3.462 $$\int (d+e x)^2 (a+c x^2)^2 \, dx$$

Optimal. Leaf size=80 $a^2 d^2 x+\frac{1}{5} c x^5 \left (2 a e^2+c d^2\right )+\frac{1}{3} a x^3 \left (a e^2+2 c d^2\right )+\frac{d e \left (a+c x^2\right )^3}{3 c}+\frac{1}{7} c^2 e^2 x^7$

[Out]

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

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Rubi [A]  time = 0.0461326, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.118, Rules used = {696, 1810} $a^2 d^2 x+\frac{1}{5} c x^5 \left (2 a e^2+c d^2\right )+\frac{1}{3} a x^3 \left (a e^2+2 c d^2\right )+\frac{d e \left (a+c x^2\right )^3}{3 c}+\frac{1}{7} c^2 e^2 x^7$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^2*(a + c*x^2)^2,x]

[Out]

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

Rule 696

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*m*d^(m - 1)*(a + c*x^2)^(p + 1))
/(2*c*(p + 1)), x] + Int[((d + e*x)^m - e*m*d^(m - 1)*x)*(a + c*x^2)^p, x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*
d^2 + a*e^2, 0] && IGtQ[p, 1] && IGtQ[m, 0] && LeQ[m, p]

Rule 1810

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin{align*} \int (d+e x)^2 \left (a+c x^2\right )^2 \, dx &=\frac{d e \left (a+c x^2\right )^3}{3 c}+\int \left (a+c x^2\right )^2 \left (-2 d e x+(d+e x)^2\right ) \, dx\\ &=\frac{d e \left (a+c x^2\right )^3}{3 c}+\int \left (a^2 d^2+a \left (2 c d^2+a e^2\right ) x^2+c \left (c d^2+2 a e^2\right ) x^4+c^2 e^2 x^6\right ) \, dx\\ &=a^2 d^2 x+\frac{1}{3} a \left (2 c d^2+a e^2\right ) x^3+\frac{1}{5} c \left (c d^2+2 a e^2\right ) x^5+\frac{1}{7} c^2 e^2 x^7+\frac{d e \left (a+c x^2\right )^3}{3 c}\\ \end{align*}

Mathematica [A]  time = 0.011535, size = 91, normalized size = 1.14 $a^2 d^2 x+a^2 d e x^2+\frac{1}{5} c x^5 \left (2 a e^2+c d^2\right )+\frac{1}{3} a x^3 \left (a e^2+2 c d^2\right )+a c d e x^4+\frac{1}{3} c^2 d e x^6+\frac{1}{7} c^2 e^2 x^7$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.048, size = 88, normalized size = 1.1 \begin{align*}{\frac{{c}^{2}{e}^{2}{x}^{7}}{7}}+{\frac{de{c}^{2}{x}^{6}}{3}}+{\frac{ \left ( 2\,{e}^{2}ac+{c}^{2}{d}^{2} \right ){x}^{5}}{5}}+acde{x}^{4}+{\frac{ \left ({a}^{2}{e}^{2}+2\,{d}^{2}ac \right ){x}^{3}}{3}}+de{a}^{2}{x}^{2}+{a}^{2}{d}^{2}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.15998, size = 117, normalized size = 1.46 \begin{align*} \frac{1}{7} \, c^{2} e^{2} x^{7} + \frac{1}{3} \, c^{2} d e x^{6} + a c d e x^{4} + a^{2} d e x^{2} + \frac{1}{5} \,{\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} x^{5} + a^{2} d^{2} x + \frac{1}{3} \,{\left (2 \, a c d^{2} + a^{2} e^{2}\right )} x^{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.66084, size = 198, normalized size = 2.48 \begin{align*} \frac{1}{7} x^{7} e^{2} c^{2} + \frac{1}{3} x^{6} e d c^{2} + \frac{1}{5} x^{5} d^{2} c^{2} + \frac{2}{5} x^{5} e^{2} c a + x^{4} e d c a + \frac{2}{3} x^{3} d^{2} c a + \frac{1}{3} x^{3} e^{2} a^{2} + x^{2} e d a^{2} + x d^{2} a^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.122765, size = 95, normalized size = 1.19 \begin{align*} a^{2} d^{2} x + a^{2} d e x^{2} + a c d e x^{4} + \frac{c^{2} d e x^{6}}{3} + \frac{c^{2} e^{2} x^{7}}{7} + x^{5} \left (\frac{2 a c e^{2}}{5} + \frac{c^{2} d^{2}}{5}\right ) + x^{3} \left (\frac{a^{2} e^{2}}{3} + \frac{2 a c d^{2}}{3}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.35347, size = 120, normalized size = 1.5 \begin{align*} \frac{1}{7} \, c^{2} x^{7} e^{2} + \frac{1}{3} \, c^{2} d x^{6} e + \frac{1}{5} \, c^{2} d^{2} x^{5} + \frac{2}{5} \, a c x^{5} e^{2} + a c d x^{4} e + \frac{2}{3} \, a c d^{2} x^{3} + \frac{1}{3} \, a^{2} x^{3} e^{2} + a^{2} d x^{2} e + a^{2} d^{2} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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