3.3.51 \(\int \frac {x^5}{(a+b x^2)^3 (c+d x^2)} \, dx\) [251]

Optimal. Leaf size=116 \[ -\frac {a^2}{4 b^2 (b c-a d) \left (a+b x^2\right )^2}+\frac {a (2 b c-a d)}{2 b^2 (b c-a d)^2 \left (a+b x^2\right )}+\frac {c^2 \log \left (a+b x^2\right )}{2 (b c-a d)^3}-\frac {c^2 \log \left (c+d x^2\right )}{2 (b c-a d)^3} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 116, 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 = {457, 90} \begin {gather*} -\frac {a^2}{4 b^2 \left (a+b x^2\right )^2 (b c-a d)}+\frac {a (2 b c-a d)}{2 b^2 \left (a+b x^2\right ) (b c-a d)^2}+\frac {c^2 \log \left (a+b x^2\right )}{2 (b c-a d)^3}-\frac {c^2 \log \left (c+d x^2\right )}{2 (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/((a + b*x^2)^3*(c + d*x^2)),x]

[Out]

-1/4*a^2/(b^2*(b*c - a*d)*(a + b*x^2)^2) + (a*(2*b*c - a*d))/(2*b^2*(b*c - a*d)^2*(a + b*x^2)) + (c^2*Log[a +
b*x^2])/(2*(b*c - a*d)^3) - (c^2*Log[c + d*x^2])/(2*(b*c - a*d)^3)

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 457

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

Rubi steps

\begin {align*} \int \frac {x^5}{\left (a+b x^2\right )^3 \left (c+d x^2\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{(a+b x)^3 (c+d x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {a^2}{b (b c-a d) (a+b x)^3}+\frac {a (-2 b c+a d)}{b (b c-a d)^2 (a+b x)^2}+\frac {b c^2}{(b c-a d)^3 (a+b x)}-\frac {c^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^2}{4 b^2 (b c-a d) \left (a+b x^2\right )^2}+\frac {a (2 b c-a d)}{2 b^2 (b c-a d)^2 \left (a+b x^2\right )}+\frac {c^2 \log \left (a+b x^2\right )}{2 (b c-a d)^3}-\frac {c^2 \log \left (c+d x^2\right )}{2 (b c-a d)^3}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 99, normalized size = 0.85 \begin {gather*} \frac {-\frac {a^2 (b c-a d)^2}{b^2 \left (a+b x^2\right )^2}+\frac {2 a (-2 b c+a d) (-b c+a d)}{b^2 \left (a+b x^2\right )}+2 c^2 \log \left (a+b x^2\right )-2 c^2 \log \left (c+d x^2\right )}{4 (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5/((a + b*x^2)^3*(c + d*x^2)),x]

[Out]

(-((a^2*(b*c - a*d)^2)/(b^2*(a + b*x^2)^2)) + (2*a*(-2*b*c + a*d)*(-(b*c) + a*d))/(b^2*(a + b*x^2)) + 2*c^2*Lo
g[a + b*x^2] - 2*c^2*Log[c + d*x^2])/(4*(b*c - a*d)^3)

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Maple [A]
time = 0.12, size = 124, normalized size = 1.07

