### 3.451 $$\int (d+e x)^4 (a+c x^2) \, dx$$

Optimal. Leaf size=57 $\frac{(d+e x)^5 \left (a e^2+c d^2\right )}{5 e^3}+\frac{c (d+e x)^7}{7 e^3}-\frac{c d (d+e x)^6}{3 e^3}$

[Out]

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

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Rubi [A]  time = 0.0632473, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.067, Rules used = {697} $\frac{(d+e x)^5 \left (a e^2+c d^2\right )}{5 e^3}+\frac{c (d+e x)^7}{7 e^3}-\frac{c d (d+e x)^6}{3 e^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^4*(a + c*x^2),x]

[Out]

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

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

Mathematica [A]  time = 0.0161587, size = 101, normalized size = 1.77 $\frac{1}{5} e^2 x^5 \left (a e^2+6 c d^2\right )+d e x^4 \left (a e^2+c d^2\right )+\frac{1}{3} d^2 x^3 \left (6 a e^2+c d^2\right )+2 a d^3 e x^2+a d^4 x+\frac{2}{3} c d e^3 x^6+\frac{1}{7} c e^4 x^7$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^4*(a + c*x^2),x]

[Out]

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

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Maple [A]  time = 0.042, size = 97, normalized size = 1.7 \begin{align*}{\frac{{e}^{4}c{x}^{7}}{7}}+{\frac{2\,d{e}^{3}c{x}^{6}}{3}}+{\frac{ \left ({e}^{4}a+6\,{d}^{2}{e}^{2}c \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,d{e}^{3}a+4\,{d}^{3}ec \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,{d}^{2}{e}^{2}a+{d}^{4}c \right ){x}^{3}}{3}}+2\,{d}^{3}ea{x}^{2}+{d}^{4}ax \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.1672, size = 126, normalized size = 2.21 \begin{align*} \frac{1}{7} \, c e^{4} x^{7} + \frac{2}{3} \, c d e^{3} x^{6} + 2 \, a d^{3} e x^{2} + a d^{4} x + \frac{1}{5} \,{\left (6 \, c d^{2} e^{2} + a e^{4}\right )} x^{5} +{\left (c d^{3} e + a d e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (c d^{4} + 6 \, a 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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.64037, size = 212, normalized size = 3.72 \begin{align*} \frac{1}{7} x^{7} e^{4} c + \frac{2}{3} x^{6} e^{3} d c + \frac{6}{5} x^{5} e^{2} d^{2} c + \frac{1}{5} x^{5} e^{4} a + x^{4} e d^{3} c + x^{4} e^{3} d a + \frac{1}{3} x^{3} d^{4} c + 2 x^{3} e^{2} d^{2} a + 2 x^{2} e d^{3} a + x d^{4} a \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.127347, size = 100, normalized size = 1.75 \begin{align*} a d^{4} x + 2 a d^{3} e x^{2} + \frac{2 c d e^{3} x^{6}}{3} + \frac{c e^{4} x^{7}}{7} + x^{5} \left (\frac{a e^{4}}{5} + \frac{6 c d^{2} e^{2}}{5}\right ) + x^{4} \left (a d e^{3} + c d^{3} e\right ) + x^{3} \left (2 a d^{2} e^{2} + \frac{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),x)

[Out]

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

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Giac [A]  time = 1.17834, size = 124, normalized size = 2.18 \begin{align*} \frac{1}{7} \, c x^{7} e^{4} + \frac{2}{3} \, c d x^{6} e^{3} + \frac{6}{5} \, c d^{2} x^{5} e^{2} + c d^{3} x^{4} e + \frac{1}{3} \, c d^{4} x^{3} + \frac{1}{5} \, a x^{5} e^{4} + a d x^{4} e^{3} + 2 \, a d^{2} x^{3} e^{2} + 2 \, a d^{3} x^{2} e + a 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),x, algorithm="giac")

[Out]

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