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

Optimal. Leaf size=157 \[ \frac{\left (-3 a^2 d^2-6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 \sqrt{c} d^{3/2} (b c-a d)^3}+\frac{a^{3/2} \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{(b c-a d)^3}+\frac{x (b c-5 a d)}{8 d \left (c+d x^2\right ) (b c-a d)^2}-\frac{c x}{4 d \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

-(c*x)/(4*d*(b*c - a*d)*(c + d*x^2)^2) + ((b*c - 5*a*d)*x)/(8*d*(b*c - a*d)^2*(c + d*x^2)) + (a^(3/2)*Sqrt[b]*
ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b*c - a*d)^3 + ((b^2*c^2 - 6*a*b*c*d - 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(
8*Sqrt[c]*d^(3/2)*(b*c - a*d)^3)

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Rubi [A]  time = 0.172041, antiderivative size = 157, 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 = {470, 527, 522, 205} \[ \frac{\left (-3 a^2 d^2-6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 \sqrt{c} d^{3/2} (b c-a d)^3}+\frac{a^{3/2} \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{(b c-a d)^3}+\frac{x (b c-5 a d)}{8 d \left (c+d x^2\right ) (b c-a d)^2}-\frac{c x}{4 d \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c*x)/(4*d*(b*c - a*d)*(c + d*x^2)^2) + ((b*c - 5*a*d)*x)/(8*d*(b*c - a*d)^2*(c + d*x^2)) + (a^(3/2)*Sqrt[b]*
ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b*c - a*d)^3 + ((b^2*c^2 - 6*a*b*c*d - 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(
8*Sqrt[c]*d^(3/2)*(b*c - a*d)^3)

Rule 470

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

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 522

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

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^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx &=-\frac{c x}{4 d (b c-a d) \left (c+d x^2\right )^2}+\frac{\int \frac{a c+(b c-4 a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx}{4 d (b c-a d)}\\ &=-\frac{c x}{4 d (b c-a d) \left (c+d x^2\right )^2}+\frac{(b c-5 a d) x}{8 d (b c-a d)^2 \left (c+d x^2\right )}+\frac{\int \frac{a c (b c+3 a d)+b c (b c-5 a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{8 c d (b c-a d)^2}\\ &=-\frac{c x}{4 d (b c-a d) \left (c+d x^2\right )^2}+\frac{(b c-5 a d) x}{8 d (b c-a d)^2 \left (c+d x^2\right )}+\frac{\left (a^2 b\right ) \int \frac{1}{a+b x^2} \, dx}{(b c-a d)^3}+\frac{\left (b^2 c^2-6 a b c d-3 a^2 d^2\right ) \int \frac{1}{c+d x^2} \, dx}{8 d (b c-a d)^3}\\ &=-\frac{c x}{4 d (b c-a d) \left (c+d x^2\right )^2}+\frac{(b c-5 a d) x}{8 d (b c-a d)^2 \left (c+d x^2\right )}+\frac{a^{3/2} \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{(b c-a d)^3}+\frac{\left (b^2 c^2-6 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 \sqrt{c} d^{3/2} (b c-a d)^3}\\ \end{align*}

Mathematica [A]  time = 0.247804, size = 154, normalized size = 0.98 \[ \frac{1}{8} \left (\frac{\left (-3 a^2 d^2-6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} d^{3/2} (b c-a d)^3}+\frac{8 a^{3/2} \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{(b c-a d)^3}+\frac{x (b c-5 a d)}{d \left (c+d x^2\right ) (b c-a d)^2}+\frac{2 c x}{d \left (c+d x^2\right )^2 (a d-b c)}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*c*x)/(d*(-(b*c) + a*d)*(c + d*x^2)^2) + ((b*c - 5*a*d)*x)/(d*(b*c - a*d)^2*(c + d*x^2)) + (8*a^(3/2)*Sqrt[
b]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(b*c - a*d)^3 + ((b^2*c^2 - 6*a*b*c*d - 3*a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]]
)/(Sqrt[c]*d^(3/2)*(b*c - a*d)^3))/8

