3.227 \(\int \frac {(c+d x^2)^3}{x^4 (a+b x^2)} \, dx\)

Optimal. Leaf size=74 \[ \frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} b^{3/2}}+\frac {c^2 (b c-3 a d)}{a^2 x}-\frac {c^3}{3 a x^3}+\frac {d^3 x}{b} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 74, 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} \[ \frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} b^{3/2}}+\frac {c^2 (b c-3 a d)}{a^2 x}-\frac {c^3}{3 a x^3}+\frac {d^3 x}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.04, size = 74, normalized size = 1.00 \[ \frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} b^{3/2}}+\frac {c^2 (b c-3 a d)}{a^2 x}-\frac {c^3}{3 a x^3}+\frac {d^3 x}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.55, size = 256, normalized size = 3.46 \[ \left [\frac {6 \, a^{3} b d^{3} x^{4} - 2 \, a^{2} b^{2} c^{3} + 3 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-a b} x^{3} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 6 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d\right )} x^{2}}{6 \, a^{3} b^{2} x^{3}}, \frac {3 \, a^{3} b d^{3} x^{4} - a^{2} b^{2} c^{3} + 3 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a b} x^{3} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 3 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d\right )} x^{2}}{3 \, a^{3} b^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(6*a^3*b*d^3*x^4 - 2*a^2*b^2*c^3 + 3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(-a*b)*x^3*l
og((b*x^2 + 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) + 6*(a*b^3*c^3 - 3*a^2*b^2*c^2*d)*x^2)/(a^3*b^2*x^3), 1/3*(3*a^3*
b*d^3*x^4 - a^2*b^2*c^3 + 3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(a*b)*x^3*arctan(sqrt(a*b)
*x/a) + 3*(a*b^3*c^3 - 3*a^2*b^2*c^2*d)*x^2)/(a^3*b^2*x^3)]

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giac [A]  time = 0.37, size = 100, normalized size = 1.35 \[ \frac {d^{3} x}{b} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2} b} + \frac {3 \, b c^{3} x^{2} - 9 \, a c^{2} d x^{2} - a c^{3}}{3 \, a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^3*x/b + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^2*b) + 1/3*(3
*b*c^3*x^2 - 9*a*c^2*d*x^2 - a*c^3)/(a^2*x^3)

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maple [B]  time = 0.01, size = 135, normalized size = 1.82 \[ -\frac {a \,d^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b}-\frac {3 b \,c^{2} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a}+\frac {b^{2} c^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{2}}+\frac {3 c \,d^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}+\frac {d^{3} x}{b}-\frac {3 c^{2} d}{a x}+\frac {b \,c^{3}}{a^{2} x}-\frac {c^{3}}{3 a \,x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.44, size = 98, normalized size = 1.32 \[ \frac {d^{3} x}{b} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2} b} - \frac {a c^{3} - 3 \, {\left (b c^{3} - 3 \, a c^{2} d\right )} x^{2}}{3 \, a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.14, size = 122, normalized size = 1.65 \[ \frac {d^3\,x}{b}-\frac {\frac {b\,c^3}{3\,a}+\frac {b\,c^2\,x^2\,\left (3\,a\,d-b\,c\right )}{a^2}}{b\,x^3}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^3}{\sqrt {a}\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{a^{5/2}\,b^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 1.13, size = 221, normalized size = 2.99 \[ \frac {\sqrt {- \frac {1}{a^{5} b^{3}}} \left (a d - b c\right )^{3} \log {\left (- \frac {a^{3} b \sqrt {- \frac {1}{a^{5} b^{3}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} - \frac {\sqrt {- \frac {1}{a^{5} b^{3}}} \left (a d - b c\right )^{3} \log {\left (\frac {a^{3} b \sqrt {- \frac {1}{a^{5} b^{3}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac {d^{3} x}{b} + \frac {- a c^{3} + x^{2} \left (- 9 a c^{2} d + 3 b c^{3}\right )}{3 a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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