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

Optimal. Leaf size=147 \[ \frac{d x \left (105 a^2 d^2-190 a b c d+81 b^2 c^2\right )}{30 b^4}+\frac{d x \left (c+d x^2\right ) (33 b c-35 a d)}{30 b^3}+\frac{(b c-7 a d) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{9/2}}-\frac{x \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{7 d x \left (c+d x^2\right )^2}{10 b^2} \]

[Out]

(d*(81*b^2*c^2 - 190*a*b*c*d + 105*a^2*d^2)*x)/(30*b^4) + (d*(33*b*c - 35*a*d)*x*(c + d*x^2))/(30*b^3) + (7*d*
x*(c + d*x^2)^2)/(10*b^2) - (x*(c + d*x^2)^3)/(2*b*(a + b*x^2)) + ((b*c - 7*a*d)*(b*c - a*d)^2*ArcTan[(Sqrt[b]
*x)/Sqrt[a]])/(2*Sqrt[a]*b^(9/2))

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Rubi [A]  time = 0.190167, antiderivative size = 147, 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 = {467, 528, 388, 205} \[ \frac{d x \left (105 a^2 d^2-190 a b c d+81 b^2 c^2\right )}{30 b^4}+\frac{d x \left (c+d x^2\right ) (33 b c-35 a d)}{30 b^3}+\frac{(b c-7 a d) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{9/2}}-\frac{x \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{7 d x \left (c+d x^2\right )^2}{10 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(81*b^2*c^2 - 190*a*b*c*d + 105*a^2*d^2)*x)/(30*b^4) + (d*(33*b*c - 35*a*d)*x*(c + d*x^2))/(30*b^3) + (7*d*
x*(c + d*x^2)^2)/(10*b^2) - (x*(c + d*x^2)^3)/(2*b*(a + b*x^2)) + ((b*c - 7*a*d)*(b*c - a*d)^2*ArcTan[(Sqrt[b]
*x)/Sqrt[a]])/(2*Sqrt[a]*b^(9/2))

Rule 467

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

Rule 528

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[
(f*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*(n*(p + q + 1) + 1)), x] + Dist[1/(b*(n*(p + q + 1) + 1)), Int[(a +
b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*
d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1
, 0]

Rule 388

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

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{x^2 \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx &=-\frac{x \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{\int \frac{\left (c+d x^2\right )^2 \left (c+7 d x^2\right )}{a+b x^2} \, dx}{2 b}\\ &=\frac{7 d x \left (c+d x^2\right )^2}{10 b^2}-\frac{x \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{\int \frac{\left (c+d x^2\right ) \left (c (5 b c-7 a d)+d (33 b c-35 a d) x^2\right )}{a+b x^2} \, dx}{10 b^2}\\ &=\frac{d (33 b c-35 a d) x \left (c+d x^2\right )}{30 b^3}+\frac{7 d x \left (c+d x^2\right )^2}{10 b^2}-\frac{x \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{\int \frac{c \left (15 b^2 c^2-54 a b c d+35 a^2 d^2\right )+d \left (81 b^2 c^2-190 a b c d+105 a^2 d^2\right ) x^2}{a+b x^2} \, dx}{30 b^3}\\ &=\frac{d \left (81 b^2 c^2-190 a b c d+105 a^2 d^2\right ) x}{30 b^4}+\frac{d (33 b c-35 a d) x \left (c+d x^2\right )}{30 b^3}+\frac{7 d x \left (c+d x^2\right )^2}{10 b^2}-\frac{x \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{\left ((b c-7 a d) (b c-a d)^2\right ) \int \frac{1}{a+b x^2} \, dx}{2 b^4}\\ &=\frac{d \left (81 b^2 c^2-190 a b c d+105 a^2 d^2\right ) x}{30 b^4}+\frac{d (33 b c-35 a d) x \left (c+d x^2\right )}{30 b^3}+\frac{7 d x \left (c+d x^2\right )^2}{10 b^2}-\frac{x \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{(b c-7 a d) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.0669277, size = 125, normalized size = 0.85 \[ \frac{d^2 x^3 (3 b c-2 a d)}{3 b^3}-\frac{x (b c-a d)^3}{2 b^4 \left (a+b x^2\right )}+\frac{3 d x (b c-a d)^2}{b^4}+\frac{(b c-7 a d) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{9/2}}+\frac{d^3 x^5}{5 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*d*(b*c - a*d)^2*x)/b^4 + (d^2*(3*b*c - 2*a*d)*x^3)/(3*b^3) + (d^3*x^5)/(5*b^2) - ((b*c - a*d)^3*x)/(2*b^4*(
a + b*x^2)) + ((b*c - 7*a*d)*(b*c - a*d)^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[a]*b^(9/2))

