3.2.66 \(\int \frac {(a+b x^2)^2 (c+d x^2)^3}{x^3} \, dx\) [166]

Optimal. Leaf size=123 \[ -\frac {a^2 c^3}{2 x^2}+\frac {1}{2} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^2+\frac {1}{4} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^4+\frac {1}{6} b d^2 (3 b c+2 a d) x^6+\frac {1}{8} b^2 d^3 x^8+a c^2 (2 b c+3 a d) \log (x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 123, 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} d x^4 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {1}{2} c x^2 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac {a^2 c^3}{2 x^2}+a c^2 \log (x) (3 a d+2 b c)+\frac {1}{6} b d^2 x^6 (2 a d+3 b c)+\frac {1}{8} b^2 d^3 x^8 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Mathematica [A]
time = 0.04, size = 120, normalized size = 0.98 \begin {gather*} \frac {4 a b d x^4 \left (18 c^2+9 c d x^2+2 d^2 x^4\right )+3 b^2 x^4 \left (4 c^3+6 c^2 d x^2+4 c d^2 x^4+d^3 x^6\right )+6 a^2 \left (-2 c^3+6 c d^2 x^4+d^3 x^6\right )}{24 x^2}+a c^2 (2 b c+3 a d) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.08, size = 130, normalized size = 1.06

method	result	size
norman	\(\frac {\left (\frac {1}{3} a b \,d^{3}+\frac {1}{2} b^{2} c \,d^{2}\right ) x^{8}+\left (\frac {1}{4} a^{2} d^{3}+\frac {3}{2} a b c \,d^{2}+\frac {3}{4} b^{2} c^{2} d \right ) x^{6}+\left (\frac {3}{2} a^{2} c \,d^{2}+3 a b \,c^{2} d +\frac {1}{2} b^{2} c^{3}\right ) x^{4}-\frac {a^{2} c^{3}}{2}+\frac {b^{2} d^{3} x^{10}}{8}}{x^{2}}+\left (3 a^{2} c^{2} d +2 a b \,c^{3}\right ) \ln \left (x \right )\)	\(127\)
default	\(\frac {b^{2} d^{3} x^{8}}{8}+\frac {a b \,d^{3} x^{6}}{3}+\frac {b^{2} c \,d^{2} x^{6}}{2}+\frac {a^{2} d^{3} x^{4}}{4}+\frac {3 a b c \,d^{2} x^{4}}{2}+\frac {3 b^{2} c^{2} d \,x^{4}}{4}+\frac {3 a^{2} c \,d^{2} x^{2}}{2}+3 a b \,c^{2} d \,x^{2}+\frac {b^{2} c^{3} x^{2}}{2}-\frac {a^{2} c^{3}}{2 x^{2}}+a \,c^{2} \left (3 a d +2 b c \right ) \ln \left (x \right )\)	\(130\)
risch	\(\frac {b^{2} d^{3} x^{8}}{8}+\frac {a b \,d^{3} x^{6}}{3}+\frac {b^{2} c \,d^{2} x^{6}}{2}+\frac {a^{2} d^{3} x^{4}}{4}+\frac {3 a b c \,d^{2} x^{4}}{2}+\frac {3 b^{2} c^{2} d \,x^{4}}{4}+\frac {3 a^{2} c \,d^{2} x^{2}}{2}+3 a b \,c^{2} d \,x^{2}+\frac {b^{2} c^{3} x^{2}}{2}-\frac {a^{2} c^{3}}{2 x^{2}}+3 \ln \left (x \right ) a^{2} c^{2} d +2 \ln \left (x \right ) a b \,c^{3}\)	\(134\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 128, normalized size = 1.04 \begin {gather*} \frac {1}{8} \, b^{2} d^{3} x^{8} + \frac {1}{6} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{6} + \frac {1}{4} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{4} - \frac {a^{2} c^{3}}{2 \, x^{2}} + \frac {1}{2} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{2} + \frac {1}{2} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} \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)^3/x^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.90, size = 131, normalized size = 1.07 \begin {gather*} \frac {3 \, b^{2} d^{3} x^{10} + 4 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} + 6 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} - 12 \, a^{2} c^{3} + 12 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} + 24 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2} \log \left (x\right )}{24 \, 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)^3/x^3,x, algorithm="fricas")

[Out]

1/24*(3*b^2*d^3*x^10 + 4*(3*b^2*c*d^2 + 2*a*b*d^3)*x^8 + 6*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^6 - 12*a^2*
c^3 + 12*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^4 + 24*(2*a*b*c^3 + 3*a^2*c^2*d)*x^2*log(x))/x^2

________________________________________________________________________________________

Sympy [A]
time = 0.21, size = 133, normalized size = 1.08 \begin {gather*} - \frac {a^{2} c^{3}}{2 x^{2}} + a c^{2} \cdot \left (3 a d + 2 b c\right ) \log {\left (x \right )} + \frac {b^{2} d^{3} x^{8}}{8} + x^{6} \left (\frac {a b d^{3}}{3} + \frac {b^{2} c d^{2}}{2}\right ) + x^{4} \left (\frac {a^{2} d^{3}}{4} + \frac {3 a b c d^{2}}{2} + \frac {3 b^{2} c^{2} d}{4}\right ) + x^{2} \cdot \left (\frac {3 a^{2} c d^{2}}{2} + 3 a b c^{2} d + \frac {b^{2} c^{3}}{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)**3/x**3,x)

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.89, size = 160, normalized size = 1.30 \begin {gather*} \frac {1}{8} \, b^{2} d^{3} x^{8} + \frac {1}{2} \, b^{2} c d^{2} x^{6} + \frac {1}{3} \, a b d^{3} x^{6} + \frac {3}{4} \, b^{2} c^{2} d x^{4} + \frac {3}{2} \, a b c d^{2} x^{4} + \frac {1}{4} \, a^{2} d^{3} x^{4} + \frac {1}{2} \, b^{2} c^{3} x^{2} + 3 \, a b c^{2} d x^{2} + \frac {3}{2} \, a^{2} c d^{2} x^{2} + \frac {1}{2} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} \log \left (x^{2}\right ) - \frac {2 \, a b c^{3} x^{2} + 3 \, a^{2} c^{2} d x^{2} + a^{2} c^{3}}{2 \, 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)^3/x^3,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.05, size = 121, normalized size = 0.98 \begin {gather*} x^2\,\left (\frac {3\,a^2\,c\,d^2}{2}+3\,a\,b\,c^2\,d+\frac {b^2\,c^3}{2}\right )+x^4\,\left (\frac {a^2\,d^3}{4}+\frac {3\,a\,b\,c\,d^2}{2}+\frac {3\,b^2\,c^2\,d}{4}\right )+\ln \left (x\right )\,\left (3\,d\,a^2\,c^2+2\,b\,a\,c^3\right )-\frac {a^2\,c^3}{2\,x^2}+\frac {b^2\,d^3\,x^8}{8}+\frac {b\,d^2\,x^6\,\left (2\,a\,d+3\,b\,c\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________