3.3.79 \(\int \frac {(d+e x^2)^2 (a+b x^2+c x^4)}{x^3} \, dx\) [279]

Optimal. Leaf size=74 \[ -\frac {a d^2}{2 x^2}+\frac {1}{2} \left (c d^2+e (2 b d+a e)\right ) x^2+\frac {1}{4} e (2 c d+b e) x^4+\frac {1}{6} c e^2 x^6+d (b d+2 a e) \log (x) \]

[Out]

-1/2*a*d^2/x^2+1/2*(c*d^2+e*(a*e+2*b*d))*x^2+1/4*e*(b*e+2*c*d)*x^4+1/6*c*e^2*x^6+d*(2*a*e+b*d)*ln(x)

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Rubi [A]
time = 0.06, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1265, 907} \begin {gather*} \frac {1}{2} x^2 \left (e (a e+2 b d)+c d^2\right )+d \log (x) (2 a e+b d)-\frac {a d^2}{2 x^2}+\frac {1}{4} e x^4 (b e+2 c d)+\frac {1}{6} c e^2 x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(a*d^2)/x^2 + ((c*d^2 + e*(2*b*d + a*e))*x^2)/2 + (e*(2*c*d + b*e)*x^4)/4 + (c*e^2*x^6)/6 + d*(b*d + 2*a*
e)*Log[x]

Rule 907

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 1265

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

Rubi steps

\begin {align*} \int \frac {\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )}{x^3} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(d+e x)^2 \left (a+b x+c x^2\right )}{x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (c d^2 \left (1+\frac {e (2 b d+a e)}{c d^2}\right )+\frac {a d^2}{x^2}+\frac {d (b d+2 a e)}{x}+e (2 c d+b e) x+c e^2 x^2\right ) \, dx,x,x^2\right )\\ &=-\frac {a d^2}{2 x^2}+\frac {1}{2} \left (c d^2+e (2 b d+a e)\right ) x^2+\frac {1}{4} e (2 c d+b e) x^4+\frac {1}{6} c e^2 x^6+d (b d+2 a e) \log (x)\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 71, normalized size = 0.96 \begin {gather*} \frac {1}{12} \left (-\frac {6 a d^2}{x^2}+6 \left (c d^2+e (2 b d+a e)\right ) x^2+3 e (2 c d+b e) x^4+2 c e^2 x^6+12 d (b d+2 a e) \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-6*a*d^2)/x^2 + 6*(c*d^2 + e*(2*b*d + a*e))*x^2 + 3*e*(2*c*d + b*e)*x^4 + 2*c*e^2*x^6 + 12*d*(b*d + 2*a*e)*L
og[x])/12

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Maple [A]
time = 0.12, size = 74, normalized size = 1.00

method	result	size
norman	\(\frac {\left (\frac {1}{4} e^{2} b +\frac {1}{2} c d e \right ) x^{6}+\left (\frac {1}{2} a \,e^{2}+d e b +\frac {1}{2} c \,d^{2}\right ) x^{4}-\frac {a \,d^{2}}{2}+\frac {c \,e^{2} x^{8}}{6}}{x^{2}}+\left (2 a d e +d^{2} b \right ) \ln \left (x \right )\)	\(73\)
default	\(\frac {c \,e^{2} x^{6}}{6}+\frac {b \,e^{2} x^{4}}{4}+\frac {c d e \,x^{4}}{2}+\frac {a \,e^{2} x^{2}}{2}+b d e \,x^{2}+\frac {c \,d^{2} x^{2}}{2}-\frac {a \,d^{2}}{2 x^{2}}+d \left (2 a e +b d \right ) \ln \left (x \right )\)	\(74\)
risch	\(\frac {c \,e^{2} x^{6}}{6}+\frac {b \,e^{2} x^{4}}{4}+\frac {c d e \,x^{4}}{2}+\frac {a \,e^{2} x^{2}}{2}+b d e \,x^{2}+\frac {c \,d^{2} x^{2}}{2}-\frac {a \,d^{2}}{2 x^{2}}+2 \ln \left (x \right ) a d e +\ln \left (x \right ) b \,d^{2}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*c*e^2*x^6+1/4*b*e^2*x^4+1/2*c*d*e*x^4+1/2*a*e^2*x^2+b*d*e*x^2+1/2*c*d^2*x^2-1/2*a*d^2/x^2+d*(2*a*e+b*d)*ln
(x)

