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

Optimal. Leaf size=161 \[ \frac{x \left (7 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{12 c^3 \left (c+d x^2\right )^2}+\frac{\left (35 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{9/2} \sqrt{d}}-\frac{a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac{x (3 b c-7 a d)^2}{24 c^4 \left (c+d x^2\right )}-\frac{a (6 b c-7 a d)}{3 c^4 x} \]

[Out]

-(a*(6*b*c - 7*a*d))/(3*c^4*x) - a^2/(3*c*x^3*(c + d*x^2)^2) + ((3*b^2*c^2 - 6*a*b*c*d + 7*a^2*d^2)*x)/(12*c^3
*(c + d*x^2)^2) + ((3*b*c - 7*a*d)^2*x)/(24*c^4*(c + d*x^2)) + ((3*b^2*c^2 - 30*a*b*c*d + 35*a^2*d^2)*ArcTan[(
Sqrt[d]*x)/Sqrt[c]])/(8*c^(9/2)*Sqrt[d])

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Rubi [A]  time = 0.191423, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {462, 456, 453, 205} \[ \frac{x \left (7 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{12 c^3 \left (c+d x^2\right )^2}+\frac{\left (35 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{9/2} \sqrt{d}}-\frac{a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac{x (3 b c-7 a d)^2}{24 c^4 \left (c+d x^2\right )}-\frac{a (6 b c-7 a d)}{3 c^4 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*(6*b*c - 7*a*d))/(3*c^4*x) - a^2/(3*c*x^3*(c + d*x^2)^2) + ((3*b^2*c^2 - 6*a*b*c*d + 7*a^2*d^2)*x)/(12*c^3
*(c + d*x^2)^2) + ((3*b*c - 7*a*d)^2*x)/(24*c^4*(c + d*x^2)) + ((3*b^2*c^2 - 30*a*b*c*d + 35*a^2*d^2)*ArcTan[(
Sqrt[d]*x)/Sqrt[c]])/(8*c^(9/2)*Sqrt[d])

Rule 462

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^2, x_Symbol] :> Simp[(c^2*(e*x)^(
m + 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*x^n)^p*Simp[b
*c^2*n*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*(m + 1)*d^2*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && Ne
Q[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && GtQ[n, 0]

Rule 456

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[((-a)^(m/2 - 1)*(b*c - a*d)*
x*(a + b*x^2)^(p + 1))/(2*b^(m/2 + 1)*(p + 1)), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[x^m*(a + b*x^2)^(p +
1)*ExpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)]
- ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &
& ILtQ[m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 453

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

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]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )^3} \, dx &=-\frac{a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac{\int \frac{a (6 b c-7 a d)+3 b^2 c x^2}{x^2 \left (c+d x^2\right )^3} \, dx}{3 c}\\ &=-\frac{a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac{\left (3 b^2 c^2-6 a b c d+7 a^2 d^2\right ) x}{12 c^3 \left (c+d x^2\right )^2}-\frac{\int \frac{-\frac{4 a (6 b c-7 a d)}{c}-3 \left (3 b^2-\frac{6 a b d}{c}+\frac{7 a^2 d^2}{c^2}\right ) x^2}{x^2 \left (c+d x^2\right )^2} \, dx}{12 c}\\ &=-\frac{a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac{\left (3 b^2 c^2-6 a b c d+7 a^2 d^2\right ) x}{12 c^3 \left (c+d x^2\right )^2}+\frac{(3 b c-7 a d)^2 x}{24 c^4 \left (c+d x^2\right )}+\frac{\int \frac{\frac{8 a (6 b c-7 a d)}{c^2}+\frac{(3 b c-7 a d)^2 x^2}{c^3}}{x^2 \left (c+d x^2\right )} \, dx}{24 c}\\ &=-\frac{a (6 b c-7 a d)}{3 c^4 x}-\frac{a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac{\left (3 b^2 c^2-6 a b c d+7 a^2 d^2\right ) x}{12 c^3 \left (c+d x^2\right )^2}+\frac{(3 b c-7 a d)^2 x}{24 c^4 \left (c+d x^2\right )}+\frac{\left (3 b^2 c^2-30 a b c d+35 a^2 d^2\right ) \int \frac{1}{c+d x^2} \, dx}{8 c^4}\\ &=-\frac{a (6 b c-7 a d)}{3 c^4 x}-\frac{a^2}{3 c x^3 \left (c+d x^2\right )^2}+\frac{\left (3 b^2 c^2-6 a b c d+7 a^2 d^2\right ) x}{12 c^3 \left (c+d x^2\right )^2}+\frac{(3 b c-7 a d)^2 x}{24 c^4 \left (c+d x^2\right )}+\frac{\left (3 b^2 c^2-30 a b c d+35 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{9/2} \sqrt{d}}\\ \end{align*}

