3.178 \(\int \frac {(a+b x^2)^2}{x^6 (c+d x^2)} \, dx\)

Optimal. Leaf size=87 \[ -\frac {a^2}{5 c x^5}-\frac {\sqrt {d} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{7/2}}-\frac {(b c-a d)^2}{c^3 x}-\frac {a (2 b c-a d)}{3 c^2 x^3} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {461, 205} \[ -\frac {a^2}{5 c x^5}-\frac {a (2 b c-a d)}{3 c^2 x^3}-\frac {(b c-a d)^2}{c^3 x}-\frac {\sqrt {d} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 461

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[((e*x)^m*(a + b*x^n)^p)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 86, normalized size = 0.99 \[ -\frac {a^2}{5 c x^5}-\frac {\sqrt {d} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{7/2}}-\frac {(b c-a d)^2}{c^3 x}+\frac {a (a d-2 b c)}{3 c^2 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*a^2/(c*x^5) + (a*(-2*b*c + a*d))/(3*c^2*x^3) - (b*c - a*d)^2/(c^3*x) - (Sqrt[d]*(b*c - a*d)^2*ArcTan[(Sqr
t[d]*x)/Sqrt[c]])/c^(7/2)

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fricas [A]  time = 0.47, size = 236, normalized size = 2.71 \[ \left [\frac {15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{5} \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{2} - 2 \, c x \sqrt {-\frac {d}{c}} - c}{d x^{2} + c}\right ) - 30 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{4} - 6 \, a^{2} c^{2} - 10 \, {\left (2 \, a b c^{2} - a^{2} c d\right )} x^{2}}{30 \, c^{3} x^{5}}, -\frac {15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{5} \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right ) + 15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{4} + 3 \, a^{2} c^{2} + 5 \, {\left (2 \, a b c^{2} - a^{2} c d\right )} x^{2}}{15 \, c^{3} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/30*(15*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^5*sqrt(-d/c)*log((d*x^2 - 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) - 30*
(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^4 - 6*a^2*c^2 - 10*(2*a*b*c^2 - a^2*c*d)*x^2)/(c^3*x^5), -1/15*(15*(b^2*c^2
- 2*a*b*c*d + a^2*d^2)*x^5*sqrt(d/c)*arctan(x*sqrt(d/c)) + 15*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^4 + 3*a^2*c^2
+ 5*(2*a*b*c^2 - a^2*c*d)*x^2)/(c^3*x^5)]

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giac [A]  time = 0.32, size = 112, normalized size = 1.29 \[ -\frac {{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} c^{3}} - \frac {15 \, b^{2} c^{2} x^{4} - 30 \, a b c d x^{4} + 15 \, a^{2} d^{2} x^{4} + 10 \, a b c^{2} x^{2} - 5 \, a^{2} c d x^{2} + 3 \, a^{2} c^{2}}{15 \, c^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*arctan(d*x/sqrt(c*d))/(sqrt(c*d)*c^3) - 1/15*(15*b^2*c^2*x^4 - 30*a*b*c*d
*x^4 + 15*a^2*d^2*x^4 + 10*a*b*c^2*x^2 - 5*a^2*c*d*x^2 + 3*a^2*c^2)/(c^3*x^5)

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maple [A]  time = 0.01, size = 143, normalized size = 1.64 \[ -\frac {a^{2} d^{3} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}\, c^{3}}+\frac {2 a b \,d^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}\, c^{2}}-\frac {b^{2} d \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}\, c}-\frac {a^{2} d^{2}}{c^{3} x}+\frac {2 a b d}{c^{2} x}-\frac {b^{2}}{c x}+\frac {a^{2} d}{3 c^{2} x^{3}}-\frac {2 a b}{3 c \,x^{3}}-\frac {a^{2}}{5 c \,x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.44, size = 107, normalized size = 1.23 \[ -\frac {{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} c^{3}} - \frac {15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{4} + 3 \, a^{2} c^{2} + 5 \, {\left (2 \, a b c^{2} - a^{2} c d\right )} x^{2}}{15 \, c^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.09, size = 129, normalized size = 1.48 \[ \frac {a^2\,d}{3\,c^2\,x^3}-\frac {b^2}{c\,x}-\frac {a^2}{5\,c\,x^5}-\frac {a^2\,d^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {d}\,x}{\sqrt {c}}\right )}{c^{7/2}}-\frac {b^2\,\sqrt {d}\,\mathrm {atan}\left (\frac {\sqrt {d}\,x}{\sqrt {c}}\right )}{c^{3/2}}-\frac {a^2\,d^2}{c^3\,x}-\frac {2\,a\,b}{3\,c\,x^3}+\frac {2\,a\,b\,d}{c^2\,x}+\frac {2\,a\,b\,d^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {d}\,x}{\sqrt {c}}\right )}{c^{5/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.76, size = 207, normalized size = 2.38 \[ \frac {\sqrt {- \frac {d}{c^{7}}} \left (a d - b c\right )^{2} \log {\left (- \frac {c^{4} \sqrt {- \frac {d}{c^{7}}} \left (a d - b c\right )^{2}}{a^{2} d^{3} - 2 a b c d^{2} + b^{2} c^{2} d} + x \right )}}{2} - \frac {\sqrt {- \frac {d}{c^{7}}} \left (a d - b c\right )^{2} \log {\left (\frac {c^{4} \sqrt {- \frac {d}{c^{7}}} \left (a d - b c\right )^{2}}{a^{2} d^{3} - 2 a b c d^{2} + b^{2} c^{2} d} + x \right )}}{2} + \frac {- 3 a^{2} c^{2} + x^{4} \left (- 15 a^{2} d^{2} + 30 a b c d - 15 b^{2} c^{2}\right ) + x^{2} \left (5 a^{2} c d - 10 a b c^{2}\right )}{15 c^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-d/c**7)*(a*d - b*c)**2*log(-c**4*sqrt(-d/c**7)*(a*d - b*c)**2/(a**2*d**3 - 2*a*b*c*d**2 + b**2*c**2*d) +
x)/2 - sqrt(-d/c**7)*(a*d - b*c)**2*log(c**4*sqrt(-d/c**7)*(a*d - b*c)**2/(a**2*d**3 - 2*a*b*c*d**2 + b**2*c*
*2*d) + x)/2 + (-3*a**2*c**2 + x**4*(-15*a**2*d**2 + 30*a*b*c*d - 15*b**2*c**2) + x**2*(5*a**2*c*d - 10*a*b*c*
*2))/(15*c**3*x**5)

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