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Maple [B]  time = 0.009, size = 299, normalized size = 1.9 \begin{align*} -{\frac{5\,{x}^{3}{a}^{2}{d}^{2}}{8\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{3\,{x}^{3}abcd}{4\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{{x}^{3}{b}^{2}{c}^{2}}{8\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{3\,{a}^{2}cdx}{8\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{ab{c}^{2}x}{4\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{{b}^{2}{c}^{3}x}{8\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}d}}+{\frac{3\,{a}^{2}d}{8\, \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{3\,abc}{4\, \left ( ad-bc \right ) ^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{{b}^{2}{c}^{2}}{8\, \left ( ad-bc \right ) ^{3}d}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{b{a}^{2}}{ \left ( ad-bc \right ) ^{3}}\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^4/(b*x^2+a)/(d*x^2+c)^3,x)

[Out]

-5/8/(a*d-b*c)^3/(d*x^2+c)^2*x^3*a^2*d^2+3/4/(a*d-b*c)^3/(d*x^2+c)^2*x^3*a*b*c*d-1/8/(a*d-b*c)^3/(d*x^2+c)^2*x
^3*b^2*c^2-3/8/(a*d-b*c)^3/(d*x^2+c)^2*a^2*c*d*x+1/4/(a*d-b*c)^3/(d*x^2+c)^2*a*b*c^2*x+1/8/(a*d-b*c)^3/(d*x^2+
c)^2*c^3/d*x*b^2+3/8/(a*d-b*c)^3*d/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a^2+3/4/(a*d-b*c)^3/(c*d)^(1/2)*arctan(
x*d/(c*d)^(1/2))*c*a*b-1/8/(a*d-b*c)^3/d/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*b^2*c^2-b/(a*d-b*c)^3*a^2/(a*b)^(
1/2)*arctan(b*x/(a*b)^(1/2))