Mathematica [A]  time = 0.0724567, size = 148, normalized size = 0.92 \[ \frac{x \left (11 a^2 d^2-14 a b c d+3 b^2 c^2\right )}{8 c^4 \left (c+d x^2\right )}+\frac{\left (35 a^2 d^2-30 a b c d+3 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{9/2} \sqrt{d}}-\frac{a^2}{3 c^3 x^3}+\frac{x (b c-a d)^2}{4 c^3 \left (c+d x^2\right )^2}+\frac{a (3 a d-2 b c)}{c^4 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-a^2/(3*c^3*x^3) + (a*(-2*b*c + 3*a*d))/(c^4*x) + ((b*c - a*d)^2*x)/(4*c^3*(c + d*x^2)^2) + ((3*b^2*c^2 - 14*a
*b*c*d + 11*a^2*d^2)*x)/(8*c^4*(c + d*x^2)) + ((3*b^2*c^2 - 30*a*b*c*d + 35*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c
]])/(8*c^(9/2)*Sqrt[d])

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Maple [A]  time = 0.014, size = 227, normalized size = 1.4 \begin{align*}{\frac{11\,{x}^{3}{a}^{2}{d}^{3}}{8\,{c}^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{7\,{x}^{3}ab{d}^{2}}{4\,{c}^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{3\,{b}^{2}d{x}^{3}}{8\,{c}^{2} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{13\,{a}^{2}{d}^{2}x}{8\,{c}^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{9\,abdx}{4\,{c}^{2} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{5\,{b}^{2}x}{8\,c \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{35\,{a}^{2}{d}^{2}}{8\,{c}^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{15\,abd}{4\,{c}^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{3\,{b}^{2}}{8\,{c}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{{a}^{2}}{3\,{c}^{3}{x}^{3}}}+3\,{\frac{{a}^{2}d}{{c}^{4}x}}-2\,{\frac{ab}{{c}^{3}x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/8/c^4/(d*x^2+c)^2*x^3*a^2*d^3-7/4/c^3/(d*x^2+c)^2*x^3*a*b*d^2+3/8/c^2/(d*x^2+c)^2*x^3*b^2*d+13/8/c^3/(d*x^2
+c)^2*a^2*d^2*x-9/4/c^2/(d*x^2+c)^2*a*b*d*x+5/8/c/(d*x^2+c)^2*b^2*x+35/8/c^4/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2
))*a^2*d^2-15/4/c^3/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a*b*d+3/8/c^2/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*b^2-
1/3*a^2/c^3/x^3+3*a^2/c^4/x*d-2*a/c^3/x*b

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.55762, size = 1131, normalized size = 7.02 \begin{align*} \left [-\frac{16 \, a^{2} c^{4} d - 6 \,{\left (3 \, b^{2} c^{3} d^{2} - 30 \, a b c^{2} d^{3} + 35 \, a^{2} c d^{4}\right )} x^{6} - 10 \,{\left (3 \, b^{2} c^{4} d - 30 \, a b c^{3} d^{2} + 35 \, a^{2} c^{2} d^{3}\right )} x^{4} + 16 \,{\left (6 \, a b c^{4} d - 7 \, a^{2} c^{3} d^{2}\right )} x^{2} + 3 \,{\left ({\left (3 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{7} + 2 \,{\left (3 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{5} +{\left (3 \, b^{2} c^{4} - 30 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{3}\right )} \sqrt{-c d} \log \left (\frac{d x^{2} - 2 \, \sqrt{-c d} x - c}{d x^{2} + c}\right )}{48 \,{\left (c^{5} d^{3} x^{7} + 2 \, c^{6} d^{2} x^{5} + c^{7} d x^{3}\right )}}, -\frac{8 \, a^{2} c^{4} d - 3 \,{\left (3 \, b^{2} c^{3} d^{2} - 30 \, a b c^{2} d^{3} + 35 \, a^{2} c d^{4}\right )} x^{6} - 5 \,{\left (3 \, b^{2} c^{4} d - 30 \, a b c^{3} d^{2} + 35 \, a^{2} c^{2} d^{3}\right )} x^{4} + 8 \,{\left (6 \, a b c^{4} d - 7 \, a^{2} c^{3} d^{2}\right )} x^{2} - 3 \,{\left ({\left (3 \, b^{2} c^{2} d^{2} - 30 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{7} + 2 \,{\left (3 \, b^{2} c^{3} d - 30 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{5} +{\left (3 \, b^{2} c^{4} - 30 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} x^{3}\right )} \sqrt{c d} \arctan \left (\frac{\sqrt{c d} x}{c}\right )}{24 \,{\left (c^{5} d^{3} x^{7} + 2 \, c^{6} d^{2} x^{5} + c^{7} d x^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/48*(16*a^2*c^4*d - 6*(3*b^2*c^3*d^2 - 30*a*b*c^2*d^3 + 35*a^2*c*d^4)*x^6 - 10*(3*b^2*c^4*d - 30*a*b*c^3*d^
2 + 35*a^2*c^2*d^3)*x^4 + 16*(6*a*b*c^4*d - 7*a^2*c^3*d^2)*x^2 + 3*((3*b^2*c^2*d^2 - 30*a*b*c*d^3 + 35*a^2*d^4
)*x^7 + 2*(3*b^2*c^3*d - 30*a*b*c^2*d^2 + 35*a^2*c*d^3)*x^5 + (3*b^2*c^4 - 30*a*b*c^3*d + 35*a^2*c^2*d^2)*x^3)
*sqrt(-c*d)*log((d*x^2 - 2*sqrt(-c*d)*x - c)/(d*x^2 + c)))/(c^5*d^3*x^7 + 2*c^6*d^2*x^5 + c^7*d*x^3), -1/24*(8
*a^2*c^4*d - 3*(3*b^2*c^3*d^2 - 30*a*b*c^2*d^3 + 35*a^2*c*d^4)*x^6 - 5*(3*b^2*c^4*d - 30*a*b*c^3*d^2 + 35*a^2*
c^2*d^3)*x^4 + 8*(6*a*b*c^4*d - 7*a^2*c^3*d^2)*x^2 - 3*((3*b^2*c^2*d^2 - 30*a*b*c*d^3 + 35*a^2*d^4)*x^7 + 2*(3
*b^2*c^3*d - 30*a*b*c^2*d^2 + 35*a^2*c*d^3)*x^5 + (3*b^2*c^4 - 30*a*b*c^3*d + 35*a^2*c^2*d^2)*x^3)*sqrt(c*d)*a
rctan(sqrt(c*d)*x/c))/(c^5*d^3*x^7 + 2*c^6*d^2*x^5 + c^7*d*x^3)]

