3.3.94 \(\int \frac {a+b x^2+c x^4}{x^6 (d+e x^2)^3} \, dx\) [294]

Optimal. Leaf size=171 \[ -\frac {a}{5 d^3 x^5}-\frac {b d-3 a e}{3 d^4 x^3}-\frac {c d^2-3 b d e+6 a e^2}{d^5 x}-\frac {e \left (c d^2-b d e+a e^2\right ) x}{4 d^4 \left (d+e x^2\right )^2}-\frac {e \left (7 c d^2-e (11 b d-15 a e)\right ) x}{8 d^5 \left (d+e x^2\right )}-\frac {\sqrt {e} \left (15 c d^2-35 b d e+63 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{11/2}} \]

[Out]

-1/5*a/d^3/x^5+1/3*(3*a*e-b*d)/d^4/x^3+(-6*a*e^2+3*b*d*e-c*d^2)/d^5/x-1/4*e*(a*e^2-b*d*e+c*d^2)*x/d^4/(e*x^2+d
)^2-1/8*e*(7*c*d^2-e*(-15*a*e+11*b*d))*x/d^5/(e*x^2+d)-1/8*(63*a*e^2-35*b*d*e+15*c*d^2)*arctan(x*e^(1/2)/d^(1/
2))*e^(1/2)/d^(11/2)

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Rubi [A]
time = 0.24, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1273, 1819, 1816, 211} \begin {gather*} -\frac {\sqrt {e} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (63 a e^2-35 b d e+15 c d^2\right )}{8 d^{11/2}}-\frac {6 a e^2-3 b d e+c d^2}{d^5 x}-\frac {e x \left (7 c d^2-e (11 b d-15 a e)\right )}{8 d^5 \left (d+e x^2\right )}-\frac {e x \left (a e^2-b d e+c d^2\right )}{4 d^4 \left (d+e x^2\right )^2}-\frac {b d-3 a e}{3 d^4 x^3}-\frac {a}{5 d^3 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)/(x^6*(d + e*x^2)^3),x]

[Out]

-1/5*a/(d^3*x^5) - (b*d - 3*a*e)/(3*d^4*x^3) - (c*d^2 - 3*b*d*e + 6*a*e^2)/(d^5*x) - (e*(c*d^2 - b*d*e + a*e^2
)*x)/(4*d^4*(d + e*x^2)^2) - (e*(7*c*d^2 - e*(11*b*d - 15*a*e))*x)/(8*d^5*(d + e*x^2)) - (Sqrt[e]*(15*c*d^2 -
35*b*d*e + 63*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(11/2))

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 1273

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[(-d)^(m
/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*((d + e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1))), x] + Dist[(-d)^(m/2 - 1)/
(2*e^(2*p)*(q + 1)), Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e*x^2))*(2*(-d)^(-m/2 + 1)*e^(2*
p)*(q + 1)*(a + b*x^2 + c*x^4)^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2))], x], x],
x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]

Rule 1816

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]

