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

Optimal. Leaf size=106 \[ \frac {(5 a d+b c) (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{7/2}}+\frac {d^2 x (3 b c-2 a d)}{b^3}+\frac {x (b c-a d)^3}{2 a b^3 \left (a+b x^2\right )}+\frac {d^3 x^3}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 0.09, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {390, 385, 205} \[ \frac {(5 a d+b c) (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{7/2}}+\frac {d^2 x (3 b c-2 a d)}{b^3}+\frac {x (b c-a d)^3}{2 a b^3 \left (a+b x^2\right )}+\frac {d^3 x^3}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^3/(a + b*x^2)^2,x]

[Out]

(d^2*(3*b*c - 2*a*d)*x)/b^3 + (d^3*x^3)/(3*b^2) + ((b*c - a*d)^3*x)/(2*a*b^3*(a + b*x^2)) + ((b*c - a*d)^2*(b*
c + 5*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(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 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 106, normalized size = 1.00 \[ \frac {(5 a d+b c) (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{7/2}}+\frac {d^2 x (3 b c-2 a d)}{b^3}+\frac {x (b c-a d)^3}{2 a b^3 \left (a+b x^2\right )}+\frac {d^3 x^3}{3 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.53, size = 442, normalized size = 4.17 \[ \left [\frac {4 \, a^{2} b^{3} d^{3} x^{5} + 4 \, {\left (9 \, a^{2} b^{3} c d^{2} - 5 \, a^{3} b^{2} d^{3}\right )} x^{3} - 3 \, {\left (a b^{3} c^{3} + 3 \, a^{2} b^{2} c^{2} d - 9 \, a^{3} b c d^{2} + 5 \, a^{4} d^{3} + {\left (b^{4} c^{3} + 3 \, a b^{3} c^{2} d - 9 \, a^{2} b^{2} c d^{2} + 5 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 6 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x}{12 \, {\left (a^{2} b^{5} x^{2} + a^{3} b^{4}\right )}}, \frac {2 \, a^{2} b^{3} d^{3} x^{5} + 2 \, {\left (9 \, a^{2} b^{3} c d^{2} - 5 \, a^{3} b^{2} d^{3}\right )} x^{3} + 3 \, {\left (a b^{3} c^{3} + 3 \, a^{2} b^{2} c^{2} d - 9 \, a^{3} b c d^{2} + 5 \, a^{4} d^{3} + {\left (b^{4} c^{3} + 3 \, a b^{3} c^{2} d - 9 \, a^{2} b^{2} c d^{2} + 5 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 3 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x}{6 \, {\left (a^{2} b^{5} x^{2} + a^{3} b^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(4*a^2*b^3*d^3*x^5 + 4*(9*a^2*b^3*c*d^2 - 5*a^3*b^2*d^3)*x^3 - 3*(a*b^3*c^3 + 3*a^2*b^2*c^2*d - 9*a^3*b*
c*d^2 + 5*a^4*d^3 + (b^4*c^3 + 3*a*b^3*c^2*d - 9*a^2*b^2*c*d^2 + 5*a^3*b*d^3)*x^2)*sqrt(-a*b)*log((b*x^2 - 2*s
qrt(-a*b)*x - a)/(b*x^2 + a)) + 6*(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 9*a^3*b^2*c*d^2 - 5*a^4*b*d^3)*x)/(a^2*b^5*x^
2 + a^3*b^4), 1/6*(2*a^2*b^3*d^3*x^5 + 2*(9*a^2*b^3*c*d^2 - 5*a^3*b^2*d^3)*x^3 + 3*(a*b^3*c^3 + 3*a^2*b^2*c^2*
d - 9*a^3*b*c*d^2 + 5*a^4*d^3 + (b^4*c^3 + 3*a*b^3*c^2*d - 9*a^2*b^2*c*d^2 + 5*a^3*b*d^3)*x^2)*sqrt(a*b)*arcta
n(sqrt(a*b)*x/a) + 3*(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 9*a^3*b^2*c*d^2 - 5*a^4*b*d^3)*x)/(a^2*b^5*x^2 + a^3*b^4)]

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giac [A]  time = 0.36, size = 152, normalized size = 1.43 \[ \frac {{\left (b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b^{3}} + \frac {b^{3} c^{3} x - 3 \, a b^{2} c^{2} d x + 3 \, a^{2} b c d^{2} x - a^{3} d^{3} x}{2 \, {\left (b x^{2} + a\right )} a b^{3}} + \frac {b^{4} d^{3} x^{3} + 9 \, b^{4} c d^{2} x - 6 \, a b^{3} d^{3} x}{3 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^3*c^3 + 3*a*b^2*c^2*d - 9*a^2*b*c*d^2 + 5*a^3*d^3)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a*b^3) + 1/2*(b^3*c
^3*x - 3*a*b^2*c^2*d*x + 3*a^2*b*c*d^2*x - a^3*d^3*x)/((b*x^2 + a)*a*b^3) + 1/3*(b^4*d^3*x^3 + 9*b^4*c*d^2*x -
6*a*b^3*d^3*x)/b^6