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Maple [A]  time = 0.01, size = 247, normalized size = 1.7 \begin{align*}{\frac{{d}^{3}{x}^{5}}{5\,{b}^{2}}}-{\frac{2\,{d}^{3}{x}^{3}a}{3\,{b}^{3}}}+{\frac{{d}^{2}{x}^{3}c}{{b}^{2}}}+3\,{\frac{{a}^{2}{d}^{3}x}{{b}^{4}}}-6\,{\frac{ac{d}^{2}x}{{b}^{3}}}+3\,{\frac{{c}^{2}dx}{{b}^{2}}}+{\frac{x{a}^{3}{d}^{3}}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,{a}^{2}c{d}^{2}x}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{3\,ax{c}^{2}d}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{x{c}^{3}}{2\,b \left ( b{x}^{2}+a \right ) }}-{\frac{7\,{a}^{3}{d}^{3}}{2\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{15\,{a}^{2}c{d}^{2}}{2\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{9\,a{c}^{2}d}{2\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{c}^{3}}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*d^3/b^2*x^5-2/3*d^3/b^3*x^3*a+d^2/b^2*x^3*c+3*d^3/b^4*a^2*x-6*d^2/b^3*c*a*x+3*d/b^2*c^2*x+1/2/b^4*x/(b*x^2
+a)*a^3*d^3-3/2/b^3*x/(b*x^2+a)*a^2*c*d^2+3/2/b^2*x/(b*x^2+a)*a*c^2*d-1/2/b*x/(b*x^2+a)*c^3-7/2/b^4/(a*b)^(1/2
)*arctan(b*x/(a*b)^(1/2))*a^3*d^3+15/2/b^3/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*a^2*c*d^2-9/2/b^2/(a*b)^(1/2)*a
rctan(b*x/(a*b)^(1/2))*a*c^2*d+1/2/b/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*c^3

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.59148, size = 1061, normalized size = 7.22 \begin{align*} \left [\frac{12 \, a b^{4} d^{3} x^{7} + 4 \,{\left (15 \, a b^{4} c d^{2} - 7 \, a^{2} b^{3} d^{3}\right )} x^{5} + 20 \,{\left (9 \, a b^{4} c^{2} d - 15 \, a^{2} b^{3} c d^{2} + 7 \, a^{3} b^{2} d^{3}\right )} x^{3} + 15 \,{\left (a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 15 \, a^{3} b c d^{2} - 7 \, a^{4} d^{3} +{\left (b^{4} c^{3} - 9 \, a b^{3} c^{2} d + 15 \, a^{2} b^{2} c d^{2} - 7 \, 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 ) - 30 \,{\left (a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 15 \, a^{3} b^{2} c d^{2} - 7 \, a^{4} b d^{3}\right )} x}{60 \,{\left (a b^{6} x^{2} + a^{2} b^{5}\right )}}, \frac{6 \, a b^{4} d^{3} x^{7} + 2 \,{\left (15 \, a b^{4} c d^{2} - 7 \, a^{2} b^{3} d^{3}\right )} x^{5} + 10 \,{\left (9 \, a b^{4} c^{2} d - 15 \, a^{2} b^{3} c d^{2} + 7 \, a^{3} b^{2} d^{3}\right )} x^{3} + 15 \,{\left (a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 15 \, a^{3} b c d^{2} - 7 \, a^{4} d^{3} +{\left (b^{4} c^{3} - 9 \, a b^{3} c^{2} d + 15 \, a^{2} b^{2} c d^{2} - 7 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) - 15 \,{\left (a b^{4} c^{3} - 9 \, a^{2} b^{3} c^{2} d + 15 \, a^{3} b^{2} c d^{2} - 7 \, a^{4} b d^{3}\right )} x}{30 \,{\left (a b^{6} x^{2} + a^{2} b^{5}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/60*(12*a*b^4*d^3*x^7 + 4*(15*a*b^4*c*d^2 - 7*a^2*b^3*d^3)*x^5 + 20*(9*a*b^4*c^2*d - 15*a^2*b^3*c*d^2 + 7*a^
3*b^2*d^3)*x^3 + 15*(a*b^3*c^3 - 9*a^2*b^2*c^2*d + 15*a^3*b*c*d^2 - 7*a^4*d^3 + (b^4*c^3 - 9*a*b^3*c^2*d + 15*
a^2*b^2*c*d^2 - 7*a^3*b*d^3)*x^2)*sqrt(-a*b)*log((b*x^2 + 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) - 30*(a*b^4*c^3 - 9
*a^2*b^3*c^2*d + 15*a^3*b^2*c*d^2 - 7*a^4*b*d^3)*x)/(a*b^6*x^2 + a^2*b^5), 1/30*(6*a*b^4*d^3*x^7 + 2*(15*a*b^4
*c*d^2 - 7*a^2*b^3*d^3)*x^5 + 10*(9*a*b^4*c^2*d - 15*a^2*b^3*c*d^2 + 7*a^3*b^2*d^3)*x^3 + 15*(a*b^3*c^3 - 9*a^
2*b^2*c^2*d + 15*a^3*b*c*d^2 - 7*a^4*d^3 + (b^4*c^3 - 9*a*b^3*c^2*d + 15*a^2*b^2*c*d^2 - 7*a^3*b*d^3)*x^2)*sqr
t(a*b)*arctan(sqrt(a*b)*x/a) - 15*(a*b^4*c^3 - 9*a^2*b^3*c^2*d + 15*a^3*b^2*c*d^2 - 7*a^4*b*d^3)*x)/(a*b^6*x^2
+ a^2*b^5)]

