3.159 \(\int x^4 (a+b x^2)^2 (c+d x^2)^3 \, dx\)

Optimal. Leaf size=127 \[ \frac{1}{11} d x^{11} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{9} c x^9 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{1}{5} a^2 c^3 x^5+\frac{1}{7} a c^2 x^7 (3 a d+2 b c)+\frac{1}{13} b d^2 x^{13} (2 a d+3 b c)+\frac{1}{15} b^2 d^3 x^{15} \]

[Out]

(a^2*c^3*x^5)/5 + (a*c^2*(2*b*c + 3*a*d)*x^7)/7 + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^9)/9 + (d*(3*b^2*c^2
+ 6*a*b*c*d + a^2*d^2)*x^11)/11 + (b*d^2*(3*b*c + 2*a*d)*x^13)/13 + (b^2*d^3*x^15)/15

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Rubi [A]  time = 0.0880314, antiderivative size = 127, 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} \[ \frac{1}{11} d x^{11} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{9} c x^9 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{1}{5} a^2 c^3 x^5+\frac{1}{7} a c^2 x^7 (3 a d+2 b c)+\frac{1}{13} b d^2 x^{13} (2 a d+3 b c)+\frac{1}{15} b^2 d^3 x^{15} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*c^3*x^5)/5 + (a*c^2*(2*b*c + 3*a*d)*x^7)/7 + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^9)/9 + (d*(3*b^2*c^2
+ 6*a*b*c*d + a^2*d^2)*x^11)/11 + (b*d^2*(3*b*c + 2*a*d)*x^13)/13 + (b^2*d^3*x^15)/15

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 x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx &=\int \left (a^2 c^3 x^4+a c^2 (2 b c+3 a d) x^6+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^8+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{10}+b d^2 (3 b c+2 a d) x^{12}+b^2 d^3 x^{14}\right ) \, dx\\ &=\frac{1}{5} a^2 c^3 x^5+\frac{1}{7} a c^2 (2 b c+3 a d) x^7+\frac{1}{9} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^9+\frac{1}{11} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{11}+\frac{1}{13} b d^2 (3 b c+2 a d) x^{13}+\frac{1}{15} b^2 d^3 x^{15}\\ \end{align*}

Mathematica [A]  time = 0.0266395, size = 127, normalized size = 1. \[ \frac{1}{11} d x^{11} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{9} c x^9 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{1}{5} a^2 c^3 x^5+\frac{1}{7} a c^2 x^7 (3 a d+2 b c)+\frac{1}{13} b d^2 x^{13} (2 a d+3 b c)+\frac{1}{15} b^2 d^3 x^{15} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*c^3*x^5)/5 + (a*c^2*(2*b*c + 3*a*d)*x^7)/7 + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^9)/9 + (d*(3*b^2*c^2
+ 6*a*b*c*d + a^2*d^2)*x^11)/11 + (b*d^2*(3*b*c + 2*a*d)*x^13)/13 + (b^2*d^3*x^15)/15

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Maple [A]  time = 0.001, size = 128, normalized size = 1. \begin{align*}{\frac{{b}^{2}{d}^{3}{x}^{15}}{15}}+{\frac{ \left ( 2\,ab{d}^{3}+3\,{b}^{2}c{d}^{2} \right ){x}^{13}}{13}}+{\frac{ \left ({a}^{2}{d}^{3}+6\,abc{d}^{2}+3\,{b}^{2}{c}^{2}d \right ){x}^{11}}{11}}+{\frac{ \left ( 3\,{a}^{2}c{d}^{2}+6\,ab{c}^{2}d+{b}^{2}{c}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ( 3\,{a}^{2}{c}^{2}d+2\,ab{c}^{3} \right ){x}^{7}}{7}}+{\frac{{a}^{2}{c}^{3}{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*b^2*d^3*x^15+1/13*(2*a*b*d^3+3*b^2*c*d^2)*x^13+1/11*(a^2*d^3+6*a*b*c*d^2+3*b^2*c^2*d)*x^11+1/9*(3*a^2*c*d
^2+6*a*b*c^2*d+b^2*c^3)*x^9+1/7*(3*a^2*c^2*d+2*a*b*c^3)*x^7+1/5*a^2*c^3*x^5

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Maxima [A]  time = 1.03304, size = 171, normalized size = 1.35 \begin{align*} \frac{1}{15} \, b^{2} d^{3} x^{15} + \frac{1}{13} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{13} + \frac{1}{11} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{11} + \frac{1}{5} \, a^{2} c^{3} x^{5} + \frac{1}{9} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{9} + \frac{1}{7} \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^2*(d*x^2+c)^3,x, algorithm="maxima")

[Out]