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maple [B]  time = 0.01, size = 205, normalized size = 1.93 \[ \frac {d^{3} x^{3}}{3 b^{2}}-\frac {a^{2} d^{3} x}{2 \left (b \,x^{2}+a \right ) b^{3}}+\frac {5 a^{2} d^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{3}}+\frac {3 a c \,d^{2} x}{2 \left (b \,x^{2}+a \right ) b^{2}}-\frac {9 a c \,d^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{2}}+\frac {c^{3} x}{2 \left (b \,x^{2}+a \right ) a}+\frac {c^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a}-\frac {3 c^{2} d x}{2 \left (b \,x^{2}+a \right ) b}+\frac {3 c^{2} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b}-\frac {2 a \,d^{3} x}{b^{3}}+\frac {3 c \,d^{2} x}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.35, size = 147, normalized size = 1.39 \[ \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x}{2 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}} + \frac {b d^{3} x^{3} + 3 \, {\left (3 \, b c d^{2} - 2 \, a d^{3}\right )} x}{3 \, b^{3}} + \frac {{\left (b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x/(a*b^4*x^2 + a^2*b^3) + 1/3*(b*d^3*x^3 + 3*(3*b*c*d^
2 - 2*a*d^3)*x)/b^3 + 1/2*(b^3*c^3 + 3*a*b^2*c^2*d - 9*a^2*b*c*d^2 + 5*a^3*d^3)*arctan(b*x/sqrt(a*b))/(sqrt(a*
b)*a*b^3)

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mupad [B]  time = 0.19, size = 182, normalized size = 1.72 \[ \frac {d^3\,x^3}{3\,b^2}-x\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )-\frac {x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{2\,a\,\left (b^4\,x^2+a\,b^3\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d+b\,c\right )}{\sqrt {a}\,\left (5\,a^3\,d^3-9\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d+b^3\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d+b\,c\right )}{2\,a^{3/2}\,b^{7/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2)^3/(a + b*x^2)^2,x)

[Out]

(d^3*x^3)/(3*b^2) - x*((2*a*d^3)/b^3 - (3*c*d^2)/b^2) - (x*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)
)/(2*a*(a*b^3 + b^4*x^2)) + (atan((b^(1/2)*x*(a*d - b*c)^2*(5*a*d + b*c))/(a^(1/2)*(5*a^3*d^3 + b^3*c^3 + 3*a*
b^2*c^2*d - 9*a^2*b*c*d^2)))*(a*d - b*c)^2*(5*a*d + b*c))/(2*a^(3/2)*b^(7/2))

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sympy [B]  time = 1.06, size = 314, normalized size = 2.96 \[ x \left (- \frac {2 a d^{3}}{b^{3}} + \frac {3 c d^{2}}{b^{2}}\right ) + \frac {x \left (- a^{3} d^{3} + 3 a^{2} b c d^{2} - 3 a b^{2} c^{2} d + b^{3} c^{3}\right )}{2 a^{2} b^{3} + 2 a b^{4} x^{2}} - \frac {\sqrt {- \frac {1}{a^{3} b^{7}}} \left (a d - b c\right )^{2} \left (5 a d + b c\right ) \log {\left (- \frac {a^{2} b^{3} \sqrt {- \frac {1}{a^{3} b^{7}}} \left (a d - b c\right )^{2} \left (5 a d + b c\right )}{5 a^{3} d^{3} - 9 a^{2} b c d^{2} + 3 a b^{2} c^{2} d + b^{3} c^{3}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{3} b^{7}}} \left (a d - b c\right )^{2} \left (5 a d + b c\right ) \log {\left (\frac {a^{2} b^{3} \sqrt {- \frac {1}{a^{3} b^{7}}} \left (a d - b c\right )^{2} \left (5 a d + b c\right )}{5 a^{3} d^{3} - 9 a^{2} b c d^{2} + 3 a b^{2} c^{2} d + b^{3} c^{3}} + x \right )}}{4} + \frac {d^{3} x^{3}}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(-2*a*d**3/b**3 + 3*c*d**2/b**2) + x*(-a**3*d**3 + 3*a**2*b*c*d**2 - 3*a*b**2*c**2*d + b**3*c**3)/(2*a**2*b*
*3 + 2*a*b**4*x**2) - sqrt(-1/(a**3*b**7))*(a*d - b*c)**2*(5*a*d + b*c)*log(-a**2*b**3*sqrt(-1/(a**3*b**7))*(a
*d - b*c)**2*(5*a*d + b*c)/(5*a**3*d**3 - 9*a**2*b*c*d**2 + 3*a*b**2*c**2*d + b**3*c**3) + x)/4 + sqrt(-1/(a**
3*b**7))*(a*d - b*c)**2*(5*a*d + b*c)*log(a**2*b**3*sqrt(-1/(a**3*b**7))*(a*d - b*c)**2*(5*a*d + b*c)/(5*a**3*
d**3 - 9*a**2*b*c*d**2 + 3*a*b**2*c**2*d + b**3*c**3) + x)/4 + d**3*x**3/(3*b**2)

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