3.460 $$\int (d+e x)^4 (a+c x^2)^2 \, dx$$

Optimal. Leaf size=117 $\frac{2 c (d+e x)^7 \left (a e^2+3 c d^2\right )}{7 e^5}-\frac{2 c d (d+e x)^6 \left (a e^2+c d^2\right )}{3 e^5}+\frac{(d+e x)^5 \left (a e^2+c d^2\right )^2}{5 e^5}+\frac{c^2 (d+e x)^9}{9 e^5}-\frac{c^2 d (d+e x)^8}{2 e^5}$

[Out]

((c*d^2 + a*e^2)^2*(d + e*x)^5)/(5*e^5) - (2*c*d*(c*d^2 + a*e^2)*(d + e*x)^6)/(3*e^5) + (2*c*(3*c*d^2 + a*e^2)
*(d + e*x)^7)/(7*e^5) - (c^2*d*(d + e*x)^8)/(2*e^5) + (c^2*(d + e*x)^9)/(9*e^5)

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Rubi [A]  time = 0.131277, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.059, Rules used = {697} $\frac{2 c (d+e x)^7 \left (a e^2+3 c d^2\right )}{7 e^5}-\frac{2 c d (d+e x)^6 \left (a e^2+c d^2\right )}{3 e^5}+\frac{(d+e x)^5 \left (a e^2+c d^2\right )^2}{5 e^5}+\frac{c^2 (d+e x)^9}{9 e^5}-\frac{c^2 d (d+e x)^8}{2 e^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^4*(a + c*x^2)^2,x]

[Out]

((c*d^2 + a*e^2)^2*(d + e*x)^5)/(5*e^5) - (2*c*d*(c*d^2 + a*e^2)*(d + e*x)^6)/(3*e^5) + (2*c*(3*c*d^2 + a*e^2)
*(d + e*x)^7)/(7*e^5) - (c^2*d*(d + e*x)^8)/(2*e^5) + (c^2*(d + e*x)^9)/(9*e^5)

Rule 697

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

Rubi steps

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

Mathematica [A]  time = 0.0188699, size = 167, normalized size = 1.43 $\frac{1}{5} x^5 \left (a^2 e^4+12 a c d^2 e^2+c^2 d^4\right )+2 a^2 d^3 e x^2+a^2 d^4 x+\frac{2}{7} c e^2 x^7 \left (a e^2+3 c d^2\right )+\frac{2}{3} c d e x^6 \left (2 a e^2+c d^2\right )+a d e x^4 \left (a e^2+2 c d^2\right )+\frac{2}{3} a d^2 x^3 \left (3 a e^2+c d^2\right )+\frac{1}{2} c^2 d e^3 x^8+\frac{1}{9} c^2 e^4 x^9$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

a^2*d^4*x + 2*a^2*d^3*e*x^2 + (2*a*d^2*(c*d^2 + 3*a*e^2)*x^3)/3 + a*d*e*(2*c*d^2 + a*e^2)*x^4 + ((c^2*d^4 + 12
*a*c*d^2*e^2 + a^2*e^4)*x^5)/5 + (2*c*d*e*(c*d^2 + 2*a*e^2)*x^6)/3 + (2*c*e^2*(3*c*d^2 + a*e^2)*x^7)/7 + (c^2*
d*e^3*x^8)/2 + (c^2*e^4*x^9)/9

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Maple [A]  time = 0.044, size = 169, normalized size = 1.4 \begin{align*}{\frac{{e}^{4}{c}^{2}{x}^{9}}{9}}+{\frac{d{e}^{3}{c}^{2}{x}^{8}}{2}}+{\frac{ \left ( 2\,{e}^{4}ac+6\,{d}^{2}{e}^{2}{c}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 8\,d{e}^{3}ac+4\,{d}^{3}e{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ({a}^{2}{e}^{4}+12\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,d{e}^{3}{a}^{2}+8\,{d}^{3}eac \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,{d}^{2}{e}^{2}{a}^{2}+2\,{d}^{4}ac \right ){x}^{3}}{3}}+2\,{d}^{3}e{a}^{2}{x}^{2}+{d}^{4}{a}^{2}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*e^4*c^2*x^9+1/2*d*e^3*c^2*x^8+1/7*(2*a*c*e^4+6*c^2*d^2*e^2)*x^7+1/6*(8*a*c*d*e^3+4*c^2*d^3*e)*x^6+1/5*(a^2
*e^4+12*a*c*d^2*e^2+c^2*d^4)*x^5+1/4*(4*a^2*d*e^3+8*a*c*d^3*e)*x^4+1/3*(6*a^2*d^2*e^2+2*a*c*d^4)*x^3+2*d^3*e*a
^2*x^2+d^4*a^2*x

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Maxima [A]  time = 1.05288, size = 220, normalized size = 1.88 \begin{align*} \frac{1}{9} \, c^{2} e^{4} x^{9} + \frac{1}{2} \, c^{2} d e^{3} x^{8} + 2 \, a^{2} d^{3} e x^{2} + \frac{2}{7} \,{\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{7} + a^{2} d^{4} x + \frac{2}{3} \,{\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (c^{2} d^{4} + 12 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{5} +{\left (2 \, a c d^{3} e + a^{2} d e^{3}\right )} x^{4} + \frac{2}{3} \,{\left (a c d^{4} + 3 \, a^{2} d^{2} e^{2}\right )} x^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*c^2*e^4*x^9 + 1/2*c^2*d*e^3*x^8 + 2*a^2*d^3*e*x^2 + 2/7*(3*c^2*d^2*e^2 + a*c*e^4)*x^7 + a^2*d^4*x + 2/3*(c
^2*d^3*e + 2*a*c*d*e^3)*x^6 + 1/5*(c^2*d^4 + 12*a*c*d^2*e^2 + a^2*e^4)*x^5 + (2*a*c*d^3*e + a^2*d*e^3)*x^4 + 2
/3*(a*c*d^4 + 3*a^2*d^2*e^2)*x^3