Rule 1819

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {a+b x^2+c x^4}{x^6 \left (d+e x^2\right )^3} \, dx &=-\frac {e \left (c d^2-b d e+a e^2\right ) x}{4 d^4 \left (d+e x^2\right )^2}-\frac {\int \frac {-4 a d^3 e^2-4 d^2 e^2 (b d-a e) x^2-4 d e^2 \left (c d^2-b d e+a e^2\right ) x^4+3 e^3 \left (c d^2-b d e+a e^2\right ) x^6}{x^6 \left (d+e x^2\right )^2} \, dx}{4 d^4 e^2}\\ &=-\frac {e \left (c d^2-b d e+a e^2\right ) x}{4 d^4 \left (d+e x^2\right )^2}-\frac {e \left (7 c d^2-e (11 b d-15 a e)\right ) x}{8 d^5 \left (d+e x^2\right )}+\frac {\int \frac {8 a d^3 e^2+8 d^2 e^2 (b d-2 a e) x^2+8 d e^2 \left (c d^2-e (2 b d-3 a e)\right ) x^4-e^3 \left (7 c d^2-e (11 b d-15 a e)\right ) x^6}{x^6 \left (d+e x^2\right )} \, dx}{8 d^5 e^2}\\ &=-\frac {e \left (c d^2-b d e+a e^2\right ) x}{4 d^4 \left (d+e x^2\right )^2}-\frac {e \left (7 c d^2-e (11 b d-15 a e)\right ) x}{8 d^5 \left (d+e x^2\right )}+\frac {\int \left (\frac {8 a d^2 e^2}{x^6}+\frac {8 d e^2 (b d-3 a e)}{x^4}+\frac {8 e^2 \left (c d^2-3 b d e+6 a e^2\right )}{x^2}-\frac {e^3 \left (15 c d^2-35 b d e+63 a e^2\right )}{d+e x^2}\right ) \, dx}{8 d^5 e^2}\\ &=-\frac {a}{5 d^3 x^5}-\frac {b d-3 a e}{3 d^4 x^3}-\frac {c d^2-3 b d e+6 a e^2}{d^5 x}-\frac {e \left (c d^2-b d e+a e^2\right ) x}{4 d^4 \left (d+e x^2\right )^2}-\frac {e \left (7 c d^2-e (11 b d-15 a e)\right ) x}{8 d^5 \left (d+e x^2\right )}-\frac {\left (e \left (15 c d^2-35 b d e+63 a e^2\right )\right ) \int \frac {1}{d+e x^2} \, dx}{8 d^5}\\ &=-\frac {a}{5 d^3 x^5}-\frac {b d-3 a e}{3 d^4 x^3}-\frac {c d^2-3 b d e+6 a e^2}{d^5 x}-\frac {e \left (c d^2-b d e+a e^2\right ) x}{4 d^4 \left (d+e x^2\right )^2}-\frac {e \left (7 c d^2-e (11 b d-15 a e)\right ) x}{8 d^5 \left (d+e x^2\right )}-\frac {\sqrt {e} \left (15 c d^2-35 b d e+63 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 173, normalized size = 1.01 \begin {gather*} -\frac {a}{5 d^3 x^5}+\frac {-b d+3 a e}{3 d^4 x^3}+\frac {-c d^2+3 b d e-6 a e^2}{d^5 x}-\frac {e \left (c d^2-b d e+a e^2\right ) x}{4 d^4 \left (d+e x^2\right )^2}-\frac {\left (7 c d^2 e-11 b d e^2+15 a e^3\right ) x}{8 d^5 \left (d+e x^2\right )}-\frac {\sqrt {e} \left (15 c d^2-35 b d e+63 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)/(x^6*(d + e*x^2)^3),x]

[Out]

-1/5*a/(d^3*x^5) + (-(b*d) + 3*a*e)/(3*d^4*x^3) + (-(c*d^2) + 3*b*d*e - 6*a*e^2)/(d^5*x) - (e*(c*d^2 - b*d*e +
a*e^2)*x)/(4*d^4*(d + e*x^2)^2) - ((7*c*d^2*e - 11*b*d*e^2 + 15*a*e^3)*x)/(8*d^5*(d + e*x^2)) - (Sqrt[e]*(15*
c*d^2 - 35*b*d*e + 63*a*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(11/2))

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Maple [A]
time = 0.14, size = 151, normalized size = 0.88

method	result	size
default	\(-\frac {e \left (\frac {\left (\frac {15}{8} a \,e^{3}-\frac {11}{8} d \,e^{2} b +\frac {7}{8} c \,d^{2} e \right ) x^{3}+\frac {d \left (17 a \,e^{2}-13 d e b +9 c \,d^{2}\right ) x}{8}}{\left (e \,x^{2}+d \right )^{2}}+\frac {\left (63 a \,e^{2}-35 d e b +15 c \,d^{2}\right ) \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \sqrt {d e}}\right )}{d^{5}}-\frac {a}{5 x^{5} d^{3}}-\frac {-3 a e +b d}{3 d^{4} x^{3}}-\frac {6 a \,e^{2}-3 d e b +c \,d^{2}}{d^{5} x}\)	\(151\)
risch	\(\frac {-\frac {e^{2} \left (63 a \,e^{2}-35 d e b +15 c \,d^{2}\right ) x^{8}}{8 d^{5}}-\frac {5 e \left (63 a \,e^{2}-35 d e b +15 c \,d^{2}\right ) x^{6}}{24 d^{4}}-\frac {\left (63 a \,e^{2}-35 d e b +15 c \,d^{2}\right ) x^{4}}{15 d^{3}}+\frac {\left (9 a e -5 b d \right ) x^{2}}{15 d^{2}}-\frac {a}{5 d}}{x^{5} \left (e \,x^{2}+d \right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (d^{11} \textit {\_Z}^{2}+3969 a^{2} e^{5}-4410 a b d \,e^{4}+1890 a c \,d^{2} e^{3}+1225 b^{2} d^{2} e^{3}-1050 b c \,d^{3} e^{2}+225 c^{2} d^{4} e \right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} d^{11}+7938 a^{2} e^{5}-8820 a b d \,e^{4}+3780 a c \,d^{2} e^{3}+2450 b^{2} d^{2} e^{3}-2100 b c \,d^{3} e^{2}+450 c^{2} d^{4} e \right ) x +\left (63 a \,d^{6} e^{2}-35 b \,d^{7} e +15 c \,d^{8}\right ) \textit {\_R} \right )\right )}{16}\)	\(287\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)/x^6/(e*x^2+d)^3,x,method=_RETURNVERBOSE)