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.63587, size = 3168, normalized size = 20.18 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*(2*(b^2*c^3*d^2 - 6*a*b*c^2*d^3 + 5*a^2*c*d^4)*x^3 - 8*(a*c*d^4*x^4 + 2*a*c^2*d^3*x^2 + a*c^3*d^2)*sqrt(
-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) - (b^2*c^4 - 6*a*b*c^3*d - 3*a^2*c^2*d^2 + (b^2*c^2*d^2 -
6*a*b*c*d^3 - 3*a^2*d^4)*x^4 + 2*(b^2*c^3*d - 6*a*b*c^2*d^2 - 3*a^2*c*d^3)*x^2)*sqrt(-c*d)*log((d*x^2 - 2*sqrt
(-c*d)*x - c)/(d*x^2 + c)) - 2*(b^2*c^4*d + 2*a*b*c^3*d^2 - 3*a^2*c^2*d^3)*x)/(b^3*c^6*d^2 - 3*a*b^2*c^5*d^3 +
3*a^2*b*c^4*d^4 - a^3*c^3*d^5 + (b^3*c^4*d^4 - 3*a*b^2*c^3*d^5 + 3*a^2*b*c^2*d^6 - a^3*c*d^7)*x^4 + 2*(b^3*c^
5*d^3 - 3*a*b^2*c^4*d^4 + 3*a^2*b*c^3*d^5 - a^3*c^2*d^6)*x^2), 1/8*((b^2*c^3*d^2 - 6*a*b*c^2*d^3 + 5*a^2*c*d^4
)*x^3 + (b^2*c^4 - 6*a*b*c^3*d - 3*a^2*c^2*d^2 + (b^2*c^2*d^2 - 6*a*b*c*d^3 - 3*a^2*d^4)*x^4 + 2*(b^2*c^3*d -
6*a*b*c^2*d^2 - 3*a^2*c*d^3)*x^2)*sqrt(c*d)*arctan(sqrt(c*d)*x/c) - 4*(a*c*d^4*x^4 + 2*a*c^2*d^3*x^2 + a*c^3*d
^2)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) - (b^2*c^4*d + 2*a*b*c^3*d^2 - 3*a^2*c^2*d^3)*x)/
(b^3*c^6*d^2 - 3*a*b^2*c^5*d^3 + 3*a^2*b*c^4*d^4 - a^3*c^3*d^5 + (b^3*c^4*d^4 - 3*a*b^2*c^3*d^5 + 3*a^2*b*c^2*
d^6 - a^3*c*d^7)*x^4 + 2*(b^3*c^5*d^3 - 3*a*b^2*c^4*d^4 + 3*a^2*b*c^3*d^5 - a^3*c^2*d^6)*x^2), 1/16*(2*(b^2*c^
3*d^2 - 6*a*b*c^2*d^3 + 5*a^2*c*d^4)*x^3 + 16*(a*c*d^4*x^4 + 2*a*c^2*d^3*x^2 + a*c^3*d^2)*sqrt(a*b)*arctan(sqr
t(a*b)*x/a) - (b^2*c^4 - 6*a*b*c^3*d - 3*a^2*c^2*d^2 + (b^2*c^2*d^2 - 6*a*b*c*d^3 - 3*a^2*d^4)*x^4 + 2*(b^2*c^
3*d - 6*a*b*c^2*d^2 - 3*a^2*c*d^3)*x^2)*sqrt(-c*d)*log((d*x^2 - 2*sqrt(-c*d)*x - c)/(d*x^2 + c)) - 2*(b^2*c^4*
d + 2*a*b*c^3*d^2 - 3*a^2*c^2*d^3)*x)/(b^3*c^6*d^2 - 3*a*b^2*c^5*d^3 + 3*a^2*b*c^4*d^4 - a^3*c^3*d^5 + (b^3*c^
4*d^4 - 3*a*b^2*c^3*d^5 + 3*a^2*b*c^2*d^6 - a^3*c*d^7)*x^4 + 2*(b^3*c^5*d^3 - 3*a*b^2*c^4*d^4 + 3*a^2*b*c^3*d^
5 - a^3*c^2*d^6)*x^2), 1/8*((b^2*c^3*d^2 - 6*a*b*c^2*d^3 + 5*a^2*c*d^4)*x^3 + 8*(a*c*d^4*x^4 + 2*a*c^2*d^3*x^2
+ a*c^3*d^2)*sqrt(a*b)*arctan(sqrt(a*b)*x/a) + (b^2*c^4 - 6*a*b*c^3*d - 3*a^2*c^2*d^2 + (b^2*c^2*d^2 - 6*a*b*
c*d^3 - 3*a^2*d^4)*x^4 + 2*(b^2*c^3*d - 6*a*b*c^2*d^2 - 3*a^2*c*d^3)*x^2)*sqrt(c*d)*arctan(sqrt(c*d)*x/c) - (b
^2*c^4*d + 2*a*b*c^3*d^2 - 3*a^2*c^2*d^3)*x)/(b^3*c^6*d^2 - 3*a*b^2*c^5*d^3 + 3*a^2*b*c^4*d^4 - a^3*c^3*d^5 +
(b^3*c^4*d^4 - 3*a*b^2*c^3*d^5 + 3*a^2*b*c^2*d^6 - a^3*c*d^7)*x^4 + 2*(b^3*c^5*d^3 - 3*a*b^2*c^4*d^4 + 3*a^2*b
*c^3*d^5 - a^3*c^2*d^6)*x^2)]

