3.128 \(\int (d+e x^2)^3 (a+c x^4)^2 \, dx\)

Optimal. Leaf size=133 \[ a^2 d^2 e x^3+a^2 d^3 x+\frac{1}{11} c e x^{11} \left (2 a e^2+3 c d^2\right )+\frac{1}{9} c d x^9 \left (6 a e^2+c d^2\right )+\frac{1}{7} a e x^7 \left (a e^2+6 c d^2\right )+\frac{1}{5} a d x^5 \left (3 a e^2+2 c d^2\right )+\frac{3}{13} c^2 d e^2 x^{13}+\frac{1}{15} c^2 e^3 x^{15} \]

[Out]

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

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Rubi [A]  time = 0.106869, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {1154} \[ a^2 d^2 e x^3+a^2 d^3 x+\frac{1}{11} c e x^{11} \left (2 a e^2+3 c d^2\right )+\frac{1}{9} c d x^9 \left (6 a e^2+c d^2\right )+\frac{1}{7} a e x^7 \left (a e^2+6 c d^2\right )+\frac{1}{5} a d x^5 \left (3 a e^2+2 c d^2\right )+\frac{3}{13} c^2 d e^2 x^{13}+\frac{1}{15} c^2 e^3 x^{15} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)^3*(a + c*x^4)^2,x]

[Out]

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

Rule 1154

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a
+ c*x^4)^p, x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps

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

Mathematica [A]  time = 0.0216422, size = 133, normalized size = 1. \[ a^2 d^2 e x^3+a^2 d^3 x+\frac{1}{11} c e x^{11} \left (2 a e^2+3 c d^2\right )+\frac{1}{9} c d x^9 \left (6 a e^2+c d^2\right )+\frac{1}{7} a e x^7 \left (a e^2+6 c d^2\right )+\frac{1}{5} a d x^5 \left (3 a e^2+2 c d^2\right )+\frac{3}{13} c^2 d e^2 x^{13}+\frac{1}{15} c^2 e^3 x^{15} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.042, size = 130, normalized size = 1. \begin{align*}{\frac{{c}^{2}{e}^{3}{x}^{15}}{15}}+{\frac{3\,{c}^{2}d{e}^{2}{x}^{13}}{13}}+{\frac{ \left ( 2\,ac{e}^{3}+3\,{c}^{2}{d}^{2}e \right ){x}^{11}}{11}}+{\frac{ \left ( 6\,ad{e}^{2}c+{c}^{2}{d}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ({e}^{3}{a}^{2}+6\,ac{d}^{2}e \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,d{e}^{2}{a}^{2}+2\,{d}^{3}ac \right ){x}^{5}}{5}}+{a}^{2}{d}^{2}e{x}^{3}+{a}^{2}{d}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00803, size = 174, normalized size = 1.31 \begin{align*} \frac{1}{15} \, c^{2} e^{3} x^{15} + \frac{3}{13} \, c^{2} d e^{2} x^{13} + \frac{1}{11} \,{\left (3 \, c^{2} d^{2} e + 2 \, a c e^{3}\right )} x^{11} + \frac{1}{9} \,{\left (c^{2} d^{3} + 6 \, a c d e^{2}\right )} x^{9} + a^{2} d^{2} e x^{3} + \frac{1}{7} \,{\left (6 \, a c d^{2} e + a^{2} e^{3}\right )} x^{7} + a^{2} d^{3} x + \frac{1}{5} \,{\left (2 \, a c d^{3} + 3 \, a^{2} d e^{2}\right )} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.64227, size = 304, normalized size = 2.29 \begin{align*} \frac{1}{15} x^{15} e^{3} c^{2} + \frac{3}{13} x^{13} e^{2} d c^{2} + \frac{3}{11} x^{11} e d^{2} c^{2} + \frac{2}{11} x^{11} e^{3} c a + \frac{1}{9} x^{9} d^{3} c^{2} + \frac{2}{3} x^{9} e^{2} d c a + \frac{6}{7} x^{7} e d^{2} c a + \frac{1}{7} x^{7} e^{3} a^{2} + \frac{2}{5} x^{5} d^{3} c a + \frac{3}{5} x^{5} e^{2} d a^{2} + x^{3} e d^{2} a^{2} + x d^{3} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.083892, size = 144, normalized size = 1.08 \begin{align*} a^{2} d^{3} x + a^{2} d^{2} e x^{3} + \frac{3 c^{2} d e^{2} x^{13}}{13} + \frac{c^{2} e^{3} x^{15}}{15} + x^{11} \left (\frac{2 a c e^{3}}{11} + \frac{3 c^{2} d^{2} e}{11}\right ) + x^{9} \left (\frac{2 a c d e^{2}}{3} + \frac{c^{2} d^{3}}{9}\right ) + x^{7} \left (\frac{a^{2} e^{3}}{7} + \frac{6 a c d^{2} e}{7}\right ) + x^{5} \left (\frac{3 a^{2} d e^{2}}{5} + \frac{2 a c d^{3}}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.11997, size = 173, normalized size = 1.3 \begin{align*} \frac{1}{15} \, c^{2} x^{15} e^{3} + \frac{3}{13} \, c^{2} d x^{13} e^{2} + \frac{3}{11} \, c^{2} d^{2} x^{11} e + \frac{1}{9} \, c^{2} d^{3} x^{9} + \frac{2}{11} \, a c x^{11} e^{3} + \frac{2}{3} \, a c d x^{9} e^{2} + \frac{6}{7} \, a c d^{2} x^{7} e + \frac{2}{5} \, a c d^{3} x^{5} + \frac{1}{7} \, a^{2} x^{7} e^{3} + \frac{3}{5} \, a^{2} d x^{5} e^{2} + a^{2} d^{2} x^{3} e + a^{2} d^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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