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

Optimal. Leaf size=77 \[ -\frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} b^{5/2}}+\frac {d^2 x (3 b c-a d)}{b^2}-\frac {c^3}{a x}+\frac {d^3 x^3}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 77, 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} \[ -\frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} b^{5/2}}+\frac {d^2 x (3 b c-a d)}{b^2}-\frac {c^3}{a x}+\frac {d^3 x^3}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.03, size = 76, normalized size = 0.99 \[ \frac {(a d-b c)^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} b^{5/2}}+\frac {d^2 x (3 b c-a d)}{b^2}-\frac {c^3}{a x}+\frac {d^3 x^3}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.49, size = 253, normalized size = 3.29 \[ \left [\frac {2 \, a^{2} b^{2} d^{3} x^{4} - 6 \, a b^{3} c^{3} + 3 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-a b} x \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 6 \, {\left (3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{2}}{6 \, a^{2} b^{3} x}, \frac {a^{2} b^{2} d^{3} x^{4} - 3 \, a b^{3} c^{3} - 3 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a b} x \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 3 \, {\left (3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{2}}{3 \, a^{2} b^{3} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.36, size = 104, normalized size = 1.35 \[ -\frac {c^{3}}{a x} - \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a b^{2}} + \frac {b^{2} d^{3} x^{3} + 9 \, b^{2} c d^{2} x - 3 \, a b d^{3} x}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 135, normalized size = 1.75 \[ \frac {d^{3} x^{3}}{3 b}+\frac {a^{2} d^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{2}}-\frac {3 a c \,d^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b}-\frac {b \,c^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a}+\frac {3 c^{2} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}-\frac {a \,d^{3} x}{b^{2}}+\frac {3 c \,d^{2} x}{b}-\frac {c^{3}}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.37, size = 101, normalized size = 1.31 \[ -\frac {c^{3}}{a x} + \frac {b d^{3} x^{3} + 3 \, {\left (3 \, b c d^{2} - a d^{3}\right )} x}{3 \, b^{2}} - \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.07, size = 118, normalized size = 1.53 \[ \frac {d^3\,x^3}{3\,b}-\frac {c^3}{a\,x}-x\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^3}{\sqrt {a}\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{a^{3/2}\,b^{5/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 0.81, size = 221, normalized size = 2.87 \[ x \left (- \frac {a d^{3}}{b^{2}} + \frac {3 c d^{2}}{b}\right ) - \frac {\sqrt {- \frac {1}{a^{3} b^{5}}} \left (a d - b c\right )^{3} \log {\left (- \frac {a^{2} b^{2} \sqrt {- \frac {1}{a^{3} b^{5}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{a^{3} b^{5}}} \left (a d - b c\right )^{3} \log {\left (\frac {a^{2} b^{2} \sqrt {- \frac {1}{a^{3} b^{5}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac {d^{3} x^{3}}{3 b} - \frac {c^{3}}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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