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Sympy [B]  time = 65.0699, size = 3390, normalized size = 21.59 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(-a**3*b)*log(x + (-192*a**8*c*d**11*(-a**3*b)**(3/2)/(a*d - b*c)**9 + 256*a**7*b*c**2*d**10*(-a**3*b)**(3
/2)/(a*d - b*c)**9 + 2560*a**6*b**2*c**3*d**9*(-a**3*b)**(3/2)/(a*d - b*c)**9 - 27*a**6*d**6*sqrt(-a**3*b)/(a*
d - b*c)**3 - 9984*a**5*b**3*c**4*d**8*(-a**3*b)**(3/2)/(a*d - b*c)**9 - 162*a**5*b*c*d**5*sqrt(-a**3*b)/(a*d
- b*c)**3 + 16000*a**4*b**4*c**5*d**7*(-a**3*b)**(3/2)/(a*d - b*c)**9 - 809*a**4*b**2*c**2*d**4*sqrt(-a**3*b)/
(a*d - b*c)**3 - 13568*a**3*b**5*c**6*d**6*(-a**3*b)**(3/2)/(a*d - b*c)**9 - 108*a**3*b**3*c**3*d**3*sqrt(-a**
3*b)/(a*d - b*c)**3 + 6144*a**2*b**6*c**7*d**5*(-a**3*b)**(3/2)/(a*d - b*c)**9 + 99*a**2*b**4*c**4*d**2*sqrt(-
a**3*b)/(a*d - b*c)**3 - 1280*a*b**7*c**8*d**4*(-a**3*b)**(3/2)/(a*d - b*c)**9 - 18*a*b**5*c**5*d*sqrt(-a**3*b
)/(a*d - b*c)**3 + 64*b**8*c**9*d**3*(-a**3*b)**(3/2)/(a*d - b*c)**9 + b**6*c**6*sqrt(-a**3*b)/(a*d - b*c)**3)
/(27*a**4*b*d**3 + 51*a**3*b**2*c*d**2 - 15*a**2*b**3*c**2*d + a*b**4*c**3))/(2*(a*d - b*c)**3) - sqrt(-a**3*b
)*log(x + (192*a**8*c*d**11*(-a**3*b)**(3/2)/(a*d - b*c)**9 - 256*a**7*b*c**2*d**10*(-a**3*b)**(3/2)/(a*d - b*
c)**9 - 2560*a**6*b**2*c**3*d**9*(-a**3*b)**(3/2)/(a*d - b*c)**9 + 27*a**6*d**6*sqrt(-a**3*b)/(a*d - b*c)**3 +
9984*a**5*b**3*c**4*d**8*(-a**3*b)**(3/2)/(a*d - b*c)**9 + 162*a**5*b*c*d**5*sqrt(-a**3*b)/(a*d - b*c)**3 - 1
6000*a**4*b**4*c**5*d**7*(-a**3*b)**(3/2)/(a*d - b*c)**9 + 809*a**4*b**2*c**2*d**4*sqrt(-a**3*b)/(a*d - b*c)**
3 + 13568*a**3*b**5*c**6*d**6*(-a**3*b)**(3/2)/(a*d - b*c)**9 + 108*a**3*b**3*c**3*d**3*sqrt(-a**3*b)/(a*d - b
*c)**3 - 6144*a**2*b**6*c**7*d**5*(-a**3*b)**(3/2)/(a*d - b*c)**9 - 99*a**2*b**4*c**4*d**2*sqrt(-a**3*b)/(a*d
- b*c)**3 + 1280*a*b**7*c**8*d**4*(-a**3*b)**(3/2)/(a*d - b*c)**9 + 18*a*b**5*c**5*d*sqrt(-a**3*b)/(a*d - b*c)
**3 - 64*b**8*c**9*d**3*(-a**3*b)**(3/2)/(a*d - b*c)**9 - b**6*c**6*sqrt(-a**3*b)/(a*d - b*c)**3)/(27*a**4*b*d
**3 + 51*a**3*b**2*c*d**2 - 15*a**2*b**3*c**2*d + a*b**4*c**3))/(2*(a*d - b*c)**3) + sqrt(-1/(c*d**3))*(3*a**2
*d**2 + 6*a*b*c*d - b**2*c**2)*log(x + (-3*a**8*c*d**11*(-1/(c*d**3))**(3/2)*(3*a**2*d**2 + 6*a*b*c*d - b**2*c
