e + a*b**2*d - b**
3*c)/(a**4*f - a**3*b*e + a**2*b**2*d - a*b**3*c) + x)/2 - sqrt(-a**3/b**11)*(a**3*f - a**2*b*e + a*b**2*d - b
**3*c)*log(b**5*sqrt(-a**3/b**11)*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)/(a**4*f - a**3*b*e + a**2*b**2*d - a
*b**3*c) + x)/2 + f*x**9/(9*b)

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