3.3.81 \(\int \frac {x^3 (c+d x^2)^3}{(a+b x^2)^2} \, dx\) [281]

Optimal. Leaf size=117 \[ \frac {3 d (b c-a d)^2 x^2}{2 b^4}+\frac {d^2 (3 b c-2 a d) x^4}{4 b^3}+\frac {d^3 x^6}{6 b^2}+\frac {a (b c-a d)^3}{2 b^5 \left (a+b x^2\right )}+\frac {(b c-4 a d) (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^5} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 117, 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, 78} \begin {gather*} \frac {a (b c-a d)^3}{2 b^5 \left (a+b x^2\right )}+\frac {(b c-4 a d) (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^5}+\frac {3 d x^2 (b c-a d)^2}{2 b^4}+\frac {d^2 x^4 (3 b c-2 a d)}{4 b^3}+\frac {d^3 x^6}{6 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

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

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Mathematica [A]
time = 0.07, size = 106, normalized size = 0.91 \begin {gather*} \frac {18 b d (b c-a d)^2 x^2+3 b^2 d^2 (3 b c-2 a d) x^4+2 b^3 d^3 x^6-\frac {6 a (-b c+a d)^3}{a+b x^2}+6 (b c-4 a d) (b c-a d)^2 \log \left (a+b x^2\right )}{12 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(18*b*d*(b*c - a*d)^2*x^2 + 3*b^2*d^2*(3*b*c - 2*a*d)*x^4 + 2*b^3*d^3*x^6 - (6*a*(-(b*c) + a*d)^3)/(a + b*x^2)
+ 6*(b*c - 4*a*d)*(b*c - a*d)^2*Log[a + b*x^2])/(12*b^5)

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Maple [A]
time = 0.09, size = 137, normalized size = 1.17

method	result	size
default	\(\frac {d \left (\frac {b^{2} d^{2} x^{6}}{6}+\frac {\left (-2 a b \,d^{2}+3 b^{2} c d \right ) x^{4}}{4}+\frac {\left (3 a^{2} d^{2}-6 a b c d +3 b^{2} c^{2}\right ) x^{2}}{2}\right )}{b^{4}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {\left (4 a d -b c \right ) \ln \left (b \,x^{2}+a \right )}{b}+\frac {a \left (a d -b c \right )}{b \left (b \,x^{2}+a \right )}\right )}{2 b^{4}}\)	\(137\)
norman	\(\frac {\frac {d^{3} x^{8}}{6 b}+\frac {d \left (4 a^{2} d^{2}-9 a b c d +6 b^{2} c^{2}\right ) x^{4}}{4 b^{3}}-\frac {d^{2} \left (4 a d -9 b c \right ) x^{6}}{12 b^{2}}+\frac {\left (4 a^{4} d^{3}-9 a^{3} b c \,d^{2}+6 a^{2} b^{2} c^{2} d -a \,b^{3} c^{3}\right ) x^{2}}{2 b^{4} a}}{b \,x^{2}+a}-\frac {\left (4 a^{3} d^{3}-9 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b \,x^{2}+a \right )}{2 b^{5}}\)	\(177\)
risch	\(\frac {d^{3} x^{6}}{6 b^{2}}-\frac {d^{3} a \,x^{4}}{2 b^{3}}+\frac {3 d^{2} c \,x^{4}}{4 b^{2}}+\frac {3 d^{3} a^{2} x^{2}}{2 b^{4}}-\frac {3 d^{2} a c \,x^{2}}{b^{3}}+\frac {3 d \,c^{2} x^{2}}{2 b^{2}}-\frac {a^{4} d^{3}}{2 b^{5} \left (b \,x^{2}+a \right )}+\frac {3 a^{3} c \,d^{2}}{2 b^{4} \left (b \,x^{2}+a \right )}-\frac {3 a^{2} c^{2} d}{2 b^{3} \left (b \,x^{2}+a \right )}+\frac {a \,c^{3}}{2 b^{2} \left (b \,x^{2}+a \right )}-\frac {2 \ln \left (b \,x^{2}+a \right ) a^{3} d^{3}}{b^{5}}+\frac {9 \ln \left (b \,x^{2}+a \right ) a^{2} c \,d^{2}}{2 b^{4}}-\frac {3 \ln \left (b \,x^{2}+a \right ) a \,c^{2} d}{b^{3}}+\frac {\ln \left (b \,x^{2}+a \right ) c^{3}}{2 b^{2}}\)	\(229\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 174, normalized size = 1.49 \begin {gather*} \frac {a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}}{2 \, {\left (b^{6} x^{2} + a b^{5}\right )}} + \frac {2 \, b^{2} d^{3} x^{6} + 3 \, {\left (3 \, b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{4} + 18 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}}{12 \, b^{4}} + \frac {{\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)/(b^6*x^2 + a*b^5) + 1/12*(2*b^2*d^3*x^6 + 3*(3*b^2
*c*d^2 - 2*a*b*d^3)*x^4 + 18*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*x^2)/b^4 + 1/2*(b^3*c^3 - 6*a*b^2*c^2*d + 9*a
^2*b*c*d^2 - 4*a^3*d^3)*log(b*x^2 + a)/b^5

