3.3.20 \(\int \frac {x^3 (c+d x^2)^3}{a+b x^2} \, dx\) [220]

Optimal. Leaf size=115 \[ \frac {(b c-a d)^3 x^2}{2 b^4}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^4}{4 b^3}+\frac {d^2 (3 b c-a d) x^6}{6 b^2}+\frac {d^3 x^8}{8 b}-\frac {a (b c-a d)^3 \log \left (a+b x^2\right )}{2 b^5} \]

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 115, 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 {d x^4 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{4 b^3}-\frac {a (b c-a d)^3 \log \left (a+b x^2\right )}{2 b^5}+\frac {x^2 (b c-a d)^3}{2 b^4}+\frac {d^2 x^6 (3 b c-a d)}{6 b^2}+\frac {d^3 x^8}{8 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.04, size = 125, normalized size = 1.09 \begin {gather*} \frac {b x^2 \left (-12 a^3 d^3+6 a^2 b d^2 \left (6 c+d x^2\right )-2 a b^2 d \left (18 c^2+9 c d x^2+2 d^2 x^4\right )+3 b^3 \left (4 c^3+6 c^2 d x^2+4 c d^2 x^4+d^3 x^6\right )\right )+12 a (-b c+a d)^3 \log \left (a+b x^2\right )}{24 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x^2*(-12*a^3*d^3 + 6*a^2*b*d^2*(6*c + d*x^2) - 2*a*b^2*d*(18*c^2 + 9*c*d*x^2 + 2*d^2*x^4) + 3*b^3*(4*c^3 +
6*c^2*d*x^2 + 4*c*d^2*x^4 + d^3*x^6)) + 12*a*(-(b*c) + a*d)^3*Log[a + b*x^2])/(24*b^5)

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Maple [A]
time = 0.09, size = 177, normalized size = 1.54

method	result	size
norman	\(-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{2}}{2 b^{4}}+\frac {d^{3} x^{8}}{8 b}-\frac {d^{2} \left (a d -3 b c \right ) x^{6}}{6 b^{2}}+\frac {d \left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) x^{4}}{4 b^{3}}+\frac {a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b \,x^{2}+a \right )}{2 b^{5}}\)	\(157\)
default	\(-\frac {-\frac {1}{4} d^{3} x^{8} b^{3}+\frac {1}{3} a \,b^{2} d^{3} x^{6}-b^{3} c \,d^{2} x^{6}-\frac {1}{2} a^{2} b \,d^{3} x^{4}+\frac {3}{2} a \,b^{2} c \,d^{2} x^{4}-\frac {3}{2} b^{3} c^{2} d \,x^{4}+a^{3} d^{3} x^{2}-3 a^{2} b c \,d^{2} x^{2}+3 a \,b^{2} c^{2} d \,x^{2}-b^{3} c^{3} x^{2}}{2 b^{4}}+\frac {a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b \,x^{2}+a \right )}{2 b^{5}}\)	\(177\)
risch	\(\frac {d^{3} x^{8}}{8 b}-\frac {a \,d^{3} x^{6}}{6 b^{2}}+\frac {c \,d^{2} x^{6}}{2 b}+\frac {a^{2} d^{3} x^{4}}{4 b^{3}}-\frac {3 a c \,d^{2} x^{4}}{4 b^{2}}+\frac {3 c^{2} d \,x^{4}}{4 b}-\frac {a^{3} d^{3} x^{2}}{2 b^{4}}+\frac {3 a^{2} c \,d^{2} x^{2}}{2 b^{3}}-\frac {3 a \,c^{2} d \,x^{2}}{2 b^{2}}+\frac {c^{3} x^{2}}{2 b}+\frac {a^{4} \ln \left (b \,x^{2}+a \right ) d^{3}}{2 b^{5}}-\frac {3 a^{3} \ln \left (b \,x^{2}+a \right ) c \,d^{2}}{2 b^{4}}+\frac {3 a^{2} \ln \left (b \,x^{2}+a \right ) c^{2} d}{2 b^{3}}-\frac {a \ln \left (b \,x^{2}+a \right ) c^{3}}{2 b^{2}}\)	\(205\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 168, normalized size = 1.46 \begin {gather*} \frac {3 \, b^{3} d^{3} x^{8} + 4 \, {\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{6} + 6 \, {\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} + 12 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2}}{24 \, b^{4}} - \frac {{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} 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),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.38, size = 169, normalized size = 1.47 \begin {gather*} \frac {3 \, b^{4} d^{3} x^{8} + 4 \, {\left (3 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{6} + 6 \, {\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{4} + 12 \, {\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 )} x^{2} - 12 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \log \left (b x^{2} + a\right )}{24 \, 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),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.31, size = 144, normalized size = 1.25 \begin {gather*} \frac {a \left (a d - b c\right )^{3} \log {\left (a + b x^{2} \right )}}{2 b^{5}} + x^{6} \left (- \frac {a d^{3}}{6 b^{2}} + \frac {c d^{2}}{2 b}\right ) + x^{4} \left (\frac {a^{2} d^{3}}{4 b^{3}} - \frac {3 a c d^{2}}{4 b^{2}} + \frac {3 c^{2} d}{4 b}\right ) + x^{2} \left (- \frac {a^{3} d^{3}}{2 b^{4}} + \frac {3 a^{2} c d^{2}}{2 b^{3}} - \frac {3 a c^{2} d}{2 b^{2}} + \frac {c^{3}}{2 b}\right ) + \frac {d^{3} x^{8}}{8 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),x)

[Out]

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

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Giac [A]
time = 0.58, size = 180, normalized size = 1.57 \begin {gather*} \frac {3 \, b^{3} d^{3} x^{8} + 12 \, b^{3} c d^{2} x^{6} - 4 \, a b^{2} d^{3} x^{6} + 18 \, b^{3} c^{2} d x^{4} - 18 \, a b^{2} c d^{2} x^{4} + 6 \, a^{2} b d^{3} x^{4} + 12 \, b^{3} c^{3} x^{2} - 36 \, a b^{2} c^{2} d x^{2} + 36 \, a^{2} b c d^{2} x^{2} - 12 \, a^{3} d^{3} x^{2}}{24 \, b^{4}} - \frac {{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \log \left ({\left | b x^{2} + a \right |}\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),x, algorithm="giac")

[Out]

1/24*(3*b^3*d^3*x^8 + 12*b^3*c*d^2*x^6 - 4*a*b^2*d^3*x^6 + 18*b^3*c^2*d*x^4 - 18*a*b^2*c*d^2*x^4 + 6*a^2*b*d^3
*x^4 + 12*b^3*c^3*x^2 - 36*a*b^2*c^2*d*x^2 + 36*a^2*b*c*d^2*x^2 - 12*a^3*d^3*x^2)/b^4 - 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)*log(abs(b*x^2 + a))/b^5

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Mupad [B]
time = 0.03, size = 178, normalized size = 1.55 \begin {gather*} x^2\,\left (\frac {c^3}{2\,b}-\frac {a\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )}{2\,b}\right )-x^6\,\left (\frac {a\,d^3}{6\,b^2}-\frac {c\,d^2}{2\,b}\right )+x^4\,\left (\frac {3\,c^2\,d}{4\,b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{4\,b}\right )+\frac {\ln \left (b\,x^2+a\right )\,\left (a^4\,d^3-3\,a^3\,b\,c\,d^2+3\,a^2\,b^2\,c^2\,d-a\,b^3\,c^3\right )}{2\,b^5}+\frac {d^3\,x^8}{8\,b} \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),x)

[Out]

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

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