3.167 \(\int \frac{(a+b x^2)^2 (c+d x^2)^3}{x^4} \, dx\)

Optimal. Leaf size=120 \[ \frac{1}{3} d x^3 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+c x \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac{a^2 c^3}{3 x^3}-\frac{a c^2 (3 a d+2 b c)}{x}+\frac{1}{5} b d^2 x^5 (2 a d+3 b c)+\frac{1}{7} b^2 d^3 x^7 \]

[Out]

-(a^2*c^3)/(3*x^3) - (a*c^2*(2*b*c + 3*a*d))/x + c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x + (d*(3*b^2*c^2 + 6*a*b
*c*d + a^2*d^2)*x^3)/3 + (b*d^2*(3*b*c + 2*a*d)*x^5)/5 + (b^2*d^3*x^7)/7

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Rubi [A]  time = 0.0601588, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {448} \[ \frac{1}{3} d x^3 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+c x \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac{a^2 c^3}{3 x^3}-\frac{a c^2 (3 a d+2 b c)}{x}+\frac{1}{5} b d^2 x^5 (2 a d+3 b c)+\frac{1}{7} b^2 d^3 x^7 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^2*(c + d*x^2)^3)/x^4,x]

[Out]

-(a^2*c^3)/(3*x^3) - (a*c^2*(2*b*c + 3*a*d))/x + c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x + (d*(3*b^2*c^2 + 6*a*b
*c*d + a^2*d^2)*x^3)/3 + (b*d^2*(3*b*c + 2*a*d)*x^5)/5 + (b^2*d^3*x^7)/7

Rule 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{x^4} \, dx &=\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^4}+\frac{a c^2 (2 b c+3 a d)}{x^2}+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^2+b d^2 (3 b c+2 a d) x^4+b^2 d^3 x^6\right ) \, dx\\ &=-\frac{a^2 c^3}{3 x^3}-\frac{a c^2 (2 b c+3 a d)}{x}+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x+\frac{1}{3} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^3+\frac{1}{5} b d^2 (3 b c+2 a d) x^5+\frac{1}{7} b^2 d^3 x^7\\ \end{align*}

Mathematica [A]  time = 0.0420917, size = 120, normalized size = 1. \[ \frac{1}{3} d x^3 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+c x \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac{a^2 c^3}{3 x^3}-\frac{a c^2 (3 a d+2 b c)}{x}+\frac{1}{5} b d^2 x^5 (2 a d+3 b c)+\frac{1}{7} b^2 d^3 x^7 \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^2*(c + d*x^2)^3)/x^4,x]

[Out]

-(a^2*c^3)/(3*x^3) - (a*c^2*(2*b*c + 3*a*d))/x + c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x + (d*(3*b^2*c^2 + 6*a*b
*c*d + a^2*d^2)*x^3)/3 + (b*d^2*(3*b*c + 2*a*d)*x^5)/5 + (b^2*d^3*x^7)/7

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Maple [A]  time = 0.005, size = 124, normalized size = 1. \begin{align*}{\frac{{b}^{2}{d}^{3}{x}^{7}}{7}}+{\frac{2\,{x}^{5}ab{d}^{3}}{5}}+{\frac{3\,{x}^{5}{b}^{2}c{d}^{2}}{5}}+{\frac{{x}^{3}{a}^{2}{d}^{3}}{3}}+2\,{x}^{3}abc{d}^{2}+{x}^{3}{b}^{2}{c}^{2}d+3\,{a}^{2}c{d}^{2}x+6\,ab{c}^{2}dx+{b}^{2}{c}^{3}x-{\frac{{a}^{2}{c}^{3}}{3\,{x}^{3}}}-{\frac{a{c}^{2} \left ( 3\,ad+2\,bc \right ) }{x}} \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^4,x)

[Out]

1/7*b^2*d^3*x^7+2/5*x^5*a*b*d^3+3/5*x^5*b^2*c*d^2+1/3*x^3*a^2*d^3+2*x^3*a*b*c*d^2+x^3*b^2*c^2*d+3*a^2*c*d^2*x+
6*a*b*c^2*d*x+b^2*c^3*x-1/3*a^2*c^3/x^3-a*c^2*(3*a*d+2*b*c)/x