[Out]

-e/d^5*(((15/8*a*e^3-11/8*d*e^2*b+7/8*c*d^2*e)*x^3+1/8*d*(17*a*e^2-13*b*d*e+9*c*d^2)*x)/(e*x^2+d)^2+1/8*(63*a*
e^2-35*b*d*e+15*c*d^2)/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2)))-1/5*a/x^5/d^3-1/3*(-3*a*e+b*d)/d^4/x^3-(6*a*e^2-3*
b*d*e+c*d^2)/d^5/x

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Maxima [A]
time = 0.51, size = 175, normalized size = 1.02 \begin {gather*} -\frac {15 \, {\left (15 \, c d^{2} e^{2} - 35 \, b d e^{3} + 63 \, a e^{4}\right )} x^{8} + 25 \, {\left (15 \, c d^{3} e - 35 \, b d^{2} e^{2} + 63 \, a d e^{3}\right )} x^{6} + 24 \, a d^{4} + 8 \, {\left (15 \, c d^{4} - 35 \, b d^{3} e + 63 \, a d^{2} e^{2}\right )} x^{4} + 8 \, {\left (5 \, b d^{4} - 9 \, a d^{3} e\right )} x^{2}}{120 \, {\left (d^{5} x^{9} e^{2} + 2 \, d^{6} x^{7} e + d^{7} x^{5}\right )}} - \frac {{\left (15 \, c d^{2} e - 35 \, b d e^{2} + 63 \, a e^{3}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{8 \, d^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^6/(e*x^2+d)^3,x, algorithm="maxima")

[Out]

-1/120*(15*(15*c*d^2*e^2 - 35*b*d*e^3 + 63*a*e^4)*x^8 + 25*(15*c*d^3*e - 35*b*d^2*e^2 + 63*a*d*e^3)*x^6 + 24*a
*d^4 + 8*(15*c*d^4 - 35*b*d^3*e + 63*a*d^2*e^2)*x^4 + 8*(5*b*d^4 - 9*a*d^3*e)*x^2)/(d^5*x^9*e^2 + 2*d^6*x^7*e
+ d^7*x^5) - 1/8*(15*c*d^2*e - 35*b*d*e^2 + 63*a*e^3)*arctan(x*e^(1/2)/sqrt(d))*e^(-1/2)/d^(11/2)

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Fricas [A]
time = 0.36, size = 534, normalized size = 3.12 \begin {gather*} \left [-\frac {1890 \, a x^{8} e^{4} + 240 \, c d^{4} x^{4} + 80 \, b d^{4} x^{2} + 48 \, a d^{4} - 15 \, {\left (63 \, a x^{9} e^{4} + 15 \, c d^{4} x^{5} - 7 \, {\left (5 \, b d x^{9} - 18 \, a d x^{7}\right )} e^{3} + {\left (15 \, c d^{2} x^{9} - 70 \, b d^{2} x^{7} + 63 \, a d^{2} x^{5}\right )} e^{2} + 5 \, {\left (6 \, c d^{3} x^{7} - 7 \, b d^{3} x^{5}\right )} e\right )} \sqrt {-\frac {e}{d}} \log \left (\frac {x^{2} e - 2 \, d x \sqrt {-\frac {e}{d}} - d}{x^{2} e + d}\right ) - 1050 \, {\left (b d x^{8} - 3 \, a d x^{6}\right )} e^{3} + 2 \, {\left (225 \, c d^{2} x^{8} - 875 \, b d^{2} x^{6} + 504 \, a d^{2} x^{4}\right )} e^{2} + 2 \, {\left (375 \, c d^{3} x^{6} - 280 \, b d^{3} x^{4} - 72 \, a d^{3} x^{2}\right )} e}{240 \, {\left (d^{5} x^{9} e^{2} + 2 \, d^{6} x^{7} e + d^{7} x^{5}\right )}}, -\frac {945 \, a x^{8} e^{4} + 120 \, c d^{4} x^{4} + 40 \, b d^{4} x^{2} + 24 \, a d^{4} + \frac {15 \, {\left (63 \, a x^{9} e^{4} + 15 \, c d^{4} x^{5} - 7 \, {\left (5 \, b d x^{9} - 18 \, a d x^{7}\right )} e^{3} + {\left (15 \, c d^{2} x^{9} - 70 \, b d^{2} x^{7} + 63 \, a d^{2} x^{5}\right )} e^{2} + 5 \, {\left (6 \, c d^{3} x^{7} - 7 \, b d^{3} x^{5}\right )} e\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\frac {1}{2}}}{\sqrt {d}} - 525 \, {\left (b d x^{8} - 3 \, a d x^{6}\right )} e^{3} + {\left (225 \, c d^{2} x^{8} - 875 \, b d^{2} x^{6} + 504 \, a d^{2} x^{4}\right )} e^{2} + {\left (375 \, c d^{3} x^{6} - 280 \, b d^{3} x^{4} - 72 \, a d^{3} x^{2}\right )} e}{120 \, {\left (d^{5} x^{9} e^{2} + 2 \, d^{6} x^{7} e + d^{7} x^{5}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^6/(e*x^2+d)^3,x, algorithm="fricas")

