3.2.55 \(\int \frac {(a+b x^2)^2 (c+d x^2)^2}{x} \, dx\) [155]

Optimal. Leaf size=80 \[ a c (b c+a d) x^2+\frac {1}{4} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^4+\frac {1}{3} b d (b c+a d) x^6+\frac {1}{8} b^2 d^2 x^8+a^2 c^2 \log (x) \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 80, 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 {1}{4} x^4 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 \log (x)+\frac {1}{3} b d x^6 (a d+b c)+a c x^2 (a d+b c)+\frac {1}{8} b^2 d^2 x^8 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a*c*(b*c + a*d)*x^2 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4)/4 + (b*d*(b*c + a*d)*x^6)/3 + (b^2*d^2*x^8)/8 + a^
2*c^2*Log[x]

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

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Mathematica [A]
time = 0.02, size = 80, normalized size = 1.00 \begin {gather*} a c (b c+a d) x^2+\frac {1}{4} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^4+\frac {1}{3} b d (b c+a d) x^6+\frac {1}{8} b^2 d^2 x^8+a^2 c^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a*c*(b*c + a*d)*x^2 + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4)/4 + (b*d*(b*c + a*d)*x^6)/3 + (b^2*d^2*x^8)/8 + a^
2*c^2*Log[x]

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Maple [A]
time = 0.13, size = 90, normalized size = 1.12

method	result	size
norman	\(\left (\frac {1}{3} a b \,d^{2}+\frac {1}{3} b^{2} c d \right ) x^{6}+\left (\frac {1}{4} a^{2} d^{2}+a b c d +\frac {1}{4} b^{2} c^{2}\right ) x^{4}+\left (a^{2} c d +a b \,c^{2}\right ) x^{2}+\frac {b^{2} d^{2} x^{8}}{8}+a^{2} c^{2} \ln \left (x \right )\)	\(84\)
default	\(\frac {b^{2} d^{2} x^{8}}{8}+\frac {a b \,d^{2} x^{6}}{3}+\frac {b^{2} c d \,x^{6}}{3}+\frac {a^{2} d^{2} x^{4}}{4}+a b c d \,x^{4}+\frac {b^{2} c^{2} x^{4}}{4}+a^{2} c d \,x^{2}+a b \,c^{2} x^{2}+a^{2} c^{2} \ln \left (x \right )\)	\(90\)
risch	\(-\frac {b^{2} c^{4}}{24 d^{2}}-\frac {d^{2} a^{4}}{24 b^{2}}+\frac {b a \,c^{3}}{3 d}+\frac {d \,a^{3} c}{3 b}+\frac {3 a^{2} c^{2}}{4}+\frac {a b \,d^{2} x^{6}}{3}+\frac {b^{2} c d \,x^{6}}{3}+\frac {a^{2} d^{2} x^{4}}{4}+\frac {b^{2} c^{2} x^{4}}{4}+\frac {b^{2} d^{2} x^{8}}{8}+a b c d \,x^{4}+a^{2} c d \,x^{2}+a b \,c^{2} x^{2}+a^{2} c^{2} \ln \left (x \right )\)	\(140\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 85, normalized size = 1.06 \begin {gather*} \frac {1}{8} \, b^{2} d^{2} x^{8} + \frac {1}{3} \, {\left (b^{2} c d + a b d^{2}\right )} x^{6} + \frac {1}{4} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} + \frac {1}{2} \, a^{2} c^{2} \log \left (x^{2}\right ) + {\left (a b c^{2} + a^{2} c d\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.85, size = 82, normalized size = 1.02 \begin {gather*} \frac {1}{8} \, b^{2} d^{2} x^{8} + \frac {1}{3} \, {\left (b^{2} c d + a b d^{2}\right )} x^{6} + \frac {1}{4} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} \log \left (x\right ) + {\left (a b c^{2} + a^{2} c d\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.07, size = 85, normalized size = 1.06 \begin {gather*} a^{2} c^{2} \log {\left (x \right )} + \frac {b^{2} d^{2} x^{8}}{8} + x^{6} \left (\frac {a b d^{2}}{3} + \frac {b^{2} c d}{3}\right ) + x^{4} \left (\frac {a^{2} d^{2}}{4} + a b c d + \frac {b^{2} c^{2}}{4}\right ) + x^{2} \left (a^{2} c d + a b c^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.28, size = 92, normalized size = 1.15 \begin {gather*} \frac {1}{8} \, b^{2} d^{2} x^{8} + \frac {1}{3} \, b^{2} c d x^{6} + \frac {1}{3} \, a b d^{2} x^{6} + \frac {1}{4} \, b^{2} c^{2} x^{4} + a b c d x^{4} + \frac {1}{4} \, a^{2} d^{2} x^{4} + a b c^{2} x^{2} + a^{2} c d x^{2} + \frac {1}{2} \, a^{2} c^{2} \log \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.02, size = 74, normalized size = 0.92 \begin {gather*} x^4\,\left (\frac {a^2\,d^2}{4}+a\,b\,c\,d+\frac {b^2\,c^2}{4}\right )+\frac {b^2\,d^2\,x^8}{8}+a^2\,c^2\,\ln \left (x\right )+a\,c\,x^2\,\left (a\,d+b\,c\right )+\frac {b\,d\,x^6\,\left (a\,d+b\,c\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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