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Sympy [A]  time = 1.79279, size = 240, normalized size = 1.49 \begin{align*} - \frac{\sqrt{- \frac{1}{c^{9} d}} \left (35 a^{2} d^{2} - 30 a b c d + 3 b^{2} c^{2}\right ) \log{\left (- c^{5} \sqrt{- \frac{1}{c^{9} d}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{c^{9} d}} \left (35 a^{2} d^{2} - 30 a b c d + 3 b^{2} c^{2}\right ) \log{\left (c^{5} \sqrt{- \frac{1}{c^{9} d}} + x \right )}}{16} + \frac{- 8 a^{2} c^{3} + x^{6} \left (105 a^{2} d^{3} - 90 a b c d^{2} + 9 b^{2} c^{2} d\right ) + x^{4} \left (175 a^{2} c d^{2} - 150 a b c^{2} d + 15 b^{2} c^{3}\right ) + x^{2} \left (56 a^{2} c^{2} d - 48 a b c^{3}\right )}{24 c^{6} x^{3} + 48 c^{5} d x^{5} + 24 c^{4} d^{2} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-1/(c**9*d))*(35*a**2*d**2 - 30*a*b*c*d + 3*b**2*c**2)*log(-c**5*sqrt(-1/(c**9*d)) + x)/16 + sqrt(-1/(c*
*9*d))*(35*a**2*d**2 - 30*a*b*c*d + 3*b**2*c**2)*log(c**5*sqrt(-1/(c**9*d)) + x)/16 + (-8*a**2*c**3 + x**6*(10
5*a**2*d**3 - 90*a*b*c*d**2 + 9*b**2*c**2*d) + x**4*(175*a**2*c*d**2 - 150*a*b*c**2*d + 15*b**2*c**3) + x**2*(
56*a**2*c**2*d - 48*a*b*c**3))/(24*c**6*x**3 + 48*c**5*d*x**5 + 24*c**4*d**2*x**7)

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Giac [A]  time = 1.18574, size = 204, normalized size = 1.27 \begin{align*} \frac{{\left (3 \, b^{2} c^{2} - 30 \, a b c d + 35 \, a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \, \sqrt{c d} c^{4}} + \frac{3 \, b^{2} c^{2} d x^{3} - 14 \, a b c d^{2} x^{3} + 11 \, a^{2} d^{3} x^{3} + 5 \, b^{2} c^{3} x - 18 \, a b c^{2} d x + 13 \, a^{2} c d^{2} x}{8 \,{\left (d x^{2} + c\right )}^{2} c^{4}} - \frac{6 \, a b c x^{2} - 9 \, a^{2} d x^{2} + a^{2} c}{3 \, c^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(3*b^2*c^2 - 30*a*b*c*d + 35*a^2*d^2)*arctan(d*x/sqrt(c*d))/(sqrt(c*d)*c^4) + 1/8*(3*b^2*c^2*d*x^3 - 14*a*
b*c*d^2*x^3 + 11*a^2*d^3*x^3 + 5*b^2*c^3*x - 18*a*b*c^2*d*x + 13*a^2*c*d^2*x)/((d*x^2 + c)^2*c^4) - 1/3*(6*a*b
*c*x^2 - 9*a^2*d*x^2 + a^2*c)/(c^4*x^3)