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]

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

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Rubi [A]  time = 0.06, antiderivative size = 120, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.04, size = 120, normalized size = 1.00 \[ \frac {1}{3} d x^3 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+c x \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac {a^2 c^3}{3 x^3}-\frac {a c^2 (3 a d+2 b c)}{x}+\frac {1}{5} b d^2 x^5 (2 a d+3 b c)+\frac {1}{7} b^2 d^3 x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(a^2*c^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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fricas [A]  time = 0.44, size = 129, normalized size = 1.08 \[ \frac {15 \, b^{2} d^{3} x^{10} + 21 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} + 35 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} - 35 \, a^{2} c^{3} + 105 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} - 105 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}}{105 \, x^{3}} \]

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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giac [A]  time = 0.28, size = 129, normalized size = 1.08 \[ \frac {1}{7} \, b^{2} d^{3} x^{7} + \frac {3}{5} \, b^{2} c d^{2} x^{5} + \frac {2}{5} \, a b d^{3} x^{5} + b^{2} c^{2} d x^{3} + 2 \, a b c d^{2} x^{3} + \frac {1}{3} \, a^{2} d^{3} x^{3} + b^{2} c^{3} x + 6 \, a b c^{2} d x + 3 \, a^{2} c d^{2} x - \frac {6 \, a b c^{3} x^{2} + 9 \, a^{2} c^{2} d x^{2} + a^{2} c^{3}}{3 \, x^{3}} \]

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

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maple [A]  time = 0.01, size = 124, normalized size = 1.03 \[ \frac {b^{2} d^{3} x^{7}}{7}+\frac {2 a b \,d^{3} x^{5}}{5}+\frac {3 b^{2} c \,d^{2} x^{5}}{5}+\frac {a^{2} d^{3} x^{3}}{3}+2 a b c \,d^{2} x^{3}+b^{2} c^{2} d \,x^{3}+3 a^{2} c \,d^{2} x +6 a b \,c^{2} d x +b^{2} c^{3} x -\frac {a^{2} c^{3}}{3 x^{3}}-\frac {\left (3 a d +2 b c \right ) a \,c^{2}}{x} \]

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

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maxima [A]  time = 1.09, size = 126, normalized size = 1.05 \[ \frac {1}{7} \, b^{2} d^{3} x^{7} + \frac {1}{5} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{5} + \frac {1}{3} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} + {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x - \frac {a^{2} c^{3} + 3 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}}{3 \, x^{3}} \]

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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mupad [B]  time = 0.04, size = 121, normalized size = 1.01 \[ x^3\,\left (\frac {a^2\,d^3}{3}+2\,a\,b\,c\,d^2+b^2\,c^2\,d\right )-\frac {x^2\,\left (3\,d\,a^2\,c^2+2\,b\,a\,c^3\right )+\frac {a^2\,c^3}{3}}{x^3}+x\,\left (3\,a^2\,c\,d^2+6\,a\,b\,c^2\,d+b^2\,c^3\right )+\frac {b^2\,d^3\,x^7}{7}+\frac {b\,d^2\,x^5\,\left (2\,a\,d+3\,b\,c\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x^2)^2*(c + d*x^2)^3)/x^4,x)

[Out]

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

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sympy [A]  time = 0.35, size = 131, normalized size = 1.09 \[ \frac {b^{2} d^{3} x^{7}}{7} + x^{5} \left (\frac {2 a b d^{3}}{5} + \frac {3 b^{2} c d^{2}}{5}\right ) + x^{3} \left (\frac {a^{2} d^{3}}{3} + 2 a b c d^{2} + b^{2} c^{2} d\right ) + x \left (3 a^{2} c d^{2} + 6 a b c^{2} d + b^{2} c^{3}\right ) + \frac {- a^{2} c^{3} + x^{2} \left (- 9 a^{2} c^{2} d - 6 a b c^{3}\right )}{3 x^{3}} \]

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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