**2)**3/(8*(a*d - b*c)**9) + a**7*b*c**2*d**10*(-1/(c*d**3))**(3/2)*(3*a**2*d**2 + 6*a*b*c*d - b**2*c**2)**3/(
2*(a*d - b*c)**9) + 5*a**6*b**2*c**3*d**9*(-1/(c*d**3))**(3/2)*(3*a**2*d**2 + 6*a*b*c*d - b**2*c**2)**3/(a*d -
b*c)**9 - 27*a**6*d**6*sqrt(-1/(c*d**3))*(3*a**2*d**2 + 6*a*b*c*d - b**2*c**2)/(8*(a*d - b*c)**3) - 39*a**5*b
**3*c**4*d**8*(-1/(c*d**3))**(3/2)*(3*a**2*d**2 + 6*a*b*c*d - b**2*c**2)**3/(2*(a*d - b*c)**9) - 81*a**5*b*c*d
**5*sqrt(-1/(c*d**3))*(3*a**2*d**2 + 6*a*b*c*d - b**2*c**2)/(4*(a*d - b*c)**3) + 125*a**4*b**4*c**5*d**7*(-1/(
c*d**3))**(3/2)*(3*a**2*d**2 + 6*a*b*c*d - b**2*c**2)**3/(4*(a*d - b*c)**9) - 809*a**4*b**2*c**2*d**4*sqrt(-1/
(c*d**3))*(3*a**2*d**2 + 6*a*b*c*d - b**2*c**2)/(8*(a*d - b*c)**3) - 53*a**3*b**5*c**6*d**6*(-1/(c*d**3))**(3/
2)*(3*a**2*d**2 + 6*a*b*c*d - b**2*c**2)**3/(2*(a*d - b*c)**9) - 27*a**3*b**3*c**3*d**3*sqrt(-1/(c*d**3))*(3*a
**2*d**2 + 6*a*b*c*d - b**2*c**2)/(2*(a*d - b*c)**3) + 12*a**2*b**6*c**7*d**5*(-1/(c*d**3))**(3/2)*(3*a**2*d**
2 + 6*a*b*c*d - b**2*c**2)**3/(a*d - b*c)**9 + 99*a**2*b**4*c**4*d**2*sqrt(-1/(c*d**3))*(3*a**2*d**2 + 6*a*b*c
*d - b**2*c**2)/(8*(a*d - b*c)**3) - 5*a*b**7*c**8*d**4*(-1/(c*d**3))**(3/2)*(3*a**2*d**2 + 6*a*b*c*d - b**2*c
**2)**3/(2*(a*d - b*c)**9) - 9*a*b**5*c**5*d*sqrt(-1/(c*d**3))*(3*a**2*d**2 + 6*a*b*c*d - b**2*c**2)/(4*(a*d -
b*c)**3) + b**8*c**9*d**3*(-1/(c*d**3))**(3/2)*(3*a**2*d**2 + 6*a*b*c*d - b**2*c**2)**3/(8*(a*d - b*c)**9) +
b**6*c**6*sqrt(-1/(c*d**3))*(3*a**2*d**2 + 6*a*b*c*d - b**2*c**2)/(8*(a*d - b*c)**3))/(27*a**4*b*d**3 + 51*a**
3*b**2*c*d**2 - 15*a**2*b**3*c**2*d + a*b**4*c**3))/(16*(a*d - b*c)**3) - sqrt(-1/(c*d**3))*(3*a**2*d**2 + 6*a
*b*c*d - b**2*c**2)*log(x + (3*a**8*c*d**11*(-1/(c*d**3))**(3/2)*(3*a**2*d**2 + 6*a*b*c*d - b**2*c**2)**3/(8*(
a*d - b*c)**9) - a**7*b*c**2*d**10*(-1/(c*d**3))**(3/2)*(3*a**2*d**2 + 6*a*b*c*d - b**2*c**2)**3/(2*(a*d - b*c
)**9) - 5*a**6*b**2*c**3*d**9*(-1/(c*d**3))**(3/2)*(3*a**2*d**2 + 6*a*b*c*d - b**2*c**2)**3/(a*d - b*c)**9 + 2
7*a**6*d**6*sqrt(-1/(c*d**3))*(3*a**2*d**2 + 6*a*b*c*d - b**2*c**2)/(8*(a*d - b*c)**3) + 39*a**5*b**3*c**4*d**
8*(-1/(c*d**3))**(3/2)*(3*a**2*d**2 + 6*a*b*c*d - b**2*c**2)**3/(2*(a*d - b*c)**9) + 81*a**5*b*c*d**5*sqrt(-1/
(c*d**3))*(3*a**2*d**2 + 6*a*b*c*d - b**2*c**2)/(4*(a*d - b*c)**3) - 125*a**4*b**4*c**5*d**7*(-1/(c*d**3))**(3