[Out]

[-1/240*(1890*a*x^8*e^4 + 240*c*d^4*x^4 + 80*b*d^4*x^2 + 48*a*d^4 - 15*(63*a*x^9*e^4 + 15*c*d^4*x^5 - 7*(5*b*d
*x^9 - 18*a*d*x^7)*e^3 + (15*c*d^2*x^9 - 70*b*d^2*x^7 + 63*a*d^2*x^5)*e^2 + 5*(6*c*d^3*x^7 - 7*b*d^3*x^5)*e)*s
qrt(-e/d)*log((x^2*e - 2*d*x*sqrt(-e/d) - d)/(x^2*e + d)) - 1050*(b*d*x^8 - 3*a*d*x^6)*e^3 + 2*(225*c*d^2*x^8
- 875*b*d^2*x^6 + 504*a*d^2*x^4)*e^2 + 2*(375*c*d^3*x^6 - 280*b*d^3*x^4 - 72*a*d^3*x^2)*e)/(d^5*x^9*e^2 + 2*d^
6*x^7*e + d^7*x^5), -1/120*(945*a*x^8*e^4 + 120*c*d^4*x^4 + 40*b*d^4*x^2 + 24*a*d^4 + 15*(63*a*x^9*e^4 + 15*c*
d^4*x^5 - 7*(5*b*d*x^9 - 18*a*d*x^7)*e^3 + (15*c*d^2*x^9 - 70*b*d^2*x^7 + 63*a*d^2*x^5)*e^2 + 5*(6*c*d^3*x^7 -
7*b*d^3*x^5)*e)*arctan(x*e^(1/2)/sqrt(d))*e^(1/2)/sqrt(d) - 525*(b*d*x^8 - 3*a*d*x^6)*e^3 + (225*c*d^2*x^8 -
875*b*d^2*x^6 + 504*a*d^2*x^4)*e^2 + (375*c*d^3*x^6 - 280*b*d^3*x^4 - 72*a*d^3*x^2)*e)/(d^5*x^9*e^2 + 2*d^6*x^
7*e + d^7*x^5)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 330 vs. \(2 (163) = 326\).
time = 1.95, size = 330, normalized size = 1.93 \begin {gather*} \frac {\sqrt {- \frac {e}{d^{11}}} \cdot \left (63 a e^{2} - 35 b d e + 15 c d^{2}\right ) \log {\left (- \frac {d^{6} \sqrt {- \frac {e}{d^{11}}} \cdot \left (63 a e^{2} - 35 b d e + 15 c d^{2}\right )}{63 a e^{3} - 35 b d e^{2} + 15 c d^{2} e} + x \right )}}{16} - \frac {\sqrt {- \frac {e}{d^{11}}} \cdot \left (63 a e^{2} - 35 b d e + 15 c d^{2}\right ) \log {\left (\frac {d^{6} \sqrt {- \frac {e}{d^{11}}} \cdot \left (63 a e^{2} - 35 b d e + 15 c d^{2}\right )}{63 a e^{3} - 35 b d e^{2} + 15 c d^{2} e} + x \right )}}{16} + \frac {- 24 a d^{4} + x^{8} \left (- 945 a e^{4} + 525 b d e^{3} - 225 c d^{2} e^{2}\right ) + x^{6} \left (- 1575 a d e^{3} + 875 b d^{2} e^{2} - 375 c d^{3} e\right ) + x^{4} \left (- 504 a d^{2} e^{2} + 280 b d^{3} e - 120 c d^{4}\right ) + x^{2} \cdot \left (72 a d^{3} e - 40 b d^{4}\right )}{120 d^{7} x^{5} + 240 d^{6} e x^{7} + 120 d^{5} e^{2} x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)/x**6/(e*x**2+d)**3,x)