method	result	size
default	\(-\frac {c^{2} \ln \left (b \,x^{2}+a \right )-\frac {a^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 b^{2} \left (b \,x^{2}+a \right )^{2}}+\frac {a \left (a^{2} d^{2}-3 a b c d +2 b^{2} c^{2}\right )}{b^{2} \left (b \,x^{2}+a \right )}}{2 \left (a d -b c \right )^{3}}+\frac {c^{2} \ln \left (d \,x^{2}+c \right )}{2 \left (a d -b c \right )^{3}}\)	\(124\)
norman	\(\frac {\frac {\left (-a d +3 b c \right ) a^{2}}{4 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {a \left (-a d +2 b c \right ) x^{2}}{2 b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b \,x^{2}+a \right )^{2}}-\frac {c^{2} \ln \left (b \,x^{2}+a \right )}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {c^{2} \ln \left (d \,x^{2}+c \right )}{2 a^{3} d^{3}-6 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -2 b^{3} c^{3}}\)	\(196\)
risch	\(\frac {-\frac {a \left (a d -2 b c \right ) x^{2}}{2 b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {a^{2} \left (a d -3 b c \right )}{4 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b \,x^{2}+a \right )^{2}}+\frac {c^{2} \ln \left (d \,x^{2}+c \right )}{2 a^{3} d^{3}-6 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -2 b^{3} c^{3}}-\frac {c^{2} \ln \left (-b \,x^{2}-a \right )}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\)	\(197\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x^2+a)^3/(d*x^2+c),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (108) = 216\).
time = 0.30, size = 236, normalized size = 2.03 \begin {gather*} \frac {c^{2} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} - \frac {c^{2} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} + \frac {3 \, a^{2} b c - a^{3} d + 2 \, {\left (2 \, a b^{2} c - a^{2} b d\right )} x^{2}}{4 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2} + {\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} x^{4} + 2 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a)^3/(d*x^2+c),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (108) = 216\).
time = 1.13, size = 290, normalized size = 2.50 \begin {gather*} \frac {3 \, a^{2} b^{2} c^{2} - 4 \, a^{3} b c d + a^{4} d^{2} + 2 \, {\left (2 \, a b^{3} c^{2} - 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2} + 2 \, {\left (b^{4} c^{2} x^{4} + 2 \, a b^{3} c^{2} x^{2} + a^{2} b^{2} c^{2}\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left (b^{4} c^{2} x^{4} + 2 \, a b^{3} c^{2} x^{2} + a^{2} b^{2} c^{2}\right )} \log \left (d x^{2} + c\right )}{4 \, {\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3} + {\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} x^{4} + 2 \, {\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a)^3/(d*x^2+c),x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (97) = 194\).
time = 3.07, size = 418, normalized size = 3.60 \begin {gather*} \frac {c^{2} \log {\left (x^{2} + \frac {- \frac {a^{4} c^{2} d^{4}}{\left (a d - b c\right )^{3}} + \frac {4 a^{3} b c^{3} d^{3}}{\left (a d - b c\right )^{3}} - \frac {6 a^{2} b^{2} c^{4} d^{2}}{\left (a d - b c\right )^{3}} + \frac {4 a b^{3} c^{5} d}{\left (a d - b c\right )^{3}} + a c^{2} d - \frac {b^{4} c^{6}}{\left (a d - b c\right )^{3}} + b c^{3}}{2 b c^{2} d} \right )}}{2 \left (a d - b c\right )^{3}} - \frac {c^{2} \log {\left (x^{2} + \frac {\frac {a^{4} c^{2} d^{4}}{\left (a d - b c\right )^{3}} - \frac {4 a^{3} b c^{3} d^{3}}{\left (a d - b c\right )^{3}} + \frac {6 a^{2} b^{2} c^{4} d^{2}}{\left (a d - b c\right )^{3}} - \frac {4 a b^{3} c^{5} d}{\left (a d - b c\right )^{3}} + a c^{2} d + \frac {b^{4} c^{6}}{\left (a d - b c\right )^{3}} + b c^{3}}{2 b c^{2} d} \right )}}{2 \left (a d - b c\right )^{3}} + \frac {- a^{3} d + 3 a^{2} b c + x^{2} \left (- 2 a^{2} b d + 4 a b^{2} c\right )}{4 a^{4} b^{2} d^{2} - 8 a^{3} b^{3} c d + 4 a^{2} b^{4} c^{2} + x^{4} \cdot \left (4 a^{2} b^{4} d^{2} - 8 a b^{5} c d + 4 b^{6} c^{2}\right ) + x^{2} \cdot \left (8 a^{3} b^{3} d^{2} - 16 a^{2} b^{4} c d + 8 a b^{5} c^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b*x**2+a)**3/(d*x**2+c),x)