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Maxima [A]  time = 0.984019, size = 170, normalized size = 1.42 \begin{align*} \frac{1}{7} \, b^{2} d^{3} x^{7} + \frac{1}{5} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{5} + \frac{1}{3} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} +{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x - \frac{a^{2} c^{3} + 3 \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}}{3 \, x^{3}} \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^4,x, algorithm="maxima")

[Out]

1/7*b^2*d^3*x^7 + 1/5*(3*b^2*c*d^2 + 2*a*b*d^3)*x^5 + 1/3*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^3 + (b^2*c^3
+ 6*a*b*c^2*d + 3*a^2*c*d^2)*x - 1/3*(a^2*c^3 + 3*(2*a*b*c^3 + 3*a^2*c^2*d)*x^2)/x^3

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Fricas [A]  time = 1.28312, size = 281, normalized size = 2.34 \begin{align*} \frac{15 \, b^{2} d^{3} x^{10} + 21 \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} + 35 \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} - 35 \, a^{2} c^{3} + 105 \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} - 105 \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}}{105 \, x^{3}} \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^4,x, algorithm="fricas")

[Out]

1/105*(15*b^2*d^3*x^10 + 21*(3*b^2*c*d^2 + 2*a*b*d^3)*x^8 + 35*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^6 - 35*
a^2*c^3 + 105*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^4 - 105*(2*a*b*c^3 + 3*a^2*c^2*d)*x^2)/x^3

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Sympy [A]  time = 0.472482, size = 129, normalized size = 1.08 \begin{align*} \frac{b^{2} d^{3} x^{7}}{7} + x^{5} \left (\frac{2 a b d^{3}}{5} + \frac{3 b^{2} c d^{2}}{5}\right ) + x^{3} \left (\frac{a^{2} d^{3}}{3} + 2 a b c d^{2} + b^{2} c^{2} d\right ) + x \left (3 a^{2} c d^{2} + 6 a b c^{2} d + b^{2} c^{3}\right ) - \frac{a^{2} c^{3} + x^{2} \left (9 a^{2} c^{2} d + 6 a b c^{3}\right )}{3 x^{3}} \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**4,x)

[Out]

b**2*d**3*x**7/7 + x**5*(2*a*b*d**3/5 + 3*b**2*c*d**2/5) + x**3*(a**2*d**3/3 + 2*a*b*c*d**2 + b**2*c**2*d) + x
*(3*a**2*c*d**2 + 6*a*b*c**2*d + b**2*c**3) - (a**2*c**3 + x**2*(9*a**2*c**2*d + 6*a*b*c**3))/(3*x**3)

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Giac [A]  time = 1.14904, size = 174, normalized size = 1.45 \begin{align*} \frac{1}{7} \, b^{2} d^{3} x^{7} + \frac{3}{5} \, b^{2} c d^{2} x^{5} + \frac{2}{5} \, a b d^{3} x^{5} + b^{2} c^{2} d x^{3} + 2 \, a b c d^{2} x^{3} + \frac{1}{3} \, a^{2} d^{3} x^{3} + b^{2} c^{3} x + 6 \, a b c^{2} d x + 3 \, a^{2} c d^{2} x - \frac{6 \, a b c^{3} x^{2} + 9 \, a^{2} c^{2} d x^{2} + a^{2} c^{3}}{3 \, x^{3}} \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^4,x, algorithm="giac")

[Out]

1/7*b^2*d^3*x^7 + 3/5*b^2*c*d^2*x^5 + 2/5*a*b*d^3*x^5 + b^2*c^2*d*x^3 + 2*a*b*c*d^2*x^3 + 1/3*a^2*d^3*x^3 + b^
2*c^3*x + 6*a*b*c^2*d*x + 3*a^2*c*d^2*x - 1/3*(6*a*b*c^3*x^2 + 9*a^2*c^2*d*x^2 + a^2*c^3)/x^3