/2)*(3*a**2*d**2 + 6*a*b*c*d - b**2*c**2)**3/(4*(a*d - b*c)**9) + 809*a**4*b**2*c**2*d**4*sqrt(-1/(c*d**3))*(3
*a**2*d**2 + 6*a*b*c*d - b**2*c**2)/(8*(a*d - b*c)**3) + 53*a**3*b**5*c**6*d**6*(-1/(c*d**3))**(3/2)*(3*a**2*d
**2 + 6*a*b*c*d - b**2*c**2)**3/(2*(a*d - b*c)**9) + 27*a**3*b**3*c**3*d**3*sqrt(-1/(c*d**3))*(3*a**2*d**2 + 6
*a*b*c*d - b**2*c**2)/(2*(a*d - b*c)**3) - 12*a**2*b**6*c**7*d**5*(-1/(c*d**3))**(3/2)*(3*a**2*d**2 + 6*a*b*c*
d - b**2*c**2)**3/(a*d - b*c)**9 - 99*a**2*b**4*c**4*d**2*sqrt(-1/(c*d**3))*(3*a**2*d**2 + 6*a*b*c*d - b**2*c*
*2)/(8*(a*d - b*c)**3) + 5*a*b**7*c**8*d**4*(-1/(c*d**3))**(3/2)*(3*a**2*d**2 + 6*a*b*c*d - b**2*c**2)**3/(2*(
a*d - b*c)**9) + 9*a*b**5*c**5*d*sqrt(-1/(c*d**3))*(3*a**2*d**2 + 6*a*b*c*d - b**2*c**2)/(4*(a*d - b*c)**3) -
b**8*c**9*d**3*(-1/(c*d**3))**(3/2)*(3*a**2*d**2 + 6*a*b*c*d - b**2*c**2)**3/(8*(a*d - b*c)**9) - b**6*c**6*sq
rt(-1/(c*d**3))*(3*a**2*d**2 + 6*a*b*c*d - b**2*c**2)/(8*(a*d - b*c)**3))/(27*a**4*b*d**3 + 51*a**3*b**2*c*d**
2 - 15*a**2*b**3*c**2*d + a*b**4*c**3))/(16*(a*d - b*c)**3) - (x**3*(5*a*d**2 - b*c*d) + x*(3*a*c*d + b*c**2))
/(8*a**2*c**2*d**3 - 16*a*b*c**3*d**2 + 8*b**2*c**4*d + x**4*(8*a**2*d**5 - 16*a*b*c*d**4 + 8*b**2*c**2*d**3)
+ x**2*(16*a**2*c*d**4 - 32*a*b*c**2*d**3 + 16*b**2*c**3*d**2))

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Giac [A]  time = 1.14527, size = 275, normalized size = 1.75 \begin{align*} \frac{a^{2} b \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{{\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}} + \frac{{\left (b^{2} c^{2} - 6 \, a b c d - 3 \, a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \,{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} \sqrt{c d}} + \frac{b c d x^{3} - 5 \, a d^{2} x^{3} - b c^{2} x - 3 \, a c d x}{8 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}{\left (d x^{2} + c\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*b*arctan(b*x/sqrt(a*b))/((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)) + 1/8*(b^2*c^2 - 6
*a*b*c*d - 3*a^2*d^2)*arctan(d*x/sqrt(c*d))/((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*sqrt(c*d)
) + 1/8*(b*c*d*x^3 - 5*a*d^2*x^3 - b*c^2*x - 3*a*c*d*x)/((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*(d*x^2 + c)^2)