[Out]

c**2*log(x**2 + (-a**4*c**2*d**4/(a*d - b*c)**3 + 4*a**3*b*c**3*d**3/(a*d - b*c)**3 - 6*a**2*b**2*c**4*d**2/(a
*d - b*c)**3 + 4*a*b**3*c**5*d/(a*d - b*c)**3 + a*c**2*d - b**4*c**6/(a*d - b*c)**3 + b*c**3)/(2*b*c**2*d))/(2
*(a*d - b*c)**3) - c**2*log(x**2 + (a**4*c**2*d**4/(a*d - b*c)**3 - 4*a**3*b*c**3*d**3/(a*d - b*c)**3 + 6*a**2
*b**2*c**4*d**2/(a*d - b*c)**3 - 4*a*b**3*c**5*d/(a*d - b*c)**3 + a*c**2*d + b**4*c**6/(a*d - b*c)**3 + b*c**3
)/(2*b*c**2*d))/(2*(a*d - b*c)**3) + (-a**3*d + 3*a**2*b*c + x**2*(-2*a**2*b*d + 4*a*b**2*c))/(4*a**4*b**2*d**
2 - 8*a**3*b**3*c*d + 4*a**2*b**4*c**2 + x**4*(4*a**2*b**4*d**2 - 8*a*b**5*c*d + 4*b**6*c**2) + x**2*(8*a**3*b
**3*d**2 - 16*a**2*b**4*c*d + 8*a*b**5*c**2))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (108) = 216\).
time = 1.35, size = 232, normalized size = 2.00 \begin {gather*} \frac {b c^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}} - \frac {c^{2} d \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\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 )}} - \frac {3 \, b^{4} c^{2} x^{4} + 2 \, a b^{3} c^{2} x^{2} + 6 \, a^{2} b^{2} c d x^{2} - 2 \, a^{3} b d^{2} x^{2} + 4 \, a^{3} b c d - a^{4} d^{2}}{4 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} {\left (b x^{2} + a\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x^2+a)^3/(d*x^2+c),x, algorithm="giac")

[Out]

1/2*b*c^2*log(abs(b*x^2 + a))/(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3) - 1/2*c^2*d*log(abs(d*x^
2 + c))/(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4) - 1/4*(3*b^4*c^2*x^4 + 2*a*b^3*c^2*x^2 + 6*a^2
*b^2*c*d*x^2 - 2*a^3*b*d^2*x^2 + 4*a^3*b*c*d - a^4*d^2)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*
d^3)*(b*x^2 + a)^2)

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Mupad [B]
time = 0.25, size = 370, normalized size = 3.19 \begin {gather*} \frac {b^3\,\left (4\,a\,c^2\,x^2+a\,c^2\,x^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,8{}\mathrm {i}\right )+b\,\left (2\,a^3\,d^2\,x^2-4\,a^3\,c\,d\right )+a^4\,d^2+b^2\,\left (3\,a^2\,c^2-6\,a^2\,c\,d\,x^2+a^2\,c^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,4{}\mathrm {i}\right )+b^4\,c^2\,x^4\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,4{}\mathrm {i}}{-4\,a^5\,b^2\,d^3+12\,a^4\,b^3\,c\,d^2-8\,a^4\,b^3\,d^3\,x^2-12\,a^3\,b^4\,c^2\,d+24\,a^3\,b^4\,c\,d^2\,x^2-4\,a^3\,b^4\,d^3\,x^4+4\,a^2\,b^5\,c^3-24\,a^2\,b^5\,c^2\,d\,x^2+12\,a^2\,b^5\,c\,d^2\,x^4+8\,a\,b^6\,c^3\,x^2-12\,a\,b^6\,c^2\,d\,x^4+4\,b^7\,c^3\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^3*(4*a*c^2*x^2 + a*c^2*x^2*atan((a*d*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c*x^2))*8i) + b*(2*a^3*d^2*x
^2 - 4*a^3*c*d) + a^4*d^2 + b^2*(3*a^2*c^2 + a^2*c^2*atan((a*d*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c*x^2
))*4i - 6*a^2*c*d*x^2) + b^4*c^2*x^4*atan((a*d*x^2*1i - b*c*x^2*1i)/(2*a*c + a*d*x^2 + b*c*x^2))*4i)/(4*a^2*b^
5*c^3 - 4*a^5*b^2*d^3 + 4*b^7*c^3*x^4 - 12*a^3*b^4*c^2*d + 12*a^4*b^3*c*d^2 + 8*a*b^6*c^3*x^2 - 8*a^4*b^3*d^3*
x^2 - 4*a^3*b^4*d^3*x^4 - 12*a*b^6*c^2*d*x^4 - 24*a^2*b^5*c^2*d*x^2 + 24*a^3*b^4*c*d^2*x^2 + 12*a^2*b^5*c*d^2*
x^4)

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