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (107) = 214\).
time = 1.06, size = 254, normalized size = 2.17 \begin {gather*} \frac {2 \, b^{4} d^{3} x^{8} + 6 \, a b^{3} c^{3} - 18 \, a^{2} b^{2} c^{2} d + 18 \, a^{3} b c d^{2} - 6 \, a^{4} d^{3} + {\left (9 \, b^{4} c d^{2} - 4 \, a b^{3} d^{3}\right )} x^{6} + 3 \, {\left (6 \, b^{4} c^{2} d - 9 \, a b^{3} c d^{2} + 4 \, a^{2} b^{2} d^{3}\right )} x^{4} + 18 \, {\left (a b^{3} c^{2} d - 2 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x^{2} + 6 \, {\left (a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} - 4 \, a^{4} d^{3} + {\left (b^{4} c^{3} - 6 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - 4 \, a^{3} b d^{3}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{12 \, {\left (b^{6} x^{2} + a b^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(2*b^4*d^3*x^8 + 6*a*b^3*c^3 - 18*a^2*b^2*c^2*d + 18*a^3*b*c*d^2 - 6*a^4*d^3 + (9*b^4*c*d^2 - 4*a*b^3*d^3
)*x^6 + 3*(6*b^4*c^2*d - 9*a*b^3*c*d^2 + 4*a^2*b^2*d^3)*x^4 + 18*(a*b^3*c^2*d - 2*a^2*b^2*c*d^2 + a^3*b*d^3)*x
^2 + 6*(a*b^3*c^3 - 6*a^2*b^2*c^2*d + 9*a^3*b*c*d^2 - 4*a^4*d^3 + (b^4*c^3 - 6*a*b^3*c^2*d + 9*a^2*b^2*c*d^2 -
4*a^3*b*d^3)*x^2)*log(b*x^2 + a))/(b^6*x^2 + a*b^5)

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Sympy [A]
time = 0.95, size = 163, normalized size = 1.39 \begin {gather*} x^{4} \left (- \frac {a d^{3}}{2 b^{3}} + \frac {3 c d^{2}}{4 b^{2}}\right ) + x^{2} \cdot \left (\frac {3 a^{2} d^{3}}{2 b^{4}} - \frac {3 a c d^{2}}{b^{3}} + \frac {3 c^{2} d}{2 b^{2}}\right ) + \frac {- a^{4} d^{3} + 3 a^{3} b c d^{2} - 3 a^{2} b^{2} c^{2} d + a b^{3} c^{3}}{2 a b^{5} + 2 b^{6} x^{2}} + \frac {d^{3} x^{6}}{6 b^{2}} - \frac {\left (a d - b c\right )^{2} \cdot \left (4 a d - b c\right ) \log {\left (a + b x^{2} \right )}}{2 b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 249 vs. \(2 (107) = 214\).
time = 0.67, size = 249, normalized size = 2.13 \begin {gather*} \frac {\frac {{\left (2 \, d^{3} + \frac {3 \, {\left (3 \, b^{2} c d^{2} - 4 \, a b d^{3}\right )}}{{\left (b x^{2} + a\right )} b} + \frac {18 \, {\left (b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + 2 \, a^{2} b^{2} d^{3}\right )}}{{\left (b x^{2} + a\right )}^{2} b^{2}}\right )} {\left (b x^{2} + a\right )}^{3}}{b^{4}} - \frac {6 \, {\left (b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 4 \, a^{3} d^{3}\right )} \log \left (\frac {{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2} {\left | b \right |}}\right )}{b^{4}} + \frac {6 \, {\left (\frac {a b^{6} c^{3}}{b x^{2} + a} - \frac {3 \, a^{2} b^{5} c^{2} d}{b x^{2} + a} + \frac {3 \, a^{3} b^{4} c d^{2}}{b x^{2} + a} - \frac {a^{4} b^{3} d^{3}}{b x^{2} + a}\right )}}{b^{7}}}{12 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*((2*d^3 + 3*(3*b^2*c*d^2 - 4*a*b*d^3)/((b*x^2 + a)*b) + 18*(b^4*c^2*d - 3*a*b^3*c*d^2 + 2*a^2*b^2*d^3)/((
b*x^2 + a)^2*b^2))*(b*x^2 + a)^3/b^4 - 6*(b^3*c^3 - 6*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 4*a^3*d^3)*log(abs(b*x^2 +
a)/((b*x^2 + a)^2*abs(b)))/b^4 + 6*(a*b^6*c^3/(b*x^2 + a) - 3*a^2*b^5*c^2*d/(b*x^2 + a) + 3*a^3*b^4*c*d^2/(b*
x^2 + a) - a^4*b^3*d^3/(b*x^2 + a))/b^7)/b

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Mupad [B]
time = 0.10, size = 194, normalized size = 1.66 \begin {gather*} x^2\,\left (\frac {3\,c^2\,d}{2\,b^2}+\frac {a\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )}{b}-\frac {a^2\,d^3}{2\,b^4}\right )-x^4\,\left (\frac {a\,d^3}{2\,b^3}-\frac {3\,c\,d^2}{4\,b^2}\right )-\frac {\ln \left (b\,x^2+a\right )\,\left (4\,a^3\,d^3-9\,a^2\,b\,c\,d^2+6\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{2\,b^5}-\frac {a^4\,d^3-3\,a^3\,b\,c\,d^2+3\,a^2\,b^2\,c^2\,d-a\,b^3\,c^3}{2\,b\,\left (b^5\,x^2+a\,b^4\right )}+\frac {d^3\,x^6}{6\,b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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