\(\int (d+e x)^2 (a+c x^2)^2 \, dx\) [462]

Optimal result

Integrand size = 17, antiderivative size = 80 \[ \int (d+e x)^2 \left (a+c x^2\right )^2 \, dx=a^2 d^2 x+\frac {1}{3} a \left (2 c d^2+a e^2\right ) x^3+\frac {1}{5} c \left (c d^2+2 a e^2\right ) x^5+\frac {1}{7} c^2 e^2 x^7+\frac {d e \left (a+c x^2\right )^3}{3 c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {710, 1824} \[ \int (d+e x)^2 \left (a+c x^2\right )^2 \, dx=a^2 d^2 x+\frac {1}{5} c x^5 \left (2 a e^2+c d^2\right )+\frac {1}{3} a x^3 \left (a e^2+2 c d^2\right )+\frac {d e \left (a+c x^2\right )^3}{3 c}+\frac {1}{7} c^2 e^2 x^7 \]

[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 710

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 1824

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*} \text {integral}& = \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] (verified)

Time = 0.01 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.14 \[ \int (d+e x)^2 \left (a+c x^2\right )^2 \, dx=a^2 d^2 x+a^2 d e x^2+\frac {1}{3} a \left (2 c d^2+a e^2\right ) x^3+a c d e x^4+\frac {1}{5} c \left (c d^2+2 a e^2\right ) x^5+\frac {1}{3} c^2 d e x^6+\frac {1}{7} c^2 e^2 x^7 \]

[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

Maple [A] (verified)

Time = 2.16 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.10

method	result	size
default	\(\frac {c^{2} e^{2} x^{7}}{7}+\frac {x^{6} d e \,c^{2}}{3}+\frac {\left (2 a c \,e^{2}+c^{2} d^{2}\right ) x^{5}}{5}+a c d e \,x^{4}+\frac {\left (a^{2} e^{2}+2 a c \,d^{2}\right ) x^{3}}{3}+a^{2} d e \,x^{2}+a^{2} d^{2} x\)	\(88\)
norman	\(\frac {c^{2} e^{2} x^{7}}{7}+\frac {x^{6} d e \,c^{2}}{3}+\left (\frac {2}{5} a c \,e^{2}+\frac {1}{5} c^{2} d^{2}\right ) x^{5}+a c d e \,x^{4}+\left (\frac {1}{3} a^{2} e^{2}+\frac {2}{3} a c \,d^{2}\right ) x^{3}+a^{2} d e \,x^{2}+a^{2} d^{2} x\)	\(88\)
gosper	\(\frac {1}{7} c^{2} e^{2} x^{7}+\frac {1}{3} x^{6} d e \,c^{2}+\frac {2}{5} x^{5} a c \,e^{2}+\frac {1}{5} c^{2} d^{2} x^{5}+a c d e \,x^{4}+\frac {1}{3} x^{3} a^{2} e^{2}+\frac {2}{3} a c \,d^{2} x^{3}+a^{2} d e \,x^{2}+a^{2} d^{2} x\)	\(90\)
risch	\(\frac {1}{7} c^{2} e^{2} x^{7}+\frac {1}{3} x^{6} d e \,c^{2}+\frac {2}{5} x^{5} a c \,e^{2}+\frac {1}{5} c^{2} d^{2} x^{5}+a c d e \,x^{4}+\frac {1}{3} x^{3} a^{2} e^{2}+\frac {2}{3} a c \,d^{2} x^{3}+a^{2} d e \,x^{2}+a^{2} d^{2} x\)	\(90\)
parallelrisch	\(\frac {1}{7} c^{2} e^{2} x^{7}+\frac {1}{3} x^{6} d e \,c^{2}+\frac {2}{5} x^{5} a c \,e^{2}+\frac {1}{5} c^{2} d^{2} x^{5}+a c d e \,x^{4}+\frac {1}{3} x^{3} a^{2} e^{2}+\frac {2}{3} a c \,d^{2} x^{3}+a^{2} d e \,x^{2}+a^{2} d^{2} x\)	\(90\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.09 \[ \int (d+e x)^2 \left (a+c x^2\right )^2 \, dx=\frac {1}{7} \, c^{2} e^{2} x^{7} + \frac {1}{3} \, c^{2} d e x^{6} + a c d e x^{4} + a^{2} d e x^{2} + \frac {1}{5} \, {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} x^{5} + a^{2} d^{2} x + \frac {1}{3} \, {\left (2 \, a c d^{2} + a^{2} e^{2}\right )} x^{3} \]

[In]

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

[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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.19 \[ \int (d+e x)^2 \left (a+c x^2\right )^2 \, dx=a^{2} d^{2} x + a^{2} d e x^{2} + a c d e x^{4} + \frac {c^{2} d e x^{6}}{3} + \frac {c^{2} e^{2} x^{7}}{7} + x^{5} \cdot \left (\frac {2 a c e^{2}}{5} + \frac {c^{2} d^{2}}{5}\right ) + x^{3} \left (\frac {a^{2} e^{2}}{3} + \frac {2 a c d^{2}}{3}\right ) \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.09 \[ \int (d+e x)^2 \left (a+c x^2\right )^2 \, dx=\frac {1}{7} \, c^{2} e^{2} x^{7} + \frac {1}{3} \, c^{2} d e x^{6} + a c d e x^{4} + a^{2} d e x^{2} + \frac {1}{5} \, {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} x^{5} + a^{2} d^{2} x + \frac {1}{3} \, {\left (2 \, a c d^{2} + a^{2} e^{2}\right )} x^{3} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.11 \[ \int (d+e x)^2 \left (a+c x^2\right )^2 \, dx=\frac {1}{7} \, c^{2} e^{2} x^{7} + \frac {1}{3} \, c^{2} d e x^{6} + \frac {1}{5} \, c^{2} d^{2} x^{5} + \frac {2}{5} \, a c e^{2} x^{5} + a c d e x^{4} + \frac {2}{3} \, a c d^{2} x^{3} + \frac {1}{3} \, a^{2} e^{2} x^{3} + a^{2} d e x^{2} + a^{2} d^{2} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.32 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.09 \[ \int (d+e x)^2 \left (a+c x^2\right )^2 \, dx=x^3\,\left (\frac {a^2\,e^2}{3}+\frac {2\,c\,a\,d^2}{3}\right )+x^5\,\left (\frac {c^2\,d^2}{5}+\frac {2\,a\,c\,e^2}{5}\right )+a^2\,d^2\,x+\frac {c^2\,e^2\,x^7}{7}+a^2\,d\,e\,x^2+\frac {c^2\,d\,e\,x^6}{3}+a\,c\,d\,e\,x^4 \]

[In]

int((a + c*x^2)^2*(d + e*x)^2,x)

[Out]

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