1/15*b^2*d^3*x^15 + 1/13*(3*b^2*c*d^2 + 2*a*b*d^3)*x^13 + 1/11*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^11 + 1/
5*a^2*c^3*x^5 + 1/9*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^9 + 1/7*(2*a*b*c^3 + 3*a^2*c^2*d)*x^7

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Fricas [A]  time = 1.12534, size = 323, normalized size = 2.54 \begin{align*} \frac{1}{15} x^{15} d^{3} b^{2} + \frac{3}{13} x^{13} d^{2} c b^{2} + \frac{2}{13} x^{13} d^{3} b a + \frac{3}{11} x^{11} d c^{2} b^{2} + \frac{6}{11} x^{11} d^{2} c b a + \frac{1}{11} x^{11} d^{3} a^{2} + \frac{1}{9} x^{9} c^{3} b^{2} + \frac{2}{3} x^{9} d c^{2} b a + \frac{1}{3} x^{9} d^{2} c a^{2} + \frac{2}{7} x^{7} c^{3} b a + \frac{3}{7} x^{7} d c^{2} a^{2} + \frac{1}{5} x^{5} c^{3} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^2*(d*x^2+c)^3,x, algorithm="fricas")

[Out]

1/15*x^15*d^3*b^2 + 3/13*x^13*d^2*c*b^2 + 2/13*x^13*d^3*b*a + 3/11*x^11*d*c^2*b^2 + 6/11*x^11*d^2*c*b*a + 1/11
*x^11*d^3*a^2 + 1/9*x^9*c^3*b^2 + 2/3*x^9*d*c^2*b*a + 1/3*x^9*d^2*c*a^2 + 2/7*x^7*c^3*b*a + 3/7*x^7*d*c^2*a^2
+ 1/5*x^5*c^3*a^2

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Sympy [A]  time = 0.086249, size = 143, normalized size = 1.13 \begin{align*} \frac{a^{2} c^{3} x^{5}}{5} + \frac{b^{2} d^{3} x^{15}}{15} + x^{13} \left (\frac{2 a b d^{3}}{13} + \frac{3 b^{2} c d^{2}}{13}\right ) + x^{11} \left (\frac{a^{2} d^{3}}{11} + \frac{6 a b c d^{2}}{11} + \frac{3 b^{2} c^{2} d}{11}\right ) + x^{9} \left (\frac{a^{2} c d^{2}}{3} + \frac{2 a b c^{2} d}{3} + \frac{b^{2} c^{3}}{9}\right ) + x^{7} \left (\frac{3 a^{2} c^{2} d}{7} + \frac{2 a b c^{3}}{7}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b*x**2+a)**2*(d*x**2+c)**3,x)

[Out]

a**2*c**3*x**5/5 + b**2*d**3*x**15/15 + x**13*(2*a*b*d**3/13 + 3*b**2*c*d**2/13) + x**11*(a**2*d**3/11 + 6*a*b
*c*d**2/11 + 3*b**2*c**2*d/11) + x**9*(a**2*c*d**2/3 + 2*a*b*c**2*d/3 + b**2*c**3/9) + x**7*(3*a**2*c**2*d/7 +
2*a*b*c**3/7)

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Giac [A]  time = 1.14775, size = 182, normalized size = 1.43 \begin{align*} \frac{1}{15} \, b^{2} d^{3} x^{15} + \frac{3}{13} \, b^{2} c d^{2} x^{13} + \frac{2}{13} \, a b d^{3} x^{13} + \frac{3}{11} \, b^{2} c^{2} d x^{11} + \frac{6}{11} \, a b c d^{2} x^{11} + \frac{1}{11} \, a^{2} d^{3} x^{11} + \frac{1}{9} \, b^{2} c^{3} x^{9} + \frac{2}{3} \, a b c^{2} d x^{9} + \frac{1}{3} \, a^{2} c d^{2} x^{9} + \frac{2}{7} \, a b c^{3} x^{7} + \frac{3}{7} \, a^{2} c^{2} d x^{7} + \frac{1}{5} \, a^{2} c^{3} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^2*(d*x^2+c)^3,x, algorithm="giac")

[Out]

1/15*b^2*d^3*x^15 + 3/13*b^2*c*d^2*x^13 + 2/13*a*b*d^3*x^13 + 3/11*b^2*c^2*d*x^11 + 6/11*a*b*c*d^2*x^11 + 1/11
*a^2*d^3*x^11 + 1/9*b^2*c^3*x^9 + 2/3*a*b*c^2*d*x^9 + 1/3*a^2*c*d^2*x^9 + 2/7*a*b*c^3*x^7 + 3/7*a^2*c^2*d*x^7
+ 1/5*a^2*c^3*x^5