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Sympy [B]  time = 1.44394, size = 337, normalized size = 2.29 \begin{align*} \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 b^{4} + 2 b^{5} x^{2}} + \frac{\sqrt{- \frac{1}{a b^{9}}} \left (a d - b c\right )^{2} \left (7 a d - b c\right ) \log{\left (- \frac{a b^{4} \sqrt{- \frac{1}{a b^{9}}} \left (a d - b c\right )^{2} \left (7 a d - b c\right )}{7 a^{3} d^{3} - 15 a^{2} b c d^{2} + 9 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{4} - \frac{\sqrt{- \frac{1}{a b^{9}}} \left (a d - b c\right )^{2} \left (7 a d - b c\right ) \log{\left (\frac{a b^{4} \sqrt{- \frac{1}{a b^{9}}} \left (a d - b c\right )^{2} \left (7 a d - b c\right )}{7 a^{3} d^{3} - 15 a^{2} b c d^{2} + 9 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{4} + \frac{d^{3} x^{5}}{5 b^{2}} - \frac{x^{3} \left (2 a d^{3} - 3 b c d^{2}\right )}{3 b^{3}} + \frac{x \left (3 a^{2} d^{3} - 6 a b c d^{2} + 3 b^{2} c^{2} d\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*b**4 + 2*b**5*x**2) + sqrt(-1/(a*b**9))*(a*
d - b*c)**2*(7*a*d - b*c)*log(-a*b**4*sqrt(-1/(a*b**9))*(a*d - b*c)**2*(7*a*d - b*c)/(7*a**3*d**3 - 15*a**2*b*
c*d**2 + 9*a*b**2*c**2*d - b**3*c**3) + x)/4 - sqrt(-1/(a*b**9))*(a*d - b*c)**2*(7*a*d - b*c)*log(a*b**4*sqrt(
-1/(a*b**9))*(a*d - b*c)**2*(7*a*d - b*c)/(7*a**3*d**3 - 15*a**2*b*c*d**2 + 9*a*b**2*c**2*d - b**3*c**3) + x)/
4 + d**3*x**5/(5*b**2) - x**3*(2*a*d**3 - 3*b*c*d**2)/(3*b**3) + x*(3*a**2*d**3 - 6*a*b*c*d**2 + 3*b**2*c**2*d
)/b**4

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Giac [A]  time = 1.12739, size = 248, normalized size = 1.69 \begin{align*} \frac{{\left (b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 15 \, a^{2} b c d^{2} - 7 \, a^{3} d^{3}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{4}} - \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 )} b^{4}} + \frac{3 \, b^{8} d^{3} x^{5} + 15 \, b^{8} c d^{2} x^{3} - 10 \, a b^{7} d^{3} x^{3} + 45 \, b^{8} c^{2} d x - 90 \, a b^{7} c d^{2} x + 45 \, a^{2} b^{6} d^{3} x}{15 \, b^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^3*c^3 - 9*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 7*a^3*d^3)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^4) - 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)*b^4) + 1/15*(3*b^8*d^3*x^5 + 15*b^8*c*d^2*x^
3 - 10*a*b^7*d^3*x^3 + 45*b^8*c^2*d*x - 90*a*b^7*c*d^2*x + 45*a^2*b^6*d^3*x)/b^10