[Out]

sqrt(-e/d**11)*(63*a*e**2 - 35*b*d*e + 15*c*d**2)*log(-d**6*sqrt(-e/d**11)*(63*a*e**2 - 35*b*d*e + 15*c*d**2)/
(63*a*e**3 - 35*b*d*e**2 + 15*c*d**2*e) + x)/16 - sqrt(-e/d**11)*(63*a*e**2 - 35*b*d*e + 15*c*d**2)*log(d**6*s
qrt(-e/d**11)*(63*a*e**2 - 35*b*d*e + 15*c*d**2)/(63*a*e**3 - 35*b*d*e**2 + 15*c*d**2*e) + x)/16 + (-24*a*d**4
+ x**8*(-945*a*e**4 + 525*b*d*e**3 - 225*c*d**2*e**2) + x**6*(-1575*a*d*e**3 + 875*b*d**2*e**2 - 375*c*d**3*e
) + x**4*(-504*a*d**2*e**2 + 280*b*d**3*e - 120*c*d**4) + x**2*(72*a*d**3*e - 40*b*d**4))/(120*d**7*x**5 + 240
*d**6*e*x**7 + 120*d**5*e**2*x**9)

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Giac [A]
time = 4.01, size = 164, normalized size = 0.96 \begin {gather*} -\frac {{\left (15 \, c d^{2} e - 35 \, b d e^{2} + 63 \, a e^{3}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{8 \, d^{\frac {11}{2}}} - \frac {7 \, c d^{2} x^{3} e^{2} - 11 \, b d x^{3} e^{3} + 9 \, c d^{3} x e + 15 \, a x^{3} e^{4} - 13 \, b d^{2} x e^{2} + 17 \, a d x e^{3}}{8 \, {\left (x^{2} e + d\right )}^{2} d^{5}} - \frac {15 \, c d^{2} x^{4} - 45 \, b d x^{4} e + 90 \, a x^{4} e^{2} + 5 \, b d^{2} x^{2} - 15 \, a d x^{2} e + 3 \, a d^{2}}{15 \, d^{5} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)/x^6/(e*x^2+d)^3,x, algorithm="giac")

[Out]

-1/8*(15*c*d^2*e - 35*b*d*e^2 + 63*a*e^3)*arctan(x*e^(1/2)/sqrt(d))*e^(-1/2)/d^(11/2) - 1/8*(7*c*d^2*x^3*e^2 -
11*b*d*x^3*e^3 + 9*c*d^3*x*e + 15*a*x^3*e^4 - 13*b*d^2*x*e^2 + 17*a*d*x*e^3)/((x^2*e + d)^2*d^5) - 1/15*(15*c
*d^2*x^4 - 45*b*d*x^4*e + 90*a*x^4*e^2 + 5*b*d^2*x^2 - 15*a*d*x^2*e + 3*a*d^2)/(d^5*x^5)

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Mupad [B]
time = 0.41, size = 168, normalized size = 0.98 \begin {gather*} -\frac {\frac {a}{5\,d}-\frac {x^2\,\left (9\,a\,e-5\,b\,d\right )}{15\,d^2}+\frac {x^4\,\left (15\,c\,d^2-35\,b\,d\,e+63\,a\,e^2\right )}{15\,d^3}+\frac {5\,e\,x^6\,\left (15\,c\,d^2-35\,b\,d\,e+63\,a\,e^2\right )}{24\,d^4}+\frac {e^2\,x^8\,\left (15\,c\,d^2-35\,b\,d\,e+63\,a\,e^2\right )}{8\,d^5}}{d^2\,x^5+2\,d\,e\,x^7+e^2\,x^9}-\frac {\sqrt {e}\,\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (15\,c\,d^2-35\,b\,d\,e+63\,a\,e^2\right )}{8\,d^{11/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)/(x^6*(d + e*x^2)^3),x)

[Out]

- (a/(5*d) - (x^2*(9*a*e - 5*b*d))/(15*d^2) + (x^4*(63*a*e^2 + 15*c*d^2 - 35*b*d*e))/(15*d^3) + (5*e*x^6*(63*a
*e^2 + 15*c*d^2 - 35*b*d*e))/(24*d^4) + (e^2*x^8*(63*a*e^2 + 15*c*d^2 - 35*b*d*e))/(8*d^5))/(d^2*x^5 + e^2*x^9
+ 2*d*e*x^7) - (e^(1/2)*atan((e^(1/2)*x)/d^(1/2))*(63*a*e^2 + 15*c*d^2 - 35*b*d*e))/(8*d^(11/2))

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