3.166 \(\int \frac{(a+b x^2)^2 (c+d x^2)^3}{x^3} \, dx\)

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

[Out]

-(a^2*c^3)/(2*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]

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Rubi [A]  time = 0.100873, antiderivative size = 123, normalized size of antiderivative = 1., 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 = {446, 88} \[ \frac{1}{4} d x^4 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{2} c x^2 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac{a^2 c^3}{2 x^2}+a c^2 \log (x) (3 a d+2 b c)+\frac{1}{6} b d^2 x^6 (2 a d+3 b c)+\frac{1}{8} b^2 d^3 x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2*c^3)/(2*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 446

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]]

Rule 88

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]))

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} \operatorname{Subst}\left (\int \frac{(a+b x)^2 (c+d x)^3}{x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{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 = 0.0476805, size = 120, normalized size = 0.98 \[ \frac{6 a^2 \left (-2 c^3+6 c d^2 x^4+d^3 x^6\right )+4 a b d x^4 \left (18 c^2+9 c d x^2+2 d^2 x^4\right )+3 b^2 x^4 \left (6 c^2 d x^2+4 c^3+4 c d^2 x^4+d^3 x^6\right )}{24 x^2}+a c^2 \log (x) (3 a d+2 b c) \]

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]

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Maple [A]  time = 0.007, size = 134, normalized size = 1.1 \begin{align*}{\frac{{b}^{2}{d}^{3}{x}^{8}}{8}}+{\frac{{x}^{6}ab{d}^{3}}{3}}+{\frac{{x}^{6}{b}^{2}c{d}^{2}}{2}}+{\frac{{x}^{4}{a}^{2}{d}^{3}}{4}}+{\frac{3\,{x}^{4}abc{d}^{2}}{2}}+{\frac{3\,{x}^{4}{b}^{2}{c}^{2}d}{4}}+{\frac{3\,{x}^{2}{a}^{2}c{d}^{2}}{2}}+3\,{x}^{2}ab{c}^{2}d+{\frac{{x}^{2}{b}^{2}{c}^{3}}{2}}+3\,\ln \left ( x \right ){a}^{2}{c}^{2}d+2\,\ln \left ( x \right ) ab{c}^{3}-{\frac{{a}^{2}{c}^{3}}{2\,{x}^{2}}} \end{align*}

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)

[Out]

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

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Maxima [A]  time = 0.988683, size = 173, normalized size = 1.41 \begin{align*} \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{align*}

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)

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Fricas [A]  time = 1.25571, size = 282, normalized size = 2.29 \begin{align*} \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{align*}

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

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Sympy [A]  time = 0.453486, size = 133, normalized size = 1.08 \begin{align*} - \frac{a^{2} c^{3}}{2 x^{2}} + a c^{2} \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} \left (\frac{3 a^{2} c d^{2}}{2} + 3 a b c^{2} d + \frac{b^{2} c^{3}}{2}\right ) \end{align*}

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)

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Giac [A]  time = 1.15674, size = 216, normalized size = 1.76 \begin{align*} \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{align*}

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