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Maxima [A]
time = 0.29, size = 73, normalized size = 0.99 \begin {gather*} \frac {1}{6} \, c x^{6} e^{2} + \frac {1}{4} \, {\left (2 \, c d e + b e^{2}\right )} x^{4} + \frac {1}{2} \, {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{2} + \frac {1}{2} \, {\left (b d^{2} + 2 \, a d e\right )} \log \left (x^{2}\right ) - \frac {a d^{2}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*c*x^6*e^2 + 1/4*(2*c*d*e + b*e^2)*x^4 + 1/2*(c*d^2 + 2*b*d*e + a*e^2)*x^2 + 1/2*(b*d^2 + 2*a*d*e)*log(x^2)
- 1/2*a*d^2/x^2

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Fricas [A]
time = 0.37, size = 83, normalized size = 1.12 \begin {gather*} \frac {6 \, c d^{2} x^{4} - 6 \, a d^{2} + {\left (2 \, c x^{8} + 3 \, b x^{6} + 6 \, a x^{4}\right )} e^{2} + 6 \, {\left (c d x^{6} + 2 \, b d x^{4}\right )} e + 12 \, {\left (b d^{2} x^{2} + 2 \, a d x^{2} e\right )} \log \left (x\right )}{12 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(6*c*d^2*x^4 - 6*a*d^2 + (2*c*x^8 + 3*b*x^6 + 6*a*x^4)*e^2 + 6*(c*d*x^6 + 2*b*d*x^4)*e + 12*(b*d^2*x^2 +
2*a*d*x^2*e)*log(x))/x^2

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Sympy [A]
time = 0.12, size = 71, normalized size = 0.96 \begin {gather*} - \frac {a d^{2}}{2 x^{2}} + \frac {c e^{2} x^{6}}{6} + d \left (2 a e + b d\right ) \log {\left (x \right )} + x^{4} \left (\frac {b e^{2}}{4} + \frac {c d e}{2}\right ) + x^{2} \left (\frac {a e^{2}}{2} + b d e + \frac {c d^{2}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*d**2/(2*x**2) + c*e**2*x**6/6 + d*(2*a*e + b*d)*log(x) + x**4*(b*e**2/4 + c*d*e/2) + x**2*(a*e**2/2 + b*d*e
+ c*d**2/2)

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Giac [A]
time = 6.22, size = 97, normalized size = 1.31 \begin {gather*} \frac {1}{6} \, c x^{6} e^{2} + \frac {1}{2} \, c d x^{4} e + \frac {1}{4} \, b x^{4} e^{2} + \frac {1}{2} \, c d^{2} x^{2} + b d x^{2} e + \frac {1}{2} \, a x^{2} e^{2} + \frac {1}{2} \, {\left (b d^{2} + 2 \, a d e\right )} \log \left (x^{2}\right ) - \frac {b d^{2} x^{2} + 2 \, a d x^{2} e + a d^{2}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*c*x^6*e^2 + 1/2*c*d*x^4*e + 1/4*b*x^4*e^2 + 1/2*c*d^2*x^2 + b*d*x^2*e + 1/2*a*x^2*e^2 + 1/2*(b*d^2 + 2*a*d
*e)*log(x^2) - 1/2*(b*d^2*x^2 + 2*a*d*x^2*e + a*d^2)/x^2

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Mupad [B]
time = 0.04, size = 70, normalized size = 0.95 \begin {gather*} x^2\,\left (\frac {c\,d^2}{2}+b\,d\,e+\frac {a\,e^2}{2}\right )+x^4\,\left (\frac {b\,e^2}{4}+\frac {c\,d\,e}{2}\right )+\ln \left (x\right )\,\left (b\,d^2+2\,a\,e\,d\right )-\frac {a\,d^2}{2\,x^2}+\frac {c\,e^2\,x^6}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*((a*e^2)/2 + (c*d^2)/2 + b*d*e) + x^4*((b*e^2)/4 + (c*d*e)/2) + log(x)*(b*d^2 + 2*a*d*e) - (a*d^2)/(2*x^2)
+ (c*e^2*x^6)/6

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