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Fricas [A]  time = 1.60657, size = 375, normalized size = 3.21 \begin{align*} \frac{1}{9} x^{9} e^{4} c^{2} + \frac{1}{2} x^{8} e^{3} d c^{2} + \frac{6}{7} x^{7} e^{2} d^{2} c^{2} + \frac{2}{7} x^{7} e^{4} c a + \frac{2}{3} x^{6} e d^{3} c^{2} + \frac{4}{3} x^{6} e^{3} d c a + \frac{1}{5} x^{5} d^{4} c^{2} + \frac{12}{5} x^{5} e^{2} d^{2} c a + \frac{1}{5} x^{5} e^{4} a^{2} + 2 x^{4} e d^{3} c a + x^{4} e^{3} d a^{2} + \frac{2}{3} x^{3} d^{4} c a + 2 x^{3} e^{2} d^{2} a^{2} + 2 x^{2} e d^{3} a^{2} + x d^{4} a^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^9*e^4*c^2 + 1/2*x^8*e^3*d*c^2 + 6/7*x^7*e^2*d^2*c^2 + 2/7*x^7*e^4*c*a + 2/3*x^6*e*d^3*c^2 + 4/3*x^6*e^3*
d*c*a + 1/5*x^5*d^4*c^2 + 12/5*x^5*e^2*d^2*c*a + 1/5*x^5*e^4*a^2 + 2*x^4*e*d^3*c*a + x^4*e^3*d*a^2 + 2/3*x^3*d
^4*c*a + 2*x^3*e^2*d^2*a^2 + 2*x^2*e*d^3*a^2 + x*d^4*a^2

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Sympy [A]  time = 0.138325, size = 182, normalized size = 1.56 \begin{align*} a^{2} d^{4} x + 2 a^{2} d^{3} e x^{2} + \frac{c^{2} d e^{3} x^{8}}{2} + \frac{c^{2} e^{4} x^{9}}{9} + x^{7} \left (\frac{2 a c e^{4}}{7} + \frac{6 c^{2} d^{2} e^{2}}{7}\right ) + x^{6} \left (\frac{4 a c d e^{3}}{3} + \frac{2 c^{2} d^{3} e}{3}\right ) + x^{5} \left (\frac{a^{2} e^{4}}{5} + \frac{12 a c d^{2} e^{2}}{5} + \frac{c^{2} d^{4}}{5}\right ) + x^{4} \left (a^{2} d e^{3} + 2 a c d^{3} e\right ) + x^{3} \left (2 a^{2} d^{2} e^{2} + \frac{2 a c d^{4}}{3}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*d**4*x + 2*a**2*d**3*e*x**2 + c**2*d*e**3*x**8/2 + c**2*e**4*x**9/9 + x**7*(2*a*c*e**4/7 + 6*c**2*d**2*e*
*2/7) + x**6*(4*a*c*d*e**3/3 + 2*c**2*d**3*e/3) + x**5*(a**2*e**4/5 + 12*a*c*d**2*e**2/5 + c**2*d**4/5) + x**4
*(a**2*d*e**3 + 2*a*c*d**3*e) + x**3*(2*a**2*d**2*e**2 + 2*a*c*d**4/3)

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Giac [A]  time = 1.33699, size = 224, normalized size = 1.91 \begin{align*} \frac{1}{9} \, c^{2} x^{9} e^{4} + \frac{1}{2} \, c^{2} d x^{8} e^{3} + \frac{6}{7} \, c^{2} d^{2} x^{7} e^{2} + \frac{2}{3} \, c^{2} d^{3} x^{6} e + \frac{1}{5} \, c^{2} d^{4} x^{5} + \frac{2}{7} \, a c x^{7} e^{4} + \frac{4}{3} \, a c d x^{6} e^{3} + \frac{12}{5} \, a c d^{2} x^{5} e^{2} + 2 \, a c d^{3} x^{4} e + \frac{2}{3} \, a c d^{4} x^{3} + \frac{1}{5} \, a^{2} x^{5} e^{4} + a^{2} d x^{4} e^{3} + 2 \, a^{2} d^{2} x^{3} e^{2} + 2 \, a^{2} d^{3} x^{2} e + a^{2} d^{4} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*c^2*x^9*e^4 + 1/2*c^2*d*x^8*e^3 + 6/7*c^2*d^2*x^7*e^2 + 2/3*c^2*d^3*x^6*e + 1/5*c^2*d^4*x^5 + 2/7*a*c*x^7*
e^4 + 4/3*a*c*d*x^6*e^3 + 12/5*a*c*d^2*x^5*e^2 + 2*a*c*d^3*x^4*e + 2/3*a*c*d^4*x^3 + 1/5*a^2*x^5*e^4 + a^2*d*x
^4*e^3 + 2*a^2*d^2*x^3*e^2 + 2*a^2*d^3*x^2*e + a^2*d^4*x